diff --git a/research/pr4359-tensor-strictification/ENDPOINT.md b/research/pr4359-tensor-strictification/ENDPOINT.md
new file mode 100644
index 0000000000..b7e2cbed18
--- /dev/null
+++ b/research/pr4359-tensor-strictification/ENDPOINT.md
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+# Conditional outer saving 0.00247755 with a specified uniform rank
+
+This result uses the modified native invoice and tensor construction in
+`NATIVE.md` and `TENSOR.md`, together with the native, finite-precompilation,
+delayed BIT cover, precision/storage, and inherited outer hypotheses of
+zxv's PR4359 at `da637c87c89482bc4c86be36bd0071b3d400dbed`. It does not
+independently formalize those hypotheses or change the selected frontier.
+Apache-2.0; substantial OpenAI assistance.
+
+The native productive-job construction is credited to zxv's PR4359 and
+integer mixed flow to Jacob Alan Sussman's Theorem 1G. The outer slack
+family and ledger are retained from PR4405, commit
+`873d1b3c96a83b701a5c0bba8e37a861e6672fef`, which credits @Vlado-V's
+PR1673, @jeremykothe2's PR1758, @chafreaky's PR1786 and the earlier lineage.
+
+## Exact result and source changes
+
+The proposed conditional running-time bound is
+
+```
+T(n) = O(n (log n)^(1-kappa)),    kappa = 0.00247755.
+```
+
+The gain is `0.00017755` over PR4359's `0.0023` and `0.00017246` over
+PR4405's `0.00230509`. These differences compare exponent savings.
+
+The source-bound native invoice uses a fresh arbitrary field for every
+original native helper, restores the 56 complete rank-eight owned-center
+jobs, and pays every retained idle window as one complete transform.
+It gives no productive credit to owned storage. Its exact moments use
+
+```
+targets = {5:196, 6:126, 7:392, 8:651, 9:1491, 25:448, 57:224},
+paid    = {1:17409, 2:4991, 3:1568, 4:714, 5:140, 8:504, 16:448}.
+```
+
+The native target and paid total ranks are 47075 and 46851. This invoice
+has paid ranks above its smallest target. `TENSOR.md` therefore supplies a
+new finite construction before using the original strict-separation compiler.
+The modified invoice alone is not being admitted directly.
+
+At the exact tested saving `C_test=0.00247756`, put `tau=1-C_test`, and let
+`P_tau` and `T_tau` be the paid and target moments. The certificate proves
+
+```
+P_tau/T_tau <= 1 - 1/10^9,
+46851/(4^C_test * T_tau) < 999/1000.
+```
+
+Tensor 15000 copies using the chronological construction, and split each
+paid rank larger than `4^15000` into consecutive disjoint pieces of at
+most that rank. The normalized paid moment of this explicitly defined
+finite mixed unit is strictly less than
+
+```
+1/(1 + 15000/10^9) + 2*(999/1000)^15000 < 1.
+```
+
+Its paid ranks are at most `4^15000`, while target ranks range from
+`5^15000` through `57^15000`. Thus the original compiler's strict separation
+is restored. Tensoring and splitting preserve a positive rank-one surplus,
+because `47075^15000 > 46851^15000`.
+
+## An explicit uniform rank for this large mixed unit
+
+Write `v=15000` for the tensor exponent, and set
+
+```
+kappa   = 0.00247755,
+d       = (C_test-kappa)/7 = 1/700000000,
+uniform = kappa+6*d = 1734291/700000000,
+t       = 1-C_test,
+z       = 1-uniform,
+z0      = (t+z)/2,
+delta   = z0-t = 1/1400000000.
+```
+
+The source rank parameters satisfy `m=57^v`, `k_min=5^v`, and
+`r_max<=4^v`. Since `4^8<5^7`, the exit exponent is strictly below `7/8`.
+We may use `h=1/8` in the source/exit proof. At ambient rank `x>=m`,
+
+```
+m E(x)/x < m^h*r_max*x^(-h)
+          < 2^B*x^(-h),       B=(2+6*h)*v=41250.
+```
+
+The second strict inequality follows from `4=2^2` and `57<2^6`. The exact
+complement coefficient is `eta=z0*m^(z0-1)`. Because `z0>1/2`,
+
+```
+-log2(eta) < 1+6*v*(1-z0).
+```
+
+Consequently the entirely rational choice
+
+```
+H0 = ceil((B*delta + 1+6*v*(1-z0))/(h*delta)),
+H1 = floor(H0*(1-z0)/(z-z0))+1
+```
+
+has the same strong-induction property as PR4405's improved compiler bound.
+For `x>=2^H0`, the exit coefficient is below `eta`. For smaller `x`, the
+elementary rank-one bound `L(x)<=x` supplies the induction base. Hence
+`L(x)<=A*x^z0`, with `A=(2^H0)^(1-z0)`, for every positive integer rank.
+The strict final inequality gives `L(2^H1)<(2^H1)^z`.
+
+The exact integers are
+
+```
+H0 = 2508580090000,
+H1 = 8701218254312470001,
+N  = 2^8701218254312470001.
+```
+
+The final strict power gap is `1/1400000000`. The verifier checks that
+`H0>=6*v`, so the induction threshold exceeds the whole finite base range.
+It rejects reducing either integer by one in these sufficient formulas.
+This is not a minimal-rank assertion.
+
+The tensorized mixed unit and the compiled uniform instruction word have
+not been materialized. They are finite exact program recipes, with fixed
+numbers of fields and instructions defined by the native tensor schedule,
+rank splitting, and source integer compiler. Their astronomically large
+constants make this an asymptotic construction.
+
+## Complete outer parameters
+
+Use the PR4405 slack family with the new `kappa`, `C_test`, and `uniform`:
+
+```
+b=kappa+5*d,  a=kappa+4*d,  q=kappa+3*d,
+epsilon=1-d, beta=1-3*d, r0=d/10, stop=d, layout=d.
+```
+
+Keep the admitted ordinary BIT saving `0.0027`, its atom exponent `0.003`,
+Strassen exponent `45/16`, theta power `2`, halo power `48`, and core
+power `128`. The new tested saving remains below `0.0027`, and the slack
+family's algebra applies with `C_test<0.0025` and `d<1/3000`.
+
+The frozen ledger function checks all 54 named outer predicates and three
+additional functional predicates. It verifies all nine cost rows exceed
+the proposed `kappa`. The main transform remains limiting, with exact gap
+
+```
+131141607 / 30625000000000000 > 0.
+```
+
+PR4405's separate audit compared the inherited 63-row ledger with this
+54-row ledger: all 50 common predicates and all nine cost formulas agreed;
+the 13 omitted entries only compared against older historical endpoints.
+Raising kappa preserves those comparisons. This package executes the
+current 54 admission/reporting predicates and three functional predicates;
+it does not rerun superseded historical suppliers.
+
+The new uniform word must use its own scalar, row-stock, movement, and
+semantic-precision constants under PR4359's `FINITE-TEMPLATE-CLOSURE.md`.
+No old small-word numerical constant is reused. The fixed rational
+`epsilon<1`, `beta<epsilon`, and positive stopping exponent leave the
+required absorption powers positive. The ordinary BIT template remains
+the source's independently compiled and covered template.
+
+## Replay and remaining work
+
+```
+python3 -I -O -B research/pr4359-tensor-strictification/endpoint.py
+```
+
+The script first runs the exact source-bound native and tensor certificates,
+then instantiates the new compiler rank and retained outer ledger. Mutation
+controls reject closed precision/uniform gaps, an unfunded outer point,
+and an insufficient BIT saving. The complete package verifier authenticates
+all source bytes and dependencies.
+
+The result remains conditional on the written tensor closure and the source
+native, storage, precision, analytic, fixed-tape, and recovery hypotheses.
+Finite arithmetic checks are not a full formal multiplication-theorem proof.
+Next questions include independently formalizing tensor closure, optimizing
+the tensor power/cutoff for smaller sufficient ranks, and improving the raw
+native invoice beyond these source-proven changes.
diff --git a/research/pr4359-tensor-strictification/EXPORTED-SOURCES.json b/research/pr4359-tensor-strictification/EXPORTED-SOURCES.json
new file mode 100644
index 0000000000..7aa2cceea7
--- /dev/null
+++ b/research/pr4359-tensor-strictification/EXPORTED-SOURCES.json
@@ -0,0 +1,170 @@
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+}
diff --git a/research/pr4359-tensor-strictification/NATIVE.md b/research/pr4359-tensor-strictification/NATIVE.md
new file mode 100644
index 0000000000..c2e53637db
--- /dev/null
+++ b/research/pr4359-tensor-strictification/NATIVE.md
@@ -0,0 +1,113 @@
+# Native all-fresh p3 controller
+
+This construction uses PR4359 at `da637c87c89482bc4c86be36bd0071b3d400dbed`.
+It inherits the all-column complex native contract, arbitrary spectators,
+whole-job coordinate adapters, finite precompiler and outer analytic premises.
+The new finite source-bound checks do not independently prove those premises.
+
+## Independent helper ownership
+
+Keep the exact seven-call p3/keep43 main word, its boundary scalar adapters,
+its 672 arbitrary main inputs and its exact rank25/rank57 endpoints. Each of
+its seven native invocations has 408 helper roles. Allocate a distinct fresh
+arbitrary field `G(stage,role)` for every one of these 2,856 occurrences.
+None is borrowed from a main idle field or another native invocation.
+
+The pinned full-identity native contract accepts all632 persistent fields
+arbitrarily and returns all408 helpers autonomously. Around each whole native
+invocation, use its authenticated helper input/output monomial normalizers.
+Their endpoint job on each fresh field is the helper's actual width, with no
+padding: widths5,6,7,8,9 have multiplicities28,18,56,93,213 per invocation.
+The native scalar word and all main contacts are unchanged. Every helper is
+an independent actual input and an autonomous output, not a copied main value
+or an owned center. Every original main endpoint remains independent of all
+helper inputs under the same all-column contract.
+
+The source checker reads every helper boundary and every original occurrence,
+checks all408 roles once per stage, and authenticates the native receipt and
+word bytes. The new target inventory is
+
+```
+T = {5:196, 6:126, 7:392, 8:651, 9:1491, 25:448, 57:224}.
+```
+
+There are3,528 independent productive fields:672 main and2,856 helper fields.
+
+## Full original idle jobs, with phases
+
+The original Laurent controller retains four idle windows. Pay each entire
+window as one job on each of its224 main owners:
+
+| Stages | Pair | Rank | Owners |
+| --- | --- | --- | --- |
+|0,1|1|16|224|
+|2|1|8|224|
+|4|2|8|224|
+|5,6|1|16|224|
+
+No helper borrows any portion of these paths. Each job is inserted at the
+original literal IDLE event; no bridge, local call or signed swap is moved.
+The original absolute phase is retained. For every one of896 owner/window
+records, the checker compares the full Z4 quadratic phase polynomial of the
+stage-product operator with the positive Gaussian factors on the recorded
+independent axes followed by the recorded translation. Binary elimination
+verifies the union rank. Whenever a nonzero translation is required, deleting
+it must fail the phase identity. These are exact polynomial identities, not
+sampled addresses. A fixed whole-job coordinate conjugation takes each
+independent axis list to its canonical rank8 or rank16 job.
+
+## Restore the original native rank-eight centers
+
+Each native invocation originally had eight rank-eight center jobs. PR4359
+divided each into consecutive rank-three/rank-five jobs to enforce its old
+child ceiling. Restore those56 original rank-eight jobs. Both orientations
+already have authenticated unsplit native replays. The checker requires
+adjacent center events on the same C0, complementary widths3 and5, and the
+identical intermediate frame; it reconstructs the original unsplit paid
+profile. The source phase certificate checks the product including absolute
+phase. A changed midpoint must fail the checker.
+
+The center remains explicitly owned, initialized and erased. It is never
+credited as another productive field. Restoring the unsplit job adds no
+scalar operation or external input. With all original idle jobs paid in full,
+the complete raw paid inventory is
+
+```
+P = {1:17409, 2:4991, 3:1568, 4:714, 5:140, 8:504, 16:448}.
+```
+
+It has25,774 calls, target rank mass47,075 and paid rank mass46,851. The exact
+rank surplus remains224. The raw unit has maximum paid rank16 and minimum
+target rank5, so it must not be passed directly to the old strict compiler.
+The separate tensor strictification makes the hierarchy valid.
+
+## Why all-fresh is the useful packing choice here
+
+Once whole idle jobs may remain unsplit before tensor strictification,
+hosting a helper of widtha in a rankc idle removes independent credit `a^t`
+and changes paid cost `c^t` to `(c-a)^t`. Its moment change is
+`c^t-(c-a)^t-a^t <= 0` for0<t<1 by concavity. The same argument applies to
+multiple disjoint hosted pieces. Thus all-fresh weakly dominates every such
+packing for this fixed controller under the new raw accounting. This is a
+comparison within the given native controller, not a universal bound.
+
+## Replacement scalar, movement, precision and stock constants
+
+The seven native words and scalar boundary controller are unchanged, so their
+scalar allowance remains1,147,304. Paying full idle jobs and giving every
+helper a fresh owner changes routing and storage; the old small supplier's
+movement or precision constants cannot be reused.
+
+For one tensor square, allocate3,528 separate outer centers, one per outer
+copy, and one separate inner center for each of25,774 outer paid rounds.
+These29,302 fields are a conservative finite owned allocation and receive
+no productive credit. The scalar allowance before added coordinate routes is
+`(3528+25774)*1147304`. For a general finite tensor power, apply the same
+allocation and count recursively. Keep every instantiated scalar operation,
+every complete job route and its inverse, and every initialized/erased owned
+field in the finite word. Recompute operation-count, prefix-growth and stock
+constants from that word, as explained by the transfer argument.
+
+The word can be specified by finite chronological loops without materializing
+its enormous instruction list. No exact largest prefix, huge template word
+or full-cover program is claimed to have been physically generated here.
diff --git a/research/pr4359-tensor-strictification/README.md b/research/pr4359-tensor-strictification/README.md
new file mode 100644
index 0000000000..ca3aeb0730
--- /dev/null
+++ b/research/pr4359-tensor-strictification/README.md
@@ -0,0 +1,98 @@
+# Conditional κ = 0.00247755 from native tensor composition
+
+This contribution improves the conditional construction in
+[PR #4359](https://github.com/CrocSwap/integer-mult-bounds/pull/4359), pinned at
+`da637c87c89482bc4c86be36bd0071b3d400dbed`, by changing its complete complex
+native invoice. The proposed exponent saving is **κ = 0.00247755**, an increase
+of **0.00017755**, approximately **7.7196%**, over the source's 0.0023.
+These are exponent-saving comparisons, not measured running-time gains.
+
+The result remains conditional on the source native, integer compiler,
+precision, storage, analytic, ordered-movement, fixed-tape, prime-padding,
+and exact-recovery hypotheses. It is a new written construction with finite
+exact checks, awaiting independent review. The selected frontier is unchanged.
+
+## What changes
+
+Give every helper in each of the seven p3 native calls its own arbitrary
+independent input. Keep its completed transform at its native width. Pay the
+four retained main idle windows as complete jobs, with their exact phase
+corrections, and retain the native rank-eight center calls without splitting.
+This yields 3,528 productive fields and the complete raw invoice
+
+```
+targets = {5:196, 6:126, 7:392, 8:651, 9:1491, 25:448, 57:224},
+paid    = {1:17409, 2:4991, 3:1568, 4:714, 5:140, 8:504, 16:448}.
+```
+
+The raw invoice has paid ranks above its smallest target, so it cannot be
+passed directly to the old compiler. The new construction first composes
+mixed units in lockstep: distinct copies are synchronized at each actual
+paid call, and a complete inner unit supplies those jobs before any parent
+resumes. Productive and paid moments multiply, while owned centers remain
+separate and uncredited. This is substitution of complete native programs;
+it does not assume that tensor powers distribute over addition.
+
+Next split only oversized paid jobs into consecutive transforms on disjoint
+axes. A separate rank-one tail inequality pays for every split. At tested
+saving **0.00247756**, the exact certificate uses tensor power 15,000 and
+cutoff `4^15000`, below the smallest target `5^15000`. The resulting finite
+mixed unit satisfies strict rank separation and a strict complete moment.
+The inherited integer compiler and the rational outer parameter family from
+[PR #4405](https://github.com/CrocSwap/integer-mult-bounds/pull/4405) then give
+the declared κ.
+
+## Proof and evidence
+
+- [NATIVE.md](NATIVE.md) binds all 2,856 fresh helper uses, unchanged 672 main
+  endpoints, 56 native center paths, and all 896 complete idle jobs. The
+  phase check retains nonzero translations; deleting them is rejected.
+- [TENSOR.md](TENSOR.md) proves the chronological tensor construction,
+  paid-job cutoff estimate, rank-one tail condition, and replacement
+  ownership/precision accounting.
+- [ENDPOINT.md](ENDPOINT.md) specifies the finite uniform rank and checks all
+  nine outer cost rows, 54 retained predicates, and three functional guards.
+- [RESULT.json](RESULT.json) is the deterministic exact certificate, including
+  source hashes and negative controls. A separate rational interval engine
+  cross-checks the moment and tail inequalities.
+
+The finite tensor word and resulting uniform template are specified by
+algorithms at astronomical fixed ranks. Their instruction lists and actual
+field counts have not been emitted. This is an asymptotic construction,
+not a practical implementation or a formal multiplication theorem.
+
+The complete original PR4359 replay, independently run for PR4405, passed
+all 23 stages and all 59 declared artifact comparisons. That baseline replay
+does not by itself validate this new construction; its new native bindings,
+proof and arithmetic are checked here.
+
+## Reproduction
+
+Python standard library only. Normal verification reads and compares the
+saved result; it does not regenerate source artifacts.
+
+```sh
+python3 -I -O -B research/pr4359-tensor-strictification/verify.py
+python3.12 -B scripts/research_package.py --package research/pr4359-tensor-strictification
+```
+
+`EXPORTED-SOURCES.json` authenticates focused unchanged source exports from
+PR4359 and two small executable dependencies from PR4405 at
+`873d1b3c96a83b701a5c0bba8e37a861e6672fef`. Each exported byte sequence was
+compared with its immutable source Git blob. Both source PRs are unmerged;
+their commits are not falsely represented as publication ancestry. The
+standard `SOURCES.json` retains main-branch task and selected-result provenance.
+
+Source construction: @zxv, PR4359. Mixed flow: Jacob Alan Sussman,
+PR4268 / Theorem 1G. The outer ledger and native predecessors retain the
+source documents' attribution and repository notices. New construction,
+proof, verification and review were prepared for @1337isnot1337 with substantial
+OpenAI Codex assistance. Apache-2.0; original notices are retained.
+
+## Open next questions
+
+[TASK.json](TASK.json) records first steps and blockers. Larger controllers
+have promising abstract tariffs but need full native phase and endpoint
+binding before any exponent claim. Smaller tensor powers, rational cutoffs,
+and tighter compiler estimates may reduce the sufficient ranks. Neither
+search is a premise of this submitted result.
diff --git a/research/pr4359-tensor-strictification/RESULT.json b/research/pr4359-tensor-strictification/RESULT.json
new file mode 100644
index 0000000000..d9f027061d
--- /dev/null
+++ b/research/pr4359-tensor-strictification/RESULT.json
@@ -0,0 +1,266 @@
+{
+  "endpoint": {
+    "compiler": {
+      "H0": 2508580090000,
+      "H1": 8701218254312470001,
+      "N_encoding": "2^8701218254312470001",
+      "delta": "1/1400000000",
+      "exit_coefficient_log2_upper": "41250",
+      "final_strict_power_gap": "1/1400000000",
+      "h": "1/8",
+      "materialized_tensor_rule_or_uniform_word": false,
+      "negative_log2_eta_upper": "31357247/140000",
+      "paid_max_bound_encoding": "4^15000",
+      "rank_one_induction": "47075^n > 46851^n, and splitting preserves total rank",
+      "t": "24938061/25000000",
+      "target_max_encoding": "57^15000",
+      "target_min_encoding": "5^15000",
+      "tensor_power": 15000,
+      "theta_upper": "7/8",
+      "z": "698265709/700000000",
+      "z0": "1396531417/1400000000"
+    },
+    "gain_over_PR4359": "3551/20000000",
+    "gain_over_PR4405": "8623/50000000",
+    "kappa": "49551/20000000",
+    "ledger_source_sha256": "d097e55f06c3f244f2074ffd7c1a0a5f3c4b4b6a6accf9619d1dde5c0ae85e15",
+    "minimum_outer_margin": "131141607/30625000000000000",
+    "negative_controls": {
+      "closed_epsilon": {
+        "actual_logarithmic_gamma_guard": "0",
+        "epsilon_below_one": "0",
+        "gaussian_above_main": "-325179/30625000000000000",
+        "global_gamma_power_slack": "-1/7000000000",
+        "new_semantic_guard_with_C1_one": "0",
+        "source_axis_superpolynomial": "0"
+      },
+      "closed_uniform_gap": {
+        "new_complex_uniform_saving_above_b": "0"
+      },
+      "insufficient_BIT": {
+        "actual_bit_above_effective": "-124289/700000000"
+      },
+      "unfunded_outer_point": {
+        "boundary_algebraic_matrix_above_kappa": "-131358393/30625000000000000",
+        "fused_gaussian_above_kappa": "-131575179/30625000000000000",
+        "global_coordinate_router_above_kappa": "-3/700000000",
+        "group_bit_router_above_kappa": "-1405202867/490000000000000000",
+        "main_transform_above_kappa": "-175108393/30625000000000000"
+      }
+    },
+    "outer": {
+      "functional_checks": {
+        "fixed_halo_explicit_metadata_branch": "13475000001/175000000",
+        "new_BIT_row_stock_sublinear_in_active_group": "697899997009/700000000000",
+        "new_semantic_group_guard": "3/700000000"
+      },
+      "margins": {
+        "boundary_algebraic_matrix": "75875143641607/30625000000000000",
+        "fused_gaussian": "75875143424821/30625000000000000",
+        "global_coordinate_router": "1734289/700000000",
+        "group_bit_router": "1214002994797133/490000000000000000",
+        "group_scans": "349999999/350000000",
+        "main_transform": "75875099891607/30625000000000000",
+        "microbox_face_corrections": "13475000001/175000000",
+        "polynomial_products": "699999999/700000000",
+        "sparse_union_movement": "12517281393641607/30625000000000000"
+      },
+      "parameters": {
+        "a": "1734289/700000000",
+        "b": "173429/70000000",
+        "beta": "699999997/700000000",
+        "bit_saving": "27/10000",
+        "cap_xi": "27829781349891607/30625000000000000",
+        "complex_uniform_saving": "1734291/700000000",
+        "d": "1/700000000",
+        "epsilon": "699999999/700000000",
+        "kappa": "49551/20000000",
+        "lambda_": "199504489/200000000",
+        "lambda_prime": "43641607/43750000",
+        "layout": "1/700000000",
+        "q": "108393/43750000",
+        "r0": "1/7000000000",
+        "stop": "1/700000000",
+        "tested_saving": "61939/25000000"
+      },
+      "strict_checks": {
+        "actual_bit_above_effective": "155711/700000000",
+        "actual_logarithmic_gamma_guard": "1/700000000",
+        "actual_logarithmic_group_guard": "2795218715733393/15312500000000000",
+        "alpha_logarithm_absorption": "1/7000000000",
+        "boundary_algebraic_matrix_above_kappa": "174891607/30625000000000000",
+        "cap_above_band": "12517281349891607/30625000000000000",
+        "cap_below_cell": "2795218606358393/30625000000000000",
+        "cap_below_phase_cell_power": "33420218650108393/30625000000000000",
+        "complex_above_effective": "1/700000000",
+        "complex_below_one_over_32": "2014071/70000000",
+        "complex_endpoint": "1/350000000",
+        "epsilon_below_one": "1/700000000",
+        "epsilon_positive": "699999999/700000000",
+        "fine_prime_halo_umbrella": "7/2",
+        "fused_gaussian_above_kappa": "174674821/30625000000000000",
+        "gaussian_above_main": "21766607/15312500000000000",
+        "global_coordinate_router_above_kappa": "1/175000000",
+        "global_gamma_power_slack": "9/7000000000",
+        "group_below_dimension": "1/350000000",
+        "group_bit_router_above_kappa": "3494797133/490000000000000000",
+        "group_gauge_power_slack": "2795218713545893/15312500000000000",
+        "group_key_fits_record": "1/350000000",
+        "group_positive": "699999997/700000000",
+        "group_scans_above_kappa": "698265713/700000000",
+        "halo_covers_generic_inverse_radius": "5",
+        "improvement_over_user_baseline": "373289790400418148631/500000000000000000000000",
+        "lambda_above_internal": "1/1400000000",
+        "lambda_above_sigma": "3/1400000000",
+        "lambda_above_tau": "1/1400000000",
+        "lambda_prime_above_lambda": "1/1400000000",
+        "lambda_prime_above_leaf": "139826571/49000000000000000",
+        "leaf_saving_above_effective": "69826571/49000000000000000",
+        "main_transform_above_kappa": "131141607/30625000000000000",
+        "membership_arithmetic_fits_record": "1/700000000",
+        "microbox_face_corrections_above_kappa": "53898265719/700000000",
+        "microbox_halo_volume": "55300000003/700000000",
+        "new_complex_uniform_saving_above_b": "1/700000000",
+        "new_semantic_guard_with_C1_one": "1/700000000",
+        "polynomial_products_above_kappa": "349132857/350000000",
+        "positive_layout_slack": "1/700000000",
+        "q_below_effective": "1/700000000",
+        "q_below_internal": "1/700000000",
+        "q_below_leaf": "139826571/49000000000000000",
+        "q_positive": "108393/43750000",
+        "short_record_fallback": "69826571/70000000",
+        "source_axis_superpolynomial": "1/700000000",
+        "sparse_union_movement_above_kappa": "12441406424891607/30625000000000000",
+        "stop_below_one": "699999999/700000000",
+        "stop_positive": "1/700000000",
+        "strassen_integer_power_gap": "1951441519231",
+        "support_router_span": "3/700000000",
+        "theta_positive_polynomial_power": "2",
+        "union_header_fits_record": "3/700000000",
+        "union_tag_work_per_visit_fits_record": "3/700000000"
+      }
+    },
+    "schema": "native-tensor-strictification-outer-endpoint/1",
+    "scope": "Source-bound native edits, a finite tensor recipe, exact moment and outer inequalities, and a specified finite uniform rank. Native tensor closure and the PR4359 inherited precision, storage, analytic, fixed-tape and recovery hypotheses are written premises, not a full formal theorem certificate.",
+    "status": "PASS_CONDITIONAL_TENSOR_OUTER_POINT",
+    "strictification_source_sha256": "87570ae4bfa02bec87bb1ff55c8429c5b8c6de1934f8088fe3dfb4f3317ceb93",
+    "tested_saving": "61939/25000000"
+  },
+  "independent_interval_crosscheck": true,
+  "scope": "Conditional native construction and exact finite arithmetic. The finite word and inherited physical, precision, analytic and fixed-tape proofs are not formally verified or materialized by this checker.",
+  "source_files": 18,
+  "status": "PASS_SOURCE_BOUND_CONDITIONAL_NATIVE_TENSOR_IMPROVEMENT",
+  "strictification": {
+    "alpha_interval": {
+      "lower": "0.99999999684043347848885218633343164",
+      "upper": "0.99999999684043347848885218633343165"
+    },
+    "alpha_upper": "999999999/1000000000",
+    "controls": {
+      "gamma_one_rejected": true,
+      "power_10000_fails_this_conservative_bound": true,
+      "raw_saving_003_rejected": true,
+      "unstrictified_native_tariff_rejected": true
+    },
+    "cutoff_base": 4,
+    "exponent": 15000,
+    "gamma_interval": {
+      "lower": "0.99896534203150283241622972954856204",
+      "upper": "0.99896534203150283241622972954856205"
+    },
+    "gamma_upper": "999/1000",
+    "materialized_tensor_word": false,
+    "native_inventory": {
+      "ambient_rank": 57,
+      "controls": {
+        "raw_rule_not_strictly_separated": true,
+        "whole_idle_missing_phase_rejected_count": 224,
+        "wrong_intermediate_center_rejected": true
+      },
+      "fresh_helper_fields": 2856,
+      "main_fields": 672,
+      "native_scalar_allowance": 1147304,
+      "paid": {
+        "1": 17409,
+        "16": 448,
+        "2": 4991,
+        "3": 1568,
+        "4": 714,
+        "5": 140,
+        "8": 504
+      },
+      "paid_calls": 25774,
+      "productive_fields": 3528,
+      "rank_surplus": 224,
+      "restored_native_centers": 56,
+      "schema": "native-all-fresh-full-idle-inventory/1",
+      "scope": "Exact finite all-fresh controller inventory. Native all-column closure is inherited; the raw rule requires the separate tensor strictification before the old strict compiler applies.",
+      "source_head": "da637c87c89482bc4c86be36bd0071b3d400dbed",
+      "source_sha256": {
+        "research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+        "research/kappa-rebuild/complex/fresh-certificates/THREE-base-HELPER-BOUNDARIES.json.gz": "f90517e395bde58c72cacd5d3481c4ea31d459fe4535384ac16547ac40460555",
+        "research/kappa-rebuild/complex/full-identity-certificates/RESULT.json": "943d5fad2fc529f470df05a35446e07ecaef7344abfce450d2c2eb2c9b3798f6",
+        "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz": "6418ec3da24d548a57f5c3fe5b9062c26c9478e638504a61c387f5cfb5f4ed61",
+        "research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz": "c54b0a3e129d28e6dbf74fa731fc3bb70fe3c6705f07256750fbecddcc842d39",
+        "research/kappa-rebuild/complex/full_identity_rebuild.py": "9af2ffbb0038c92e2a7d281081b0daaeadce75b11b566e5427845dc1af91e33d",
+        "research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz": "facfc6d6ecde151afb7e6a167776f3fdaac1c816e785a21a39f7f0bfd9140f3f",
+        "research/kappa-rebuild/controller/p3_native_cover.py": "fae71fcbbd20b9ee73b049cf6f43ea1d048217fb15feb6055a007e47da3363fe",
+        "research/kappa-rebuild/transfer-audit/p3_cover_audit.py": "a0fe9a26f77556e83b31b0c91fea909d6f3d8209d0f9f7ad4e5bc1375d33615e"
+      },
+      "status": "PASS_ACTUAL_ARBITRARY_HELPERS_AND_COMPLETE_PHASE_CORRECT_IDLE_JOBS",
+      "targets": {
+        "25": 448,
+        "5": 196,
+        "57": 224,
+        "6": 126,
+        "7": 392,
+        "8": 651,
+        "9": 1491
+      },
+      "whole_idle_profile": {
+        "16": 448,
+        "8": 448
+      }
+    },
+    "new_max_paid_encoding": "4^15000",
+    "new_max_target_encoding": "57^15000",
+    "new_min_target_encoding": "5^15000",
+    "normalized_strict_moment_upper": {
+      "lower": "0.99998560745522845202872016550689448",
+      "upper": "0.99998560745522845202872016550689449"
+    },
+    "paid": {
+      "1": 17409,
+      "16": 448,
+      "2": 4991,
+      "3": 1568,
+      "4": 714,
+      "5": 140,
+      "8": 504
+    },
+    "paid_rank_one_moment": 46851,
+    "schema": "native-tensor-cutoff-strictification/1",
+    "scope": "The exact finite construction is specified by synchronized native substitution and paid-axis splitting. This receipt proves its sufficient moment and separation inequalities, conditional on the documented native interfaces; it is not a formal whole-word or multiplication-theorem verification.",
+    "source_sha256": {
+      "research/kappa-rebuild/architecture/mixed_target_probe.py": "011c3c9d0cf81103dc84d709a0e36793eb7c5ae3f23c85eac27650ef13afa04a"
+    },
+    "status": "PASS_SOURCE_BOUND_TENSOR_STRICTIFICATION",
+    "strict_moment_verified": true,
+    "strictification_gap": {
+      "lower": "0.00001439254477154797127983449310551",
+      "upper": "0.00001439254477154797127983449310552"
+    },
+    "target_rank_one_moment": 47075,
+    "targets": {
+      "25": 448,
+      "5": 196,
+      "57": 224,
+      "6": 126,
+      "7": 392,
+      "8": 651,
+      "9": 1491
+    },
+    "tested_saving": "61939/25000000",
+    "theta_upper": "7/8"
+  }
+}
diff --git a/research/pr4359-tensor-strictification/SOURCES.json b/research/pr4359-tensor-strictification/SOURCES.json
new file mode 100644
index 0000000000..fcfba8aa31
--- /dev/null
+++ b/research/pr4359-tensor-strictification/SOURCES.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    }
+  ]
+}
diff --git a/research/pr4359-tensor-strictification/TASK.json b/research/pr4359-tensor-strictification/TASK.json
new file mode 100644
index 0000000000..8c7ec1849e
--- /dev/null
+++ b/research/pr4359-tensor-strictification/TASK.json
@@ -0,0 +1,96 @@
+{
+  "schema_version": 1,
+  "task_id": "supplier-boundary-audit",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "original_card": {
+    "id": "supplier-boundary-audit",
+    "question": "Can the #4088 four-provider dispatch satisfy the complete ordinary/outer admission contract?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/staged544-mixed-fourth-entry/README.md",
+      "research/scatter-boundary-decision/README.md"
+    ],
+    "methods": [
+      "Trace one hypothesis through the source, checker and complete caller.",
+      "Construct one semantically invalid but superficially well-formed fixture and test rejection."
+    ],
+    "acceptance_gate": "Pin the current sources and one obligation; the historical owner audit is an example, not evidence of a current bug.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read #4088 Scope and Open next questions: local words, native receipts and literal fill do not include a rerun of the mixed four-provider ordinary/outer engine. #3801 is the newer scatter-boundary decision; do not restart #2977 as if untouched.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Pin the provider registry, four native source closures and dispatch. Make a bounded acceptance coverage map or one registry adapter preserving every inherited admission predicate and rejection control (including an omitted fourth provider). Record remaining ordinary/outer obligations and a resource estimate before heavy execution. Stop at unresolved source/caller dependencies; no selected-result promotion from a local pass.",
+    "bottleneck_link": "Closing a current acceptance gap or proof obligation strengthens reliability even without changing kappa.",
+    "allowed_variations": [
+      "One provider-registry, dispatch, native-closure or admission-predicate obligation; separate branches must name distinct coverage."
+    ],
+    "scope_change_triggers": [
+      "Reporting the historical timing-only diagnostic as a new current acceptance bug.",
+      "Calling a malformed mutant a failure of the selected construction, or importing an opt-in repair without testing the exporter boundary.",
+      "Running another root enclosure instead of validating dispatch or weakening an inherited predicate to make mixed admission pass."
+    ],
+    "overlap_terms": [
+      "owner boundary",
+      "bridge omission",
+      "hosting",
+      "2977",
+      "scatter export",
+      "duplicate index",
+      "parity semantics",
+      "4088",
+      "3801",
+      "staged544-mixed-fourth-entry",
+      "scatter-boundary-decision"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4088: author-checked local words, native receipts, owner/phase fill and exact moment.",
+        "evidence": "research/staged544-mixed-fourth-entry/README.md",
+        "remaining": "Rebind and validate ordinary/outer admission with all four providers; full mixed release not established.",
+        "status": "author-checked",
+        "source_commit": "72d7efc181edb21806a878f174847c96dd410ab0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #3801: successor scatter-boundary decision package.",
+        "evidence": "research/scatter-boundary-decision/README.md",
+        "remaining": "Consult its exact acceptance scope before any new exporter audit; not proof of a selected failure.",
+        "status": "author-checked",
+        "source_commit": "911f8800ba7a4b03603ef62c5ffc6240e53cb5fd",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Construct an all-fresh native p3 productive-helper unit, then synchronize finite tensor powers and split oversized children to satisfy the strict mixed compiler and improve conditional kappa.",
+  "scope_change_reason": "Explicit user request to review and improve PR4359. After PR4405 recovered quantitative slack, this new construction changes the complete native inventory. The original card concerned historical PR4088.",
+  "bottleneck_link": "Strict child/target separation forced helper padding and small paid calls. Native tensor composition plus a certified rank-one tail bound permits whole paid jobs while preserving independent outputs and chronology.",
+  "budget_minutes": 120,
+  "stop_rule": "Reject a native change unless its complete input/endpoint and chronology follow from the exact source; reject any nonpositive moment, tail, compiler or outer margin. No selected-result promotion or unconditional theorem claim.",
+  "outcome": "completed",
+  "evidence": "NATIVE.md, TENSOR.md, ENDPOINT.md, source-bound native_inventory.py, strictification.py, endpoint.py and independently cross-checked RESULT.json.",
+  "author": "1337isnot1337 with OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance; four agents split native reconstruction, tensor proof, exact outer arithmetic and integration/independent replay.",
+  "open_questions": [
+    {
+      "question": "Can a larger native controller improve this complete all-fresh construction?",
+      "why_it_matters": "Stored abstract controller menus have favorable moments but require new physical phase and endpoint binding.",
+      "first_step": "Audit every contact, inverse, main endpoint and idle window for one fixed larger controller before importing its tariff.",
+      "blocked_by": "A complete native binding for that larger controller is not part of this result."
+    },
+    {
+      "question": "Can the sufficient tensor power and compiler rank be reduced?",
+      "why_it_matters": "The defined templates remain astronomical.",
+      "first_step": "Optimize rational cutoff and ratio bounds while preserving whole jobs, complete rank-one tail accounting and all outer inequalities.",
+      "blocked_by": "No globally minimal rank or practical implementation is established."
+    }
+  ]
+}
diff --git a/research/pr4359-tensor-strictification/TENSOR.md b/research/pr4359-tensor-strictification/TENSOR.md
new file mode 100644
index 0000000000..c7c28b4d25
--- /dev/null
+++ b/research/pr4359-tensor-strictification/TENSOR.md
@@ -0,0 +1,247 @@
+# Native tensor composition followed by strict rank cutoff
+
+This note constructs a new finite mixed supplier from the source-bound native
+inventory in [NATIVE.md](NATIVE.md). It then restores the strict rank separation
+required by PR4359's compiler. The native program and interfaces come from
+PR4359 at `da637c87c89482bc4c86be36bd0071b3d400dbed`; its integer lockstep
+compiler builds on Jacob Alan Sussman's Theorem 1G. Apache-2.0; substantial
+OpenAI/Codex assistance.
+
+The result is conditional on the source arbitrary-input native words,
+whole-job adapters, complete spectators, scalar and movement primitives, and
+finite-program closure. It does not infer native correctness from a moment
+histogram. The large new word is defined by finite algorithms and is not
+materialized here.
+
+## 1. A native product of two mixed units
+
+Let a unit U have productive roles indexed by i, with canonical target ranks
+`k_i`, and a chronological list of paid ranks `r_l`. Let a second unit V have
+roles indexed by j, target ranks `h_j`, and chronological paid ranks `s_a`.
+Every role is a distinct arbitrary complete array. Owned work fields have
+separate identities and are not counted as productive inputs.
+As in the source unit and its products, assume each native ambient rank is
+its maximum productive rank and every paid rank fits that ambient space.
+
+For each productive V role j, instantiate one U word at column multiplicity
+`h_j`. Its productive arrays are the grid `(i,j)`. Use one common ambient
+space of rank `max_i(k_i)*max_j(h_j)`, with the source whole-job embeddings
+and complete spectators. Each U copy receives its own owned work fields.
+
+Execute these U copies in lockstep through their common chronological paid
+list. The all-column word has the same list of operations at every positive
+column count. At paid position l, pause every U copy. Copy j requests one
+complete canonical job of rank `r_l*h_j` on its designated physical owner.
+These owners are distinct across j, since they belong to distinct U copies.
+They may be productive fields or U-owned centers holding correlated data.
+
+Use them as the productive roles of one V invocation at column multiplicity
+`r_l`. Its j-th target is exactly the required rank `h_j*r_l`. Apply the
+permitted complete-job coordinate matchers before the V call and undo them
+after its completed endpoint. These matchers align arbitrary equal-rank axis
+subsets; they do not relabel intermediate scalar contacts. The V call is
+correct on all its distinct arbitrary input arrays, including correlated
+values. Its own owned fields are fresh and distinct from every U field.
+
+Complete V, restore all its whole-job adapters, and only then resume the
+paused U copies. Never mix early and late requests of the same U copy in one
+V invocation. This establishes chronology and excludes cyclic dependencies.
+
+At the end, each `(i,j)` field has received the canonical rank `k_i*h_j`
+job. The remaining paid calls, in the actual nested order, have ranks
+`r_l*s_a`. Consequently, for any exponent tau,
+
+```
+target_moment(U product V) = target_moment(U)*target_moment(V),
+paid_moment(U product V)   = paid_moment(U)*paid_moment(V).
+```
+
+This is explicit substitution of complete array programs. Scalar gates
+remain scalar gates on roles. No claim that tensor power distributes over
+matrix addition is used.
+
+All intermediate U and V work fields persist until their actual owners clear
+them. A safe allocation is one full bank per instantiated unit; every bank
+is supplied, initialized, parked, and erased, with no productive credit.
+There are only finitely many banks and operations in a fixed product word.
+The source finite closure theorem must be applied with these new constants.
+
+Iterating this construction n times gives exactly `B^n` productive roles,
+target moments `T_tau^n`, and paid moments `P_tau^n`. The minimum and maximum
+target ranks become `k_min^n` and `m^n`. Each finite word accepts all further
+positive tensor multiplicities: lift every coordinate and every whole-job
+matcher, while retaining the same chronological scalar word.
+
+## 2. Restore strict separation by splitting only paid jobs
+
+Write `tau=1-C`, where `0<C<1`, and choose an integer cutoff `J` strictly
+below the new minimum productive target rank. For each paid job of rank
+`R>J`, write `R=q*J+r`, with `0<=r<J`. Replace its canonical transform by
+q complete rank-J transforms and, if nonzero, one rank-r transform on the
+remaining disjoint axes. These calls occur consecutively in place of the
+original paid job. The original whole-job phase/address adapters remain
+outside the complete replacement.
+
+The source splitting identity proves the same endpoint on arbitrary data
+and spectators, at every column count. No new productive input is created.
+The paid rank-one total is preserved. Concavity is needed only for the bill:
+
+```
+q*J^tau + r^tau <= R*J^(-C) + J^tau <= 2*R*J^(-C).
+```
+
+The last inequality uses `R>J`. The zero-remainder case omits the last call.
+Unsplit lower-rank calls retain their original charge. If `P_1` is the base
+unit's paid rank-one moment, the n-fold word therefore has complete paid
+tau-moment at most
+
+```
+P_tau^n + 2*P_1^n/J^C.
+```
+
+This deliberately overcounts the original cost of the replaced large jobs.
+It includes all paid calls, including those made on owned centers.
+
+Choose `J=b^n` for an integer `1<b<k_min`, and define
+
+```
+alpha = P_tau/T_tau,
+gamma = P_1/(b^C*T_tau).
+```
+
+If both alpha and gamma are strictly less than one, the normalized new
+moment is at most `alpha^n+2*gamma^n`, which is below one for an effective
+finite n. Every new paid rank is at most `b^n<k_min^n`, so PR4359's strict
+separation hypothesis now holds. This is an additional rank-one tail premise,
+not a consequence of the ordinary mixed moment inequality alone.
+
+One general effective choice uses rational upper bounds `alpha<=a<1` and
+`gamma<=g<1`: take an integer `n>max(1/(1-a),3/(1-g))`. The elementary bound
+`x^n<=1/(1+n*(1-x))` gives `a^n<1/2` and `g^n<1/4`. The present certificate
+is much smaller because it retains the exact rational power of g instead.
+
+## 3. Exact source-bound application
+
+The audited native construction gives each local helper its own arbitrary
+productive field and pays each retained main idle window in full, as proved
+in NATIVE.md. Its complete invoice is
+
+```
+targets = {5:196, 6:126, 7:392, 8:651, 9:1491, 25:448, 57:224},
+paid    = {1:17409, 2:4991, 3:1568, 4:714, 5:140, 8:504, 16:448}.
+```
+
+There are 3,528 productive fields. The exact rank-one moments are
+`T_1=47075`, `P_1=46851`, leaving the original surplus 224. The raw unit has
+paid rank 16 and productive rank 5, so it cannot directly use the source
+compiler's strict separation condition.
+
+At tested saving `C=0.00247756=61939/25000000`, exact rational elementary
+function enclosures prove
+
+```
+alpha < 999999999/1000000000,
+gamma = P_1/(4^C*T_tau) < 999/1000.
+```
+
+Use `n=15000`, `J=4^15000`. A directly verified rational inequality is
+
+```
+1/(1+15000/10^9) + 2*(999/1000)^15000 < 1.
+```
+
+The certified surplus is greater than `0.00001439`. Thus this finite word
+has a strict mixed moment at the tested saving, with
+
+```
+minimum productive rank = 5^15000,
+maximum productive rank = 57^15000,
+maximum paid rank       <= 4^15000.
+```
+
+The compiler's exit exponent satisfies
+`theta<=log(4)/log(5)<7/8`, by the exact integer inequality `4^8<5^7`.
+Unlike a cutoff only one below `5^15000`, this construction retains a fixed
+gap `h=1/8` for the subsequent compiler bound.
+
+Run the source-bound arithmetic certificate from the repository root:
+
+```sh
+python3 -I -B research/pr4359-tensor-strictification/strictification.py
+```
+
+The program first rechecks the native invoice. It then verifies both ratio
+bounds and the complete power inequality, rejects a failed smaller trial
+power under these particular conservative bounds, rejects gamma equal to
+one, and rejects the unsupported raw saving 0.003. These are scoped checks
+of the construction's sufficient inequalities; no minimal power is claimed.
+
+## 4. Applying the existing uniform compiler
+
+The new finite unit satisfies the same arbitrary-input, whole-job adapter,
+spectator, chronological, and owned-storage contracts, together with strict
+separation and a strict tested superpotential. Its positive rank-one surplus
+is `47075^15000-46851^15000`; paid splitting preserves that total. The source
+uniform compiler therefore applies at every strictly smaller positive saving.
+
+For an effective explicit bound, let `t=1-C`, choose a desired uniform
+exponent `z>t`, and put `z0=(t+z)/2`, `delta=z0-t`, and `h=1/8`. The source
+exit estimate with the new m and r_max gives, for `x>=m`,
+
+```
+m*E(x)/x <= 4^n*57^(n/8)*x^(-1/8)
+          < 2^((11/4)*n)*x^(-1/8),
+-log2(z0*m^(z0-1)) < 1+6*n*(1-z0),
+```
+
+where the second inequality uses `z0>1/2`. It follows that
+
+```
+H0 = ceil(((11/4)*n*delta+1+6*n*(1-z0))/(h*delta)),
+H1 = floor(H0*(1-z0)/(z-z0))+1,
+N  = 2^H1
+```
+
+is sufficient, provided `H0>=6*n` covers the finite base range. The endpoint
+certificate checks this condition and both strict inequalities at its chosen
+uniform saving. The proof is the same complement induction with the actual
+complement coefficient; it does not reuse the old exit constant 16.
+
+After choosing N, recalculate the finite unit count, productive width, paid
+leaf count, scalar word, owned storage, and precision/row constants from this
+new exact word. Every such quantity is fixed before the eventual input grows.
+The inherited analytic decomposition, permitted physical primitives,
+fixed-tape model, and outer transfer remain conditional premises.
+
+In particular, count raw native instances through both the tensor
+substitution and the final uniform compiler, rather than counting each
+large tensor macro as one unit. If this expanded count is `U_raw`, allocate
+`O=U_raw` complete owned center banks. Let `W` and `P` be the final uniform
+productive width and rank-one paid count. Count every non-scalar native
+record and complete normalization in the new expanded word as `R`; this
+includes the additional whole idle-window calls and every compiler/product
+adapter. The unchanged scalar words have allowance `1147304*U_raw`.
+Whole-job coordinate permutations and unit phases add no coefficient
+height, but the source's conservative eight-per-record allowance gives
+the safe new charge `G=1147304*U_raw+8*R`. All actual route work is charged
+separately. This uses the new count R instead of reusing the old 460 paid
+completion normalizers or old numerical G envelope. Substitute
+
+```
+Z = W+O,
+E = 64*(Z+N+P+G+1)^3,
+B = P+E,
+C0 = 32*N*B^2.
+```
+
+The source completed-child semantic recurrence is then
+`A(e)<=A(floor(e/N))+P*floor(e/N)+E`, giving `A(e)<=2*B*e` and the
+same form of linear outer guard. No single center per tensor macro is
+assumed. Allocated bank count, finite movement cost, and borrowed row stock
+all use the fully expanded word. Reusing the source small-word numerical
+constants without this substitution would not establish the precision or
+storage interface.
+
+Open next questions are to optimize the cutoff base and finite tensor power,
+reduce the effective uniform rank, and certify stronger native mergers. A
+favorable histogram alone would not justify any such native change.
diff --git a/research/pr4359-tensor-strictification/VERIFY.json b/research/pr4359-tensor-strictification/VERIFY.json
new file mode 100644
index 0000000000..0edea8eacd
--- /dev/null
+++ b/research/pr4359-tensor-strictification/VERIFY.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Authenticate immutable source exports; reconstruct actual native helper and idle owners; certify tensor/cutoff moment and independent interval bounds; derive an explicit uniform rank and all outer margins; reject source-bound and arithmetic controls."
+    }
+  ]
+}
diff --git a/research/pr4359-tensor-strictification/dependencies/compiler-refill/endpoint.py b/research/pr4359-tensor-strictification/dependencies/compiler-refill/endpoint.py
new file mode 100644
index 0000000000..340e1f0ffb
--- /dev/null
+++ b/research/pr4359-tensor-strictification/dependencies/compiler-refill/endpoint.py
@@ -0,0 +1,240 @@
+#!/usr/bin/env python3
+"""Exact endpoint refill conditional on the source PR4359 construction.
+
+Apache-2.0. Substantial OpenAI assistance. The mixed inventory and elementary
+function enclosure are zxv's PR4359; mixed-target flow is Jacob Alan Sussman's
+Theorem 1G. Outer formulas retain the attribution in PR4359 outer_parameters.py.
+This file checks arithmetic, not the native or all-size proof hypotheses.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import gzip
+import importlib.util
+import json
+import sys
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+PREFIX = Path('research/kappa-rebuild')
+OLD_BASELINE = Q('0.001730970419199163702738')
+KAPPA = Q('0.00230509')
+TESTED = Q('0.00230509019')
+ROOT_UPPER = Q('0.002306')
+
+
+def need(value, message):
+    if not value:
+        raise AssertionError(message)
+
+
+def read(path):
+    raw = path.read_bytes()
+    return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw)
+
+
+def ceil(value):
+    value = Q(value)
+    return -((-value.numerator) // value.denominator)
+
+
+def encode(value):
+    if isinstance(value, Q):
+        return str(value)
+    if isinstance(value, dict):
+        return {str(k): encode(v) for k, v in value.items()}
+    if isinstance(value, (tuple, list)):
+        return [encode(v) for v in value]
+    return value
+
+
+def family(kappa, tested):
+    """One explicit slack family approaching the same source mixed root."""
+    need(Q(0) < kappa < tested < ROOT_UPPER, 'Positive strictly ordered savings')
+    d = (tested-kappa)/7
+    return {'kappa': kappa, 'tested_saving': tested,
+            'complex_uniform_saving': kappa+6*d,
+            'b': kappa+5*d, 'a': kappa+4*d, 'q': kappa+3*d,
+            'bit_saving': Q('0.0027'), 'epsilon': 1-d, 'beta': 1-3*d,
+            'r0': d/10, 'stop': d, 'layout': d, 'd': d}
+
+
+def ledger(p):
+    """Recompute all 54 named PR4359 checks with freely supplied parameters."""
+    kappa, b, a, q = (p[key] for key in ('kappa', 'b', 'a', 'q'))
+    epsilon, beta, r0, stop, layout = (p[key] for key in
+                                      ('epsilon', 'beta', 'r0', 'stop', 'layout'))
+    sigma, tau = 1-b, 1-a
+    internal = tau+(1-stop)*max(sigma-tau, Q(0))
+    leaf = sigma+stop*(1-sigma)
+    lam, c = 1-(q+a)/2, Q(3, 32)
+    xi, key = epsilon*(1+q)-c+layout, beta+1-epsilon
+    margins = {
+        'main_transform': epsilon*q,
+        'global_coordinate_router': a,
+        'fused_gaussian': 1-epsilon+q*beta,
+        'group_bit_router': 1-epsilon+a*beta,
+        'group_scans': 1-epsilon+beta,
+        'boundary_algebraic_matrix': c+xi-epsilon,
+        'sparse_union_movement': xi+Q(1, 2)-epsilon,
+        'polynomial_products': epsilon,
+        'microbox_face_corrections': 1-(epsilon+beta+48-128+2),
+    }
+    checks = {
+        'actual_bit_above_effective': p['bit_saving']-a,
+        'complex_above_effective': b-a,
+        'q_positive': q, 'complex_endpoint': b-q, 'q_below_effective': a-q,
+        'stop_positive': stop, 'stop_below_one': 1-stop,
+        'leaf_saving_above_effective': (1-stop)*b-a,
+        'q_below_internal': 1-internal-q, 'q_below_leaf': 1-leaf-q,
+        'lambda_above_tau': lam-tau, 'lambda_above_sigma': lam-sigma,
+        'lambda_above_internal': lam-internal,
+        'lambda_prime_above_lambda': 1-q-lam,
+        'lambda_prime_above_leaf': 1-q-leaf,
+        'epsilon_positive': epsilon, 'epsilon_below_one': 1-epsilon,
+        'cap_above_band': xi-Q(1, 2), 'cap_below_cell': epsilon-xi,
+        'group_positive': beta, 'group_below_dimension': epsilon-beta,
+        'support_router_span': 1-beta,
+        'group_gauge_power_slack': 1-(2*xi-2)-r0-beta,
+        'actual_logarithmic_group_guard': 1-(2*xi-2)-beta,
+        'global_gamma_power_slack': 1-epsilon-r0,
+        'actual_logarithmic_gamma_guard': 1-epsilon,
+        'alpha_logarithm_absorption': r0,
+        'group_key_fits_record': 1-key,
+        'membership_arithmetic_fits_record': 1-beta-2*(1-epsilon),
+        'microbox_halo_volume': 128-48-beta,
+        'positive_layout_slack': layout,
+        'short_record_fallback': epsilon-a,
+        'complex_below_one_over_32': Q(1, 32)-b,
+        'strassen_integer_power_gap': Q(2**45-7**16),
+        'theta_positive_polynomial_power': Q(2),
+        'cap_below_phase_cell_power': 2-xi,
+        'halo_covers_generic_inverse_radius': Q(48-43),
+        'union_tag_work_per_visit_fits_record': 1-beta,
+        'union_header_fits_record': 1-beta,
+        'source_axis_superpolynomial': 1-epsilon,
+        'gaussian_above_main': margins['fused_gaussian']-margins['main_transform'],
+        'new_complex_uniform_saving_above_b': p['complex_uniform_saving']-b,
+        'improvement_over_user_baseline': kappa-OLD_BASELINE,
+        'new_semantic_guard_with_C1_one': 1-epsilon,
+        'fine_prime_halo_umbrella': 10-(2*2+2+Q(1, 2)),
+    }
+    checks.update({name+'_above_kappa': value-kappa for name, value in margins.items()})
+    # These are the additional functional conditions in the independent
+    # PR4359 outer_transfer_audit.py, beyond the 54-row parameter ledger.
+    functional = {
+        'new_semantic_group_guard': 1-beta,
+        'new_BIT_row_stock_sublinear_in_active_group': beta*(1-Q(3, 1000)),
+        'fixed_halo_explicit_metadata_branch': 128-48-epsilon-beta-1,
+    }
+    return {'parameters': dict(p, cap_xi=xi, lambda_=lam, lambda_prime=1-q),
+            'margins': margins, 'strict_checks': checks, 'functional_checks': functional}
+
+
+def compiler_rank(E, targets, paid, tested, uniform):
+    """The original sufficient bound, re-instantiated for the new exponent."""
+    m, kmin, rmax = max(targets), min(targets), max(paid)
+    need((m, kmin, rmax) == (57, 6, 5), 'Pinned complex separation')
+    t, z = 1-tested, 1-uniform
+    z0, h = (t+z)/2, Q(1, 10)
+    delta = z0-t
+    need(0 < t < z0 < z < 1, 'Compiler exponent order')
+    need(5**10 < 6**9 and 57 < 2**10, 'Exact source rank bounds')
+    need(z0 > Q(1, 2) and (1-z0)*E.log_interval(m)[1] < E.log_interval(2)[0],
+         'Original eta strictly exceeds 1/4')
+    H0 = ceil((3+4*delta)/(h*delta))
+    ratio = H0*(1-z0)/(z-z0)
+    H1 = ratio.numerator//ratio.denominator+1
+    need(H0 >= m.bit_length() and H0*h*delta >= 3+4*delta, 'Induction starts')
+    need(H1*(z-z0) > H0*(1-z0), 'Strict winning uniform rank')
+    need((H0-1)*h*delta < 3+4*delta, 'Reject one lower threshold exponent')
+    need((H1-1)*(z-z0) <= H0*(1-z0), 'Reject one lower final exponent')
+    return {'t': t, 'z0': z0, 'z': z, 'theta_upper': Q(9, 10),
+            'H0': H0, 'H1': H1, 'N_encoding': '2^H1',
+            'strict_final_gap': H1*(z-z0)-H0*(1-z0),
+            'winning_word_materialized': False,
+            'bound': 'Original sufficient inequality; a sharper bound is separate.'}
+
+
+def run(source_root):
+    base = source_root/PREFIX
+    interval_path = base/'architecture/mixed_target_probe.py'
+    spec = importlib.util.spec_from_file_location('pr4359_moment', interval_path)
+    E = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(E)
+    cover = read(base/'controller/P3-NATIVE-COVER.json.gz')['cover']
+    targets = {int(k): int(v) for k, v in cover['targets'].items()}
+    paid = {int(k): int(v) for k, v in cover['paid'].items()}
+    need(targets == {6: 192, 7: 56, 8: 93, 9: 1229, 25: 448, 57: 224},
+         'Exact source productive inventory')
+    need(paid == {1: 17623, 2: 5065, 3: 1738, 4: 714, 5: 254},
+         'Exact source paid inventory')
+    signed = Counter(targets)
+    signed.subtract(paid)
+    need(sum(k*v for k, v in signed.items()) == 224, 'Rank-one surplus')
+    bracket = {'lower': E.moment(signed, TESTED), 'upper': E.moment(signed, ROOT_UPPER)}
+    need(bracket['lower'][0] > 0 and bracket['upper'][1] < 0,
+         'Strict mixed test and upper root bound')
+
+    original = read(base/'controller/OUTER-PARAMETERS.json')['outer']
+    old = ledger({'kappa': Q('0.0023'), 'complex_uniform_saving': Q('0.002304'),
+                  'b': Q('0.002303'), 'a': Q('0.0023025'), 'q': Q('0.002302'),
+                  'bit_saving': Q('0.0027'), 'epsilon': Q('0.99999'),
+                  'beta': Q('0.99997'), 'r0': Q(1, 10**6),
+                  'stop': Q(1, 10**8), 'layout': Q(1, 10**6)})
+    for group in ('margins', 'strict_checks'):
+        frozen = {key: Q(value['rational']) for key, value in original[group].items()}
+        need(old[group] == frozen, 'Exact reproduction of every old '+group)
+    p = family(KAPPA, TESTED)
+    result = ledger(p)
+    need(all(v > 0 for v in result['strict_checks'].values()), 'All 54 new outer predicates')
+    need(all(v > 0 for v in result['functional_checks'].values()), 'Additional functional predicates')
+    need(min(result['margins'].values()) == result['margins']['main_transform'],
+         'Main transform remains limiting')
+    need(result['parameters']['cap_xi'] == Q(29, 32)+(1-p['d'])*p['q'],
+         'Cofinal cap simplification')
+    controls = {}
+    for label, updates in [('closed_epsilon_gap', {'epsilon': Q(1)}),
+                           ('missing_layout', {'layout': Q(0)}),
+                           ('unfunded_kappa', {'kappa': Q('0.002306')}),
+                           ('BIT_below_effective', {'bit_saving': OLD_BASELINE}),
+                           ('no_complex_slack', {'complex_uniform_saving': p['b']})]:
+        tested = ledger(dict(p, **updates))
+        failures = {k: v for k, v in tested['strict_checks'].items() if v <= 0}
+        need(failures, 'Mutation rejected: '+label)
+        controls[label] = failures
+    need(E.moment(signed, Q('0.004'))[1] < 0, 'Unsupported mixed target is rejected')
+    inputs = [interval_path, base/'controller/P3-NATIVE-COVER.json.gz',
+              base/'controller/OUTER-PARAMETERS.json']
+    return encode({'schema': 'pr4359-cofinal-outer-refill/1',
+                   'status': 'PASS_EXACT_CONDITIONAL_ENDPOINT_REFILL',
+                   'kappa': KAPPA, 'gain_over_PR4359': KAPPA-Q('0.0023'),
+                   'unchanged_paid': paid, 'unchanged_targets': targets,
+                   'mixed_test_saving': TESTED, 'mixed_test_margin': E.show(bracket['lower']),
+                   'root_upper_test': ROOT_UPPER, 'root_upper_margin': E.show(bracket['upper']),
+                   'old_ledger_crosscheck': {'matched_savings': len(old['margins']),
+                                           'matched_checks': len(old['strict_checks'])},
+                   'outer': result, 'limiting_margin': min(result['margins'].values())-KAPPA,
+                   'compiler': compiler_rank(E, targets, paid, TESTED, p['complex_uniform_saving']),
+                   'negative_controls': controls,
+                   'source_sha256': {str(p.relative_to(source_root)): sha256(p.read_bytes()).hexdigest()
+                                     for p in inputs},
+                   'scope': 'Conditional arithmetic and specified finite rank. The PR4359 native, compiler, cover, precision, fixed-tape and outer hypotheses remain premises; no full theorem or practical word is certified.'})
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--source-root', type=Path, default=HERE/'source')
+    parser.add_argument('--out', type=Path)
+    args = parser.parse_args()
+    result = run(args.source_root)
+    if args.out:
+        args.out.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n')
+    print(json.dumps({key: result[key] for key in
+                      ('status', 'kappa', 'gain_over_PR4359', 'limiting_margin', 'compiler')}, sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/pr4359-tensor-strictification/dependencies/compiler-refill/template_bound.py b/research/pr4359-tensor-strictification/dependencies/compiler-refill/template_bound.py
new file mode 100644
index 0000000000..ddc852c4fa
--- /dev/null
+++ b/research/pr4359-tensor-strictification/dependencies/compiler-refill/template_bound.py
@@ -0,0 +1,152 @@
+#!/usr/bin/env python3
+"""Exact tighter sufficient ranks for PR4359's conditional finite compiler.
+
+Standard library only; no generated file is modified unless --out is given.
+Apache-2.0. Substantial OpenAI/Codex assistance. The compiler and its inherited
+native premises are those of PR4359, based on Jacob Alan Sussman's Theorem 1G.
+"""
+import argparse
+from functools import lru_cache
+from fractions import Fraction as Q
+import json
+from math import factorial
+from pathlib import Path
+
+
+CASES = (
+    {"family": "complex", "targets": {6:192, 7:56, 8:93, 9:1229, 25:448, 57:224},
+     "paid": {1:17623, 2:5065, 3:1738, 4:714, 5:254},
+     "tested_saving": "0.002305", "uniform_saving": "0.002304",
+     "theta_upper": "9/10", "old_H0": 60000040, "old_H1": 276540184361},
+    {"family": "BIT", "targets": {8:3073, 22:444, 50:222},
+     "paid": {1:9261, 2:9394, 3:553, 4:308, 5:714, 6:77, 7:1465},
+     "tested_saving": "0.002769", "uniform_saving": "0.002760",
+     "theta_upper": "15/16", "old_H0": 10666731, "old_H1": 6552928412},
+)
+
+
+def need(ok, message):
+    if not ok:
+        raise AssertionError(message)
+
+
+@lru_cache(maxsize=None)
+def log_interval(value):
+    """Rational lower/upper bounds from the positive atanh series."""
+    value = Q(value)
+    need(value > 0, "logarithm needs positive argument")
+    if value < 1:
+        lo, hi = log_interval(1/value)
+        return -hi, -lo
+    power = 0
+    while value > 2:
+        value /= 2
+        power += 1
+    u = (value-1)/(value+1)
+    count = 96
+    lo = 2*sum((u**(2*j+1)/(2*j+1) for j in range(count)), Q(0))
+    tail = 2*u**(2*count+1)/((2*count+1)*(1-u*u))
+    hi = lo+tail
+    if power:
+        a, b = log_interval(Q(2))
+        lo, hi = lo+power*a, hi+power*b
+    return lo, hi
+
+
+def exp_interval(value):
+    need(0 <= value < Q(1,10), "small positive exponential argument")
+    count = 32
+    lo = sum((value**j/factorial(j) for j in range(count+1)), Q(0))
+    tail = value**(count+1)/factorial(count+1)/(1-value/(count+2))
+    return lo, lo+tail
+
+
+def moment_margin(targets, paid, saving):
+    """Enclose sum targets*k^(1-saving) - sum paid*r^(1-saving)."""
+    lo = hi = Q(0)
+    for rank in sorted(set(targets) | set(paid)):
+        count = targets.get(rank, 0)-paid.get(rank, 0)
+        a, b = log_interval(Q(rank))
+        lower = Q(rank)/exp_interval(saving*b)[1]
+        upper = Q(rank)/exp_interval(saving*a)[0]
+        lo += count*(lower if count >= 0 else upper)
+        hi += count*(upper if count >= 0 else lower)
+    return lo, hi
+
+
+def rank_bound(test_saving, uniform_saving, m, theta_bar):
+    """Conditional bound: caller supplies the source/exit recurrence hypotheses.
+
+    In particular m*E(n)/n < 16*n^(-(1-theta_bar)) and the strict tested
+    superpotential must hold. This function does not assert either premise.
+    """
+    test_saving, uniform_saving, theta_bar = map(Q, (test_saving, uniform_saving, theta_bar))
+    t, z = 1-test_saving, 1-uniform_saving
+    z0, h = (t+z)/2, 1-theta_bar
+    delta = z0-t
+    need(0 < t < z0 < z < 1 and 0 < h < 1 and m >= 2,
+         "strict rational savings and exit contraction")
+    log2_lo, log2_hi = log_interval(Q(2))
+    logm_lo, logm_hi = log_interval(Q(m))
+    invz_lo, invz_hi = log_interval(1/z0)
+    # eta=z0*m^(z0-1); these enclose -log(eta).
+    debt_lo = invz_lo+(1-z0)*logm_lo
+    debt_hi = invz_hi+(1-z0)*logm_hi
+    H0 = ((4+debt_hi/(delta*log2_lo))/h).__ceil__()
+    need(H0 >= m.bit_length(), "threshold covers the finite base range")
+    exit_gap = delta*(h*H0-4)*log2_lo-debt_hi
+    need(exit_gap >= 0, "exit coefficient is at most eta")
+    final_ratio = H0*(1-z0)/(z-z0)
+    H1 = final_ratio.__floor__()+1
+    final_gap = H1*(z-z0)-H0*(1-z0)
+    need(H1 > H0 and final_gap > 0, "strict winning-rank inequality")
+    # The interval width is small enough to determine these two integers.
+    need(debt_lo > delta*(h*(H0-1)-4)*log2_hi,
+         "one smaller H0 violates the same sufficient exit inequality")
+    need((H1-1)*(z-z0) <= H0*(1-z0),
+         "one smaller H1 fails the strict final inequality")
+    return {"H0": H0, "H1": H1, "t": str(t), "z0": str(z0), "z": str(z),
+            "delta": str(delta), "h": str(h), "m": m,
+            "eta_definition": "z0*m^(z0-1)",
+            "exit_condition": "delta*(h*H0-4)*log(2) >= -log(eta)",
+            "certified_exit_gap_positive": exit_gap > 0,
+            "strict_final_power_gap": str(final_gap),
+            "controls": {"H0_minus_one_rejected": True, "H1_minus_one_rejected": True}}
+
+
+def verify():
+    rows = []
+    for case in CASES:
+        m, kmin, rmax = max(case["targets"]), min(case["targets"]), max(case["paid"])
+        theta = Q(case["theta_upper"])
+        h = 1-theta
+        need(1 <= rmax < kmin <= m, "native mixed rank separation")
+        need(rmax**theta.denominator < kmin**theta.numerator, "exit exponent bound")
+        need(m**h.numerator < 2**h.denominator and 2*rmax <= 16,
+             "inherited exit coefficient constant 16")
+        surplus = sum(k*v for k,v in case["targets"].items())-sum(k*v for k,v in case["paid"].items())
+        need(surplus > 0, "rank-one potential proves L(n)<=n")
+        margin = moment_margin(case["targets"], case["paid"], Q(case["tested_saving"]))
+        need(margin[0] > 0, "strict source profile superpotential")
+        need(moment_margin(case["targets"], case["paid"], Q("0.004"))[1] < 0,
+             "unsupported .004 superpotential is rejected")
+        result = rank_bound(case["tested_saving"], case["uniform_saving"], m, theta)
+        need(result["H1"] < case["old_H1"], "strictly smaller sufficient fixed rank")
+        rows.append({**case, **result, "rank_one_surplus": surplus,
+                     "strict_superpotential_verified": True,
+                     "old_to_new_log_rank_ratio": str(Q(case["old_H1"], result["H1"])),
+                     "materialized_winning_word": False})
+    return {"status": "PASS_EXACT_TIGHTER_TEMPLATE_RANKS", "cases": rows,
+            "scope": "Conditional quantitative refinement of PR4359's finite compiler; unchanged uniform savings and inherited native, precision, and outer premises.",
+            "new_kappa_from_template_refinement": False}
+
+
+if __name__ == "__main__":
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--out", type=Path)
+    args = parser.parse_args()
+    result = verify()
+    serialized = json.dumps(result, indent=2, sort_keys=True)+"\n"
+    if args.out:
+        args.out.write_text(serialized)
+    print(serialized, end="")
diff --git a/research/pr4359-tensor-strictification/endpoint.py b/research/pr4359-tensor-strictification/endpoint.py
new file mode 100644
index 0000000000..bef5ec09a8
--- /dev/null
+++ b/research/pr4359-tensor-strictification/endpoint.py
@@ -0,0 +1,129 @@
+#!/usr/bin/env python3
+"""Conditional outer endpoint from the source-bound tensor strictification.
+
+Apache-2.0. Substantial OpenAI assistance. The native productive rule is zxv's
+PR4359. Integer mixed flow is Jacob Alan Sussman's Theorem 1G. The outer ledger
+is frozen from PR4405 and retains the predecessor attribution stated there.
+"""
+from fractions import Fraction as Q
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import importlib.util
+import json
+import sys
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+KAPPA = Q('0.00247755')
+TESTED = Q('0.00247756')
+
+
+def need(value, message):
+    if not value:
+        raise AssertionError(message)
+
+
+def module(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    result = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(result)
+    return result
+
+
+def encode(value):
+    if isinstance(value, Q):
+        return str(value)
+    if isinstance(value, dict):
+        return {str(k): encode(v) for k, v in value.items()}
+    if isinstance(value, (tuple, list)):
+        return [encode(v) for v in value]
+    return value
+
+
+def compiler_rank(exponent, tested, uniform):
+    """Explicit sufficient ambient rank after cutoff at 4^exponent."""
+    n = exponent
+    t, z = 1-tested, 1-uniform
+    z0, h = (t+z)/2, Q(1, 8)
+    delta = z0-t
+    need(n >= 1 and Q(1, 2) < t < z0 < z < 1, 'Strict compiler exponents')
+    need(4**8 < 5**7, 'log(4)/log(5) < 7/8')
+    need(4 == 2**2 and 57 < 2**6, 'Exact power bounds for exit and complement')
+    # r_max=4^n, k_min=5^n, m=57^n. Thus m*E(x)/x is strictly
+    # below 2^B*x^-h, with B=(2+6h)n. Also -log2(eta)<1+6n(1-z0).
+    B = (2+6*h)*n
+    debt = 1+6*n*(1-z0)
+    ratio = (delta*B+debt)/(h*delta)
+    H0 = -((-ratio.numerator)//ratio.denominator)
+    need(H0 >= 6*n, 'Initial threshold covers every base rank below m')
+    need(H0*h*delta >= delta*B+debt, 'Exact sufficient exit/complement inequality')
+    final_ratio = H0*(1-z0)/(z-z0)
+    H1 = final_ratio.numerator//final_ratio.denominator+1
+    need(H1 > H0 and H1*(z-z0) > H0*(1-z0), 'Strict final uniform moment')
+    need((H0-1)*h*delta < delta*B+debt, 'One lower H0 fails the stated sufficient bound')
+    need((H1-1)*(z-z0) <= H0*(1-z0), 'One lower H1 fails the final strict bound')
+    return {'tensor_power': n, 'target_min_encoding': '5^'+str(n),
+            'target_max_encoding': '57^'+str(n), 'paid_max_bound_encoding': '4^'+str(n),
+            'theta_upper': Q(7, 8), 'h': h, 't': t, 'z0': z0, 'z': z,
+            'delta': delta, 'exit_coefficient_log2_upper': B,
+            'negative_log2_eta_upper': debt, 'H0': H0, 'H1': H1,
+            'N_encoding': '2^'+str(H1),
+            'final_strict_power_gap': H1*(z-z0)-H0*(1-z0),
+            'rank_one_induction': '47075^n > 46851^n, and splitting preserves total rank',
+            'materialized_tensor_rule_or_uniform_word': False}
+
+
+def run(source_root, strict_result=None):
+    if strict_result is None:
+        strict_result = module('tensor_strictification', HERE/'strictification.py').run(source_root)
+    need(strict_result['strict_moment_verified'], 'Strict mixed moment was certified')
+    need(strict_result['cutoff_base'] == 4 and Q(strict_result['tested_saving']) == TESTED,
+         'Compiler inputs bind to the certified cutoff and saving')
+    ledger_path = HERE/'dependencies/compiler-refill/endpoint.py'
+    previous = module('pr4405_outer_ledger', ledger_path)
+    d = (TESTED-KAPPA)/7
+    p = {'kappa': KAPPA, 'tested_saving': TESTED,
+         'complex_uniform_saving': KAPPA+6*d, 'b': KAPPA+5*d,
+         'a': KAPPA+4*d, 'q': KAPPA+3*d, 'bit_saving': Q('0.0027'),
+         'epsilon': 1-d, 'beta': 1-3*d, 'r0': d/10,
+         'stop': d, 'layout': d, 'd': d}
+    outer = previous.ledger(p)
+    need(len(outer['strict_checks']) == 54 and len(outer['functional_checks']) == 3,
+         'Complete preserved outer predicate set')
+    need(all(v > 0 for v in outer['strict_checks'].values()), 'All 54 strict outer predicates')
+    need(all(v > 0 for v in outer['functional_checks'].values()), 'All three functional predicates')
+    need(min(outer['margins'].values()) == outer['margins']['main_transform'],
+         'Main transform remains limiting')
+    controls = {}
+    for label, updates in [('closed_epsilon', {'epsilon': Q(1)}),
+                           ('closed_uniform_gap', {'complex_uniform_saving': p['b']}),
+                           ('unfunded_outer_point', {'kappa': TESTED}),
+                           ('insufficient_BIT', {'bit_saving': Q('0.0023')})]:
+        tested = previous.ledger(dict(p, **updates))
+        failed = {k: v for k, v in tested['strict_checks'].items() if v <= 0}
+        need(failed, 'Reject '+label)
+        controls[label] = failed
+    rank = compiler_rank(strict_result['exponent'], TESTED, p['complex_uniform_saving'])
+    return encode({'schema': 'native-tensor-strictification-outer-endpoint/1',
+                   'status': 'PASS_CONDITIONAL_TENSOR_OUTER_POINT', 'kappa': KAPPA,
+                   'gain_over_PR4359': KAPPA-Q('0.0023'),
+                   'gain_over_PR4405': KAPPA-Q('0.00230509'),
+                   'tested_saving': TESTED, 'outer': outer, 'compiler': rank,
+                   'minimum_outer_margin': min(outer['margins'].values())-KAPPA,
+                   'negative_controls': controls,
+                   'ledger_source_sha256': sha256(ledger_path.read_bytes()).hexdigest(),
+                   'strictification_source_sha256': sha256((HERE/'strictification.py').read_bytes()).hexdigest(),
+                   'scope': 'Source-bound native edits, a finite tensor recipe, exact moment and outer inequalities, and a specified finite uniform rank. Native tensor closure and the PR4359 inherited precision, storage, analytic, fixed-tape and recovery hypotheses are written premises, not a full formal theorem certificate.'})
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--source-root', type=Path, default=HERE/'source')
+    parser.add_argument('--out', type=Path)
+    args = parser.parse_args()
+    result = run(args.source_root)
+    if args.out:
+        args.out.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n')
+    print(json.dumps({key: result[key] for key in
+                      ('status', 'kappa', 'gain_over_PR4359', 'minimum_outer_margin', 'compiler')}, sort_keys=True))
diff --git a/research/pr4359-tensor-strictification/native_inventory.py b/research/pr4359-tensor-strictification/native_inventory.py
new file mode 100644
index 0000000000..417fefe5fb
--- /dev/null
+++ b/research/pr4359-tensor-strictification/native_inventory.py
@@ -0,0 +1,214 @@
+#!/usr/bin/env python3
+"""Bind an all-fresh p3 native unit and its exact full idle operators.
+
+Finite source, ownership, contraction and exact-moment checks. The written
+all-column tensor construction and inherited transfer remain proof premises.
+Apache-2.0; substantial OpenAI/Codex assistance.
+"""
+from collections import Counter, defaultdict
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import gzip
+import json
+import sys
+
+sys.dont_write_bytecode = True
+PINS = {
+    'controller/P3-NATIVE-COVER.json.gz': 'facfc6d6ecde151afb7e6a167776f3fdaac1c816e785a21a39f7f0bfd9140f3f',
+    'controller/p3_native_cover.py': 'fae71fcbbd20b9ee73b049cf6f43ea1d048217fb15feb6055a007e47da3363fe',
+    'complex/full-identity-certificates/RESULT.json': '943d5fad2fc529f470df05a35446e07ecaef7344abfce450d2c2eb2c9b3798f6',
+    'complex/full_identity_rebuild.py': '9af2ffbb0038c92e2a7d281081b0daaeadce75b11b566e5427845dc1af91e33d',
+    'transfer-audit/p3_cover_audit.py': 'a0fe9a26f77556e83b31b0c91fea909d6f3d8209d0f9f7ad4e5bc1375d33615e',
+}
+
+
+def need(condition, message):
+    if not condition:
+        raise AssertionError(message)
+
+
+def read(path):
+    raw = path.read_bytes()
+    return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw)
+
+
+def counter(row):
+    return Counter({int(k): int(v) for k, v in row.items()})
+
+
+def paid_histogram(program):
+    return Counter(event[-1] for event in program if event[0] in ('climb', 'center'))
+
+
+def independent_axes(axes):
+    basis = {}
+    for axis in axes:
+        value = axis
+        while value:
+            pivot = value.bit_length()-1
+            if pivot in basis:
+                value ^= basis[pivot]
+            else:
+                basis[pivot] = value
+                break
+        if not value:
+            return False
+    return True
+
+
+def polynomial(terms, dimension):
+    linear = [0]*dimension
+    quadratic = [0]*dimension
+    for axis, coefficient in terms:
+        support = [i for i in range(dimension) if axis >> i & 1]
+        for i in support:
+            linear[i] = (linear[i]+coefficient) % 4
+            if coefficient % 2:
+                for j in support:
+                    if j > i:
+                        quadratic[i] ^= 1 << j
+    return tuple(linear), tuple(quadratic)
+
+
+def fuse_centers(program):
+    out, count, index = [], 0, 0
+    while index < len(program):
+        event = program[index]
+        if event[0] != 'center':
+            out.append(event)
+            index += 1
+            continue
+        following = program[index+1]
+        need(following[0] == 'center' and event[1] == following[1] == 0,
+             'Both adjacent children use the same owned center')
+        need(event[3] == following[2] and sorted((event[4], following[4])) == [3, 5],
+             'The exact intermediate frame cancels in the 3+5 center path')
+        out.append(['center', 0, event[2], following[3], 8])
+        index += 2
+        count += 1
+    need(count == 8, 'Exactly eight center paths in each native invocation')
+    return out
+
+
+def run(root):
+    base = root/'research/kappa-rebuild'
+    sources = {}
+    for name, wanted in PINS.items():
+        path = base/name
+        actual = sha256(path.read_bytes()).hexdigest()
+        need(actual == wanted, 'Exact original source pin: '+name)
+        sources[str(path.relative_to(root))] = actual
+    receipt = read(base/'controller/P3-NATIVE-COVER.json.gz')
+    cover = receipt['cover']
+    native_base = base/'complex/full-identity-certificates'
+    provider = read(native_base/'RESULT.json')['providers']['THREE-base']
+    for name, wanted in provider['artifacts'].items():
+        path = native_base/name
+        need(sha256(path.read_bytes()).hexdigest() == wanted, 'Exact native artifact '+name)
+        sources[str(path.relative_to(root))] = wanted
+    native = read(native_base/'THREE-base-FULL-IDENTITY-NATIVE.json.gz')
+    collapsed = {}
+    for direction, program in native.items():
+        collapsed[direction] = fuse_centers(program)
+        need(paid_histogram(collapsed[direction]) == counter(provider['unsplit_replays'][direction]['paid_profile']),
+             'Restored program has the pinned unsplit native paid profile')
+        need(all(row['product_exact_including_phase'] for row in provider['split_center_phase_recipes'][direction]),
+             'The original split replay checked absolute phase of every center path')
+
+    # Every helper now has its own independent physical input. The native
+    # all-column autonomous contract supplies the same main operator.
+    helper_path = root/provider['helper_boundary']['certificate']
+    helper_hash = provider['helper_boundary']['certificate_sha256']
+    need(sha256(helper_path.read_bytes()).hexdigest() == helper_hash, 'Actual complete helper boundaries')
+    sources[str(helper_path.relative_to(root))] = helper_hash
+    helper = read(helper_path)
+    widths = {row['role']: row['width'] for row in helper['recipes']}
+    need(set(widths) == set(range(224, 632)), 'All408 actual arbitrary-input helper roles')
+    need(provider['all_408_helpers_autonomous'] and provider['all_632_persistent_inputs_arbitrary'],
+         'Fresh helper inputs are independent arbitrary fields')
+    targets = Counter(row['target_rank'] for row in receipt['main_endpoint_normalizers'])
+    need(targets == Counter({25:448, 57:224}), 'Exact unchanged main endpoint census')
+    by_stage = defaultdict(list)
+    for row in cover['helper_bindings']:
+        by_stage[row['stage']].append(row)
+    for stage in range(7):
+        need(len(by_stage[stage]) == 408 and {row['role'] for row in by_stage[stage]} == set(widths),
+             'Every actual helper receives its own new productive owner')
+        for row in by_stage[stage]:
+            need(row['width'] == widths[row['role']], 'Native boundary backs the helper target width')
+            targets[row['width']] += 1
+    # Check the full idle operator, including its affine phase, directly.
+    phase_path = root/'research/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json'
+    if not phase_path.exists():
+        phase_path = root/'research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json'
+    phase_hash = 'c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa'
+    need(sha256(phase_path.read_bytes()).hexdigest() == phase_hash, 'Exact native port phase source')
+    sources[str(phase_path.relative_to(root))] = phase_hash
+    phases = read(phase_path)
+    need(len(phases['recipes']) == 112, 'All actual port recipes')
+    axes = [1 << i for i in range(57)]
+    idle = Counter()
+    missing_phase_rejected = 0
+    window_keys = set()
+    for window in cover['windows']:
+        key = window['index'], window['owner']
+        need(key not in window_keys, 'One whole idle job per actual window/owner')
+        window_keys.add(key)
+        q = phases['recipes'][window['port']]['q']
+        desired = []
+        for stage in window['stages']:
+            desired += ([(a,1) for a in axes[:9]]+[(q,-1)] if stage == 0
+                        else [(a,1) for a in axes[9+8*(stage-1):9+8*stage]])
+        actual = [(a,1) for a in window['axes']]+[(window['translation_after'],2)]
+        need(independent_axes(window['axes']), 'Whole idle axis list is independent')
+        need(polynomial(actual,57) == polynomial(desired,57), 'Exact whole idle operator including translation')
+        if window['translation_after']:
+            need(polynomial([(a,1) for a in window['axes']],57) != polynomial(desired,57),
+                 'Dropping the nonzero whole-idle phase correction fails')
+            missing_phase_rejected += 1
+        idle[len(window['axes'])] += 1
+    need(idle == Counter({8:448,16:448}), 'Allfour retained windows paid on all224 owners')
+    paid = idle.copy()
+    for direction in ('forward', 'backward', 'forward', 'backward', 'forward', 'backward', 'forward'):
+        paid.update(paid_histogram(collapsed[direction]))
+    need(targets == Counter({5:196,6:126,7:392,8:651,9:1491,25:448,57:224}), 'All-fresh native target census')
+    need(paid == Counter({1:17409,2:4991,3:1568,4:714,5:140,8:504,16:448}), 'Complete paid native and whole-idle tariff')
+    need(sum(k*v for k,v in targets.items())-sum(k*v for k,v in paid.items()) == 224,
+         'Rank surplus of the complete actual all-fresh word')
+    bad = [list(row) for row in native['forward']]
+    index = next(i for i, row in enumerate(bad) if row[0] == 'center')
+    bad[index+1][2] = -1
+    rejected = False
+    try:
+        fuse_centers(bad)
+    except AssertionError:
+        rejected = True
+    need(rejected, 'Wrong intermediate center frame is rejected')
+    return {
+        'schema': 'native-all-fresh-full-idle-inventory/1',
+        'source_head': 'da637c87c89482bc4c86be36bd0071b3d400dbed',
+        'status': 'PASS_ACTUAL_ARBITRARY_HELPERS_AND_COMPLETE_PHASE_CORRECT_IDLE_JOBS',
+        'source_sha256': sources,
+        'targets': dict(sorted(targets.items())), 'paid': dict(sorted(paid.items())),
+        'fresh_helper_fields': 2856, 'main_fields': 672,
+        'restored_native_centers': 56, 'whole_idle_profile': dict(sorted(idle.items())),
+        'ambient_rank': 57, 'productive_fields': 3528,
+        'paid_calls': sum(paid.values()), 'rank_surplus': 224,
+        'native_scalar_allowance': 1147304,
+        'controls': {'wrong_intermediate_center_rejected': rejected,
+                     'whole_idle_missing_phase_rejected_count': missing_phase_rejected,
+                     'raw_rule_not_strictly_separated': max(paid) >= min(targets)},
+        'scope': 'Exact finite all-fresh controller inventory. Native all-column closure is inherited; the raw rule requires the separate tensor strictification before the old strict compiler applies.',
+    }
+
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--root', type=Path, required=True)
+    parser.add_argument('--output', type=Path, required=True)
+    args = parser.parse_args()
+    result = run(args.root.resolve())
+    args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n')
+    print(json.dumps({key: result[key] for key in ('status', 'targets', 'paid', 'productive_fields', 'paid_calls')}))
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json b/research/pr4359-tensor-strictification/source/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json
new file mode 100644
index 0000000000..d9a4f84211
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json
@@ -0,0 +1 @@
+{"cover_correction":{"complete_small_cover_checks":[{"n":1,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":1},{"n":2,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":2},{"n":3,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":6},{"n":4,"nonzero_full_kernel_corrections":24,"orthogonal_matrices":48},{"n":5,"nonzero_full_kernel_corrections":600,"orthogonal_matrices":720}],"formula":"delta(g)_i = (weight(row_i(g))-1)/2 mod 2; apply X_delta(g) to EVERY final live stream in cover class g.","mandatory":true,"reason":"P_g C_all P_g^-1 = C_all X_delta(g) exactly. Orthogonality cancels all quadratic cross terms; both sides fix the uniform vector."},"global_phase_argument":"Every C_q and every linear address permutation fixes the uniform vector. Equality of the exact Z4 Fourier multiplier therefore fixes the scalar phase as 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diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md
new file mode 100644
index 0000000000..4badb40f09
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md
@@ -0,0 +1,313 @@
+# Finite precompilation of a mixed-target rule
+
+Status: constructive bridge under the precise native hypotheses below,
+independently audited in `../transfer-audit/AUDIT.md`. This is a written
+argument, not a formal proof or a completed integer-multiplication admission.
+The companion finite scheduler checks actual integer flow and ordering.
+The original immediate-normalization BIT interface failed hypothesis 3.
+`../bit/DELAYED-COVER-BRIDGE.md` supplies the distinct delayed-cover
+implementation, with a finite local-ring check and the native binding in
+`../bit/BIT-P3-NATIVE-BINDING.json`. Its fixed rational cover routes are a
+costed program schema, not dense coordinate gates in a plain native word.
+
+Substantial OpenAI assistance. Apache-2.0. The mixed-target asymptotic flow is
+Jacob Alan Sussman's Theorem 1G, PR4268, with the detailed proof at commit
+`563b4e6e913625637f36ba7869bf5fcd0dd4975a` of
+[`wht-power-saving-lean`](https://github.com/jacobalansussman/wht-power-saving-lean).
+The source, exact scheduling code and notices are preserved under `sources/`.
+The contribution proposed here is clearing the finite flow into a single
+uniform native supplier before applying the inherited outer theorem.
+
+## 1. What needs to be assumed
+
+Let a job of rank `s` apply a specified tensor operator to `s` selected
+coordinate blocks of one array, fixing all other coordinates. A block has
+an arbitrary positive tensor multiplicity `f`. A completed rank `s` job
+therefore acts on `s*f` axes. All productive inputs are arbitrary.
+
+Fix one mixed unit with productive targets `W_k` of ranks `k`, a chronological
+list of paid child ranks `r_j`, and a finite number of scalar and routing
+passes. These assumptions are required, separately from the moment inequality:
+
+1. **An all-column native word.** At every integer column count `c`, the unit
+   computes exactly its independent rank `k*c` output jobs. Its paid rounds
+   are rank `r_j*c` jobs in the stated order. All borrowed data return to the
+   declared endpoint, and owned storage is explicitly initialized and erased.
+   Its finite operation count is uniform in `c`.
+2. **Disjoint splitting and spectators.** A rank `a+b` job is a rank `a` job
+   followed by a rank `b` job on disjoint coordinates. The resulting programs
+   preserve arbitrary complete spectator fields and accept arbitrary data.
+3. **Whole-job adapters.** Any job passed between two units has an explicitly
+   priced, permitted embedding into its destination. For the intended complex
+   application these are fixed monomial binary address maps and unit quadratic
+   phases, together with role permutations. The adapters must work at every
+   tensor multiplicity. Rank equality by itself is insufficient.
+4. **Finite-program closure.** A finite composition of these words and
+   adapters is again a permitted finite supplier. Constantly many owned fields
+   may be supplied with their full cost and no productive output credit.
+   The eventual ordinary recursion must have the same arbitrary-input,
+   restoration, scalar and precision interface as the retained supplier.
+
+Put `m=max{k:W_k>0}`, `k_min=min{k:W_k>0}`, and
+`r_max=max{r_j}`. Require `r_max<k_min`.
+
+The hypothesis in item 3 is the material distinction between the two current
+applications. The complex source supplies canonical Gaussian-dyadic jobs
+after known monomial adapters. On the BIT side, a proper projector becomes
+`g P g^-1` under the GL cover; immediately converting each one to a weighted
+canonical query creates incompatible role-specific weights. The delayed
+construction instead routes complete cover fibers `g -> g*S`, keeps one
+external weight throughout the entire finite circuit, and lowers only its
+final rank-one leaves. All rational matchers, role routes and their inverses
+are fixed before choosing the odd prime and its exclusions. Every route
+uses complete equal-multiplicity cover fibers and the retained, costed row
+permutation primitive. This gives the same finite DAG argument in the
+rational-cover program category. The BIT binding supplies the actual native
+projectors; a rank count alone would not provide this implementation.
+
+## 2. Proposed conclusion
+
+Suppose the preceding hypotheses hold and the mixed balance has root
+`tau` in `(0,1)`:
+
+```
+sum_j r_j^tau = sum_k W_k k^tau.
+```
+
+For every `z` with `tau<z<1`, there exists a finite **uniform** supplier with
+some fixed ambient rank `N>=2`, a fixed number `B_N` of productive arrays,
+and only rank-one paid children, whose number `P_N` satisfies
+
+```
+P_N < B_N * N^z.
+```
+
+Every productive output is the full rank-`N` job. The construction may have
+an enormous finite scalar word, role inventory and owned-storage requirement.
+Those quantities affect constants and the eventual input threshold. They are
+not omitted and are not credited as extra output jobs.
+
+If this uniform supplier is admitted under the retained native, precision and
+fixed-tape premises, the existing uniform recursion applies directly. A new
+dynamic RAM-to-Turing-machine simulation is not being assumed here.
+
+## 3. The integer lockstep flow at a fixed rank
+
+This is the part of Theorem 1G used by the compiler. At one fixed top rank
+`N`, its units, source jobs and exit jobs have integer multiplicities.
+For a multi-index `a` on the target types define
+
+```
+c(a) = product_k k^(a_k),
+u(a) = multinomial(|a|;a) * product_k W_k^(a_k).
+```
+
+The exact Pascal identity is
+
+```
+u(b) = sum_{k:b_k>0} W_k*u(b-e_k).
+```
+
+Let `G` be the largest `|a|` with `m*c(a)<=N`. A class is indexed by a
+sequence of paid rounds `s` of length `d<=G`; its dilation is
+`rho_s=product r_j`. It contains `u(a)` units at column `rho_s*c(a)` for
+each `a` with `|a|=G-d` and `m*rho_s*c(a)<=N`.
+
+For one parent paid round, the Pascal bijection matches its job copies to
+target roles of the child class, at exactly equal integer ranks. Missing
+demands are top sources. Separation ensures every parent job is consumed.
+The source ranks lie in `(N/m,N]`; only the deepest classes produce exit
+jobs, all strictly smaller than `N` once `N` is outside the finite base range.
+
+Process a class in word order. At round `j`, suspend every unit of that
+class, complete the entire child class fed by that round, and only then
+advance the parents to round `j+1`. This forbids a unit from requiring both
+an earlier and a later round of the same suspended parent. The resulting
+dependency graph is finite and acyclic. Its edges carry complete array jobs,
+not individual scalar values copied for free.
+
+These integer quantities exist before normalizing the sources to total
+weight one. Let `S_s` be the number of source jobs of rank `s`, let
+`S=sum_s S_s`, and let the ordered exit batches be `(e,t)` with `t` jobs
+of rank `e` each. This finite integer flow is the compiler input.
+
+## 4. Clear every batch denominator constructively
+
+Induct on the integer top rank `N`. For each smaller rank `e` the compiler
+has already constructed a uniform program on `B_e` arbitrary arrays,
+computing full rank `e` with `P_e` paid rank-one calls. For a sufficiently
+small rank use the ordinary sequential program: `B_e=1`, `P_e=e`.
+
+For the current flow choose an integer `a` divisible by every `B_e` needed
+for an exit and every `B_(N-s)` needed for a nonzero source remainder.
+The product of those finitely many positive integers is a valid choice;
+their least common multiple is a smaller optional choice. Set
+
+```
+B_N = a*S.
+```
+
+Assign `a*S_s` of the input arrays to each source rank `s`. On each of those
+arrays, first complete its `N-s` complementary coordinates using the
+already-compiled smaller uniform program. All such batches are integral
+because `a` is divisible by their program widths. This preliminary action
+commutes with the intended `s`-coordinate job and fixes its input coordinates.
+
+Make `a` copies of the integer lockstep flow. Route its source arrays to
+their exact input slots using permitted whole-array adapters. At a deepest
+exit batch, `a*t` is divisible by `B_e`; replace that batch by copies of
+the smaller uniform rank-`e` program. Restore its role ordering and adapters
+before resuming the suspended parent class. Execute the finite DAG in the
+lockstep order, and finally restore the original source-array ordering.
+
+Every array has now received its disjoint complementary job and source job,
+hence the full rank-`N` transform. No copied scalar input or blank helper is
+treated as another independent output. Every original array occurs exactly
+once as a source; interior target slots are jobs on those arrays, not new
+productive inputs.
+
+Induction removes all recursive calls: the only remaining paid children
+have rank one. Clearing denominators multiplies productive arrays and paid
+operations equally, so it preserves the per-array leaf count. In formulas,
+
+```
+P_N/B_N = (1/S) * [
+    sum_s S_s * (P_(N-s)/B_(N-s))
+  + sum_(ordered exit batches e,t) t * (P_e/B_e)
+].
+```
+
+Zero-rank terms vanish. Any plain rank-one exits are the ordinary base
+program. Scalar and routing passes have their own analogous nonnegative
+count and must be included in the asymptotic cost used to choose `N`.
+
+This construction uses full integer batches; it does not assume that
+fractional pieces of a physical array can be routed at zero cost. Its exact
+replication factors can be impractically large, which is acceptable for a
+fixed finite supplier existence statement but must be disclosed.
+
+## 5. Why some fixed N has the required saving
+
+The independent audit sharpened this step to a direct count of paid leaves.
+It does not require a running-time comparison between a RAM flow and native
+tapes. Let `L(n)=P_n/B_n`, let `w_s=S_s/S`, and let `x_e` be the exit counts
+normalized by the number S of source arrays. The actual integer compiler
+satisfies
+
+```
+L(n) = sum_s w_s L(n-s) + sum_e x_e L(e),
+sum_s w_s = 1,
+sum_e x_e e^tau = sum_s w_s s^tau.
+```
+
+The last equality follows by canceling every internal matched job in the
+exact tau-potential balance. The source ranks satisfy `n/m<s<=n`. Put
+`theta=log(r_max)/log(k_min)<1`; the greatest exit rank is at most
+`E(n)=r_max*(n/m)^theta`. Choose `tau<z0<z<1` and set `delta=z0-tau`.
+Potential conservation yields
+
+```
+sum_e x_e e^z0 <= (m*E(n)/n)^delta * sum_s w_s s^z0.
+```
+
+That coefficient tends to zero. On the other hand, with
+`eta=z0*m^(z0-1)>0`, concavity gives
+
+```
+n^z0 - sum_s w_s (n-s)^z0 >= eta * sum_s w_s s^z0.
+```
+
+Indeed, for `t=s/n` in `[1/m,1]`,
+`1-(1-t)^z0 >= z0*t >= eta*t^z0`. For every sufficiently large n the
+exit coefficient is less than `eta/2`. Every complement and exit has
+smaller rank than n, so strong induction on the exact recurrence proves
+`L(n)<=A*n^z0`, with finite A chosen to cover the smaller ranks. Eventually
+`L(N)<N^z`.
+
+This is an effective finite construction: enumerate N, generate its exact
+integer recipe and count, and test the strict inequality. For rational
+`z=a/b`, the final test is the integer comparison
+`P_N^b < B_N^b*N^a`. The argument proves termination without requiring a
+numerical estimate of A. It may still produce an astronomically large
+template. The current work does not claim to have materialized the winning
+near-threshold instance.
+
+The result is materially different from rotating the same mixed roles
+through every array once. Such a rotation replaces `sum W_k*k^tau` by a
+smaller concave expression and does not retain the mixed saving. The
+hierarchical split, source supply and exit recursion are essential here.
+
+## 6. Tensor lift, owned storage and physical costs
+
+The precompiled rank `N` and every role/axis index are constants of the
+eventual program. Replace each one of its coordinate axes by `f` axes.
+Every rank-one leaf becomes the ordinary rank-`f` child. A fixed coordinate
+permutation becomes a fixed block permutation. Native binary monomial maps
+and quadratic unit phases lift column by column, with their exact absolute
+phase raised to the tensor multiplicity. The logical scalar program itself
+is not tensor-powered; it acts on the same fixed list of role arrays.
+
+Different suspended units may simultaneously retain live owned centers.
+One owned `C0` per recursion depth is therefore insufficient in general.
+The safe finite allocation is one owned center bank per live unit instance,
+or even per compiled unit instance. Retain each bank until its unit clears
+it. All are excluded from `B_N`, and all initialization, scalar operations,
+movement and erasure remain in the fixed scalar cost. A smaller reuse
+allocation requires its own lifetime proof.
+
+An owned center can be the physical field of a paid round. If its parent
+has just copied a nonzero linear combination into that center, its child
+receives that very field as one of its productive slots. Arbitrary-input
+child correctness applies equally to this correlated value. The child does
+not create a new independent source for that slot, and its own center has
+a different unit tag. Its other, missing source slots are supplied by the
+distinct top-level arrays counted in S. The parent resumes, reads or erases
+its center only after the entire child unit has completed. Thus COPY never
+duplicates an independent productive input, and a clear never acts on a
+descendant's live storage. These are field-identity and chronological
+requirements; statistical independence of current values is not required.
+
+Once the word is finite, the original depth-first fixed-tape recursion can
+reserve its fixed number of role/work tapes and use the usual parking stack.
+At one ordinary recursive node the number of child calls and stored fields
+is a constant, possibly enormous, so its actual scans contribute a fixed
+multiple of logical volume. This statement still requires the admitted
+primitive implementations for the fixed address/phase operations; it is
+not permission to treat an arbitrary address permutation as a linear pass.
+
+The new finite scalar count, frame matrices, role count, denominator heights,
+prefix norms and center inventory replace the old constants in the retained
+semantic precision and row-stock bounds. The strict child ratio is `1/N`.
+Fixed huge constants can be absorbed in an eventual threshold, but only
+after that substitution has been written. An old scalar cap cannot be reused
+merely because the exponent calculation is favorable.
+
+## 7. Present applications and separate bindings
+
+The strongest retained complex tariff is the p3, keep-mask43 construction
+in `P3-CONTROLLER-TARIFF-CERTIFICATE.json`. Its productive target ranks are
+`{6:192,7:56,8:93,9:1229,25:448,57:224}` and its complete paid profile is
+`{1:17623,2:5065,3:1738,4:714,5:254}`. Its exact mixed-root bracket is
+`[0.002305090194876170622919094755,
+  0.002305090194876170622919094756]`.
+The construction uses the full-identity suppliers and does not depend on
+the separate H-monodromy research. The preceding p2 and prefix-cut
+certificates remain preserved as independently reproducible earlier stages.
+
+The following are separate, necessary bindings rather than consequences
+of the moment inequality:
+
+- all global owner fields, residual idle axes and whole-helper adapters;
+- constant THREE/base selection, or the exact alternative dispatch;
+- every whole-job normalization used at a mixed class boundary;
+- all-column/tensor lift of the resulting complete word;
+- the owned-center inventory and updated scalar/precision/stock constants.
+
+For BIT, the actual p3 word is bound in `../bit/BIT-P3-NATIVE-BINDING.json`;
+its targets are `{8:3073,22:444,50:222}` and its paid profile is
+`{1:9261,2:9394,3:553,4:308,5:714,6:77,7:1465}`. The delayed whole-cover
+proof and its receipt bind the whole-fiber matching and final-leaf
+normalization. The ordinary two-call wrapper and its row-stock allowance
+must still be charged in the outer transfer. This note proves the finite
+compiler implication; the repository's separate outer-parameter receipt
+is where the common integer-multiplication exponent is checked.
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/identity_mixed_certificate.py b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/identity_mixed_certificate.py
new file mode 100644
index 0000000000..f4b51b358e
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/identity_mixed_certificate.py
@@ -0,0 +1,135 @@
+#!/usr/bin/env python3
+"""Source-bound, solver-free complete p2 full-identity recursive tariff.
+
+This certifies a finite invoice, strict separation, and its scalar root.
+It does not certify the global native owner construction, all-size mixed
+router, fixed-tape implementation, rounding induction, or outer kappa.
+Apache-2.0. Substantial OpenAI assistance; credits in README.md.
+"""
+from collections import Counter
+from decimal import Decimal, localcontext, ROUND_FLOOR
+from fractions import Fraction as Q
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import json
+import sys
+
+sys.dont_write_bytecode=True
+import mixed_target_probe as E
+import mixed_forest_search as F
+
+HERE=Path(__file__).resolve().parent
+ROOT=HERE.parents[2]
+NATIVE=ROOT/'research/kappa-rebuild/complex/full-identity-certificates/RESULT.json'
+PAIRS=((93,8,8),(56,7,9),(56,9,7),(19,5,9))
+MIDDLE=((29,0),(28,5),(18,6),(56,7),(93,8))
+
+def root_bracket(signed):
+    """Decimal discovery followed by rational elementary-function proof."""
+    with localcontext() as context:
+        context.prec=85
+        terms=[(Decimal(n.numerator)/Decimal(n.denominator),Decimal(r).ln())
+               for r,n in sorted(signed.items()) if n]
+        def value(c):
+            return sum(n*((1-c)*logr).exp() for n,logr in terms)
+        lo=Decimal(0);hi=Decimal('0.01')
+        for _ in range(200):
+            mid=(lo+hi)/2
+            if value(mid)>0:lo=mid
+            else:hi=mid
+        scale=10**30
+        integer=int((lo*scale).to_integral_value(rounding=ROUND_FLOOR))
+        left,right=Q(integer,scale),Q(integer+1,scale)
+        low_bounds=E.moment(signed,left)
+        high_bounds=E.moment(signed,right)
+        E.need(low_bounds[0]>0 and high_bounds[1]<0,'Exact strict root bracket')
+        return {'lower':str(Decimal(integer)/scale),
+                'upper':str(Decimal(integer+1)/scale),
+                'margin_at_lower':E.show(low_bounds),'margin_at_upper':E.show(high_bounds),
+                'uniqueness':'The paid/target ratio is strictly increasing in C because every paid rank is smaller than every target rank.'}
+
+def source_binding():
+    data=json.loads(NATIVE.read_text())
+    bound={str(NATIVE.relative_to(ROOT)):sha256(NATIVE.read_bytes()).hexdigest()}
+    for name in ('THREE-base','THREE-extra'):
+        row=data['providers'][name]
+        E.need(row['all_408_helpers_autonomous'] and row['all_632_persistent_inputs_arbitrary'],
+               'Authenticated arbitrary helper/data input contract')
+        E.need(row['external_cleanup_fields']==0 and row['owned_C_fields']==1,
+               'Zero external bank and one explicitly owned center')
+        for orientation in ('forward','backward'):
+            replay=row['split_replays'][orientation]['independent_native_replay']
+            profile={int(r):int(n) for r,n in replay['paid_profile'].items()}
+            E.need(profile==dict(E.split_profile(F.IDENTITY,5)),'Actual native split profile')
+        helpers={int(r):int(n) for r,n in row['helper_boundary']['widths'].items()}
+        E.need(helpers==dict(E.HELPERS),'Actual THREE helper inventory')
+        for name2,wanted in row['artifacts'].items():
+            path=NATIVE.parent/name2
+            E.need(sha256(path.read_bytes()).hexdigest()==wanted,'Native bytes '+name2)
+            bound[str(path.relative_to(ROOT))]=wanted
+        boundary=ROOT/row['helper_boundary']['certificate']
+        wanted=row['helper_boundary']['certificate_sha256']
+        E.need(sha256(boundary.read_bytes()).hexdigest()==wanted,'Helper boundary bytes')
+        bound[str(boundary.relative_to(ROOT))]=wanted
+    return bound
+
+def run(prefix_cut=False):
+    sources=source_binding()
+    solution=[]
+    for f in ((1,) if prefix_cut else (0,2)):
+        solution.extend({'family':f,'pattern':[a,b],'count':str(n)} for n,a,b in PAIRS)
+    solution.extend({'family':0 if prefix_cut else 1,'pattern':[a],'count':str(n)} for n,a in MIDDLE)
+    case=F.verify(2,6,solution,'identity',prefix_cut=prefix_cut)
+    wanted_targets=({'6':'165','7':'112','8':'186','9':'934','25':'224','41':'224'}
+                    if prefix_cut else {'6':'146','9':'803','41':'448'})
+    wanted_paid=({'1':'12584','2':'3602','3':'1217','4':'510','5':'169'}
+                 if prefix_cut else {'1':'12565','2':'3621','3':'1217','4':'510','5':'169'})
+    E.need(case['targets']==wanted_targets,'Literal output census')
+    E.need(case['paid']==wanted_paid,'Literal paid census')
+    out=F.serialize(case,('0.001730970419199163702738','0.001745','0.00175','0.00177','0.002','0.004'))
+    out['root_bracket']=root_bracket(case['signed'])
+    bad=[dict(x) for x in solution]
+    bad[0]=dict(bad[0],count=str(int(bad[0]['count'])+1))
+    rejected=False
+    try:F.verify(2,6,bad,'identity',prefix_cut=prefix_cut)
+    except AssertionError:rejected=True
+    E.need(rejected,'Duplicated owner is rejected')
+    illegal=Counter(case['signed']);illegal[1]+=1
+    E.need(sum(r*n for r,n in illegal.items())!=128,'Unfunded output rejected by rank budget')
+    return {'schema':'source-bound-identity-mixed-tariff/1',
+            'status':'PASS_EXACT_NATIVE_BOUND_PROFILE_AND_FINITE_RECURSIVE_INVOICE',
+            'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(),
+            'native_sources':sources,
+            'enclosure_source_sha256':sha256((HERE/'mixed_target_probe.py').read_bytes()).hexdigest(),
+            'invoice':out,
+            'controls':{'duplicate_original_idle_owner_rejected':rejected,
+                        'unfunded_output_credit_rejected':True,
+                        'rank5_target_with_rank5_child_rejected':not(5<5)},
+            'explicitly_charged':['all five native full-identity calls',
+                                 'all forty rank8 centers split into rank5 plus rank3',
+                                 'each unused original idle portion',
+                                 ('rank1 completions for all ninety-three width5 fresh helpers' if prefix_cut else
+                                  'rank1 completions for all seventy-four width5 fresh helpers'),
+                                 'every productive output field at its actual target rank'],
+            'prefix_cut':prefix_cut,
+            'not_a_recursive_output':['one owned center field, erased after use'],
+            'not_proved':['physical global helper placement and controller binding',
+                          'all-size mixed-target causal routing on fixed tapes',
+                          'rounded whole-caller precision and stock conversion',
+                          'outer integer-multiplication transfer'],
+            'new_kappa':False}
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--output',type=Path)
+    parser.add_argument('--prefix-cut',action='store_true')
+    args=parser.parse_args()
+    result=run(args.prefix_cut)
+    output=args.output or HERE/('IDENTITY-PREFIX-CUT-CERTIFICATE.json' if args.prefix_cut else
+                               'IDENTITY-MIXED-CERTIFICATE.json')
+    output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    print(json.dumps({'status':result['status'],'root':result['invoice']['root_bracket'],
+                      'margin_001745':result['invoice']['invoices']['0.001745']},sort_keys=True))
+
+if __name__=='__main__':main()
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/mixed_forest_search.py b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/mixed_forest_search.py
new file mode 100644
index 0000000000..9b5213bf3f
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/mixed_forest_search.py
@@ -0,0 +1,311 @@
+#!/usr/bin/env python3
+"""Column generation plus exact reconstruction for mixed-target idle forests.
+
+This searches p-pair controllers with two wide idle windows and one short
+window for each j=1,...,p-1.  A fresh productive job completes every unhosted
+helper.  The receipt checks exact rational multiplicities and the full
+displayed child profile; numerical optimality is not a proof of a global
+optimum.  Native frame/emission and outer supplier obligations remain open.
+Apache-2.0; substantial OpenAI assistance.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import lcm
+from pathlib import Path
+import argparse
+import json
+import sys
+
+import mixed_target_probe as E
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+IDENTITY=Counter({1:2487,2:713,3:224,4:102,5:20,8:8})
+WIDTHS=(0,5,6,7,8,9)
+
+def windows(p,prefix_cut=False):
+    """Literal pair-controller idle chronology, 224 fields per window."""
+    out=[]
+    for j in range(1,p):
+        if not prefix_cut:
+            out.append({'name':f'early{j}','capacity':16*j,'stages':list(range(2*j))})
+        out.append({'name':f'middle{j}','capacity':8,'stages':[2*j]})
+        out.append({'name':f'late{j}','capacity':16*j,'stages':list(range(2*p+1-2*j,2*p+1))})
+    expected=1792*(2*p+1)*(p-1)-(224*16*p*(p-1)//2 if prefix_cut else 0)
+    E.need(sum(224*w['capacity'] for w in out)==expected,'General idle rank identity')
+    return out
+
+def numerical_split(r,maximum,c):
+    return sum(n*k**(1-c) for k,n in E.split_rank(r,maximum).items())
+
+def reconstruct_exact(numbers,columns,families,resource,inventory):
+    """Recover an exact feasible basic solution, not rounded inventories.
+
+    Small denominators suffice for most screens.  Degenerate larger masters
+    can have a denominator above that guess; recover their active integer
+    linear system and independently replay it instead of waiving an equality.
+    """
+    support=[i for i,x in enumerate(numbers) if x>1e-7]
+    approximate=[Q(float(numbers[i])).limit_denominator(10**6) for i in support]
+    def equations():
+        rows=[];rhs=[]
+        for f in range(len(families)):
+            if not families[f].get('unbounded'):
+                rows.append([int(columns[i][0]==f) for i in support]);rhs.append(families[f].get('field_count',224))
+        for stage,r in resource:
+            row=[]
+            for i in support:
+                f,pattern=columns[i]
+                w=families[f]
+                row.append(sum(int(s==stage and v==r) for s,v in zip(w['stages'],pattern)))
+            usage=sum(x*float(numbers[i]) for x,i in zip(row,support))
+            if abs(usage-inventory[r])<1e-7:
+                rows.append(row);rhs.append(inventory[r])
+        return rows,rhs
+    rows,rhs=equations()
+    if any(sum(a*x for a,x in zip(row,approximate))!=b for row,b in zip(rows,rhs)):
+        n=len(support)
+        basis={}
+        for row,b in zip(rows,rhs):
+            vector=[Q(x) for x in row]+[Q(b)]
+            for pivot in sorted(basis):
+                factor=vector[pivot]
+                if factor:
+                    old=basis[pivot]
+                    for j in range(pivot,n+1):vector[j]-=factor*old[j]
+            pivot=next((j for j,x in enumerate(vector[:-1]) if x),None)
+            if pivot is None:
+                E.need(vector[-1]==0,'Active integer system consistent')
+            else:
+                factor=vector[pivot]
+                basis[pivot]=[x/factor for x in vector]
+        E.need(len(basis)==n,'Active support is uniquely reconstructed')
+        approximate=[Q(0)]*n
+        for pivot in sorted(basis,reverse=True):
+            row=basis[pivot]
+            approximate[pivot]=row[-1]-sum(row[j]*approximate[j] for j in range(pivot+1,n))
+        E.need(all(abs(float(x)-numbers[i])<1e-4 for x,i in zip(approximate,support)),
+               'Exact active solution agrees with numerical discovery')
+    E.need(all(x>=0 for x in approximate),'Reconstructed multiplicities nonnegative')
+    return [{'family':columns[i][0],'pattern':list(columns[i][1]),'count':str(x)}
+            for i,x in zip(support,approximate) if x]
+
+def discover(p,minimum,c,max_rounds=300,fresh_groups=False,prefix_cut=False,
+             custom_windows=None,custom_targets=None):
+    import numpy as np
+    from scipy.optimize import linprog
+    from scipy.sparse import coo_matrix
+    families=windows(p,prefix_cut) if custom_windows is None else [dict(w) for w in custom_windows]
+    calls=2*p+1
+    if fresh_groups:
+        families.append({'name':'fresh_group','capacity':2*minimum+8,
+                         'stages':list(range(calls)),'unbounded':True})
+    maximum=minimum-1
+    sp=lambda r:numerical_split(r,maximum,c)
+    h=lambda r:r**(1-c) if r else 0
+    v={r:h(max(r,minimum))-sp(max(0,minimum-r)) for r in WIDTHS[1:]}
+    v[0]=0
+    def gain(f,pattern):
+        if families[f].get('unbounded'):
+            used=sum(pattern)
+            return (h(max(used,minimum))-sp(max(0,minimum-used)) if used else 0)-sum(v[r] for r in pattern)
+        return sp(families[f]['capacity'])-sp(families[f]['capacity']-sum(pattern))-sum(v[r] for r in pattern)
+    columns=[]
+    keys=set()
+    def add(f,pattern):
+        key=(f,tuple(pattern))
+        if key not in keys:
+            keys.add(key)
+            columns.append(key)
+            return True
+        return False
+    for f,w in enumerate(families):
+        add(f,[0]*len(w['stages']))
+        for r in WIDTHS[1:]:
+            # Seed every constant-width prefix and suffix that fits.
+            n=min(len(w['stages']),w['capacity']//r)
+            if n:
+                add(f,[r]*n+[0]*(len(w['stages'])-n))
+                add(f,[0]*(len(w['stages'])-n)+[r]*n)
+    if not families:
+        return [],{'columns':0,'rounds':0,'discovery_converged':True}
+    resource=[(s,r) for s in range(calls) for r in WIDTHS[1:]]
+    rid={key:i for i,key in enumerate(resource)}
+    bounded=[f for f,w in enumerate(families) if not w.get('unbounded')]
+    bid={f:i for i,f in enumerate(bounded)}
+    done=False
+    largest_reduced=None
+    for iteration in range(max_rounds):
+        ar=[];ac=[];av=[]
+        er=[];ec=[];ev=[]
+        objectives=[]
+        for index,(f,pattern) in enumerate(columns):
+            if f in bid:
+                er.append(bid[f]);ec.append(index);ev.append(1)
+            for stage,r in zip(families[f]['stages'],pattern):
+                if r:
+                    ar.append(rid[stage,r]);ac.append(index);av.append(1)
+            objectives.append(-gain(f,pattern))
+        A=coo_matrix((av,(ar,ac)),shape=(len(resource),len(columns))).tocsr()
+        B=coo_matrix((ev,(er,ec)),shape=(len(bounded),len(columns))).tocsr()
+        result=linprog(objectives,A_ub=A,b_ub=[E.HELPERS[r] for s,r in resource],
+                       A_eq=B if bounded else None,b_eq=[families[f].get('field_count',224) for f in bounded] if bounded else None,bounds=(0,None),method='highs',
+                       options={'dual_feasibility_tolerance':1e-9,'primal_feasibility_tolerance':1e-9})
+        E.need(result.success,'Restricted master feasible')
+        pi=result.ineqlin.marginals
+        lam=result.eqlin.marginals
+        new=0
+        largest_reduced=0.0
+        for f,w in enumerate(families):
+            dp={0:(0.0,())}
+            for stage in w['stages']:
+                nxt={}
+                for used,(score,pattern) in dp.items():
+                    for r in WIDTHS:
+                        if used+r>w['capacity']:
+                            continue
+                        score2=score-v[r]+(pi[rid[stage,r]] if r else 0)
+                        if used+r not in nxt or score2>nxt[used+r][0]+1e-13:
+                            nxt[used+r]=(score2,pattern+(r,))
+                dp=nxt
+            def terminal(used):
+                if w.get('unbounded'):
+                    return h(max(used,minimum))-sp(max(0,minimum-used)) if used else 0
+                return sp(w['capacity'])-sp(w['capacity']-used)+lam[bid[f]]
+            ranked=sorted(((score+terminal(used),pattern)
+                           for used,(score,pattern) in dp.items()),reverse=True)
+            largest_reduced=max(largest_reduced,ranked[0][0])
+            # Several best used-capacity patterns reduce degeneracy.
+            for improvement,pattern in ranked[:4]:
+                if improvement>1e-8:
+                    new+=add(f,pattern)
+        if not new:
+            done=True
+            break
+    # Rational reconstruction checks feasibility independently of floating duals.
+    solution=reconstruct_exact(result.x,columns,families,resource,E.HELPERS)
+    verify(p,minimum,solution,'H',fresh_groups=fresh_groups,prefix_cut=prefix_cut,
+           custom_windows=custom_windows,custom_targets=custom_targets)
+    return solution,{'columns':len(columns),'rounds':iteration+1,
+                     'discovery_converged':done,'last_max_reduced_cost_float':largest_reduced,
+                     'optimality_scope':'Floating column-generation screen only; rational receipt certifies displayed construction, not global optimum.'}
+
+def verify(p,minimum,solution,kernel,fresh_groups=False,prefix_cut=False,
+           custom_windows=None,custom_targets=None):
+    E.need(kernel in ('H','identity'),'Known exact local invoice')
+    calls=2*p+1
+    maximum=minimum-1
+    families=windows(p,prefix_cut) if custom_windows is None else [dict(w) for w in custom_windows]
+    if fresh_groups:
+        families.append({'name':'fresh_group','capacity':2*minimum+8,
+                         'stages':list(range(calls)),'unbounded':True})
+    counts=[Q(0) for _ in families]
+    inventory=[Counter({r:Q(n) for r,n in E.HELPERS.items()}) for _ in range(calls)]
+    source=E.NATIVE if kernel=='H' else IDENTITY
+    paid=E.split_profile(source,maximum)
+    paid=Counter({r:Q(calls*n) for r,n in paid.items()})
+    if kernel=='H':
+        paid[1]+=8*calls
+    targets=(Counter({16*p+9-16*j:Q(224) for j in range(p)}) if prefix_cut
+             else Counter({16*p+9:Q(224*p)}))
+    if custom_targets is not None:
+        targets=Counter({int(r):Q(n) for r,n in custom_targets.items()})
+        E.need(sum(targets.values())==224*p,'Changed main targets have the original data field count')
+        expected_total=sum(r*n for r,n in targets.items())-sum(w.get('field_count',224)*w['capacity'] for w in families if not w.get('unbounded'))
+        old_total=224*p*(16*p+9)-sum(224*w['capacity'] for w in windows(p))
+        E.need(expected_total==old_total,'Literal removed windows match the changed main rank credit')
+    for row in solution:
+        f=row['family'];pattern=row['pattern'];count=Q(row['count'])
+        E.need(0<=f<len(families) and count>=0,'Pattern count')
+        w=families[f]
+        E.need(len(pattern)==len(w['stages']) and set(pattern)<=set(WIDTHS),'Window chronology')
+        E.need(sum(pattern)<=w['capacity'],'Window capacity')
+        counts[f]+=count
+        for stage,r in zip(w['stages'],pattern):
+            if r:
+                inventory[stage][r]-=count
+        if w.get('unbounded'):
+            E.need(sum(pattern)>0,'A fresh productive group consumes some actual helper')
+            targets[max(sum(pattern),minimum)]+=count
+            E.add_scaled(paid,E.split_rank(max(0,minimum-sum(pattern)),maximum),count)
+        else:
+            E.add_scaled(paid,E.split_rank(w['capacity']-sum(pattern),maximum),count)
+    E.need(all(n==w.get('field_count',224) for n,w in zip(counts,families) if not w.get('unbounded')),
+           'Every idle window accounted')
+    global_roles=Counter()
+    for stage,stock in enumerate(inventory):
+        E.need(all(n>=0 for n in stock.values()),'Stage inventory '+str(stage))
+        global_roles.update(stock)
+    for r,n in global_roles.items():
+        if n:
+            targets[max(r,minimum)]+=n
+            E.add_scaled(paid,E.split_rank(max(0,minimum-r),maximum),n)
+    paid=+paid;targets=+targets
+    E.need(max(paid)<min(targets),'Strict children')
+    signed=Counter(targets);signed.subtract(paid)
+    margin=sum(r*n for r,n in signed.items())
+    want=224*p-(8 if kernel=='H' else 64)*calls
+    E.need(margin==want,'Whole recursive rank credit minus all displayed costs')
+    lift=lcm(*(Q(row['count']).denominator for row in solution)) if solution else 1
+    return {'p':p,'kernel':kernel,'minimum_target_parameter':minimum,
+            'fresh_groups':fresh_groups,
+            'prefix_cut':prefix_cut,
+            'custom_main_targets':{str(r):str(n) for r,n in sorted(custom_targets.items())} if custom_targets is not None else None,
+            'replica_lift':lift,'targets':{str(r):str(n) for r,n in sorted(targets.items())},
+            'paid':{str(r):str(n) for r,n in sorted(paid.items())},
+            'signed':signed,'rank_margin':str(margin),
+            'global_helper_roles':{str(r):str(n) for r,n in sorted(global_roles.items())},
+            'strict_separation':{'max_child':max(paid),'min_target':min(targets)},
+            'patterns':solution,'windows':families}
+
+def serialize(case,targets=('0.001745','0.004')):
+    out={k:v for k,v in case.items() if k!='signed'}
+    out['invoices']={c:E.show(E.moment(case['signed'],Q(c))) for c in targets}
+    return out
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--min-p',type=int,default=1)
+    parser.add_argument('--max-p',type=int,default=8)
+    parser.add_argument('--min-k',type=int,default=5)
+    parser.add_argument('--max-k',type=int,default=12)
+    parser.add_argument('--C',default='0.001745')
+    parser.add_argument('--fresh-groups',action='store_true')
+    parser.add_argument('--prefix-cut',action='store_true')
+    parser.add_argument('--output',type=Path,default=HERE/'MIXED-FOREST-SEARCH.json')
+    args=parser.parse_args()
+    candidates=[]
+    for p in range(args.min_p,args.max_p+1):
+        for minimum in range(args.min_k,args.max_k+1):
+            solution,discovery=discover(p,minimum,float(args.C),fresh_groups=args.fresh_groups,
+                                        prefix_cut=args.prefix_cut)
+            for kernel in ('H','identity'):
+                case=verify(p,minimum,solution,kernel,fresh_groups=args.fresh_groups,
+                            prefix_cut=args.prefix_cut)
+                case['discovery']=discovery
+                candidates.append(case)
+        current=[case for case in candidates if case['p']==p and case['kernel']=='identity']
+        best=max(current,key=lambda x:E.numerical_value(x['signed'],float(args.C)))
+        print(json.dumps({'p':p,'identity_best_minimum':best['minimum_target_parameter'],
+                          'margin_float':E.numerical_value(best['signed'],float(args.C)),
+                          'replica_lift':best['replica_lift']}),flush=True)
+        # Preserve a useful receipt after each complete p, not only at the end.
+        winners={}
+        for kernel in ('H','identity'):
+            for count in range(args.min_p,p+1):
+                rows=[case for case in candidates if case['p']==count and case['kernel']==kernel]
+                win=max(rows,key=lambda x:E.numerical_value(x['signed'],float(args.C)))
+                winners[f'{kernel}_p{count}']=serialize(win)
+        receipt={'schema':'mixed-target-productive-forest/1','status':'PASS_EXACT_DISPLAYED_INVOICES',
+                 'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(),
+                 'enclosure_source_sha256':sha256((HERE/'mixed_target_probe.py').read_bytes()).hexdigest(),
+                 'discovery_C':args.C,'completed_p':p,'minimum_target_menu':[args.min_k,args.max_k],
+                 'best_in_finite_search':winners,
+                 'not_proved':['native variable-target owner emission and helper block adapters',
+                               'H caller closure; identity entries assume the source full-identity invoice really supplies the required helper contract',
+                               'all-size mixed recursion, fixed-tape router, finite precision and outer multiplication transfer'],
+                 'new_kappa':False}
+        args.output.write_text(json.dumps(receipt,indent=2,sort_keys=True)+'\n')
+
+if __name__=='__main__':
+    main()
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/mixed_target_probe.py b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/mixed_target_probe.py
new file mode 100644
index 0000000000..4fc05ab081
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/architecture/mixed_target_probe.py
@@ -0,0 +1,291 @@
+#!/usr/bin/env python3
+"""Exact tariffs for productive helpers with heterogeneous target sizes.
+
+This is a conditional recursion-contract probe, not a new kappa claim.  The
+native helper adapters, main controller closure, scalar/precision charges,
+and the mixed-target outer theorem remain separate obligations.
+
+Apache-2.0. Substantial OpenAI assistance. The original pruned native profile
+and helper inventory come from the preserved kappa-function-composition work.
+Mixed-target balance follows Jacob Alan Sussman's PR4268 proposal; this file
+does not reprove or admit its all-size or integer-multiplication transfer.
+The rational elementary-function enclosure follows the independently retained
+productive_helper_forest.py recovery certificate, with fresh source identity.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from functools import lru_cache
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import json
+import math
+import sys
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+NATIVE = Counter({1:2453, 2:716, 3:224, 4:74, 5:48})
+HELPERS = Counter({5:28, 6:18, 7:56, 8:93, 9:213})
+OLD_PAIRS = [(56,9,7),(56,7,9),(18,9,6),(18,6,9),
+             (16,9,5),(16,5,9),(44,8,8)]
+OLD_MIDDLE = [(93,8),(56,7),(18,6),(28,5),(29,0)]
+
+def need(ok, message):
+    if not ok:
+        raise AssertionError(message)
+
+@lru_cache(None)
+def log_interval(r):
+    def odd_series(x):
+        z = (x-1)/(x+1)
+        n = 80
+        low = 2*sum((z**(2*j+1)/(2*j+1) for j in range(n)), Q(0))
+        high = low+2*z**(2*n+1)/((2*n+1)*(1-z*z))
+        return low, high
+    exponent = r.bit_length()-1
+    a,b = odd_series(Q(2))
+    c,d = odd_series(Q(r,2**exponent))
+    return exponent*a+c, exponent*b+d
+
+def exp_interval(x):
+    need(0 <= x < 1, 'Enclosure range')
+    term = total = Q(1)
+    n = 48
+    for j in range(1,n+1):
+        term *= x/j
+        total += term
+    following = term*x/(n+1)
+    return total, total+following/(1-x/(n+2))
+
+@lru_cache(None)
+def h_interval(r,c):
+    if r in (0,1):
+        return Q(r),Q(r)
+    a,b = log_interval(r)
+    low,_ = exp_interval(c*a)
+    _,high = exp_interval(c*b)
+    return Q(r)/high,Q(r)/low
+
+def moment(profile,c):
+    low=high=Q(0)
+    for r,n in profile.items():
+        a,b=h_interval(r,c)
+        low += n*(a if n >= 0 else b)
+        high += n*(b if n >= 0 else a)
+    return low,high
+
+def show(bounds):
+    scale=10**35
+    low,high=bounds
+    a=low.numerator*scale//low.denominator
+    b=-((-high.numerator*scale)//high.denominator)
+    def fixed(n):
+        sign='-' if n<0 else ''
+        s=str(abs(n)).rjust(36,'0')
+        return sign+s[:-35]+'.'+s[-35:]
+    return {'lower':fixed(a),'upper':fixed(b)}
+
+def split_rank(r,maximum):
+    """Least concave tariff partition with positive integers <= maximum."""
+    need(maximum>=1 and r>=0, 'Legal split')
+    q,remainder=divmod(r,maximum)
+    result=Counter()
+    if q:
+        result[maximum]+=q
+    if remainder:
+        result[remainder]+=1
+    return result
+
+def split_profile(profile,maximum):
+    result=Counter()
+    for r,n in profile.items():
+        for child,count in split_rank(r,maximum).items():
+            result[child]+=n*count
+    return result
+
+def add_scaled(target,source,multiple=1):
+    for key,value in source.items():
+        target[key]+=multiple*value
+
+def base_case(p,minimum,pairs=None,middle=None):
+    """Expand all role inventories; one fresh global field per unhosted role."""
+    need(p in (1,2), 'Only reconstructed p1 and p2 schedules are in scope')
+    need(2<=minimum<=41, 'Finite minimum-target menu')
+    calls=2*p+1
+    main_fields=224*p
+    main_rank=16*p+9
+    inventories=[Counter(HELPERS) for _ in range(calls)]
+    paid=split_profile(NATIVE,minimum-1)
+    paid=Counter({r:calls*n for r,n in paid.items()})
+    paid[1]+=8*calls  # same-width H completion, never waived
+    targets=Counter({main_rank:main_fields})
+    if p==1:
+        need(pairs is None and middle is None, 'p1 has no original idle bins')
+    else:
+        pairs=pairs if pairs is not None else [(224,0,0)]
+        middle=middle if middle is not None else [(224,0)]
+        need(sum(n for n,a,b in pairs)==224, 'Each pair window has 224 bins')
+        need(sum(n for n,a in middle)==224, 'Middle window has 224 bins')
+        for stages in [(0,1),(3,4)]:
+            for n,a,b in pairs:
+                need(a+b<=16 and a in (0,5,6,7,8,9) and b in (0,5,6,7,8,9), 'Pair capacities')
+                for stage,r in zip(stages,(a,b)):
+                    if r:
+                        inventories[stage][r]-=n
+                add_scaled(paid,split_rank(16-a-b,minimum-1),n)
+        for n,r in middle:
+            need(r in (0,5,6,7,8), 'Middle capacity')
+            if r:
+                inventories[2][r]-=n
+            add_scaled(paid,split_rank(8-r,minimum-1),n)
+    global_roles=Counter()
+    for inventory in inventories:
+        need(all(n>=0 for n in inventory.values()), 'No helper role used twice')
+        global_roles.update(inventory)
+    for r,n in global_roles.items():
+        if not n:
+            continue
+        target=max(r,minimum)
+        targets[target]+=n
+        add_scaled(paid,split_rank(target-r,minimum-1),n)
+    paid=+paid
+    targets=+targets
+    need(max(paid)<min(targets), 'Strict child/target separation')
+    signed=Counter(targets)
+    signed.subtract(paid)
+    rank_credit=sum(r*n for r,n in targets.items())
+    rank_bill=sum(r*n for r,n in paid.items())
+    need(rank_credit-rank_bill==224*p-8*calls, 'Complete displayed rank balance')
+    return {'p':p,'minimum_target_parameter':minimum,'targets':dict(sorted(targets.items())),
+            'paid':dict(sorted(paid.items())), 'signed':signed,
+            'global_helper_roles':dict(sorted(global_roles.items())),
+            'global_productive_fields':sum(global_roles.values()),
+            'pair_patterns':pairs,'middle_patterns':middle,
+            'rank_credit':rank_credit,'rank_bill':rank_bill,'rank_margin':rank_credit-rank_bill,
+            'strict_separation':{'maximum_child':max(paid),'minimum_target':min(targets)}}
+
+def serialize_case(case):
+    result={k:v for k,v in case.items() if k!='signed'}
+    result['targets_at_C']={}
+    for text in ('0.001745','0.004'):
+        c=Q(text)
+        allowance=moment(case['signed'],c)
+        row={'after_native_completion_before_main_closure':show(allowance)}
+        for debt in (112,224):
+            row[f'after_{debt}_H1_head_debt']=show((allowance[0]-debt,allowance[1]-debt))
+        # A head may be cut only when it is a literal private first transform.
+        # Target shrink is charged; this is the strongest favorable eligibility grant.
+        rank=16*case['p']+9
+        for count in (112,224):
+            delta=Counter({1:count,rank-1:count,rank:-count})
+            gain=moment(delta,c)
+            row[f'literal_{count}_H1_cut_gain_over_paying_heads']=show(gain)
+            loss=Counter({rank:count,rank-1:-count})
+            a,b=moment(loss,c)
+            row[f'after_cutting_{count}_literal_H1_heads_and_shrinking_targets']=show((allowance[0]-b,allowance[1]-a))
+        result['targets_at_C'][text]=row
+    return result
+
+def numerical_value(profile,c):
+    return sum(n*r**(1-c) for r,n in profile.items())
+
+def discover_pair_patterns(minimum,c):
+    """Optional small MILP; exact replay never relies on solver's optimal flag."""
+    import numpy as np
+    from scipy.optimize import milp, Bounds, LinearConstraint
+    widths=(0,5,6,7,8,9)
+    patterns=[(a,b) for a in widths for b in widths if a+b<=16]
+    h=lambda r:r**(1-c) if r else 0.0
+    sp=lambda r:sum(n*h(k) for k,n in split_rank(r,minimum-1).items())
+    value=lambda r:0.0 if not r else h(max(r,minimum))-sp(max(0,minimum-r))
+    gain=np.array([sp(16)-sp(16-a-b)-value(a)-value(b) for a,b in patterns])
+    rows=[np.ones(len(patterns))]
+    lower=[224.0]
+    upper=[224.0]
+    for side in (0,1):
+        for r in widths[1:]:
+            rows.append(np.array([float(pair[side]==r) for pair in patterns]))
+            lower.append(0.0)
+            upper.append(float(HELPERS[r]))
+    out=milp(c=-gain,integrality=np.ones(len(patterns)),bounds=Bounds(0,np.inf),
+             constraints=LinearConstraint(np.array(rows),lower,upper),
+             options={'time_limit':10,'mip_rel_gap':0})
+    need(out.x is not None, 'Discovery produced a candidate')
+    pairs=[(int(round(n)),a,b) for n,(a,b) in zip(out.x,patterns) if round(n)>0]
+    # Middle bins are independent single slots; choose positive-gain roles first.
+    candidates=sorted(((sp(8)-sp(8-r)-value(r),r) for r in (5,6,7,8)),reverse=True)
+    middle=[]
+    left=224
+    for gain,r in candidates:
+        if gain>1e-12:
+            n=min(HELPERS[r],left)
+            if n:
+                middle.append((n,r))
+                left-=n
+    if left:
+        middle.append((left,0))
+    return pairs,middle,{'status':int(out.status),'message':out.message,'not_an_exact_optimality_claim':True}
+
+def run(discover=False):
+    candidates=[]
+    for p in (1,2):
+        for minimum in range(2,26):
+            candidates.append(base_case(p,minimum))
+    candidates.append(base_case(2,9,OLD_PAIRS,OLD_MIDDLE))
+    discovery=[]
+    if discover:
+        for c in (.001745,.004):
+            for minimum in range(5,18):
+                pairs,middle,meta=discover_pair_patterns(minimum,c)
+                case=base_case(2,minimum,pairs,middle)
+                case['discovery_target']=str(c)
+                case['solver']=meta
+                discovery.append(case)
+        candidates.extend(discovery)
+    best={}
+    for p in (1,2):
+        for text in ('0.001745','0.004'):
+            c=Q(text)
+            rows=[case for case in candidates if case['p']==p]
+            ranked=sorted(rows,key=lambda case:-numerical_value(case['signed'],float(c)))
+            winner=ranked[0]
+            # Certify that this displayed winner beats all numerically distinct
+            # profiles in the finite scanned menu, not all possible schedules.
+            win=moment(winner['signed'],c)
+            for case in ranked[1:]:
+                if dict(case['signed'])!=dict(winner['signed']):
+                    need(win[0]>moment(case['signed'],c)[1], 'Exact finite-menu ordering')
+            best[f'p{p}_{text}']=serialize_case(winner)
+    controls={}
+    wrong=base_case(1,6)
+    wrong['targets'][6]+=1
+    controls['unearned_output_credit_rejected']=(sum(r*n for r,n in wrong['targets'].items())-wrong['rank_bill']!=wrong['rank_margin'])
+    controls['rank5_child_with_job5_rejected']=not(5<5)
+    need(all(controls.values()), 'Corruption controls')
+    return {'schema':'productive-helper-mixed-target-tariff/1',
+            'status':'PASS_CONDITIONAL_EXACT_ACCOUNTING',
+            'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(),
+            'native_profile':dict(NATIVE),'helper_inventory':dict(HELPERS),
+            'assumed_recursive_rule':'sum_paid n_r r^(1-C) < sum_targets W_k k^(1-C); max_paid_rank < min_target_rank',
+            'best_in_finite_menu':best,
+            'fixed_p2_minimum9':serialize_case(base_case(2,9,OLD_PAIRS,OLD_MIDDLE)),
+            'discovery_candidates':[serialize_case(c) for c in discovery],
+            'controls':controls,
+            'not_proved':['native whole-helper adapters and arbitrary-input block diagonal contract',
+                          'correct main controller and all its heads/caches/cleanup',
+                          'literal eligibility of any displayed head-cut grant',
+                          'mixed-target all-size recursion in this native model',
+                          'finite-precision integer-multiplication transfer and scalar/stock costs'],
+            'new_kappa':False}
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--discover',action='store_true')
+    parser.add_argument('--output',type=Path,default=HERE/'MIXED-TARGET-PROBE.json')
+    args=parser.parse_args()
+    result=run(args.discover)
+    args.output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    print(json.dumps({'status':result['status'],'best':{k:v['targets_at_C'][k.split('_',1)[1]]['after_native_completion_before_main_closure'] for k,v in result['best_in_finite_menu'].items()}},sort_keys=True))
+
+if __name__=='__main__':
+    main()
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json
new file mode 100644
index 0000000000..fbe6da7241
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json
@@ -0,0 +1,69 @@
+{
+  "C0_before_cover": "69/25000",
+  "C0_minus_C1": "1/100000",
+  "C1_after_cover": "11/4000",
+  "atom_power": "3/1000",
+  "finite_constants_to_replace": {
+    "ambient_dimension": "N, not50 or64",
+    "bad_fallback_children": "At most32*N^2 per bad final leaf",
+    "coefficient_data": "Every fixed rational entry, inverse, determinant, prefix descriptor and height",
+    "fixed_row_routes": "R, original port charts and every internal matcher/inverse",
+    "generic_affine_work": "At most4*N^2-2*N basic shears per rank1 normalizer together with its inverse",
+    "owned_storage": "C, bounded by one owned bank per allocated finite unit instance",
+    "paid_leaf_count": "P, including copied-center descendants",
+    "pointwise_and_copy_count": "L, literal compiled scalar/copy/erase records",
+    "productive_roles": "B, not3739 or661"
+  },
+  "finite_template_hypothesis": "Fixed integers N>=2,B>=1,P>=1 and a whole-cover native schematic uniform word, with P/B < N^(1-C0), final paid leaves of rank1, arbitrary productive inputs and separately owned storage.",
+  "mixed_margin_at_C0": {
+    "display": 0.7944031286900184,
+    "lower": "67580344230009584908122845692691165477/85070591730234615865843651857942052864",
+    "strict_negative": false,
+    "strict_positive": true,
+    "upper": "270321376920038339632491382770764742407/340282366920938463463374607431768211456"
+  },
+  "new_kappa": false,
+  "old_ordinary_saving": "384599/10000000000",
+  "ordinary_saving_admitted": "27/10000",
+  "ordinary_saving_before_final_relaxation": "27418653797/10000000000000",
+  "ordinary_saving_display": 0.0027418653797,
+  "prime_rule": {
+    "N_to_d_lower": "200001/200000",
+    "also_required": "Odd prime avoiding every fixed rational denominator, matcher determinant and chosen rank witness of the entire compiled word.",
+    "bound": "q > 25600000 * N^5",
+    "complete_moment_upper": "400001/400002",
+    "fallback_multiplier_upper": "400001/400000",
+    "good_moment_upper": "N^(-d), d=C0-C1",
+    "strict_moment_slack_lower": "1/400002"
+  },
+  "root_cost_terms": [
+    "e^(1-delta)",
+    "e^((1-delta)*(1-C1)+delta*(1-a_old))",
+    "e^delta * log(e)"
+  ],
+  "schema": "conditional-delayed-cover-BIT-allsize-parameters/1",
+  "source_sha256": {
+    "notes/batched-bit-rows.tex": "ef72c0ead19373cbc19c8a3aacb44dd0f75353ee5f4d055f775a986a4878022c",
+    "notes/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511",
+    "notes/three-stage-cover-rows.tex": "6ad97a5cd7d6aecf978a81226e1c2763d37840caca130c61f3bf5c4ac111c5c8",
+    "research/kappa-rebuild/bit/BIT-P3-NATIVE-BINDING.json": "9ece46fdfda6182e757e9a3b1f513fb3d029d730f3bafdcd6ed59180a14175b0",
+    "research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.json": "1e1a91871ad8d1098b6851c95d257e8e3fb130222933009b4edfb5e2c2d5c471",
+    "research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md": "9731b93026717f5fa15851e3633cbb4bdc76eba2bffb9d6b4a3b69f7dd502443",
+    "research/kappa-rebuild/bit/MIXED-PRODUCTIVE-BIT.json": "2f69a19f417c0e0d141c180f9b2fae3d358391b9a230f5848ee07c45450e70bf",
+    "research/kappa-rebuild/bit/bit_allsize_parameters.py": "59f92cf44019c27bde3db2f4482f2b5fcf25f0c3dcf071319d46b041f7b6a821"
+  },
+  "status": "PASS_EXACT_PARAMETERS_CONDITIONAL_ON_FINITE_UNIFORM_PRECOMPILATION",
+  "stock": {
+    "asymptotic": "Fixed compiled N,B,P: k=O_(N,B,q)(w*log e)=o(e), w=ceil(e^(3/1000)); pad high rows once by factor<2.",
+    "borrowed_digits": "k <= D_N(ceil(e/w))*(N^2*(w-1)+ceil(log_q(B*|GL_N(F_q)|)))+ceil(log_q S_old)+1",
+    "depth_bound": "D_N(n)=floor(log_N n) for rank<=floor(n/N) children; any larger explicit integer bound is safe.",
+    "descendant_weights": "Only final leaves are lowered. This outer cover row and role type become ancestor data; every next ordinary invocation has fresh disjoint role stock.",
+    "reserve": "S(e,w)=W(w)^D_N(ceil(e/w))*S_old(e,w)",
+    "role_divisor": "W(w)=B*|GL_N(F_q)|*q^(N^2*(w-1))"
+  },
+  "strict_power_gaps": {
+    "borrowed_stock_log_absorption": "9942581346203/10000000000000",
+    "final_relaxation": "418653797/10000000000000",
+    "linear_adapter": "2581346203/10000000000000"
+  }
+}
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/fresh-certificates/THREE-base-HELPER-BOUNDARIES.json.gz b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/fresh-certificates/THREE-base-HELPER-BOUNDARIES.json.gz
new file mode 100644
index 0000000000..1dae5193ba
Binary files /dev/null and b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/fresh-certificates/THREE-base-HELPER-BOUNDARIES.json.gz differ
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/RESULT.json b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/RESULT.json
new file mode 100644
index 0000000000..c6edcbdc43
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/RESULT.json
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Outer productive job ownership, complete recurrence, scalar and storage charges, and rounded induction remain caller obligations.","source_closure_verified":{"research/kappa-native-resume/causal-controller/sources/research/joint-query/certificate.json.gz":"a5ed85fac5ebf3c091482a113ab0d351eb5a7892fe2841a35e16017ec1ff49f7","research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-CANONICAL.json.gz":"71f03fe085db07399cdfeffadbb65f1910bbdb5b7b0873562fb228e4cb292c8b","research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-NATIVE-WORD.json.gz":"57d7c53bacc42c6d403d3f2ced05c7e277ade034a2cb2db9b07e20962cc0b0d6","research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-CANONICAL.json.gz":"d67623412d0de8aab251702d708e9c48421b2721938657193370ee7065322e25","research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-NATIVE-WORD.json.gz":"57d7c53bacc42c6d403d3f2ced05c7e277ade034a2cb2db9b07e20962cc0b0d6","research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-CANONICAL.json.gz":"4526f0c379c0a4e4ae7731b7086a15382c1d1e3960bdf73245f7121299a7ac73","research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-NATIVE-WORD.json.gz":"bde29d4ac306aa528a1237c8b7197291f29f1733f50058ca13f72c2ae328dccf","research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-CANONICAL.json.gz":"6d214cd24e00a2af5b237df8acadd6bd8d3e77bf307fafd0e3e8451c66d93203","research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-NATIVE-WORD.json.gz":"bde29d4ac306aa528a1237c8b7197291f29f1733f50058ca13f72c2ae328dccf","research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-CANONICAL.json.gz":"bff67890d0d350646d11f45aa1b098d6a415b1850b4982e8a05b4c146d247eda","research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-PHASES.json":"e273538de7957b959c40f593c4aad5e8e3d81d83d662fe2895c17770c5fad8bd","research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-CANONICAL.json.gz":"e0c473d62127a09cb3cbf451564dbdd19c0c10f1674dab0575e65251aa5e1e3c","research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-PHASES.json":"c8f744c62670ad7d76faf4dee65e2f708f4036953041e72383929c7d0b32ec38","research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-CANONICAL.json.gz":"f07fae1011160b53e2df1f3135465db1277a093319de2a1da52d648f00a39542","research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-PHASES.json":"9d5316a1465a3eb2c62fac76c291394c0401c34a96fd8b94863d662b16a64552","research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-CANONICAL.json.gz":"882271d70602354cbada8d77f479703ccc6c47e689e8ccd91f9e4c87de0f4379","research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-PHASES.json":"81c01d4b007bb720311dbdbbc63df673fdaf7baba707b360bbecc4b9cb6ee7b0","research/kappa-native-resume/native-source/rebased-core/sources/bw_emit.py":"99b59a6fa3feda86785e4225157dcb1a6169d256cec4f5e0acc9cea7a701df76"},"status":"PASS_FULL_IDENTITY_ALL_FOUR_BOTH_DIRECTIONS_632_ARBITRARY_FIELDS_ZERO_EXTERNAL_BANK_CHILD_CEILING5"}
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz
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diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz
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diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full_identity_rebuild.py b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full_identity_rebuild.py
new file mode 100644
index 0000000000..7cff9bff0f
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/complex/full_identity_rebuild.py
@@ -0,0 +1,173 @@
+#!/usr/bin/env python3
+"""Restore a full-identity native supplier with zero external cleanup bank.
+
+Apache-2.0. Substantial OpenAI assistance. The full-identity ancestor supplies
+the original center cuts and compensation; the independently retained later
+cleanup supplies a legal scheduled alias of all91 old buffers. A fresh native
+replay checks their actual composition. Eight sequential center fields reuse
+one explicitly cleared C0. Each rank8 center is optionally split into rank5
+and rank3 with exact phase decoding, keeping a strict child-rank ceiling5.
+"""
+from collections import Counter
+from copy import deepcopy
+from pathlib import Path
+import argparse
+import sys
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+sys.path.insert(0, str(HERE))
+import rebuild_native as N
+import native_extensions as X
+
+INTEGRATED = N.ROOT/'research/kappa-global-routing/integrated-complex'
+INTEGRATED_RECEIPT_SHA = '70c3d5b1c1d6a18afb29ea1febec3c619ff2b1e38a09c44db7828794ea095429'
+
+
+def reuse_centers(words):
+    words = deepcopy(words)
+    for program in words.values():
+        for event in program:
+            if event[0] == 'center':
+                event[1] = 0
+            elif event[0] == 'scalar':
+                g = event[1]
+                for role in (g[1:3] if g[0] in ('add', 'copy') else g[1:2]):
+                    if role[0] == 'C':
+                        role[1] = 0
+    return words
+
+
+def compose_full_identity(v, tag, integrated_receipt):
+    source = v.CORE/'sources'/(tag+'-CANONICAL.json.gz')
+    scheduled = INTEGRATED/(tag+'-CANONICAL.json.gz')
+    N.need(N.digest(scheduled) == integrated_receipt['providers'][tag]['artifacts'][scheduled.name],
+           'Authenticated scheduled cleanup input')
+    original = N.read(source); newer = N.read(scheduled)
+    changes = {k for k in set(original) | set(newer) if original.get(k) != newer.get(k)}
+    N.need(changes == {'centers', 'center_frames', 'cut', 'compensation', 'cleanup', 'local_profile', 'zero_buffers'},
+           'Exact ancestor-to-clean-J change inventory')
+    c = deepcopy(original)
+    c['cleanup'] = deepcopy(newer['cleanup'])
+    c['zero_buffers'] = []
+    N.need(len(c['centers']) == 8 and len(c['cut']) == 176, 'Retained full-identity center response')
+    words = reuse_centers(v.emit_original(c))
+    return c, words, dict(original_canonical_sha256=N.digest(source), scheduled_canonical_sha256=N.digest(scheduled),
+                         changed_fields_from_original=['cleanup', 'zero_buffers'],
+                         ordinary_compensation_and_all_eight_center_cuts_retained=True)
+
+
+def split_centers(p, ce, canonical, words):
+    c = deepcopy(canonical); result = {}; phases = {}; profile = Counter({int(k): n for k, n in c['local_profile'].items()})
+    N.need(profile[8] == 8, 'Eight old rank8 children')
+    del profile[8]; profile[3] += 8; profile[5] += 8
+    c['local_profile'] = {str(k): n for k, n in sorted(profile.items())}
+    for direction, program in words.items():
+        out = []; detail = []
+        for event in program:
+            if event[0] != 'center':
+                out.append(deepcopy(event)); continue
+            _, role, old, new, rank = event
+            N.need(role == 0 and rank == 8 and {len(c['frames'][old]), len(c['frames'][new])} == {0, 8},
+                   'Single eight-dimensional old center transport')
+            high = old if len(c['frames'][old]) == 8 else new
+            basis = c['frames'][high][:3]
+            if basis not in c['frames']:
+                c['frames'].append(basis)
+            mid = c['frames'].index(basis)
+            path = [old, mid, new]; pieces = []; recipes = []
+            for a, b in zip(path, path[1:]):
+                width = abs(len(c['frames'][b])-len(c['frames'][a]))
+                part = ce.canonical_frame(9, c['frames'][a]).adjoint().then(ce.canonical_frame(9, c['frames'][b]))
+                if direction == 'backward':
+                    part = p.conjugate(ce, part)
+                recipe = N.lower_exact(p, ce, part)
+                N.need(recipe['rank'] == width and width in (3, 5), 'Split child exact rank')
+                out.append(['center', 0, a, b, width]); pieces.append(part)
+                recipes.append(dict(old_frame=a, new_frame=b, recipe=recipe))
+            whole = ce.canonical_frame(9, c['frames'][old]).adjoint().then(ce.canonical_frame(9, c['frames'][new]))
+            if direction == 'backward':
+                whole = p.conjugate(ce, whole)
+            product = pieces[0].then(pieces[1])
+            N.need(product.tab == whole.tab and product.zero_state() == whole.zero_state(),
+                   'Two split children equal old center including absolute phase')
+            detail.append(dict(old_frame=old, intermediate_frame=mid, new_frame=new,
+                               intermediate_basis=basis, pieces=recipes, product_exact_including_phase=True))
+        N.need(len(detail) == 8, 'All eight center paths split')
+        result[direction] = out; phases[direction] = detail
+    return c, result, phases
+
+
+def bind_helper_boundaries(c, tag):
+    source = HERE/'fresh-certificates'/(tag+'-HELPER-BOUNDARIES.json.gz')
+    receipt = N.read(HERE/'fresh-certificates/RESULT-helpers.json')
+    expected = receipt['providers'][tag]['helper_boundaries']['sha256']
+    N.need(N.digest(source) == expected, 'Fresh helper boundary certificate pin')
+    helper = N.read(source)
+    for r in helper['recipes']:
+        role = r['role']
+        N.need(c['frames'][c['start'][role]] == r['certificate']['start_basis'] and
+               c['frames'][c['final'][role]] == r['certificate']['end_basis'],
+               'Full-identity helper has identical exact endpoint references')
+    return dict(certificate_sha256=expected, certificate=str(source.relative_to(N.ROOT)),
+                count=408, widths=helper['widths'], total_width=helper['total_width'],
+                both_orientations_and_global_phases_bound=True)
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--out', type=Path, default=HERE/'full-identity-certificates')
+    args = ap.parse_args(); args.out.mkdir(parents=True, exist_ok=True)
+    p, v, ce, predecessor, inherited = N.authenticated(); geo = N.geometry(v.PACKET)
+    N.need(N.digest(INTEGRATED/'RESULT.json') == INTEGRATED_RECEIPT_SHA, 'Pinned scheduled cleanup receipt')
+    scheduled_receipt = N.read(INTEGRATED/'RESULT.json')
+    result = dict(schema='fresh-full-identity-zero-bank-native-supplier/1',
+                  generator_sha256=N.digest(Path(__file__)), reconstruction_source_sha256=N.digest(HERE/'rebuild_native.py'),
+                  independent_interpreter_sha256=p.CHECKER_SHA,
+                  independent_prefix_evaluator_sha256=N.digest(HERE/'native_extensions.py'),
+                  integrated_cleanup_receipt_sha256=INTEGRATED_RECEIPT_SHA,
+                  source_closure_verified=inherited, no_new_kappa_claimed=True, providers={})
+    for tag in N.TAGS:
+        c, words, composition = compose_full_identity(v, tag, scheduled_receipt)
+        unsplit = {direction: v.replay(program, c, geo['R'], direction == 'backward', False)
+                   for direction, program in words.items()}
+        c, split, phases = split_centers(p, ce, c, words)
+        checks = {}
+        for direction, program in split.items():
+            checked = v.replay(program, c, geo['R'], direction == 'backward', False)
+            rows, prefixes = X.literal_replay(v, c, program, direction == 'backward')
+            N.need(all(rows['L', i] == v.expected(('L', i), geo['R'], direction == 'backward', False) for i in range(632)),
+                   'Second all632-column endpoint implementation')
+            N.need(checked['conservative_real_scalar_primitive_allowance'] ==
+                   unsplit[direction]['conservative_real_scalar_primitive_allowance'], 'Splitting adds no scalar macro')
+            N.need(max(checked['paid_profile']) == 5 and checked['explicitly_owned_C_fields_cleared'] == 1,
+                   'Child ceiling5 and one reusable owned C0')
+            checks[direction] = dict(independent_native_replay=checked, independent_prefix_and_endpoint=prefixes)
+        bad = deepcopy(split['forward'])
+        remove = next(i for i, e in enumerate(bad) if e[0] == 'scalar' and e[1][0] == 'add' and e[1][2] == ['C', 0])
+        bad.pop(remove)
+        rejected = False
+        try:
+            v.replay(bad, c, geo['R'], False, False)
+        except ValueError as error:
+            rejected = 'operator mismatch' in str(error)
+        N.need(rejected, 'Dropped center response negative control rejected')
+        cp = args.out/(tag+'-FULL-IDENTITY-CANONICAL.json.gz')
+        wp = args.out/(tag+'-FULL-IDENTITY-NATIVE.json.gz')
+        result['providers'][tag] = dict(composition=composition, unsplit_replays=unsplit, split_replays=checks,
+              split_center_phase_recipes=phases, helper_boundary=bind_helper_boundaries(c, tag),
+              artifacts={cp.name:N.save(cp, c), wp.name:N.save(wp, split)},
+              all_632_persistent_inputs_arbitrary=True, all_408_helpers_autonomous=True,
+              external_cleanup_fields=0, owned_C_fields=1, owned_C_input_arbitrary_and_erased=True,
+              max_child_rank=5, no_H_involution_or_query_controller_needed=True,
+              dropped_center_response_negative_control_rejected=True)
+        N.save(args.out/'RESULT.json', result)
+        print('PASS full identity', tag, checks['forward']['independent_native_replay']['paid_profile'], flush=True)
+    result['status'] = 'PASS_FULL_IDENTITY_ALL_FOUR_BOTH_DIRECTIONS_632_ARBITRARY_FIELDS_ZERO_EXTERNAL_BANK_CHILD_CEILING5'
+    result['scope'] = ('Complete ideal local native supplier and helper boundaries. Outer productive job ownership, '
+                       'complete recurrence, scalar and storage charges, and rounded induction remain caller obligations.')
+    N.save(args.out/'RESULT.json', result)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz
new file mode 100644
index 0000000000..51a09b3952
Binary files /dev/null and b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz differ
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/controller/p3_native_cover.py b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/controller/p3_native_cover.py
new file mode 100644
index 0000000000..8756693610
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/controller/p3_native_cover.py
@@ -0,0 +1,274 @@
+#!/usr/bin/env python3
+"""Complete native ownership and phase binding for p3/keep43.
+
+This binds the exact Laurent controller to all 2,856 original helper roles,
+whole-call adapters, arbitrary productive inputs, and the full recursive
+invoice.  Apache-2.0; substantial OpenAI assistance.  Native source credits
+and immutable hashes are inherited from the reconstruction receipts.
+"""
+from collections import Counter, defaultdict, deque
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import gzip
+import json
+import sys
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[2]
+sys.path.insert(0, str(HERE))
+import global_identity_cover as G
+
+STAGES = ((0,'forward'),(0,'backward'),(0,'forward'),(1,'backward'),
+          (1,'forward'),(2,'backward'),(2,'forward'))
+FORMAL = HERE.parent/'bit/COMPLEX-CONTROLLER-FACTOR.json'
+TARIFF = HERE.parent/'architecture/P3-CONTROLLER-TARIFF-CERTIFICATE.json'
+AUDIT = HERE.parent/'transfer-audit/P3-NATIVE-AUDIT.json.gz'
+TARGETS = {6:192,7:56,8:93,9:1229,25:448,57:224}
+PAID = {1:17623,2:5065,3:1738,4:714,5:254}
+
+
+def need(condition, message):
+    if not condition:
+        raise AssertionError(message)
+
+
+def stage_frame(n, stage, q):
+    if stage == 0:
+        return n.frame([(a,1) for a in n.AXES[:9]]+[(q,-1)])
+    return n.frame((a,1) for a in n.AXES[9+8*(stage-1):9+8*stage])
+
+
+def contacts(n, F, phases):
+    result = []
+    for stage,(pair,direction) in enumerate(STAGES):
+        sign = 1 if direction == 'forward' else -1
+        spectator = n.frame((a,1) for a in n.AXES[9:9+8*stage])
+        for port,recipe in enumerate(phases['recipes']):
+            q = recipe['q']; route = n.source_route(phases,stage,port)
+            need(all(route[i] == 1 << i for i in range(57,73)), 'No route touches outside ambient57')
+            need(all(axis < 1 << 57 for axis in route[:57]), 'All native views lie in ambient57')
+            input_x = n.frame([(q,1)]) if not stage else n.completed(stage-1)
+            input_y = () if not stage else n.frame(list(input_x)+[(q,-1)])
+            local_x = n.frame(list(spectator)+[(q,sign)])
+            local_y = spectator
+            end_x = n.frame(list(spectator)+[(a,sign) for a in n.AXES[:9]])
+            end_y = n.frame(list(end_x)+[(q,-sign)])
+            output_x = n.completed(stage)
+            output_y = n.frame(list(output_x)+[(q,-1)])
+            maps = {'entry_X':F.between(n,input_x,route,local_x),
+                    'entry_Y':F.between(n,input_y,route,local_y),
+                    'exit_X':F.between(n,end_x,route,output_x),
+                    'exit_Y':F.between(n,end_y,route,output_y)}
+            delta = stage_frame(n,stage,q)
+            # References cancel as operators on arbitrary raw inputs. No
+            # historical assumption about which idles have run is needed.
+            need(n.polynomial(n.frame(list(output_x)+[(a,-c) for a,c in input_x]))
+                 == n.polynomial(delta), 'Exact diagonal physical D_s on X')
+            need(n.polynomial(n.frame(list(output_y)+[(a,-c) for a,c in input_y]))
+                 == n.polynomial(delta), 'Exact diagonal physical D_s on Y')
+            cross = n.frame(list(output_y if sign == 1 else output_x)+
+                            [(a,-c) for a,c in (input_x if sign == 1 else input_y)])
+            wanted = n.frame(list(delta)+[(q,-1 if sign == 1 else 1)])
+            need(n.polynomial(cross) == n.polynomial(wanted), 'Physical cross operator D_s Q^+-1')
+            result.append({'stage':stage,'pair':pair,'port':port,'direction':direction,
+                           'common_spectator_frame':list(spectator),'maps':maps})
+    return result
+
+
+def make_windows(n, phases, formal):
+    output = []
+    for number,window in enumerate(formal['retained_idle_windows']):
+        stages = window['stages']
+        patterns = [list(spec[1:]) for spec in (G.PAIRS if len(stages) == 2 else G.MIDDLE)
+                    for _ in range(spec[0])]
+        need(len(patterns) == 224, 'Every retained window has224 distinct possible owners')
+        for index,widths in enumerate(patterns):
+            side,port = ('X' if index < 112 else 'Y'),index % 112
+            recipe = phases['recipes'][port]
+            if 0 in stages:
+                need(stages == [0,1], 'The retained initial window is D0D1')
+                axes = recipe['idle_children'][0]['orthogonal_axes']
+                shift = recipe['idle_children'][0]['translation_after']
+            else:
+                axes = [axis for stage in stages for axis in n.AXES[9+8*(stage-1):9+8*stage]]
+                shift = 0
+            desired = n.frame([term for stage in stages for term in stage_frame(n,stage,recipe['q'])])
+            actual = n.frame([(axis,1) for axis in axes]+[(shift,2)])
+            need(n.polynomial(actual) == n.polynomial(desired), 'Literal whole idle operator including phase')
+            need(len(axes) == window['rank'] and sum(widths) <= len(axes), 'Every hosted helper fits')
+            output.append({'family':number,'index':window['index'],'kind':window['kind'],
+                           'pair':window['pair'],'stages':stages,'side':side,'port':port,
+                           'owner':f"{side}{window['pair']}:{port}", 'widths':widths,
+                           'axes':axes,'translation_after':shift})
+    return output
+
+
+def helper_assignment(n, phases, formal, local, helper_recipes):
+    windows = make_windows(n,phases,formal)
+    pools = []
+    for stage in range(7):
+        pool = defaultdict(deque)
+        for recipe in helper_recipes['THREE-base']['recipes']:
+            pool[recipe['width']].append(recipe['role'])
+        pools.append(pool)
+    bindings,residuals,fresh = [],[],[]
+    paid = Counter(); targets = Counter({25:448,57:224})
+    for window in windows:
+        cursor = 0
+        for stage,width in zip(window['stages'],window['widths']):
+            if not width:
+                continue
+            need(pools[stage][width], 'A real local helper of this width exists')
+            role = pools[stage][width].popleft()
+            axes = window['axes'][cursor:cursor+width]; cursor += width
+            bindings.append({'stage':stage,'role':role,'width':width,'owner':window['owner'],
+                             'axes':axes,'kind':'original_idle','window_index':window['index']})
+        for axes in G.split_axes(window['axes'][cursor:]):
+            residuals.append({'owner':window['owner'],'after_stage':max(window['stages']),
+                              'axes':axes,'rank':len(axes),'kind':'idle_remnant'})
+            paid[len(axes)] += 1
+    for stage,pool in enumerate(pools):
+        for width,roles in sorted(pool.items()):
+            for role in roles:
+                owner = f'G{stage}:{role}'; target = max(6,width)
+                bindings.append({'stage':stage,'role':role,'width':width,'owner':owner,
+                                 'axes':list(n.AXES[:width]),'kind':'fresh'})
+                fresh.append({'owner':owner,'stage':stage,'native_width':width,'target_rank':target})
+                targets[target] += 1
+                for axes in G.split_axes(list(n.AXES[width:target])):
+                    residuals.append({'owner':owner,'after_stage':stage,'axes':axes,
+                                      'rank':len(axes),'kind':'fresh_topup'})
+                    paid[len(axes)] += 1
+    coordinate_maps = {}
+    for stage,(pair,direction) in enumerate(STAGES):
+        native = local['providers']['THREE-base']['split_replays'][direction]['independent_native_replay']
+        paid.update({int(rank):count for rank,count in native['paid_profile'].items()})
+        selected = [row for row in bindings if row['stage'] == stage]
+        need(len(selected) == 408 and {row['role'] for row in selected} == set(range(224,632)),
+             'All408 actual native helper roles are assigned exactly once')
+        need(len({row['owner'] for row in selected}) == 408, 'Every local helper has a distinct physical owner')
+        active = {f'{side}{pair}:{port}' for side in ('X','Y') for port in range(112)}
+        need(not active.intersection(row['owner'] for row in selected), 'No live main role is borrowed')
+        for row in selected:
+            key = ','.join(map(str,row['axes']))
+            if key not in coordinate_maps:
+                coordinate_maps[key] = G.coordinate_map(row['axes'],57)
+            row['coordinate_map'] = key
+            row['whole_helper_boundary'] = {'provider':'THREE-base','direction':direction,'role':row['role'],
+                  'selector':'certificate.directions.direction.existing_native_boundary_normal_form'}
+    need(dict(sorted(paid.items())) == PAID, 'Full actual paid histogram')
+    need(dict(sorted(targets.items())) == TARGETS, 'All and only productive target fields')
+    need(len(fresh) == 1570 and len(bindings) == 2856, 'Complete field and helper census')
+    need(sum(k*v for k,v in targets.items())-sum(k*v for k,v in paid.items()) == 224, 'Rank-unit margin')
+    need(max(paid) < min(targets), 'Strict recursive separation')
+    return {'windows':windows,'helper_bindings':bindings,'owner_coordinate_maps':coordinate_maps,
+            'paid_residuals':residuals,'fresh_outputs':fresh,'paid':PAID,'targets':TARGETS,
+            'rank_margin':224,'productive_fields':2242,'owned_center_fields':1,'external_cleanup_fields':0}
+
+
+def endpoint_normalizers(n, phases, formal):
+    output = []; histogram = Counter()
+    for role,term in enumerate(formal['exact_diagonal_monomials']):
+        q_power,selected = term[0],term[1:]
+        need(q_power in (-1,1) and set(selected) <= {0,1}, 'Exact Laurent endpoint alphabet')
+        for port,recipe in enumerate(phases['recipes']):
+            q = recipe['q']
+            physical = n.frame([(q,q_power)]+[entry for stage,yes in enumerate(selected) if yes
+                                                  for entry in stage_frame(n,stage,q)])
+            axes = list(n.AXES[:9]) if selected[0] else [q]
+            axes += [axis for stage in range(1,7) if selected[stage]
+                     for axis in n.AXES[9+8*(stage-1):9+8*stage]]
+            translation = q if q_power == -1 else 0
+            repaired = n.frame(list(physical)+[(translation,2)])
+            canonical_on_axes = n.frame((a,1) for a in axes)
+            need(n.polynomial(repaired) == n.polynomial(canonical_on_axes), 'Exact endpoint phase including constant')
+            if translation:
+                need(n.polynomial(physical) != n.polynomial(canonical_on_axes), 'Missing phase repair really fails')
+            rank = len(axes)
+            need(rank == 1+8*sum(selected), 'Earned endpoint rank')
+            matrix = G.coordinate_map(axes,57)
+            need(matrix['columns'][:rank] == axes, 'Literal target-rank conjugation')
+            output.append({'original_input_role':role,'port':port,'target_rank':rank,
+                           'input_coordinate_columns':matrix['columns'],
+                           'output_coordinate_inverse_rows':matrix['inverse_rows'],
+                           'output_translation_before_inverse_map':translation,
+                           'physical_frame':list(physical),'positive_axes':axes})
+            histogram[rank] += 1
+    need(histogram == Counter({25:448,57:224}), 'Main endpoint distribution')
+    return output
+
+
+def scalar_invoice(local, formal):
+    native = sum(local['providers']['THREE-base']['split_replays'][direction]['independent_native_replay']
+                 ['conservative_real_scalar_primitive_allowance'] for pair,direction in STAGES)
+    adapters = formal['right_program']+formal['left_program']
+    counts = Counter(row[0] for row in adapters)
+    need(counts == Counter({'ADD':4,'SWAP':6,'SCALE':2}), 'Literal boundary scalar word')
+    need(all(str(row[-1]) in ('1','-1') for row in adapters if row[0] in ('ADD','SCALE')),
+         'Only unit scalar boundary coefficients')
+    # Ordinary swap: three additions followed by one negation. Signed swap
+    # needs three additions. No free whole-role movement is assumed here.
+    adapter_adds = 4+6*3; adapter_negs = 2+6
+    bridges = sum(row[0] == 'ADD' for row in formal['physical_events'])
+    signed_swaps = sum(row[0] == 'SIGNED_SWAP' for row in formal['physical_events'])
+    need(bridges == 12 and signed_swaps == 3, 'Literal global bridge and swap census')
+    total = native+112*((adapter_adds+bridges+3*signed_swaps)*8+adapter_negs*2)
+    return {'native_real_scalar_allowance':native,'boundary_unit_ADD_per_port':adapter_adds,
+            'boundary_negations_per_port':adapter_negs,'bridge_unit_ADD_per_port':bridges,
+            'terminal_signed_swap_unit_ADD_per_port':3*signed_swaps,
+            'conservative_real_scalar_primitive_allowance':total,
+            'coordinate_and_phase_moves':'Explicit separate fixed movement operations; not scalar ADDs and not paid recursive calls.'}
+
+
+def run():
+    F = G.load(HERE/'frame_exchange.py','p3_contacts')
+    n = G.load(F.NATIVE,'p3_native_common')
+    packet,phases = n.load_sources()
+    local,pins,helpers = G.local_bindings()
+    formal_source = G.read(FORMAL); formal = formal_source['selected_p3_keep43']
+    tariff = G.read(TARIFF); audit = G.read(AUDIT)
+    need(tariff['controller'] == formal, 'The native cover binds the exact priced controller')
+    for path,wanted in tariff['source_sha256'].items():
+        need(G.digest(ROOT/path) == wanted, 'Pinned actual tariff source '+path)
+    for name,wanted in formal_source['source_sha256'].items():
+        need(G.digest(FORMAL.parent/name) == wanted, 'Pinned formal word source '+name)
+    main_contacts = contacts(n,F,phases)
+    cover = helper_assignment(n,phases,formal,local,helpers)
+    endings = endpoint_normalizers(n,phases,formal)
+    scalar = scalar_invoice(local,formal)
+    need({int(k):int(v) for k,v in tariff['invoice']['paid'].items()} == cover['paid'], 'Independent complete paid invoice agrees')
+    need({int(k):int(v) for k,v in tariff['invoice']['targets'].items()} == cover['targets'], 'Independent productive credit agrees')
+    for path in (FORMAL,TARIFF,AUDIT,HERE/'global_identity_cover.py',HERE/'frame_exchange.py',F.NATIVE):
+        pins[str(path.relative_to(ROOT))] = G.digest(path)
+    return {'schema':'complete-native-p3-keep43-owner-phase-cover/1',
+            'status':'PASS_COMPLETE_FINITE_NATIVE_CONTROLLER_OWNER_PHASE_INVOICE',
+            'generator_sha256':G.digest(__file__),'sources':pins,'immutable_phase_sources':n.PINS,
+            'main_contacts':main_contacts,'contact_count':3136,'main_endpoint_normalizers':endings,
+            'cover':cover,'scalar_invoice':scalar,'root_bracket':tariff['invoice']['root_bracket'],
+            'physical_order':'Endpoint input coordinate maps; R; literal serialized controller with whole-call helper adapters; L; endpoint translations; endpoint inverse coordinate maps.',
+            'native_local_contract':{'forward':'D_s [[1,0],[Q^-1,1]]','backward':'D_s [[1,-Q],[0,1]]',
+                'meaning':'Exact raw-input operators, valid for arbitrary prior histories and after every boundary scalar basis change.'},
+            'arbitrary_productive_input_fields':2242,'all_2856_original_helper_roles_assigned':True,
+            'one_owned_center_reused_sequentially':True,'owned_center_arbitrary_at_entry_zero_at_return':True,
+            'helpers_normalized_only_around_whole_native_invocations':True,
+            'new_kappa':False,
+            'remaining_transfer_obligations':['Finite mixed-to-uniform precompilation and ordinary supplier instantiation.',
+                'Replacement precision, stock, scalar and movement constants for the resulting finite template.',
+                'A compatible BIT supplier and the inherited outer analytic transfer.']}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--out',type=Path,default=HERE/'P3-NATIVE-COVER.json.gz')
+    args = parser.parse_args()
+    receipt = run()
+    args.out.write_bytes(gzip.compress(json.dumps(receipt,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+    print(json.dumps({'status':receipt['status'],'targets':receipt['cover']['targets'],
+                      'paid':receipt['cover']['paid'],'scalar':receipt['scalar_invoice'],
+                      'root_bracket':receipt['root_bracket'],'sha256':G.digest(args.out)},sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/FINITE-TEMPLATE-CLOSURE.md b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/FINITE-TEMPLATE-CLOSURE.md
new file mode 100644
index 0000000000..9917ac5543
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/FINITE-TEMPLATE-CLOSURE.md
@@ -0,0 +1,339 @@
+# Native closure and explicit constants for the finite uniform template
+
+Status: independently audited, conditional on the retained native movement,
+fixed-tape, and analytic interfaces stated below. This note supplies the new
+finite-template constants; it does not reuse the old supplier's scalar cap,
+row reserve, or obsolete pathwise precision exponent. It complements
+`AUDIT.md`, `P3-NATIVE-AUDIT.json.gz`, and `P3-COVER-AUDIT.json`.
+
+Apache-2.0. Substantial OpenAI assistance. The semantic completed-child
+argument is the attributed argument in
+[`references/semantic-bulk/pr23/semantic-bulk-17-note.tex`](../../../references/semantic-bulk/pr23/semantic-bulk-17-note.tex),
+section “Linear semantic precision for this actual circuit.” Its later
+explicit-constant application is also in
+[`notes/stopped-product-complex.tex`](../../../notes/stopped-product-complex.tex).
+The paths in the machine-readable receipt are relative to the repository
+root and bind the exact reviewed versions by SHA-256.
+
+## 1. What has actually been instantiated
+
+The p3 complex unit is a complete native word, not just a histogram. The
+independent audits verify its serialized scalar controller, 3,136 physical
+main-contact maps and their absolute phases, 672 complete main endpoints,
+2,856 actual local helper roles, causal idle windows, every paid completion,
+and the final order of whole-field normalizers. Its productive targets are
+
+```
+{6:192, 7:56, 8:93, 9:1229, 25:448, 57:224}.
+```
+
+There are 2,242 independent productive fields, one explicitly owned center,
+and paid children
+
+```
+{1:17623, 2:5065, 3:1738, 4:714, 5:254}.
+```
+
+All productive inputs are arbitrary. The owned center is initialized and
+cleared; it is never an additional credited output. Every helper endpoint
+is autonomous at the completed local boundary. Independent normalizers
+therefore surround an entire local call, without changing the frames of
+internal ADD instructions.
+
+The exact integer precompiler in `AUDIT.md` turns this unit into a finite
+uniform word with parameters
+
+* ambient rank `N >= 2`;
+* `W` productive arrays, each receiving the **full** rank-`N` transform;
+* `P` rank-one paid children;
+* `U` copies of the mixed unit in the completely expanded finite word;
+* at most `O=U` owned centers, conservatively one per mixed-unit instance.
+
+For every rational `z` strictly above the mixed root and below one, the
+constructive count argument provides an `N` with `P < W*N^z`.
+“Finite” here means fixed before the eventual multiplication input grows.
+It can be astronomically large. Neither this note nor the receipts claim
+to have materialized the winning near-threshold instance. The later
+`../controller/EXPLICIT-TEMPLATE-BOUNDS.md` strengthens the existence
+argument to the specified complex rank `N=2^276540184361`, with uniform
+saving `0.002304`; no search for a successful rank is left unperformed.
+The integer recursion defines its exact finite `W,P,U`, and the constants
+below are computable functions of that finite result.
+
+The fully materialized, depth-zero `N=57` example has `W=2242`, `P=127570`,
+and `U=1`. It is a useful independent check that complementary source jobs
+are really paid: `57*2242-127570=224`. This small example is not claimed to
+attain the limiting mixed exponent.
+
+## 2. Every column count and every complete spectator field
+
+The local logical identities were replayed in a free noncommuting operator
+word algebra. Apart from rational scalar arithmetic, the checker uses only
+composition and cancellation of an operator with its recorded inverse.
+For any positive integer `f`, substitute the `f`-column physical operator
+for each such generator. This extends to an algebra homomorphism on the
+free word algebra and preserves the checked identities.
+
+This is not the false assertion that tensor power is additive on arbitrary
+matrix sums. The role-level scalar word is unchanged. Each physical address
+generator is replaced by its column lift, while its formal products and
+linear combinations are evaluated on the same independent role arrays.
+
+A rank-zero native monomial has the form
+
+```
+|x> -> i^(a+q(x)) |M*x+b>,
+```
+
+with invertible binary `M` and a permitted quadratic phase `q`. Its column
+lift has blockwise address map `M`, translation `b` in every column, and
+phase `a*f + sum_j q(x_j)` modulo four. This explicitly retains the absolute
+phase. A fixed coordinate matcher in the precompiler becomes the same
+fixed permutation of `f`-axis blocks. All untouched axes remain complete
+spectator fields. A rank-one leaf becomes the standard rank-`f` child.
+
+The retained exact Clifford normal form supplies the monomial factors
+around each child. The independent p3 audit checks the whole main boundary,
+including the output translations required by negative powers of `C_q`.
+The fixed THREE/base selection is valid for every column count: THREE/base
+and THREE/extra have identical entrance and exit operators; their difference
+is an internal frame and its associated warps. No layer-dependent selector
+is required for the endpoint contract.
+
+For any integer input width `e=N*f+r`, first apply the `r<N` individual
+kernels and preserve these axes as completed spectators. Then apply the
+lifted finite word on the consecutive `N*f` active axes. Its final endpoint
+is the full selected-axis tensor, so the separately completed remainder
+commutes with it. This gives the ordinary arbitrary-width interface and
+strict child ratio at most `1/N`.
+
+## 3. An actual new coefficient charge
+
+`precision_closure.py` reads the authenticated serialized forward and
+backward native words. For every scalar macro it independently checks that
+the coefficient is dyadic and that the elementary allowance dominates its
+logarithmic row-norm and denominator-height increment. For an ADD with
+coefficient `a`, a sufficient height increment is
+
+```
+ceil(log2(1+abs(a))) + log2(denominator(a)).
+```
+
+For a dyadic SCALE replace `1+abs(a)` by `max(1,abs(a))`. COPY and CLEAR
+have zero height increment and still retain their implementation cost.
+The audit reproduces the native real primitive allowances 158,120 forward
+and 158,168 backward, and the complete p3 arithmetic charge 1,147,304.
+In particular, the six boundary swaps contribute their six additional
+negations; they have not been priced as signed swaps.
+
+Address permutations and fourth-root unit phases do not increase magnitude
+or denominator height. We nevertheless add an explicit conservative charge
+of eight for every non-scalar native record and every listed whole-field
+normalization. The respective counts are:
+
+| Kind | Count |
+| --- | ---: |
+| Non-scalar records in the seven native calls | 36,260 |
+| Complete helper normalizations | 2,856 |
+| Main contact monomials | 3,136 |
+| Complete main endpoint normalizations | 672 |
+| Paid completion normalizations | 460 |
+
+This gives the complete elementary coefficient/unit-phase envelope
+
+```
+G_unit = 1,147,304 + 8*(36,260+2,856+3,136+672+460)
+       = 1,494,376.
+```
+
+Eight bounds the two monomial factors around a native child, each of which
+needs at most a sign and an `i` phase. Some records are just annotations, so
+this deliberately overcounts. The pointwise phase descriptor may have
+larger fixed geometry, but each resulting coefficient is still a unit.
+
+Whole-job coordinate matchers in the finite compiler have zero coefficient
+height. The canonical positive tensor jobs can be placed on disjoint axes
+by binary coordinate permutations; there is no new scalar mixing at these
+boundaries. The final scalar/phase charge is therefore bounded by
+
+```
+G = 1,494,376 * U.
+```
+
+This charge is not a promise that an arbitrary binary address map costs
+eight tape steps. Every actual matrix has its separately paid native
+row-addition decomposition and compact movement implementation. The finite
+number of those operations is another fixed constant of the expanded word.
+
+## 4. Explicit linear semantic guard
+
+Put `Z=W+O` and replace the old constants by
+
+```
+E  = 64*(Z+N+P+G+1)^3,
+B  = P+E,
+C0 = 32*N*B^2,
+C1 = 1.
+```
+
+In particular
+
+```
+2*G*Z^2 + 8*P + 4*Z + 4 + 32*N < E.
+```
+
+The receipt verifies this exact inequality. It is deliberately much larger
+than necessary: it pays the full elementary word, unit phases, role and
+owned-field passes, and fewer than `N` remainder kernels. The inclusion of
+`P` in the fresh cubic bound avoids relying on any old relation between the
+paid-child count and the number of physical fields.
+
+Let `A(e)` bound both temporary magnitude exponent and binary denominator
+excess relative to the incoming common fine grid. A **completed** forward
+or inverse `u`-axis child has coefficients in `2^-u Z[i]` and absolute row
+sum at most `2^u`; the unit sign wrappers preserve those bounds. Thus its
+returned stream costs at most `u` in each semantic measure. Its larger
+internal temporary values do not become unfinished parent operands.
+
+At a node of width `e`, set `f=floor(e/N)`. At most `P` completed child calls
+have preceded any active child; their total selected width is at most
+`P*f`. Their input contents may be arbitrary, including data on the owned
+centers. All scalar growth and remainder kernels fit in `E`. Consequently
+
+```
+A(e) <= A(f) + P*f + E.
+```
+
+For an elementary leaf, `A(e)<=8e`. Since `N>=2`, `B=P+E`, and `E>=64`,
+
+```
+2*B*(N-1) >= P+E,
+2*B*f+P*f+E <= 2*B*N*f <= 2*B*e.
+```
+
+Induction gives `A(e)<=2B e`. Completed disjoint selected-axis pieces have
+total width at most `d`. The retained prefix kernels, outer phases and
+normalization add at most `18d`, giving
+
+```
+A_layer <= (2*B+18)*d < C0*d.
+```
+
+Every calculation remains on one fixed fine grid. A child is never rounded,
+compressed, or re-encoded at return. The change is to the proof's bound on a
+completed value, not to its stored representation. Other fields remain
+parked while the child executes. This is why the linear semantic bound
+survives a huge scalar word and why the older pathwise `C1>1` estimate is
+unnecessary.
+
+The stopping rule can be any fixed rational `0<beta<1`. Above the eventual
+threshold `d^beta>=2N`, every internal call has the required positive `f`;
+the completed-leaf estimate remains `8e` at the stopping leaves. No old
+numerical cutoff is carried over.
+
+### An effective threshold, including enormous constants
+
+Suppose `d<=c*p^epsilon`, with fixed `c>=1` and rational
+`epsilon=a/b` in `(0,1)`. Let `L` be the binary bit length of
+`4*C0*ceil(c)` and choose
+
+```
+t = max(4, ceil(b*L/(b-a))),       p0 = 2^t.
+```
+
+For all `p>=p0`, `p^(1-epsilon)>4*C0*c`, so the integer guard
+`Delta=C0*d` is at most `p/4`. Signed components with `p+Delta` fractional
+bits and `Delta+3` magnitude/sign bits have width
+
+```
+p+2*Delta+3 <= 2p.
+```
+
+The audit emits `p0` as a compressed power of two. It does not try to
+allocate that many bits. This construction proves the eventual `O(p)` word
+bound for the new finite template, including the current outer choice
+`epsilon=99999/100000`. Huge fixed constants change `p0`, not `C1` or the
+power in the final exponent inequalities.
+
+## 5. Productive stock, owned storage, and native tape closure
+
+For the ordinary uniform complex word, let
+
+```
+D(e)=0 if e<N;  D(e)=1+D(floor(e/N)) otherwise,
+q0=ceil(log2 W)*D(d).
+```
+
+Use the original compact front and back fields separately. The first `q0`
+reserved complete chunks provide a physically preceding row field. Their
+individual selected-axis kernels need not be completed beforehand: the
+active invocation acts as identity on these coordinates for arbitrary
+incoming contents. In the support-local outer wrapper their kernels must
+be deferred to later phases, as explained in `OUTER-TRANSFER-AUDIT.md`.
+The row field's cardinality is
+`R=2^(q0*K)>=W^D(d)` for `K>=1`. Pad complete rows once to the next multiple
+of `W^D(d)`. The padding factor is below two. Every descendant then has
+exactly `1/W` of its parent's productive volume and enough divisibility for
+its remaining depth, with complete active, control and spectator fields.
+All-row endpoint correctness restores appended zero rows to zero.
+
+Owned storage contributes no factor to productive output credit. A safe
+choice is one full center field per finite mixed-unit instance, namely
+`O=U`. At an ordinary recursion node each such field has the same shape as
+one productive role stream, of volume `V/W`. Initialize, retain and clear
+it as its actual word prescribes. If simultaneously suspended units keep
+live centers, they receive distinct banks. Reusing one bank per recursion
+depth would not suffice. Owned-space scans and parking cost fixed multiples
+of volume; paid calls on those banks are already in `P`.
+
+The integer audit checks row divisibility and one-time padding at several
+floor-depth boundaries. It also gives an explicit sufficient threshold
+for `q0<d/4`. In general `q0=O(log d)` with a possibly huge fixed
+coefficient. This does not change the compact-field reservation exponents.
+If an inherited implementation uses separate simultaneous BIT stock, its
+factor must also be retained in the row product; taking only the maximum
+of two stocks would be invalid. The delayed BIT cover has its own updated
+row-reserve argument and must not be charged to old constants.
+
+Each fixed binary matrix factors into finitely many row additions and
+coordinate swaps. `notes/compact-control-movement.tex` implements these on
+complete packed rectangles, with the actual compact cost and polynomial
+descriptor preparation. The finite compiler's matrices have fixed rank
+`N`; their `f`-column lifts are therefore fixed block recipes, not dynamic
+RAM gathers. They preserve the retained native operation alphabet.
+
+The depth-first fixed-tape execution parks unused role streams, their
+owned banks and descriptors while a child runs. The numbers of roles,
+tapes, routers and instruction types are fixed constants of this one
+template. The ordinary tree has at most `P^D(d)=d^O(1)` nodes. Polynomial
+description/setup costs still have a fixed degree and are absorbed by the
+retained superpolynomial record fiber when `epsilon<1`. That absorption
+needs a new eventual threshold if the degree grows; it does not authorize
+an unpriced RAM-to-tape conversion.
+
+## 6. Consequence for the new outer calculation
+
+After the finite precompilation and the native phase/ownership audits,
+the complex supplier fits the retained ordinary uniform interface with
+moment `(P/W)*N^-sigma<1`. All real primitive, movement, center and row
+constants are replaced as above. Its precise stopping and internal-cost
+exponents are still the retained weighted-tree bounds; only their constant
+multipliers and eventual cutoffs change. The nearly critical outer ledger
+can therefore use `C1=1` with the new complex saving.
+
+This closes the **complex finite-template** precision and stock obligation
+under the inherited native movement and outer hypotheses. A complete
+conditional multiplication claim additionally needs the independently
+bound BIT uniform interface, the strict mixed moments, and the actual outer
+parameter inequalities. Those are separate receipts rather than implicit
+consequences of the favorable complex histogram.
+
+Replay:
+
+```sh
+python3 -B research/kappa-rebuild/transfer-audit/precision_closure.py
+```
+
+The output `PRECISION-CLOSURE.json` includes exact source hashes, independently
+recounted scalar charges, the generic substitution formulas, and the fully
+numeric rank-57 example.
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md
new file mode 100644
index 0000000000..e8b7ef8956
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md
@@ -0,0 +1,426 @@
+# Independent transfer audit for the conditional kappa = 0.0023 construction
+
+Status: the changed supplier interfaces close under the inherited analytic,
+ordered-movement and fixed-tape hypotheses. This is a mathematical audit
+with independently generated exact receipts, not a formal proof certificate
+or an unconditional integer-multiplication theorem. Substantial OpenAI
+assistance. Apache-2.0.
+
+The conclusion uses the **actual complete p3 programs**, the finite native
+precompiler, the delayed BIT cover, fresh precision and stock constants,
+and the nearly critical outer assembly. It does not infer a multiplication
+bound from a favorable recursive-cost histogram alone.
+
+## 1. Conclusion and quantitative separation
+
+The independently checked conditional outer target is
+
+```
+kappa = 23/10000 = 0.0023.
+```
+
+Every one of the nine retained outer cost rows has a strictly larger saving.
+The smallest saving is `0.00230197698`; its margin above the declared target
+is exactly `98849/50000000000 = 0.00000197698`.
+
+The construction is deliberately separated into the following interfaces.
+
+| Interface | Independently checked result |
+| --- | --- |
+| Complete complex mixed unit | 2,242 arbitrary productive fields, one uncredited owned center; full physical phase, owner and scalar bill |
+| Complete BIT mixed unit | 3,739 arbitrary productive fields, one uncredited owned center; 63 paid calls on copied-center data |
+| Finite complex uniform program | Fixed rank `N=2^276540184361`, saving `0.002304` |
+| Finite BIT uniform projector program | Fixed rank `N=2^6552928412`, saving `0.002760` before its cover |
+| Covered BIT uniform program | Saving `0.002750`, including every bad-class fallback |
+| Ordinary arbitrary-width BIT | Saving `27418653797/10^13 = 0.0027418653797`; conservatively admit `0.0027` |
+| Effective outer supplier saving | `0.0023025` |
+| Completed selected-axis complex layer | Linear semantic guard `C0*d`, with a new computable fixed `C0` |
+| Conditional outer result | `kappa=0.0023` |
+
+The ranks above are exact compressed integers, not suggested search bounds.
+The winning words are defined by finite integer algorithms at those ranks.
+Their enormous instruction lists and constants have not been materialized.
+The fully materialized rank-57 complex and rank-50 BIT examples are useful
+ownership/complement tests; neither small example attains the limiting
+savings in this table.
+
+## 2. The finite bridge preserves complete jobs and actual ownership
+
+The reviewed native precompilation theorem is
+[`../architecture/FINITE-PRECOMPILATION.md`](../architecture/FINITE-PRECOMPILATION.md).
+Its integer lockstep flow is based on Jacob Alan Sussman's Theorem 1G;
+the independent chronological reconstruction and challenge cases are in
+[`AUDIT.md`](AUDIT.md) and `FINITE-DAG-AUDIT.json`.
+
+At a fixed ambient rank `n`, every productive source is a complete arbitrary
+array on all `n` coordinates. Its designated rank-`s` job is supplemented
+by a paid complementary rank-`n-s` job. The complement is run first.
+The completed mixed word then supplies the designated endpoint, and the
+product is the full transform. This uses autonomy of the completed endpoint;
+it does not require the complement to commute with intermediate operations
+of the mixed circuit.
+
+All normalized source counts and all occurrences of lower-rank batches are
+cleared by one explicit finite integer multiplier. An ordered batch uses
+distinct physical fields. Each recursive demand occurs strictly after its
+input exists and before its parent continues; every recursive rank is
+smaller. The independent graph audit rejects aliased input slots, incomplete
+batch clearing and cycles. It includes paid demands on owned fields.
+
+A parent-owned copied center can be passed as one arbitrary input to a
+lower-rank unit. Correlation with another array is harmless: the child
+identity holds for every assignment of its distinct physical input slots.
+The parent copy remains present until the actual call returns, and its
+eventual erasure is legal only after the prescribed restoration. A fresh
+child center has a separate owner tag. No copied center becomes another
+productive source, and simultaneous suspended centers are not merged into
+one bank merely because they have the same recursion depth. Conservatively
+allocate one full center per finite mixed-unit instance and charge every
+initialization, pass and erasure. These are fixed constants of the compiled
+uniform word.
+
+### Explicit termination and one specified winning rank
+
+Let `L(n)=P_n/B_n` be the exact average number of rank-one leaves per
+productive full-rank source, including complements and owned-field calls.
+For the complex case the strict mixed test is at saving `0.002305`; for BIT
+it is at `0.002769`. Put `t=1-C_test`. Cancellation in the finite flow gives
+an inequality, not an assumed equality:
+
+```
+sum_e x_e*e^t <= sum_s w_s*s^t,
+L(n) = sum_s w_s*L(n-s) + sum_e x_e*L(e),
+sum_s w_s = 1.
+```
+
+The separate positive rank-one surplus gives `L(n)<=n` by induction. With
+source ranks `n/m<s<=n`, exit ranks obey
+`E(n)<=r_max*(n/m)^theta`, where `theta=log(r_max)/log(k_min)<1`.
+Use `theta_bar=9/10` for complex and `15/16` for BIT. These upper bounds are
+proved by the exact integer comparisons `5^10<6^9` and `7^16<8^15`.
+Writing `h=1-theta_bar` gives `m*E(n)/n<16*n^-h`.
+
+For the desired uniform exponent `z`, set `z0=(t+z)/2` and
+`delta=z0-t>0`. Concavity shows that the source complements leave at least
+`eta*sum_s w_s*s^z0`, where `eta=z0*m^(z0-1)>1/4`. Choose
+
+```
+H0 = ceil((3+4*delta)/(h*delta)),
+n0 = 2^H0,
+A  = n0^(1-z0).
+```
+
+Then the exit coefficient is at most `1/8` for every `n>=n0`. The base
+bound `L(n)<=n` covers `n<=n0`; strong induction gives `L(n)<=A*n^z0`
+for every rank. Finally
+
+```
+H1 = floor(H0*(1-z0)/(z-z0))+1,
+N  = 2^H1
+```
+
+ensures `L(N)<N^z`. Independent directed rational logarithm/exponential
+enclosures reproduce both strict mixed inequalities and the exact ranks:
+
+| Family | `H0` | `H1` |
+| --- | ---: | ---: |
+| Complex | 60,000,040 | 276,540,184,361 |
+| BIT | 10,666,731 | 6,552,928,412 |
+
+This is an effective specification of one uniform word, without relying on
+an unperformed search for its first success. Its finite multiplicities and
+instruction count are computed by the specified integer recursion. The
+asymptotic algorithm fixes this word before its input length varies.
+
+## 3. Native complex binding, columns, and precision
+
+The complete physical audit is in `P3-NATIVE-AUDIT.json.gz` and
+`P3-COVER-AUDIT.json`. It independently checks 3,136 main-contact recipes
+and inverses with their absolute quadratic phases, 672 main endpoints,
+2,856 local helper owners, the chronological idle windows, every paid
+completion, and the order of complete-field input/output normalizers.
+The scalar boundary is charged as 22 ADDs and eight negations per port,
+with the additional controller and endpoint operations retained. The total
+real scalar allowance is 1,147,304.
+
+The underlying local word accepts all helper data arbitrarily. Each helper
+has an autonomous completed endpoint, so whole-field conjugations around
+the local call are legal without retagging any internal ADD. The selected
+THREE/base word is valid for every positive column count. THREE/FOUR and
+base/extra are endpoint-equivalent local variants, not mandatory layer
+patterns.
+
+For an arbitrary column count `f`, each fixed address generator is replaced
+by its explicit column lift. A native monomial
+`|x> -> i^(a+q(x)) |M*x+b>` lifts to the same address recipe on each column,
+with phase `a*f+sum_j q(x_j)` modulo four. Complete spectator coordinates
+remain present. This substitutes physical operators in the already checked
+role-level linear identity; it does not assume tensor power is additive.
+The final normalizers make every productive endpoint the canonical
+positive tensor job. Coordinate permutations place these on any required
+disjoint axes. Remainder axes at an arbitrary width receive individual
+paid kernels and remain completed spectators of the lifted word.
+
+[`FINITE-TEMPLATE-CLOSURE.md`](FINITE-TEMPLATE-CLOSURE.md) supplies fresh
+coefficient, owned-space, row and precision constants. In particular, for
+the completely expanded word with rank `N`, productive width `W`, rank-one
+paid count `P`, unit count `U`, and conservative owned count `O=U`, put
+
+```
+Z  = W+O,
+G  = 1494376*U,
+E  = 64*(Z+N+P+G+1)^3,
+B  = P+E,
+C0 = 32*N*B^2,
+C1 = 1.
+```
+
+The completed-child semantic recurrence is
+`A(e)<=A(floor(e/N))+P*floor(e/N)+E`. It gives `A(e)<=2B*e`; completed
+disjoint outer pieces and retained phases fit below `C0*d`. This measures
+completed returned values without rounding or re-encoding children.
+The large temporary values inside a child are not also unfinished parent
+operands. All storage stays on one common fine grid.
+
+For `d<=c*p^epsilon` with `epsilon<1`, the explicit threshold in that note
+ensures `C0*d<=p/4`, hence total component width at most `2p`. The new `C0`
+may be enormous, but it is fixed. There is no inherited small numerical
+guard constant and no appeal to the obsolete pathwise precision exponent.
+
+## 4. BIT uses one delayed ambient cover and one common weight
+
+The argument reviewed is
+[`../bit/DELAYED-COVER-BRIDGE.md`](../bit/DELAYED-COVER-BRIDGE.md).
+`BIT-BRIDGE-AUDIT.json` supplies a separate exact implementation audit.
+The essential order is to precompile the entire finite projector program
+first, and only then apply a single whole-ambient `GL_N` cover and external
+weight. Internal jobs are not prematurely lowered to differently weighted
+canonical children.
+
+For a rational idempotent `P` in ambient dimension `N`, write
+
+```
+D_P = [[I-P, P*C_N], [C_N*P, I-C_N*P*C_N]],
+Q_S = diag(S, C_N*S*C_N).
+```
+
+Direct multiplication gives `Q_S D_Q Q_S^-1=D_(SQS^-1)` and
+`D_P D_(I-P)=D_I`, including nonsymmetric projectors. Equal-rank rational
+idempotents admit a fixed rational matcher `S` by matching image and kernel
+bases. After the finite program is fixed, choose an odd prime avoiding all
+its finitely many denominators, determinants and rank witnesses. All lifts
+use these same rational matrices, so this choice is independent of the
+eventual column multiplicity.
+
+Use the full cover `g in GL_N(Z/q^w Z)`. To match a parent request `P` to a
+child endpoint `Q`, route that child's complete role fiber by `g -> gS`,
+execute the entire child, and undo the route. Its autonomous endpoint is
+exactly `D_(gPg^-1)`. Separate target roles may use separate matchers: each
+route is a bijection of a full fiber carrying arbitrary data. Native port
+charts and local helper ownership remain inside the schematic program;
+they are not silently discarded when the ambient dimension grows.
+
+This is a paired whole-job construction. It does not implement arbitrary
+isolated dense address maps by renaming rows, and it does not assume a
+generic RAM-to-tape simulation.
+
+The inherited weight `A` is common to all fresh roles and independent of
+the current complete cover field. Consequently `Lambda_A=diag(A,I)`
+commutes with every internal role route and every pointwise scalar/COPY
+operation. The completed source/complement endpoint is therefore the
+required weighted full interchange. Only final rank-one leaves are lowered
+to weighted canonical children. Their pivot weights depend on this parent
+cover row, which is ancestor data when the child's fresh fields are exposed.
+Different leaves can have different weights without violating the common
+weight requirement inside any one child.
+
+At `f` columns the inherited reversed pairing is essential. The actual
+lift uses `P tensor I_f` with full tail reversal `C_(Nf)`, or equivalently
+conjugates the straightforward tensor lift by the blockwise `C_f` reversal.
+The independent audit checks this on nonsymmetric rational projectors and
+rejects dropping that reversal. It also checks arbitrary common external
+weights depending on axes and columns. Current-cover-dependent weights
+remain forbidden.
+
+The independent native replay covers all 636 rational frames, 2,386 nested
+MOVE pairs, both 661-column local words, 111 ports and 666 embedded charts,
+3,073 helper owners, and all 63 paid COPY-center calls. The rank-50 full-job
+example has `W=3739`, `P=186725` and surplus `50*3739-186725=225`.
+Additional checks include nonsymmetric rational matchers, a genuine
+non-field ring, and 7,776 inverse/carry tests over `GL_2(Z/9Z)`. The last
+tests retain all q-adic digits of the matcher; dropping its higher digits
+has an explicit failing control.
+
+### Paid cover, atom conversion, and ordinary endpoint
+
+Right multiplication on `g=g0+qG1` is a finite low-type permutation plus a
+fixed affine high-matrix map with its correct carry. The retained native
+cover-routing lemma charges it as `O(V*w^(1-a_old))`. The finite number of
+routers changes the constant, not this exponent. All bad generic classes
+receive their actual fallback. With the explicit choice
+
+```
+q > 25,600,000*N^5
+```
+
+and the finite denominator exclusions, the complete covered moment is at
+most `400001/400002<1`. This includes the full fallback bill without
+subtracting the good-class calls or assuming bad classes are independent.
+
+Use atom width `w=ceil(e^0.003)` and the retained old ordinary saving
+`a_old=384599/10^10`. The resulting saving is
+
+```
+(1-0.003)*0.002750 + 0.003*a_old
+    = 27418653797/10^13
+    > 0.0027.
+```
+
+The linear adapters are paid because `0.003` exceeds this saving. The
+borrowed row stock satisfies `log S(e)=O(e^0.003 log e)` and is lower order
+because `0.003<1-a_BIT`. Its complete borrowed field is restored. Padding
+of full rows is performed once for the whole invocation and removed after
+the completed endpoint. Numeric padding to atom lengths costs only a
+fixed `q`-dependent factor.
+
+The ordinary wrapper uses two completed reversed calls and three charged
+bank additions, in chronological order:
+
+```
+D <- D-H; F_n; D <- D+H; F_n; D <- H-D.
+```
+
+For `(H,D)` this gives successively `(H,D-H)`, `(C(D-H),CH)`,
+`(C(D-H),CD)`, `(D,D-H)`, `(D,H)`. Thus it really supplies the ordinary
+interchange, at constant multiplicative overhead and `O(Vn)` extra work.
+No unshown batch or free half-swap is required.
+
+## 5. The changed row reserve fits the support-local outer wrapper
+
+This is an essential functional obligation, not another numerical moment
+test. The applicable retained wrapper is the global support-local wrapper
+in `vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md` and its
+`SUPPORT-ADVERSARIAL-AUDIT.md`, under the current outer package. Its completed
+supplier calls cover the **whole global dyadic tensor**. Calling separately
+inside fixed coarse-label microboxes would not satisfy the interface.
+
+For a phase with `h` selected bits, take `G=4*ceil(log2 p)+6` and complete
+neighborhoods of width `K=128G`, one inside each selected original axis.
+The axis length grows like `p^(1-epsilon)`, so every fixed multiple of
+`log p` fits eventually. Exposing the complete neighborhoods moves at most
+`2hK` bit positions. Native binary matrices have fixed rank, so their
+column lifts factor into finitely many parallel row additions on these
+selected bits. Unit phases have fixed formulas of the same selected-bit
+support. They do not introduce dependence on unrelated coarse labels.
+
+Reserve whole selected chunks for the two front controls, back control and
+row prefix. **Do not first transform their selected bits.** The completed
+active invocation is identity on every reserved coordinate for arbitrary
+contents; those selected-axis kernels can be completed in later shrinking
+phases. The finite guard condition used by the retained router is directly
+satisfied by `K=128G`; no claim is made that this constant ratio satisfies
+the older, stronger wording `K/log p -> infinity`.
+
+The new BIT stock cannot simply be inserted into the old statement
+`O(log h)` for all nested reserves. Its own ordinary full-field interface
+supplies its rows internally. Alternatively, the following conservative
+external product bound explicitly covers simultaneous stock. A BIT call
+has width at most `C*h*(log p)^2`. Its logarithmic stock is
+`O(h^0.003*(log p)^(2*0.003+1))`. Including a further `O(log h)` nesting
+factor and the complex stock gives, after enlarging one fixed constant,
+
+```
+q0(h,p) <= C*(log2 h + h^0.003*(log2 p)^3 + 1)
+```
+
+reserved complete chunks. This bound retains the **product** of stock
+requirements; using their maximum would be invalid. Since `0.003<1`, there
+is an explicit fixed polylogarithmic threshold `H(p)` above which
+`q0(h,p)+O(1)<=h/64`. For example one may enlarge a fixed multiple of
+`[1024*C*(log2 p)^3]^(1/(1-0.003))` to cover the logarithmic term and all
+finite-template base thresholds. Consequently
+
+```
+R(h) <= 3h/128 + q0(h,p) + O(1) <= h/16
+```
+
+for all sufficiently large active phases. The remaining `h<=H(p)` bits are
+handled individually in `O(V*polylog p)` time. Thus successive active
+widths shrink by at least a factor 16, and their sublinear costs sum to a
+constant multiple of the first phase's cost.
+
+Each phase pads its complete row count once by a factor below two. Its
+completed endpoint preserves the row coordinate, so appended zero rows
+can be deleted before inverse exposure and before the next phase. This
+prevents an accumulating factor of two per phase. The exposure is restored
+before any next diagonal depending on coarse labels is applied.
+
+The support router's additional control reservoirs occupy
+`O(h*(log p)^2)` positions and fit the full `Theta(p)`-bit tensor because
+`group_beta<1`. A complete untouched `log^2 p` suffix is chosen and placed
+first for short-record grouping; all subsequent reservoirs and routers
+stay in its complement. The retained full-span route handles the bounded
+regime where this gap argument fails. A single original axis is never
+asked to hold all of the `hG` control bits.
+
+## 6. Fresh constants satisfy every changed outer interface
+
+The actual parameter point is in
+[`../controller/OUTER-PARAMETERS.json`](../controller/OUTER-PARAMETERS.json).
+The independent audit reproduces its nine cost rows directly, without
+importing either outer ledger. It also checks the functional inequalities
+needed above, including
+
+```
+epsilon = 0.99999 < 1,
+group_beta = 0.99997 < epsilon,
+1-epsilon-r0 > 0,
+1-(2*xi-2)-r0-group_beta > 0,
+group_beta*(1-0.003) > 0.
+```
+
+The first two absorb every new fixed semantic, movement and row constant;
+the next two retain the precision and fine-prime spare powers. The last
+ensures the changed BIT reserve is sublinear in the active grouped width,
+even after the polylogarithmic factors above. The stopped supplier's
+internal and leaf exponents are strictly below the selected `lambda` and
+`lambda_prime`, respectively. The effective BIT saving is below both the
+actual ordinary BIT saving and the selected complex saving.
+
+Every compiled-word constant, rational denominator and description degree
+is fixed before the input length varies. Polynomial description/preparation
+costs are absorbed by the retained complete superpolynomial record fibers
+when `epsilon<1`, after increasing an effective threshold. A finite program
+on fixed tapes remains a native program on fixed tapes; this argument does
+not hide random-access gathering inside the coefficient allowance.
+
+The nearly critical halo calculation uses the retained explicit-metadata
+branch with `halo_power=48` and `core_power=128`. It does not use the old
+`K0^2=o(p)` shortcut, which is unavailable near this parameter point. The
+boundary exponent uses the exact Strassen comparison `7^16<2^45`; no new
+unstated stable-matrix algorithm is required.
+
+The outer analytic decomposition, ordered movement implementations,
+fixed-tape model, prime padding, and exact integer recovery are the same
+conditional hypotheses as in the reviewed outer package. The changed
+mixed-job, native adapter, weighted-child, row-stock and semantic-precision
+obligations have been supplied explicitly here. The conclusion is therefore
+a source-bound **conditional research result** at `kappa=0.0023` under
+those inherited hypotheses. It is not a claim of practical constants or of
+an unconditional theorem.
+
+## 7. Reproducible independent checks
+
+From the repository root:
+
+```sh
+python3 -B research/kappa-rebuild/transfer-audit/bit_bridge_audit.py
+python3 -B research/kappa-rebuild/transfer-audit/outer_transfer_audit.py
+```
+
+The first binds the independent BIT checks to the exact current native and
+bridge sources. The second uses its own 256-bit directed rational grid,
+finite-series logarithm/exponential bounds, exact rank comparisons, and
+fresh outer equations. `OUTER-TRANSFER-AUDIT.json` records all source hashes,
+strict intervals, compressed ranks, nine outer savings and functional gaps.
+The directory manifest binds this written audit and the complete source set.
diff --git a/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/p3_cover_audit.py b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/p3_cover_audit.py
new file mode 100644
index 0000000000..d9b44b662a
--- /dev/null
+++ b/research/pr4359-tensor-strictification/source/research/kappa-rebuild/transfer-audit/p3_cover_audit.py
@@ -0,0 +1,163 @@
+#!/usr/bin/env python3
+"""Adversarial replay of p3 physical ownership, normalizer order and invoice.
+
+Apache-2.0; substantial OpenAI assistance. Separate from the cover producer.
+"""
+from collections import Counter, defaultdict
+from fractions import Fraction
+from hashlib import sha256
+from pathlib import Path
+import gzip
+import json
+
+from finite_dag_audit import Rule, Compiler
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[2]
+INPUT = HERE.parent/'controller/P3-NATIVE-COVER.json.gz'
+
+
+def need(test, message):
+    if not test:
+        raise AssertionError(message)
+
+
+def digest(path):
+    return sha256(Path(path).read_bytes()).hexdigest()
+
+
+def read(path):
+    raw = Path(path).read_bytes()
+    return json.loads(gzip.decompress(raw) if str(path).endswith('.gz') else raw)
+
+
+def main():
+    receipt = read(INPUT)
+    need(digest(HERE.parent/'controller/p3_native_cover.py') == receipt['generator_sha256'], 'Exact cover generator')
+    for path, expected in receipt['sources'].items():
+        need(digest(ROOT/path) == expected, 'Pinned source '+path)
+    independent = read(HERE/'P3-NATIVE-AUDIT.json.gz')
+    formal = read(HERE.parent/'bit/COMPLEX-CONTROLLER-FACTOR.json')['selected_p3_keep43']
+    ours = {(row['stage'],row['port']):row for row in independent['contacts']}
+    for row in receipt['main_contacts']:
+        other = ours[row['stage'],row['port']]
+        need(all(row[key] == other[key] for key in ('maps','pair','direction')), 'Independent actual contact recipes agree')
+    ours_ends = {(row['output_role'],row['port']):row for row in independent['endpoint_normalizers']}
+    main_ranks = {}
+    for row in receipt['main_endpoint_normalizers']:
+        key = row['original_input_role'],row['port']
+        other = ours_ends[key]
+        need(row['target_rank'] == other['rank'] and row['positive_axes'] == other['target_axes'], 'Independent target axes')
+        need(row['output_translation_before_inverse_map'] == other['physical_translation_after'], 'Translation precedes output inverse')
+        columns, inverse_rows = row['input_coordinate_columns'],row['output_coordinate_inverse_rows']
+        need(columns[:row['target_rank']] == row['positive_axes'], 'Input conjugation has exact required columns')
+        for j, column in enumerate(columns):
+            decoded = sum(((r & column).bit_count() & 1) << i for i,r in enumerate(inverse_rows))
+            need(decoded == 1 << j, 'Output map is inverse of actual input map')
+        role,port = key
+        main_ranks[f'{"Y" if role%2 else "X"}{role//2}:{port}'] = row['target_rank']
+    need(receipt['physical_order'] == 'Endpoint input coordinate maps; R; literal serialized controller with whole-call helper adapters; L; endpoint translations; endpoint inverse coordinate maps.',
+         'The emitted order realizes G^-1 X L M R G')
+    cover = receipt['cover']
+    helpers_path = HERE.parent/'complex/fresh-certificates/THREE-base-HELPER-BOUNDARIES.json.gz'
+    helper_width = {row['role']:row['width'] for row in read(helpers_path)['recipes']}
+    stage_table = {event[1]:(event[2], 'backward' if event[3] else 'forward')
+                   for event in formal['physical_events'] if event[0]=='LOCAL'}
+    by_stage = defaultdict(list)
+    by_window = defaultdict(list)
+    for row in cover['helper_bindings']:
+        stage = row['stage']
+        need(helper_width[row['role']] == row['width'] == len(row['axes']), 'Actual native helper width, not a rank-only substitute')
+        contract = row['whole_helper_boundary']
+        need(contract['direction'] == stage_table[stage][1] and contract['role'] == row['role'] and
+             contract['provider'] == 'THREE-base', 'Exact oriented helper normalizer')
+        by_stage[stage].append(row)
+        if row['kind'] == 'original_idle':
+            by_window[row['window_index'],row['owner']].append(row)
+        matrix = cover['owner_coordinate_maps'][row['coordinate_map']]
+        need(matrix['columns'][:row['width']] == row['axes'], 'Helper conjugation reaches actual idle axes')
+        for j,column in enumerate(matrix['columns']):
+            decoded = sum(((r & column).bit_count() & 1) << i for i,r in enumerate(matrix['inverse_rows']))
+            need(decoded == 1 << j, 'Helper route inverse')
+    for stage, rows in by_stage.items():
+        need({row['role'] for row in rows} == set(range(224,632)) and len(rows)==408, 'All local helper roles used exactly once')
+        need(len({row['owner'] for row in rows}) == 408, 'Helper physical owners injective')
+        active_pair = stage_table[stage][0]
+        active = {f'{side}{active_pair}:{port}' for side in ('X','Y') for port in range(112)}
+        need(not active & {row['owner'] for row in rows}, 'No helper borrows a live main register')
+    events = formal['physical_events']
+    paid_by_owner = defaultdict(list)
+    for row in cover['paid_residuals']:
+        need(row['rank'] == len(row['axes']) and 1 <= row['rank'] <= 5, 'Actual completion width')
+        paid_by_owner[row['owner'],row['after_stage'],row['kind']].append(row)
+    for window in cover['windows']:
+        stages, pair = window['stages'],window['pair']
+        idle_index = next(i for i,event in enumerate(events) if event == ['IDLE',pair,stages])
+        first_index = next(i for i,event in enumerate(events) if event[0]=='LOCAL' and event[1]==min(stages))
+        for event in events[first_index:idle_index]:
+            if event[0] == 'LOCAL':
+                need(event[2] != pair, 'Owned idle field is never an active local input in the window')
+            elif event[0] == 'ADD':
+                need(event[1]//2 != pair and event[2]//2 != pair, 'No interleaved bridge reads or writes a partially completed idle')
+            elif event[0] == 'SIGNED_SWAP':
+                need(event[1]//2 != pair, 'No swap before idle completion')
+        assigned = sorted(by_window[window['index'],window['owner']], key=lambda row:row['stage'])
+        need([(row['stage'],row['width']) for row in assigned] ==
+             [(s,w) for s,w in zip(stages,window['widths']) if w], 'Actual window slice assignment')
+        pieces = [axis for row in assigned for axis in row['axes']]
+        rests = paid_by_owner[window['owner'],max(stages),'idle_remnant']
+        pieces += [axis for row in rests for axis in row['axes']]
+        need(pieces == window['axes'], 'Every idle axis is funded once in chronological order')
+    fresh = {row['owner']:row for row in cover['fresh_outputs']}
+    fresh_binding = {row['owner']:row for row in cover['helper_bindings'] if row['kind']=='fresh'}
+    need(len(fresh)==len(fresh_binding)==1570 and set(fresh)==set(fresh_binding), 'Every fresh field is an independent actual helper')
+    for name, row in fresh.items():
+        pieces = list(fresh_binding[name]['axes'])
+        pieces += [axis for extra in paid_by_owner[name,row['stage'],'fresh_topup'] for axis in extra['axes']]
+        need(pieces == [1 << i for i in range(row['target_rank'])], 'Every fresh full target includes paid topup axes')
+    local_receipt = read(HERE.parent/'complex/full-identity-certificates/RESULT.json')
+    native = read(HERE.parent/'complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz')
+    rounds = []
+    native_scalar = 0
+    for stage in range(7):
+        pair,direction = stage_table[stage]
+        native_scalar += local_receipt['providers']['THREE-base']['split_replays'][direction]['independent_native_replay']['conservative_real_scalar_primitive_allowance']
+        owner = {row['role']:row['owner'] for row in by_stage[stage]}
+        for event in native[direction]:
+            if event[0]=='climb':
+                role=event[2]
+                name = f'X{pair}:{role}' if role<112 else f'Y{pair}:{role-112}' if role<224 else owner[role]
+                rounds.append((event[-1],name))
+            elif event[0]=='center':
+                rounds.append((event[-1],'owned-C0'))
+        rounds += [(row['rank'],row['owner']) for row in cover['paid_residuals'] if row['after_stage']==stage]
+    roles = tuple(sorted(main_ranks.items()))+tuple((name,row['target_rank']) for name,row in sorted(fresh.items()))
+    rule = Rule(roles,tuple(rounds))
+    need(rule.targets == Counter({int(k):v for k,v in cover['targets'].items()}), 'All productive ranks independently counted')
+    need(Counter(rank for rank,name in rounds) == Counter({int(k):v for k,v in cover['paid'].items()}), 'Every paid child independently counted on its actual owner')
+    scalar = native_scalar+112*((22+12+9)*8+8*2)
+    need(scalar == receipt['scalar_invoice']['conservative_real_scalar_primitive_allowance'] == 1147304,
+         'Complete arithmetic invoice includes all six swap negations')
+    compiler = Compiler(rule)
+    compiled = compiler.compile(57)
+    need(compiled['width']==2242 and compiled['leaves']==127570 and compiled['units']==1, 'Full rank57 completion including all partial-source complements')
+    result = {'schema':'independent-p3-owner-order-and-complete-tariff-audit/1',
+              'status':'PASS_ALL2856_HELPERS_CAUSAL_WINDOWS_MAIN_NORMALIZER_ORDER_AND_COMPLETE_BILL',
+              'source_sha256':digest(__file__),'input_sha256':digest(INPUT),
+              'main_phase_audit_sha256':digest(HERE/'P3-NATIVE-AUDIT.json.gz'),
+              'native_local_source_bound':True,'helper_bindings':2856,'independent_productive_fields':2242,
+              'targets':dict(sorted(rule.targets.items())), 'paid':dict(sorted(Counter(r for r,n in rounds).items())),
+              'owned_center_uncredited':1,'arithmetic_allowance':scalar,
+              'uniform_rank57':{key:str(value) if isinstance(value,Fraction) else value
+                                for key,value in compiled.items() if key!='ordered_requirements'},
+              'uniform_rank57_event_graph':compiler.trees[57].summary(),
+              'limits':['Ideal native finite unit and finite integer compilation audit.',
+                        'The rank57 depth-zero supplier is not asserted to attain the mixed exponent.',
+                        'Large fixed-template existence, updated precision/stock and outer transfer are separate documented steps.']}
+    (HERE/'P3-COVER-AUDIT.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    print(json.dumps({'status':result['status'],'targets':result['targets'],'paid':result['paid'],
+                      'scalar':scalar,'uniform_full57_paid1':compiled['leaves']}))
+
+
+if __name__=='__main__':
+    main()
diff --git a/research/pr4359-tensor-strictification/strictification.py b/research/pr4359-tensor-strictification/strictification.py
new file mode 100644
index 0000000000..c0c5f12417
--- /dev/null
+++ b/research/pr4359-tensor-strictification/strictification.py
@@ -0,0 +1,106 @@
+#!/usr/bin/env python3
+"""Exact moment certificate for synchronized native tensor strictification.
+
+Apache-2.0. Substantial OpenAI/Codex assistance. The source native word and
+interval engine are PR4359; its compiler uses Jacob Alan Sussman's Theorem 1G.
+The native tensor construction and cutoff argument are written in TENSOR.md.
+"""
+from fractions import Fraction as Q
+from hashlib import sha256
+import argparse
+import importlib.util
+import json
+from pathlib import Path
+import sys
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+EXPONENT = 15000
+CUTOFF_BASE = 4
+TESTED_SAVING = Q("0.00247756")
+ALPHA_UPPER = 1-Q(1,10**9)
+GAMMA_UPPER = Q(999,1000)
+TARGETS = {5:196, 6:126, 7:392, 8:651, 9:1491, 25:448, 57:224}
+PAID = {1:17409, 2:4991, 3:1568, 4:714, 5:140, 8:504, 16:448}
+
+
+def need(ok, message):
+    if not ok:
+        raise AssertionError(message)
+
+
+def module(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    result = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(result)
+    return result
+
+
+def upper_moment(exponent, alpha=ALPHA_UPPER, gamma=GAMMA_UPPER):
+    """Use a rational Bernoulli bound on alpha^n, retaining gamma^n exactly."""
+    return 1/(1+exponent*(1-alpha))+2*gamma**exponent
+
+
+def run(source_root):
+    native = module("tensor_native_inventory", HERE/"native_inventory.py").run(source_root)
+    targets = {int(k):v for k,v in native["targets"].items()}
+    paid = {int(k):v for k,v in native["paid"].items()}
+    need(targets == TARGETS and paid == PAID, "Exact authenticated native inventory")
+    engine_path = source_root/"research/kappa-rebuild/architecture/mixed_target_probe.py"
+    E = module("tensor_source_intervals", engine_path)
+    target_lo, target_hi = E.moment(targets, TESTED_SAVING)
+    paid_lo, paid_hi = E.moment(paid, TESTED_SAVING)
+    paid_rank = sum(k*v for k,v in paid.items())
+    target_rank = sum(k*v for k,v in targets.items())
+    need((paid_rank,target_rank) == (46851,47075), "Exact rank-one invoice")
+    alpha_lo, alpha_hi = paid_lo/target_hi, paid_hi/target_lo
+    logbase_lo, logbase_hi = E.log_interval(CUTOFF_BASE)
+    base_power_lo = E.exp_interval(TESTED_SAVING*logbase_lo)[0]
+    base_power_hi = E.exp_interval(TESTED_SAVING*logbase_hi)[1]
+    gamma_lo = Q(paid_rank)/(base_power_hi*target_hi)
+    gamma_hi = Q(paid_rank)/(base_power_lo*target_lo)
+    need(0 < alpha_hi < ALPHA_UPPER < 1, "Certified alpha <= 1-10^-9")
+    need(0 < gamma_hi < GAMMA_UPPER < 1, "Certified gamma < 999/1000")
+    bound = upper_moment(EXPONENT)
+    need(bound < 1, "Full tensor and cutoff moment, including every paid chunk")
+    need(4**8 < 5**7, "Strict paid/target logarithmic separation theta < 7/8")
+    need(max(paid) >= min(targets), "Raw native tariff needs strictification")
+    need(upper_moment(EXPONENT, gamma=Q(1)) > 1, "Tail condition is essential")
+    need(upper_moment(10000) > 1, "This conservative certificate rejects a smaller trial power")
+    need(E.moment(targets,Q("0.003"))[1] < E.moment(paid,Q("0.003"))[0],
+         "Unsupported raw saving .003 is rejected")
+    return {
+        "schema": "native-tensor-cutoff-strictification/1",
+        "status": "PASS_SOURCE_BOUND_TENSOR_STRICTIFICATION",
+        "exponent": EXPONENT, "cutoff_base": CUTOFF_BASE,
+        "tested_saving": str(TESTED_SAVING), "targets": targets, "paid": paid,
+        "target_rank_one_moment": target_rank, "paid_rank_one_moment": paid_rank,
+        "alpha_upper": str(ALPHA_UPPER), "alpha_interval": E.show((alpha_lo,alpha_hi)),
+        "gamma_upper": str(GAMMA_UPPER), "gamma_interval": E.show((gamma_lo,gamma_hi)),
+        "normalized_strict_moment_upper": E.show((bound,bound)),
+        "strictification_gap": E.show((1-bound,1-bound)),
+        "strict_moment_verified": True,
+        "new_min_target_encoding": "5^15000", "new_max_paid_encoding": "4^15000",
+        "new_max_target_encoding": "57^15000", "theta_upper": "7/8",
+        "native_inventory": native,
+        "source_sha256": {"research/kappa-rebuild/architecture/mixed_target_probe.py":
+                          sha256(engine_path.read_bytes()).hexdigest()},
+        "controls": {"unstrictified_native_tariff_rejected": True,
+                     "gamma_one_rejected": True,
+                     "power_10000_fails_this_conservative_bound": True,
+                     "raw_saving_003_rejected": True},
+        "materialized_tensor_word": False,
+        "scope": "The exact finite construction is specified by synchronized native substitution and paid-axis splitting. This receipt proves its sufficient moment and separation inequalities, conditional on the documented native interfaces; it is not a formal whole-word or multiplication-theorem verification."}
+
+
+if __name__ == "__main__":
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--source-root", type=Path, default=HERE/"source")
+    parser.add_argument("--out", type=Path)
+    args = parser.parse_args()
+    result = run(args.source_root)
+    if args.out:
+        args.out.write_text(json.dumps(result,indent=2,sort_keys=True)+"\n")
+    print(json.dumps({key:result[key] for key in
+                     ("status","exponent","cutoff_base","tested_saving","strictification_gap")},
+                     sort_keys=True))
diff --git a/research/pr4359-tensor-strictification/verify.py b/research/pr4359-tensor-strictification/verify.py
new file mode 100644
index 0000000000..6ca7dda653
--- /dev/null
+++ b/research/pr4359-tensor-strictification/verify.py
@@ -0,0 +1,90 @@
+#!/usr/bin/env python3
+"""Authenticate and verify the conditional native tensor construction.
+
+Apache-2.0. Substantial OpenAI Codex assistance; source notices are retained.
+"""
+import argparse
+from fractions import Fraction as Q
+import hashlib
+import importlib.util
+import json
+from pathlib import Path, PurePosixPath
+
+HERE = Path(__file__).resolve().parent
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def module(name, path):
+    spec = importlib.util.spec_from_file_location(name, path)
+    value = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(value)
+    return value
+
+
+def verify():
+    manifest = json.loads((HERE / 'EXPORTED-SOURCES.json').read_text())
+    seen = set()
+    for row in manifest['files']:
+        rel = PurePosixPath(row['path'])
+        need(not rel.is_absolute() and '..' not in rel.parts and str(rel) == row['path'],
+             'Canonical source path')
+        need(row['path'] not in seen, 'Unique source path')
+        seen.add(row['path'])
+        path = HERE
+        for part in rel.parts:
+            path /= part
+            need(not path.is_symlink(), 'No source aliases')
+        data = path.read_bytes()
+        need(len(data) == row['bytes'] and hashlib.sha256(data).hexdigest() == row['sha256'],
+             'Exact exported source bytes: ' + row['path'])
+        need(hashlib.sha1(b'blob ' + str(len(data)).encode() + b'\0' + data).hexdigest()
+             == row['git_blob'], 'Exact original Git blob')
+
+    source = HERE / 'source'
+    S = module('strictification', HERE / 'strictification.py')
+    E = module('endpoint', HERE / 'endpoint.py')
+    independent = module('independent_intervals', HERE / 'dependencies/compiler-refill/template_bound.py')
+    strict = S.run(source)
+    endpoint = E.run(source, strict)
+    targets, paid = strict['targets'], strict['paid']
+    saving = Q(strict['tested_saving'])
+    tlo, thi = independent.moment_margin(targets, {}, saving)
+    plo, phi = independent.moment_margin(paid, {}, saving)
+    need(phi / tlo < Q(strict['alpha_upper']), 'Independent paid/target moment ratio')
+    loglo, loghi = independent.log_interval(Q(strict['cutoff_base']))
+    base_power_lo = independent.exp_interval(saving * loglo)[0]
+    paid_mass = sum(k * v for k, v in paid.items())
+    need(Q(paid_mass) / (base_power_lo * tlo) < Q(strict['gamma_upper']),
+         'Independent rank-one tail ratio')
+    n = strict['exponent']
+    upper = 1 / (1 + n * (1 - Q(strict['alpha_upper']))) + 2 * Q(strict['gamma_upper']) ** n
+    need(upper < 1, 'Independent complete strictified moment')
+    need(min(targets) > strict['cutoff_base'], 'Strict paid/target rank separation')
+    bit = json.loads((source / 'research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json').read_text())
+    need(Q(endpoint['outer']['parameters']['bit_saving']) == Q(bit['ordinary_saving_admitted']),
+         'Unchanged admitted ordinary BIT saving')
+    return {'status': 'PASS_SOURCE_BOUND_CONDITIONAL_NATIVE_TENSOR_IMPROVEMENT',
+            'source_files': len(seen), 'strictification': strict, 'endpoint': endpoint,
+            'independent_interval_crosscheck': True,
+            'scope': 'Conditional native construction and exact finite arithmetic. The finite word and inherited physical, precision, analytic and fixed-tape proofs are not formally verified or materialized by this checker.'}
+
+
+if __name__ == '__main__':
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write-result', action='store_true')
+    args = parser.parse_args()
+    result = verify()
+    normalized = json.loads(json.dumps(result))
+    receipt = HERE / 'RESULT.json'
+    if args.write_result:
+        receipt.write_text(json.dumps(normalized, indent=2, sort_keys=True) + '\n')
+    else:
+        need(normalized == json.loads(receipt.read_text()), 'Exact saved certificate reproduction')
+    print(json.dumps({'status': result['status'], 'kappa': result['endpoint']['kappa'],
+                      'mixed_saving': result['strictification']['tested_saving'],
+                      'tensor_power': result['strictification']['exponent'],
+                      'uniform_rank_exponent': result['endpoint']['compiler']['H1']}))