Reduce prepared P104 joint repair from 3N to 2N native calls

research/p104-full-chart-joint/PROOF.md

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+# Completing the persistent bank once reduces the matched repair from 3N to 2N
+
+## Exact same prepared physical boundary
+
+This is a refinement of PR3978, head48e2e44d8785be8204d8c4b719850332f4f9abe0.
+Its exact PROOF and RESULT are exported unchanged in inputs/. The proposed
+chronological global producer remains unproved. We change only the repair at
+that predecessor's explicitly prepared cut, with fixed named X/Y owners and
+the same two dirty helper banks. No new payload code, changed source input,
+free bank exchange or improved global caller is asserted.
+
+Write A_r=A tensor I, A_c=I tensor A for the canonical P104 A=G/2, and
+N=104^2=10816. Let U(Q)=[[I,Q],[0,I]], S(Q)=[[I,0],[Q,I]], and
+M=U(-A_c)S(A_r). As in3978, original physical input (x,y) is normalized by
+u=T^-1 x, with T=diag(C_Uij), K=T^-1, F=C_I and Gphys=diag(F,FK).
+The prepared input is Gphys M(u,y); the required output is Gphys J(u,y),
+where J=[[0,-I],[I,0]]. This retains the original input tags. It does not
+replace the physical source x by the logical normalized source u.
+
+The common-coordinate compensator is
+
+    C = J M^-1 = [[A_r,A_r A_c-I],[I,A_c]]
+      = U(A_r-I) S(I) U(A_c-I).
+
+Set D_T=diag(I,T), D_K=diag(I,K). The predecessor's exact physical map is
+
+    Cphys = D_K C D_T
+          = U((A_r-I)T) S(K) U((A_c-I)T).
+
+The new word executes the LEFT expression in chronological order:
+(1) apply T in place to every persistent Y; (2) execute the three scalar
+layers of C in the common F chart; (3) apply K in place to every resulting Y.
+Because T(FK)=F and K F=FK, the entrance and exit charts match literally.
+The original Y payload need not survive until step3: C is intentionally a
+mixed two-bank map. The proof uses only TK=KT=I and the common F commutation;
+A_r/A_c are not commuted through T. Both dirty banks are untouched by this
+repair. Completion of their preceding paths is still a prepared-cut premise.
+
+The free noncommutative checker verifies the matched endpoint, including
+Gphys and diag(K,I) on the original physical input. A separate implementation
+compares all columns of old and new maps at varying noncommuting Gaussian
+charts. Omitting either boundary conversion or exchanging their order fails.
+
+## Exact boundary primitives and phases
+
+The canonical native certificate supplies the104 odd distinct physical root
+labels r_i. The line generators u_ij=r_i tensor r_j are10816 distinct norm-one
+81-bit labels. For nonzero u choose a set-bit pivot p, e=2^p, and define
+
+    P_u(a)=a + a_p (u+e)       over F2.
+
+P_u is an involutive invertible binary basis map and P_u(e)=u. Its address
+permutation has zero phase. With the retained positive one-coordinate C_p,
+
+    T_u=P_u C_p P_u,
+    K_u=X_u P_u C_p P_u.
+
+The finite f=1 formula C_u=((1+i)I+(1-i)X_u)/2 verifies the exact phase:
+C_u^2=X_u and C_u^-1=X_u C_u. This formula is used only as a bounded
+operator check, not as an implementation that expands a large native child.
+The independent checker compares every Gaussian matrix column for all seven
+nonzero three-bit u, both directions, and rejects a global phase mutation.
+The primary checks P_u^2=I on all81 basis vectors for every actual product
+line (876096 binary basis checks).
+
+For f parallel address columns the implementation lifts the outer basis
+permutation as P_u tensor I_f and RETAINS one positive C_f child on the
+p-coordinate block. The inverse uses the corresponding translation monomial
+on all f columns. Tensoring the exact operator identity establishes this
+ideal native recipe without expanding2^f terms. Each conversion acts on the
+same full physical array volume as3978's copied transform; there is no small-
+volume donor shortcut. Binary basis permutation, translation, scalar and
+control passes remain paid physical work. This algebraic lifting is not
+signed-floor, precision, recovery or recursive supplier admission.
+
+## Complete matched operation invoice
+
+In each upper scalar layer Q=A_axis-I has4992 signed half coefficients per
+fiber and zero diagonal. For each donor in the persistent Y bank: halve it,
+perform all signed unit additions to the distinct X bank, and double it back
+before advancing to the next donor. The lower layer is one X-to-Y unit add
+per data position. The second upper layer reads the updated Y fields. No
+new initialization, private copy or erasure is needed. All donor rescalings
+are restored; the intended mixed endpoint is still C, not the original Y.
+Exact integer rows check every208 local input column of this word, and the
+separate Gaussian checker executes both upper rescaling words around the
+lower layer, with an unrestored-donor failure control.
+
+| Primitive | PR3978 copied route | New persistent full-chart route |
+|---|---:|---:|
+| Positive rank-one transforms |32448 = 3N|21632 = 2N|
+| Unit array ADD/SUB |1049152|1049152|
+| Halves |21632 on dead private donors|21632 on persistent donors|
+| Restoring doubles |0|21632|
+| Private initialization/copy/erase |32448 each|0|
+| Peak additional private arrays |1|0|
+
+Thus the positive-child rank bill decreases by10816, exactly one third of
+that particular predecessor route. This is a matched prepared-cut operation
+comparison, not a scalar/runtime optimum or a lower bound on arbitrary native
+factorizations. The explicit new boundary recipes additionally have43264
+binary basis permutation passes and10816 inverse translation passes, without
+claiming those are physical-cost minima. All scalar additions and rescalings
+remain finite paid work, not zero because they carry no positive child rank.
+
+In the declared chart ledger, the new entrance T contributes+N persistent
+progress and the final K contributes-N. Net progress remains zero, just as
+3978's repair. The in-place route therefore cannot claim the initial lift as
+new endpoint progress while dropping the final descent from its bill.
+
+## Complete conditional gate and stop
+
+Retain3978/3954's named global hypotheses:104 row and104 column bare calls,
+m=81, W=36608 persistent roles, N initial rank-one anchors; each bare local
+call has mass2368 and persistent progress at most2312, even granting the
+unproved opposite orientation. All residual native rank tariff is at least
+net persistent progress, ranks<=81 and equal physical-role volume1/W.
+The most favorable bare bill is2966080 against capacity2965248.
+Replacing only the old prepared-cut repair gives
+
+    old lower bill = 2966080+32448 = 2998528,
+    new lower bill = 2966080+21632 = 2987712,
+    new excess     = 22464 > 0.
+
+All omitted actual preparation, interstage alignment, wrappers, scalar passes,
+routing/holding/control/tapes, finite-width/precision/recovery and admission
+costs remain unknown nonnegative additions. For0<=a<1, (r/81)^(1-a)>=r/81,
+so this favorable complete native gate still fails throughout that range.
+Stop dependent native/global construction for the unchanged scaffold.
+No selected-frontier or kappa update follows.
+
+A changed raw all-FULL producer, including the new late-delivery work, requires
+its own chronological input/exit tags and progress bound. In particular, a
+single-axis route returning F T^-1 x does not implement F x on the original
+physical source. The apparent anchor credit cannot be claimed by renaming
+that normalized field. No such changed producer is incorporated here.
+
+## Remaining obligation and credit
+
+The useful successor is an actual producer that permits one boundary pass to
+be absorbed while retaining all original physical inputs, owner returns,
+arbitrary dirty helpers and a favorable complete gate. Start with one shared
+owner at an actual adjacent call boundary; do not infer availability from the
+symbolic prepared cut. This study supplies a cheaper target repair only.
+
+The comparison and original compensator retain Chris R/Clear1nsight's3978
+credit, following3954/3940/3882 and the canonical copied-center caller.
+Rohan Arun/#1972 owns the native-word/dirty source, Chafik Boukhalfa/#1991 its
+native realization, Adam Gall/#1940 delivery, and the underlying E8, compiler,
+phase and caller authors retain the attribution listed in README. Prepared
+with substantial OpenAI Codex multi-agent assistance. Apache-2.0.

research/p104-full-chart-joint/README.md

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+# A matched full-chart optimization of P104 joint compensation
+
+Completing the persistent Y bank once, doing all three scalar layers in a
+common full chart, and restoring its terminal chart reduces PR3978's prepared-
+cut positive-child bill from **32448 to 21632**, while eliminating all private
+copies and temporary arrays. Both words implement the exact same physical
+operator with the same original input tags and arbitrary dirty helpers.
+The new word adds 21632 restoring doubles, with the same 1049152 unit array additions.
+This is a reduction in native child count and private storage; scalar work
+remains paid.
+
+The identity is D_K C D_T=Cphys, where D_T=diag(I,T), D_K=diag(I,T^-1) and
+C=U(A_r-I)S(I)U(A_c-I). See [PROOF.md](PROOF.md) for chronological charts,
+phase-exact retained native-child recipes, restricted scope, full primitive
+invoice and conditional gate. [RESULT.json](RESULT.json) records the checks.
+This combines global chart completion with elementary operator factorization.
+
+The improvement is local to the **same explicitly prepared physical cut** as
+[PR3978](https://github.com/CrocSwap/integer-mult-bounds/pull/3978), exact head
+48e2e44d8785be8204d8c4b719850332f4f9abe0. Its unchanged proof/result are exported
+with hashes/Git blobs. No actual chronological P104 producer, opposite native
+orientation or global interstage alignment is supplied. Under the inherited
+named complete invoice, the excess decreases from 33280 to 22464 and still fails.
+No native optimum, runtime gain, supplier or new kappa is claimed.
+
+## Reproduce and verification scope
+
+Standard-library Python 3.12:
+
+    python3.12 -B scripts/research_package.py --package research/p104-full-chart-joint
+    python3.12 -B scripts/check_research_commit.py --package research/p104-full-chart-joint --check-source-history
+
+VERIFY.json declares bounded checks. The primary validates exact ordered
+noncommutative identities, original physical input tags, all 208 local donor
+columns, canonical 104 physical labels and 10816 tensor lines, and both
+rank-one boundary recipes on 876096 binary basis vectors. The separate checker
+compares 32 Gaussian columns and 8 arbitrary dirty columns, executes the literal
+in-place scalar word, compares 56 finite native columns and rejects physical
+order, donor-restoration and phase faults. Finite f=1 formulas are only checks;
+the general-f implementation retains one C_f child, not an exponential scalar
+expansion. Signed-floor and recursive admission remain separate obligations.
+Actual normal/optimized CLI controls validate private fixtures before rejecting
+receipt/source/predecessor mutations. These are scoped checks, not human or
+proof-assistant certification.
+
+## Selection and overlap
+
+User-assigned changed scope under revision 6 `p104-side-correction`, base
+20ecf2891d80b5acf18d980b2164fb44ea7faa90. The coordinator refreshed 702 open PR
+bodies including drafts, live handoffs and canonical main at the new round;
+worker checked the current source/head and repository helper. Canonical claim:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3954#issuecomment-6263112891
+Discussion/closed-diff coverage is bounded. The raw-full caller, source-tag and
+other parallel studies are distinct. No frozen predecessor or frontier file
+is modified.
+
+## Open next questions
+
+TASK.json asks whether one actual producer/consumer boundary can absorb a
+conversion without changing original physical source identity, helper returns
+or the complete gate. First bind named owners and exact entrance/exit charts.
+Remaining blockers include the missing prepared producer, interstage alignment,
+native orientation/phase, physical scalar/routing/control/tape costs and
+finite-width/precision/recovery/all-size admission. Stop expansion under the
+unchanged failed gate; a common full chart alone does not erase source tags.
+
+## Credit
+
+Chris R/Clear1nsight owns PR3978's original compensator and copied route, following
+PR3954/3940/3882/3943. Retain Rohan Arun/#1972, Chafik Boukhalfa/#1991,
+Adam Gall/#1940,1337isnot1337/#3894, ToikY/#3956/#3966/#3996, and 3961/3962/3965
+for their distinct boundary, coverage and copy audits. Retain Josie Elliston,
+Sagar Thakrar, Douglas Colkitt, Dominik Scholz and the canonical E8/native/
+Clifford/compiler/caller source lineage. New matched optimization prepared for
+Chris R/Clear1nsight with substantial OpenAI Codex multi-agent assistance.
+Apache-2.0. Selected frontier unchanged.

research/p104-full-chart-joint/RESULT.json

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+{
+  "schema_version": 1,
+  "scope": "same prepared physical cut as PR3978; actual producer unproved",
+  "n": 104,
+  "N": 10816,
+  "native_boundary_certificate": {
+    "physical_labels": [
+      386,
+      391,
+      405,
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+    ],
+    "product_line_count": 10816,
+    "binary_basis_checks": 876096,
+    "recipe": "P_u C_p P_u; inverse X_u P_u C_p P_u",
+    "rank_per_direction": 1,
+    "recipe_phase": 0
+  },
+  "old_route": {
+    "positive_rank_one_copy_transforms": 32448,
+    "unit_array_adds": 1049152,
+    "private_donor_halves": 21632,
+    "private_initializations": 32448,
+    "private_copies": 32448,
+    "private_erasures": 32448,
+    "peak_additional_private_arrays": 1,
+    "persistent_helper_changes": 0
+  },
+  "new_route": {
+    "positive_rank_one_transforms": 21632,
+    "unit_array_adds": 1049152,
+    "persistent_donor_halves": 21632,
+    "persistent_donor_doubles": 21632,
+    "private_initializations": 0,
+    "private_copies": 0,
+    "private_erasures": 0,
+    "peak_additional_private_arrays": 0,
+    "binary_basis_permutation_passes": 43264,
+    "inverse_translation_passes": 10816,
+    "persistent_helper_changes": 0,
+    "net_persistent_chart_progress": 0
+  },
+  "native_rank_saved": 10816,
+  "gate": {
+    "bare_lower_bill": 2966080,
+    "capacity": 2965248,
+    "old_joint_lower_bill": 2998528,
+    "new_joint_lower_bill": 2987712,
+    "new_excess": 22464
+  },
+  "selected_frontier_changed": false,
+  "global_producer_constructed": false
+}