diff --git a/research/retirement-rebalanced-query-tree/ADAPTERS.json b/research/retirement-rebalanced-query-tree/ADAPTERS.json
new file mode 100644
index 0000000000..f934706771
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/ADAPTERS.json
@@ -0,0 +1,76 @@
+{
+  "driver": [
+    [
+      "def run(write=False):",
+      "def run(write=False, extension=None):"
+    ],
+    [
+      "result=driver.run(services=services,extension=candidate)",
+      "result=driver.run(services=services,extension=candidate if extension is None else extension(candidate))"
+    ]
+  ],
+  "controller": [
+    [
+      "def construct():",
+      "def construct(data=None):"
+    ],
+    [
+      "    data = json.loads((HERE/'ROOT.json').read_text())",
+      "    if data is None:\n        data = json.loads((HERE/'ROOT.json').read_text())"
+    ]
+  ],
+  "pricing": [
+    [
+      "from controller import formula_invoice, need",
+      "from tree_invoice import formula_invoice, need"
+    ],
+    [
+      "KAPPA = Q('0.00329394')",
+      "KAPPA = Q('0.00329492')"
+    ],
+    [
+      "TESTED = Q('0.003293945')",
+      "TESTED = Q('0.003294925')"
+    ],
+    [
+      "PREVIOUS = Q('0.003288299')",
+      "PREVIOUS = Q('0.00329394')"
+    ],
+    [
+      "expected_targets.update({1:104,16:104,17:-104})",
+      "expected_targets.update({1:104,40:104,41:-104})"
+    ],
+    [
+      "Independent tree, retirement delta and emitted-event counts",
+      "Independent rebalanced tree, retirement delta and emitted-event counts"
+    ],
+    [
+      "'kappa_decimal': '0.00329394'",
+      "'kappa_decimal': '0.00329492'"
+    ],
+    [
+      "Independent Gaussian retirement changes the complete productive profile.",
+      "Rebalance the last two binary children under the fixed Gaussian retirement."
+    ],
+    [
+      "It includes all nine output contractions and their productive target debits.",
+      "It includes all nine output contractions and their productive target debits.\nThis adaptation changes the last two child trees under the PR6545 retirement."
+    ],
+    [
+      "reserve, decay = Q(1, 10**29), Q(31, 32)",
+      "reserve, decay = Q(1, 10**27), Q(97, 100)"
+    ],
+    [
+      "need(alpha[1] < 1-reserve and gamma[1] < decay and delta[1] < decay, 'All charged tail ratios')",
+      "need(alpha[1] < 1-reserve and gamma[1] < decay and delta[1] < decay, 'All charged tail ratios')\n    need(delta[0] > Q(31, 32), 'The source tail decay does not fund this tree')"
+    ],
+    [
+      "'negative_controls': {'old_completion':",
+      "'negative_controls': {'old_tail_decay': 'The source tail decay 31/32 is below the new completion ratio.',\n                                         'old_completion':"
+    ],
+    [
+      "This adapter keeps the root and changes the native source invoice.",
+      "This adapter keeps the native word and the retirement identity."
+    ]
+  ]
+}
diff --git a/research/retirement-rebalanced-query-tree/CONTROLLER.json b/research/retirement-rebalanced-query-tree/CONTROLLER.json
new file mode 100644
index 0000000000..a29bd24a25
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/CONTROLLER.json
@@ -0,0 +1,4414 @@
+{
+  "schema": "mixed-projector-query-root/2",
+  "roles": 35,
+  "stages": 18,
+  "ambient_rank": 153,
+  "nodes": [
+    {
+      "roles": [
+        6,
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+      ],
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diff --git a/research/retirement-rebalanced-query-tree/NOTICE b/research/retirement-rebalanced-query-tree/NOTICE
new file mode 100644
index 0000000000..9b9562154e
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/NOTICE
@@ -0,0 +1,26 @@
+New retirement-aware tree allocation, proof, integration and checks:
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+No subagents or independent scientific acceptance are claimed.
+
+The exact executable source is retained in source.tar at original head
+775722aa94a5e85b05d1d704d7dca10cbf016f0c (PR6545).
+RETAINED.json binds original paths, Git blobs and SHA256 values. Original
+source headers, proofs and notices remain unchanged. ADAPTERS.json records
+all changes to the source drivers, constructor and price. Runtime driver
+and constructor copies are created in a private temporary directory.
+
+PR6545: independent Gaussian retirement, exact dual pairs and complete caller.
+PR5599/5602/5611/5615: causal handoff, productive helper and basis methods.
+PR6348/6398: binary query suppliers and the six-field mixed root.
+PR6529: scaled transport fusion and the complete native word.
+PR6524: explicit complete replay interfaces.
+
+Credit Rohan Arun's PR6399/6417, Erwan L.'s PR6453, DreamingOfClouds' PR6375,
+PR6353/6372 and all retained native predecessors. Native work retains
+Cinna Davis, David Arias, Gareth Davidson and their predecessors. Productive
+recursion retains Jacob Alan Sussman's credit. Price and interval work
+retains zxv, Cao Yifan, ywz65 and Nick Petalas's credit.
+
+Other retained contributors include Mason Hale, Pierre-Loic Doulcet and
+Chris R / Clear1nsight. The source archive and its nested notices retain
+the complete native, helper, compiler and analytic source attribution.
diff --git a/research/retirement-rebalanced-query-tree/PROOF.md b/research/retirement-rebalanced-query-tree/PROOF.md
new file mode 100644
index 0000000000..2db35d993c
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/PROOF.md
@@ -0,0 +1,147 @@
+# A strict concave gain from a different child allocation
+
+## 1. Exact source and change
+
+Use PR6545 at `775722aa94a5e85b05d1d704d7dca10cbf016f0c` as the comparison.
+It uses the PR6398 mixed root and the exact PR6529 native word. Its four
+binary children have 7, 2, 2 and 7 leaves. Each leaf consumes one native call.
+The source package, proof and executable checks are retained without change.
+
+Replace only the last two trees. The new four trees are
+
+```text
+A = [[1,[1,1]],[[1,1],[1,1]]]       (7 leaves)
+B = [1,1]                         (2 leaves)
+C = [[1,1],[1,[1,1]]]              (5 leaves)
+D = [[1,1],[1,1]]                  (4 leaves).
+```
+
+The symbols A, B, C and D also denote the products of the fresh stage
+variables of these children. All 18 variables are distinct. The six-field
+root, its scalar bases and its stage masks are unchanged. The retained binary
+identity supplies each child. Full polynomial replay also checks the emitted
+34-field word directly, for every input coefficient.
+
+## 2. The same independent retirement still applies
+
+Number events from zero. The prefix through event 328, `QUERY 20 21 -1`, is
+byte-for-byte the same event list as in the old caller. The first two children
+are unchanged. At that cut, the third child has not started its stage calls.
+Its first binary query has the same operands and sign.
+
+Fields 4, 20 and 21 still have the same complete stage product C. Here
+
+```text
+C = D10 D11 D12 D13 D14.
+```
+
+Their central signs normalize exactly as in PR6545. Put
+
+```text
+lambda = (1+i)/2 + (1-i)q^2/2,
+S = lambda q^(-1),              S^2 = 1.
+```
+
+The old dual pairs and ordinary bases G and J depend only on the unchanged
+prefix and the three field labels. Their identities remain
+
+```text
+ab = cd = 1,       aP = Sc,       Pd = Sb,
+e4^T G = c,        Gd = e4,       e4^T J = a,       Jb = e4.
+```
+
+The verifier checks all these identities over the full central polynomial
+ring. It does not set `q^2=1` or specialize the stage variables.
+
+The proof in PR6545 now applies with this five-stage C. G mixes only three
+equal normalized targets SC, so it commutes with the complete target map.
+At the cut, the pivot row and column are S and have no coupling to the other
+fields. The matching suffix block is C. Return the completed pivot input
+with output S; give the suffix a fresh independent input. Rename its pivot
+to field 34. The final ordinary normalizer gives outputs `q^(-1)` and C.
+All other independent inputs keep their prescribed outputs.
+
+Thus the retirement replaces one rank-41 target by independent rank-1 and
+rank-40 targets on each of 104 ports. It does not copy one input into two
+productive targets. The complete 35-field word has 611 events. Forward and
+reverse checks independently cover all 1,225 coefficient positions.
+
+At positive tensor width f, use the retained factorwise native convention.
+The ordinary scalar coefficients stay outside tensor powers. The new two
+target ranks are f and 40f. The physical checker verifies the central signs,
+fresh orthogonal axes and all port maps. The inherited all-width and
+precision contracts remain assumptions of the conditional result.
+
+## 3. Complete signed invoice
+
+Let H_r denote one Gaussian target or paid job of rank r. A binary node with
+k leaves adds 208 targets of rank `8k+1` and pays 208 jobs of rank `8k`.
+The four old trees have internal-node counts
+
+```text
+{2:8, 3:2, 4:2, 7:2}.
+```
+
+The new counts are
+
+```text
+{2:8, 3:2, 4:2, 5:1, 7:1}.
+```
+
+Hence one seven-leaf node becomes a five-leaf node in the complete count.
+All 18 native calls, their helper targets and their paid jobs remain. All
+17 thin queries remain. The sums A+B and C+D both stay at nine leaves, and
+the second child stays at two leaves.
+
+The complete target and paid changes from PR6545 are
+
+```text
+Delta T = 208 H41 - 208 H57 + 104 H40 - 104 H16,
+Delta P = 312 H40 - 208 H56 - 104 H16.
+```
+
+The first two target terms come from the binary node. The last two target
+terms replace the old retired suffix by the new suffix. The paid terms
+include both the changed node idles and the third child's extra root idle.
+No idle is omitted. Subtract to obtain
+
+```text
+Delta(T-P) = 208 [(H41-H40) - (H57-H56)].
+```
+
+For `H_r=r^t`, `0<t<1`, the derivative `t r^(t-1)` strictly decreases.
+Integrating on [40,41] and [56,57] proves the stated gain is strictly positive.
+This is an exact all-exponent invoice comparison within the stated domain.
+
+Both target and paid rank mass decrease by 832. Their difference remains
+760. Productive fields remain 13,756 and paid jobs remain 67,096. The largest
+target rank is 145 and the largest paid rank is 56.
+
+The new word adds two ordinary Q_POWER operations relative to PR6545. They
+are included in the literal event count and the complete ordinary allowance.
+The native owned inventory remains 29 labels, 45 lifetimes and peak 13.
+The full ordinary-work upper bound is 254,932,208, below the retained 10^9
+allowance. Scalar coefficients, routes, storage and growth checks remain in
+the complete price calculation. The change introduces no unpaid channel,
+helper, copy, clear or restoration.
+
+## 4. Conditional numerical result
+
+The new claimed saving is 0.00329492. The tested saving is 0.003294925.
+Two rational engines certify a positive complete moment at the tested
+saving. They also certify both signs of the raw root enclosure in RESULT.json.
+A negative derivative bound establishes a unique crossing below the retained
+BIT cap. Both engines reject the full PR6545 invoice at the new claim.
+
+The retained completion tails, integer compiler threshold and all 54 strict
+plus four functional outer predicates are evaluated with the new profiles.
+The completion ratio is approximately 0.96892102. It exceeds the source
+decay 31/32. Use decay 97/100 and reserve 10^(-27) at the same power 1920.
+The exact moment gap funds this reserve, and `4*(97/100)^1920` satisfies
+the complete tail test. The old decay is an explicit rejection control.
+They imply the same conditional bound form `O(n (log n)^(1-kappa))` with
+`kappa=0.00329492` under the five source framework assumptions.
+
+This small increase follows from a changed tree and the exact positive
+invoice difference. No global caller optimum, unconditional multiplication
+theorem, independent review or selected-frontier change is claimed.
diff --git a/research/retirement-rebalanced-query-tree/README.md b/research/retirement-rebalanced-query-tree/README.md
new file mode 100644
index 0000000000..467d3de51c
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/README.md
@@ -0,0 +1,121 @@
+# Rebalance the binary trees after Gaussian retirement
+
+The complete conditional bound is **κ = 0.00329492**. The increment over
+[PR6545](https://github.com/CrocSwap/integer-mult-bounds/pull/6545),
+κ = 0.00329394, is **0.00000098**, or **0.0297516%**. This comparison excludes
+all inherited gains. The selected frontier is unchanged.
+
+Keep the PR6545 retirement identity and the exact PR6529 native word. Change
+the four binary child sizes from `(7,2,2,7)` to `(7,2,5,4)`. The complete
+signed invoice improves by
+
+```text
+208 * (H41 - H40 - H57 + H56).
+```
+
+For `H_r = r^t`, with `0 < t < 1`, the derivative decreases strictly with r.
+Thus this expression is positive. This is a change to the construction and
+its complete invoice. It is not a finer decimal claim for the old invoice.
+No optimum over all trees or callers is claimed.
+
+| Complete inventory | PR6545 | This package |
+|---|---:|---:|
+| Native calls / thin queries | 18 / 17 | 18 / 17 |
+| Independent caller fields per port | 35 | 35 |
+| Productive fields, including native helpers | 13,756 | 13,756 |
+| Paid Gaussian jobs | 67,096 | 67,096 |
+| Caller events | 609 | 611 |
+| Rank-mass surplus | 760 | 760 |
+
+Read [PROOF.md](PROOF.md) for the exact delta and the input-independence
+argument. [ROOT.json](ROOT.json) gives the changed trees.
+[CONTROLLER.json](CONTROLLER.json) contains every caller event.
+[WITNESS.json](WITNESS.json) contains the unchanged retirement bases and cut.
+[RESULT.json](RESULT.json) records both rational price engines, the compiler,
+the physical maps and all outer inequalities.
+
+## Verification
+
+Use CPython 3.12 with its standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/retirement-rebalanced-query-tree
+```
+
+The registered hook runs normal and optimized checks. A direct run is
+`python3.12 -I -B research/retirement-rebalanced-query-tree/verify.py`.
+The result marker is `PASS_REBALANCED_RETIREMENT_KAPPA_00329492`.
+See [VALIDATION.md](VALIDATION.md) for checks that have run and their limits.
+
+The two rational engines enclose the raw crossing in
+`(0.00329492611057136286764342, 0.00329492611057136286764343)`.
+The tested saving is 0.003294925. The old completion decay does not suffice
+for this tree. The new check uses decay 97/100 and reserve 10^(-27), with
+the same power 1920 and the full tail and compiler tests.
+
+The verifier first rebuilds the complete retained native source and checks
+its 799 columns, owned lifetimes, helper maps, absolute phases and all 7,488
+caller contacts. It then checks the full PR6545 caller and requires every
+source result to match its frozen receipt. The new caller passes full
+forward and reverse polynomial checks, the same two dual pairs, physical
+target checks and a separate invoice count. Both rational engines reject
+PR6545 at the new claim. Deletion, sign, target and unpaid-job controls must
+fail.
+
+[source.tar](source.tar) retains the exact executable PR6545 package at
+`775722aa94a5e85b05d1d704d7dca10cbf016f0c`, including its nested PR6529 and
+PR6524 source archives. [RETAINED.json](RETAINED.json) binds all 23 outer
+source files by original path, commit, Git blob and SHA256. The nested
+manifests remain unchanged. Replay needs neither network access nor side
+branch Git objects.
+
+[ADAPTERS.json](ADAPTERS.json) declares each source edit. Two driver edits add
+an explicit extension hook. Two constructor edits accept the tree data as an
+argument. The price edits select the new invoice and rational parameters.
+No source check is removed. Temporary adapted modules are separate from the
+frozen source files.
+
+## Scope and coordination
+
+This contribution answers the explicit request for another κ improvement.
+The task and changed premise are in [TASK.json](TASK.json). Canonical main is
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8.
+
+The [canonical claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/5645#discussioncomment-19205459)
+records the move from bounded native-transport searches to this fixed
+retirement and tree allocation. The public PR6545 continuation had released
+its scope, including its retirement-aware price screens. The present result
+uses its published identity and gives the complete changed caller. It does
+not claim a new retirement mechanism. Current caller claims and recent PRs
+were checked. Unpublished work, arbitrary historical diffs and omitted
+sparse-checkout notes remain outside overlap coverage.
+
+The five inherited assumptions remain: `native_all_columns`, `owned_erasure`,
+`compiler`, `precision_tape` and `analytic_recovery`. This is an author-checked
+conditional construction. It is not independent scientific acceptance, a
+materialized uniform word or a complete rounded integer multiplier. No PR
+merge, closure or selected-result change is included.
+
+## Open next questions
+
+- Can an independent reviewer confirm the tree substitution and the complete
+  invoice delta? Start with the unchanged prefix through event 328, then the
+  four histogram differences in PROOF.md. Independent review remains open.
+- Can a different exact retirement or caller give a larger gain? A candidate
+  must supply its full arbitrary-input endpoint and every paid job. This
+  package supplies one allocation and no general optimality theorem.
+
+The structured questions are in [TASK.json](TASK.json).
+
+## Credit
+
+New tree allocation, proof, integration and checks: Ali Romman with
+substantial OpenAI Codex assistance; no subagents. Apache-2.0.
+
+Credit PR6545 for independent Gaussian retirement, PR5599/5602/5611/5615 for
+the causal handoff and basis methods, PR6348/6398 for the binary suppliers
+and mixed root, PR6529 for the native word, and PR6524 for explicit replay
+services. Rohan Arun's PR6399/6417, Erwan L.'s PR6453, DreamingOfClouds'
+PR6375 and all retained predecessors keep their credit. See [NOTICE](NOTICE)
+and the original archived notices for the native, interval, compiler and
+analytic contributors.
diff --git a/research/retirement-rebalanced-query-tree/RESULT.json b/research/retirement-rebalanced-query-tree/RESULT.json
new file mode 100644
index 0000000000..38d840ae0f
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/RESULT.json
@@ -0,0 +1,356 @@
+{
+  "assumptions": [
+    "native_all_columns",
+    "owned_erasure",
+    "compiler",
+    "precision_tape",
+    "analytic_recovery"
+  ],
+  "candidate": {
+    "caller": {
+      "coefficients": 1225,
+      "event_sha256": "5f0fd5f0f8fabcc568a32c60093a0496e90409e855a47a33e33d7a74c2194d47",
+      "events": 611,
+      "inputs": 35,
+      "native_stages": 18
+    },
+    "changes": {
+      "gain_formula": "208*(H41-H40-H57+H56)",
+      "paid": {
+        "16": -104,
+        "40": 312,
+        "56": -208
+      },
+      "rank_mass_change": -832,
+      "signed": {
+        "40": -208,
+        "41": 208,
+        "56": 208,
+        "57": -208
+      },
+      "targets": {
+        "16": -104,
+        "40": 104,
+        "41": 208,
+        "57": -208
+      }
+    },
+    "child_sizes": [
+      7,
+      2,
+      5,
+      4
+    ],
+    "complete_cost": {
+      "compiler": {
+        "H": 138100200000000,
+        "N": "2^138100200000000",
+        "eta": {
+          "lower": "0.000000000000021144595296542077",
+          "upper": "0.000000000000021144595296542078"
+        },
+        "gap": "1/1400000000",
+        "integer_test_exponent": 98643,
+        "logarithm_power": 2100,
+        "theta": "63/64",
+        "uniform_saving": "2306447/700000000",
+        "word_materialized": false
+      },
+      "derivative_upper": "-709775457495552303627723996993112628386366833833215710184754101/3138550867693340381917894711603833208051177722232017256448",
+      "full_forward_calls": 18,
+      "gain": "49/50000000",
+      "kappa": "82373/25000000",
+      "kappa_decimal": "0.00329492",
+      "margin_at_previous": {
+        "lower": "0.225948435544345616827246956773",
+        "upper": "0.225948435544345616827246956774"
+      },
+      "negative_controls": {
+        "coarse_tail": "The block tail bound cannot fund this near-limit claim.",
+        "old_BIT": "An outer inequality fails at BIT saving0.0027.",
+        "old_completion": "Cutoff16 and floor17 give a ratio above one.",
+        "old_tail_decay": "The source tail decay 31/32 is below the new completion ratio."
+      },
+      "nonrecursive": {
+        "local_routes": 5384000,
+        "local_scalar": 137218,
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+        "peak_owned_fields": 13,
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+        "raw_scalar_upper": 254932208,
+        "scalar_growth_bound": "At most 2^(10^9 + R*f), where R is the full paid rank mass."
+      },
+      "ordinary_BIT": {
+        "admitted": "17/5000",
+        "after_cover": "69/20000",
+        "atom": "1/250",
+        "before_cover": "3459/1000000",
+        "ordinary_saving": "8590884599/2500000000000",
+        "prime_rule": "odd q > 2^26*N^5 avoiding all fixed denominators and rank witnesses"
+      },
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+          "fixed_halo_explicit_metadata_branch": "26950000001/350000000",
+          "new_BIT_row_stock_sublinear_in_active_group": "1395799997009/1400000000000",
+          "new_semantic_group_guard": "3/1400000000"
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+          "global_coordinate_router": "1153223/350000000",
+          "group_bit_router": "1614512546540331/490000000000000000",
+          "group_scans": "699999999/700000000",
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+          "microbox_face_corrections": "26950000001/350000000",
+          "polynomial_products": "1399999999/1400000000",
+          "sparse_union_movement": "802708048795387109/1960000000000000000"
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+        "parameters": {
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+          "b": "4612893/1400000000",
+          "beta": "1399999997/1400000000",
+          "bit_saving": "17/5000",
+          "cap_xi": "1782708047395387109/1960000000000000000",
+          "complex_uniform_saving": "2306447/700000000",
+          "d": "1/1400000000",
+          "epsilon": "1399999999/1400000000",
+          "kappa": "82373/25000000",
+          "lambda_": "398682031/400000000",
+          "lambda_prime": "1395387109/1400000000",
+          "layout": "1/1400000000",
+          "q": "4612891/1400000000",
+          "r0": "1/14000000000",
+          "stop": "1/1400000000",
+          "tested_saving": "131797/40000000"
+        },
+        "strict_checks": {
+          "actual_bit_above_effective": "36777/350000000",
+          "actual_logarithmic_gamma_guard": "1/1400000000",
+          "actual_logarithmic_group_guard": "177291954704612891/980000000000000000",
+          "alpha_logarithm_absorption": "1/14000000000",
+          "boundary_algebraic_matrix_above_kappa": "5595387109/1960000000000000000",
+          "cap_above_band": "802708047395387109/1960000000000000000",
+          "cap_below_cell": "177291951204612891/1960000000000000000",
+          "cap_below_phase_cell_power": "2137291952604612891/1960000000000000000",
+          "complex_above_effective": "1/1400000000",
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+          "complex_endpoint": "1/700000000",
+          "epsilon_below_one": "1/1400000000",
+          "epsilon_positive": "1399999999/1400000000",
+          "fine_prime_halo_umbrella": "7/2",
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+          "gaussian_above_main": "695387109/980000000000000000",
+          "global_coordinate_router_above_kappa": "1/350000000",
+          "global_gamma_power_slack": "9/14000000000",
+          "group_below_dimension": "1/700000000",
+          "group_bit_router_above_kappa": "1746540331/490000000000000000",
+          "group_gauge_power_slack": "177291954634612891/980000000000000000",
+          "group_key_fits_record": "1/700000000",
+          "group_positive": "1399999997/1400000000",
+          "group_scans_above_kappa": "139538711/140000000",
+          "halo_covers_generic_inverse_radius": "5",
+          "improvement_over_user_baseline": "781974790400418148631/500000000000000000000000",
+          "lambda_above_internal": "1/2800000000",
+          "lambda_above_sigma": "3/2800000000",
+          "lambda_above_tau": "1/2800000000",
+          "lambda_prime_above_lambda": "1/2800000000",
+          "lambda_prime_above_leaf": "2795387107/1960000000000000000",
+          "leaf_saving_above_effective": "1395387107/1960000000000000000",
+          "main_transform_above_kappa": "4195387109/1960000000000000000",
+          "membership_arithmetic_fits_record": "1/1400000000",
+          "microbox_face_corrections_above_kappa": "26948846779/350000000",
+          "microbox_halo_volume": "110600000003/1400000000",
+          "new_complex_uniform_saving_above_b": "1/1400000000",
+          "new_semantic_guard_with_C1_one": "1/1400000000",
+          "polynomial_products_above_kappa": "1395387111/1400000000",
+          "positive_layout_slack": "1/1400000000",
+          "q_below_effective": "1/1400000000",
+          "q_below_internal": "1/1400000000",
+          "q_below_leaf": "2795387107/1960000000000000000",
+          "q_positive": "4612891/1400000000",
+          "short_record_fallback": "1395387107/1400000000",
+          "source_axis_superpolynomial": "1/1400000000",
+          "sparse_union_movement_above_kappa": "796250005595387109/1960000000000000000",
+          "stop_below_one": "1399999999/1400000000",
+          "stop_positive": "1/1400000000",
+          "strassen_integer_power_gap": "1951441519231",
+          "support_router_span": "3/1400000000",
+          "theta_positive_polynomial_power": "2",
+          "union_header_fits_record": "3/1400000000",
+          "union_tag_work_per_visit_fits_record": "3/1400000000"
+        }
+      },
+      "paid": {
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+        "16": 1768,
+        "2": 8154,
+        "24": 416,
+        "3": 8838,
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+        "40": 312,
+        "5": 2754,
+        "56": 208,
+        "6": 144,
+        "7": 162,
+        "8": 18
+      },
+      "paid_calls": 67096,
+      "previous_fixed_root": "164697/50000000",
+      "productive_fields": 13756,
+      "rank_mass_surplus": 760,
+      "raw_endpoint_margins": {
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+          "first": {
+            "lower": "0.000000000000000000001917550077",
+            "upper": "0.000000000000000000001917550078"
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+            "lower": "0.000000000000000000001917550077",
+            "upper": "0.000000000000000000001917550078"
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+        "upper": {
+          "first": {
+            "lower": "-0.000000000000000000000373754706",
+            "upper": "-0.000000000000000000000373754705"
+          },
+          "second": {
+            "lower": "-0.000000000000000000000373754706",
+            "upper": "-0.000000000000000000000373754705"
+          }
+        }
+      },
+      "raw_headroom": [
+        "305528568143382171/50000000000000000000000000",
+        "611057136286764343/100000000000000000000000000"
+      ],
+      "raw_root_interval": [
+        "164746305528568143382171/50000000000000000000000000",
+        "329492611057136286764343/100000000000000000000000000"
+      ],
+      "refinement_kind": "Rebalance the last two binary children under the fixed Gaussian retirement.",
+      "relative_gain_percent": "4900/164697",
+      "strictification": {
+        "alpha": {
+          "lower": "0.999999998609925710870310373249",
+          "upper": "0.999999998609925710870310373250"
+        },
+        "coarse_tail_upper": "1/288230376151711744",
+        "cutoff": 12,
+        "delta": {
+          "lower": "0.968921015518688688351515226154",
+          "upper": "0.968921015518688688351515226155"
+        },
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+        "power": 1920,
+        "reserve": "1/1000000000000000000000000000",
+        "tail_decay": "97/100",
+        "tail_offset": "1/7",
+        "tested": "131797/40000000"
+      },
+      "targets": {
+        "1": 460,
+        "145": 104,
+        "17": 1768,
+        "2": 144,
+        "25": 416,
+        "3": 72,
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+        "5": 270,
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+        "8": 702,
+        "9": 6534
+      },
+      "thin_queries": 17,
+      "whole_idle_jobs": 3120
+    },
+    "controls": {
+      "extra_gaussian_credit": "All main target phases and ranks",
+      "missing_boundary": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_cut": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_idle": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_idle_one": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_lambda": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_local": "Every native stage exactly once",
+      "missing_q_power": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_query": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "missing_return": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient",
+      "unearned_central_target": "Every independent reverse caller coefficient",
+      "unpaid_native_job": "All independent fields and paid calls",
+      "wrong_cut_sign": "Every Q(i)[Q,D]/(Q^4-1) endpoint coefficient"
+    },
+    "dual_pair": {
+      "cut_after": 328,
+      "cut_pairing": "1",
+      "input_pairing": "1",
+      "prefix_column": "P d = S b",
+      "prefix_row": "a P = S c"
+    },
+    "internal_node_sizes": {
+      "2": 8,
+      "3": 2,
+      "4": 2,
+      "5": 1,
+      "7": 1
+    },
+    "physical": {
+      "central_square_columns": 53248,
+      "central_translations": 19,
+      "full_calls": 18,
+      "main_targets": {
+        "1": 208,
+        "145": 104,
+        "17": 1768,
+        "25": 416,
+        "33": 416,
+        "40": 104,
+        "41": 208,
+        "57": 208,
+        "73": 208
+      },
+      "new_paid_jobs": 0,
+      "ordinary_normalizations": 34,
+      "physical_contacts": 7488,
+      "ports": 104,
+      "route_payload_sha256": "79930ec6c09a8e4930556c925c354bb5431203eaf069d9373f056049c061289b",
+      "tensor_rule": "Factorwise native maps; scalar signs stay outside the tensor power.",
+      "whole_idle_jobs": {
+        "16": 1768,
+        "24": 416,
+        "32": 416,
+        "40": 312,
+        "56": 208
+      }
+    },
+    "reverse": {
+      "coefficients": 1225,
+      "rows": 35
+    },
+    "source_margin_first": {
+      "lower": "-0.223153821753303547838006215777",
+      "upper": "-0.223153821753303547838006215776"
+    },
+    "source_margin_second": {
+      "lower": "-0.223153821753303547838006215777",
+      "upper": "-0.223153821753303547838006215776"
+    }
+  },
+  "selected_frontier_changed": false,
+  "source_head": "775722aa94a5e85b05d1d704d7dca10cbf016f0c",
+  "source_result_sha256": "13639e4770964ded237b79d72ad517448de269b34d724beaa70757dbe0be6fe2",
+  "status": "PASS_REBALANCED_RETIREMENT_KAPPA_00329492"
+}
diff --git a/research/retirement-rebalanced-query-tree/RETAINED.json b/research/retirement-rebalanced-query-tree/RETAINED.json
new file mode 100644
index 0000000000..a47367418f
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/RETAINED.json
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+      ],
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+        [
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+        ],
+        [
+          "0",
+          "0",
+          "-2",
+          "-3",
+          "-2",
+          "-1"
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+      ],
+      "projector": [
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+        [
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+        ],
+        [
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+        ],
+        [
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+          "0",
+          "2",
+          "3",
+          "2",
+          "2"
+        ]
+      ]
+    }
+  ],
+  "masks": [
+    [
+      1,
+      1,
+      1,
+      1
+    ],
+    [
+      1,
+      1,
+      0,
+      0
+    ],
+    [
+      0,
+      0,
+      1,
+      1
+    ],
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+      0,
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+      0
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+      0,
+      1,
+      0
+    ],
+    [
+      0,
+      1,
+      0,
+      0
+    ]
+  ],
+  "independent_stage_coefficients": 1152,
+  "normalization": "lambda=(1+i)/2+(1-i)*q^2/2; q^4=1",
+  "scope": "Exact rational root and Gaussian normalization; full caller costs not certified",
+  "schema": "mixed-projector-query-root/2",
+  "children": [
+    [
+      [
+        1,
+        [
+          1,
+          1
+        ]
+      ],
+      [
+        [
+          1,
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+        [
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+      ]
+    ],
+    [
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+    [
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+      [
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+    ],
+    [
+      [
+        1,
+        1
+      ],
+      [
+        1,
+        1
+      ]
+    ]
+  ],
+  "child_sizes": [
+    7,
+    2,
+    5,
+    4
+  ]
+}
diff --git a/research/retirement-rebalanced-query-tree/SOURCES.json b/research/retirement-rebalanced-query-tree/SOURCES.json
new file mode 100644
index 0000000000..a1946e70e1
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/SOURCES.json
@@ -0,0 +1,27 @@
+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ],
+  "retained_predecessors": [
+    {
+      "archive": "source.tar",
+      "archive_sha256": "2481af235a2673c5f3ce1de0391027046eebd356615d09147f8034076eed7135",
+      "source_head": "775722aa94a5e85b05d1d704d7dca10cbf016f0c",
+      "manifest": "RETAINED.json",
+      "reason": "Exact executable source, proofs and notices. Nested native and replay sources retain their manifests; no side-branch history is needed."
+    }
+  ]
+}
diff --git a/research/retirement-rebalanced-query-tree/TASK.json b/research/retirement-rebalanced-query-tree/TASK.json
new file mode 100644
index 0000000000..d6fec47a3a
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/TASK.json
@@ -0,0 +1,123 @@
+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can joint helper bases and a legal causal split improve the complete P104 productive/paid invoice?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "research/p104-helper-subspace-splits/README.md",
+      "research/dyadic-helper-channel-capacity/README.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Start with the selected #5599 emitted word and review pending #5718 distinct input/output bases plus #5717/#5719 coverage before claiming work. Pick one uncovered source word, helper group and cut region. Prove the arbitrary-input relation and legal entry/exit corrections, then compare full target and paid profiles. Reject if completion, owned lifetimes or consumer costs repay the gain. A static pairing rank alone is not a physical split.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "P104",
+      "helper basis",
+      "helper split",
+      "5718",
+      "dyadic capacity"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4133: literal-slot defect",
+        "evidence": "research/p104-literal-slot-defect/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "88666f1dee93fbd85b31fd5d6f91a219e111b5f4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4136: uniform commutation gap",
+        "evidence": "research/p104-uniform-commutation-gap/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "281ce601a8be9db470de630759179c23e1dfaa88",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4140: uniform encoding gap",
+        "evidence": "research/p104-uniform-encoding-gap/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "de25c6525aa866962309df5d107b32d55e6e5628",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4153: source-chart shortcut exhausts its optimistic allowance",
+        "evidence": "research/p104-source-chart-shortcut/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "9123241fea53df77e51ffe30d9aca1625d4bf3a8",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4154: fiber-tag seam mismatch",
+        "evidence": "research/p104-fiber-tag-seam/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "4ffcdaee5d6255a9cea1fe53a5424435068089d0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Rebalance the PR6398 binary children from (7,2,2,7) to (7,2,5,4), with the exact PR6545 independent retirement and PR6529 native word.",
+  "scope_change_reason": "The explicit user asks for another complete kappa improvement of any size. The published retirement changes the preferred tree allocation. The root identity and the retirement bases stay fixed.",
+  "bottleneck_link": "The exact complete signed gain is 208*(H41-H40-H57+H56). Every native, query, idle and ordinary operation remains charged.",
+  "budget_minutes": 180,
+  "stop_rule": "Reject any candidate that fails the full arbitrary-input identity, physical maps, complete invoice, compiler or outer checks. The user requires a verified improvement before completion.",
+  "outcome": "completed",
+  "evidence": "PROOF.md; ROOT.json; CONTROLLER.json; WITNESS.json; RESULT.json; VALIDATION.md",
+  "author": "Ali Romman with substantial OpenAI Codex assistance",
+  "ai_assistance": "Substantial OpenAI Codex assistance in construction, exact arithmetic, verification, research and writing; no subagents.",
+  "open_questions": [
+    {
+      "question": "Can an independent reviewer validate the tree substitution and complete invoice?",
+      "why_it_matters": "Author checks do not establish independent scientific acceptance.",
+      "first_step": "Check the fixed prefix, equal group targets and complete histogram delta in PROOF.md, then replay the registered package.",
+      "blocked_by": "Independent review has not occurred."
+    },
+    {
+      "question": "Can a different exact caller or retirement give a larger complete gain?",
+      "why_it_matters": "The present result is a small improvement for one fixed construction.",
+      "first_step": "Supply a full arbitrary-input endpoint and every paid job before claiming a new bound.",
+      "blocked_by": "This package does not supply a different identity or global optimization theorem."
+    }
+  ]
+}
diff --git a/research/retirement-rebalanced-query-tree/VALIDATION.md b/research/retirement-rebalanced-query-tree/VALIDATION.md
new file mode 100644
index 0000000000..fe06104534
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/VALIDATION.md
@@ -0,0 +1,60 @@
+# Validation record
+
+Environment: CPython 3.12.13 on macOS, November 3, 2026 (UTC).
+
+The complete direct generation run passed:
+
+```sh
+python3.12 -I -B research/retirement-rebalanced-query-tree/verify.py --write-result
+python3.12 -B scripts/research_package.py --package research/retirement-rebalanced-query-tree --check-only
+python3.12 -W error -B scripts/research_package.py --package research/retirement-rebalanced-query-tree
+```
+
+Markers: `PASS_REBALANCED_RETIREMENT_KAPPA_00329492`,
+`PASS_PACKAGE_METADATA_AND_PINS` and `PASS_REGISTERED_RESEARCH_PACKAGE`.
+Registered normal and optimized replay both passed. The initial attempt rejected the inherited
+31/32 tail decay. The final run uses the charged 97/100 decay and 10^(-27)
+reserve, and includes the old decay as a negative control.
+
+Coverage includes full PR6529 native regeneration and PR6545 caller replay,
+all source controls, all 799 native columns, absolute phases, all 7,488
+physical contacts, both new 1,225-coefficient caller engines, the dual pairs,
+the exact complete invoice delta, both rational price engines, the compiler
+and all 58 outer predicates. Every retained source receipt is checked after
+its scientific checks. No source receipt replaces a computation.
+
+Four extra source controls rejected a false archive SHA256, a false member
+SHA256, a false Git blob and an omitted archive member. The candidate receipt
+contains 13 endpoint and invoice rejection controls, in addition to the
+unchanged source controls and analytic rejection checks.
+
+Scientific commit: `6f779707dfddcece2e964f627cbd0df6d11bf41c`. Its parent is
+canonical main `7c49fe32c55de02c2c873655c804b16e4c3fef0a`. All 17 committed
+package files pass byte and executable-mode comparison. The official export
+passes integrity and declared-input checks:
+
+```sh
+python3.12 -B scripts/check_research_commit.py --package research/retirement-rebalanced-query-tree --check-modes
+python3.12 -B scripts/export_research_package.py export --package research/retirement-rebalanced-query-tree --ref 6f779707dfddcece2e964f627cbd0df6d11bf41c --output /tmp/rebalanced-tree.tar
+python3.12 -B scripts/export_research_package.py check /tmp/rebalanced-tree.tar --check-inputs
+```
+
+Both direct normal and optimized verifiers passed from that export with
+all files read-only, no Git metadata and an empty PATH. The interpreter was
+invoked by its absolute path from outside the export. All 18 files, including
+the export manifest, kept their hashes and read-only modes after both runs.
+This record was then updated; the scientific files were not changed.
+
+Four source authentication controls also passed on CPython 3.12.13. The
+original 23 outer source files were checked against the exact PR6545 Git
+objects before their archive was made. Nested native and replay sources
+authenticate through their unchanged source manifests on every run.
+
+At this local validation freeze, hosted CI and independent scientific review
+are pending. The PR records the later hosted status separately.
+
+The checkout is shallow. The optional full-history preflight is not claimed.
+The selected record has SHA256
+`9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa`.
+All five source framework assumptions remain. No selected-result change,
+formal-kernel verification or materialized multiplier is claimed.
diff --git a/research/retirement-rebalanced-query-tree/VERIFY.json b/research/retirement-rebalanced-query-tree/VERIFY.json
new file mode 100644
index 0000000000..0dc18f4ed5
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/VERIFY.json
@@ -0,0 +1,30 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 590,
+      "checks": "Complete native source and PR6545 regeneration, all 799 native columns and phases, both 1225-coefficient new caller engines, fixed dual pairs, all physical maps, exact invoice delta, two rational prices, compiler and all outer predicates."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 590,
+      "checks": "Complete native source and PR6545 regeneration, all 799 native columns and phases, both 1225-coefficient new caller engines, fixed dual pairs, all physical maps, exact invoice delta, two rational prices, compiler and all outer predicates."
+    }
+  ]
+}
diff --git a/research/retirement-rebalanced-query-tree/WITNESS.json b/research/retirement-rebalanced-query-tree/WITNESS.json
new file mode 100644
index 0000000000..be22af6322
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/WITNESS.json
@@ -0,0 +1,104 @@
+{
+  "cut_after": 328,
+  "handoff_after": 344,
+  "pivot": 4,
+  "fresh": 34,
+  "boundary": [
+    [
+      "SCALE",
+      4,
+      "-1"
+    ],
+    [
+      "LAMBDA",
+      4,
+      -1
+    ],
+    [
+      "ADD",
+      21,
+      4,
+      "1"
+    ],
+    [
+      "LAMBDA",
+      4,
+      1
+    ],
+    [
+      "ADD",
+      4,
+      20,
+      "1"
+    ]
+  ],
+  "middle": [
+    [
+      "ADD",
+      4,
+      2,
+      "1"
+    ],
+    [
+      "ADD",
+      2,
+      4,
+      "-1"
+    ],
+    [
+      "ADD",
+      4,
+      2,
+      "1"
+    ],
+    [
+      "SCALE",
+      2,
+      "-1"
+    ],
+    [
+      "SCALE",
+      4,
+      "-1"
+    ],
+    [
+      "ADD",
+      21,
+      4,
+      "1"
+    ],
+    [
+      "ADD",
+      4,
+      1,
+      "1"
+    ],
+    [
+      "LAMBDA",
+      20,
+      1
+    ],
+    [
+      "ADD",
+      4,
+      20,
+      "-1"
+    ],
+    [
+      "LAMBDA",
+      20,
+      -1
+    ],
+    [
+      "ADD",
+      4,
+      21,
+      "1"
+    ]
+  ],
+  "group": [
+    4,
+    20,
+    21
+  ]
+}
diff --git a/research/retirement-rebalanced-query-tree/invoice.py b/research/retirement-rebalanced-query-tree/invoice.py
new file mode 100644
index 0000000000..86ce75eb55
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/invoice.py
@@ -0,0 +1,48 @@
+"""Price the fixed (7,2,5,4) tree before its Gaussian retirement.
+
+The binary supplier and mixed root are from PR6348 and PR6398. The
+independent retirement is from PR6545. All source authors keep their credit.
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from collections import Counter
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def formula_invoice(local):
+    targets = Counter({r: 18*n for r, n in local['helpers'].items()})
+    paid = Counter({r: 18*n for r, n in local['paid'].items()})
+    paid[1] += 17*104
+    # The four trees have fourteen internal nodes.
+    for size, count in {2: 8, 3: 2, 4: 2, 5: 1, 7: 1}.items():
+        targets[1+8*size] += 208*count
+        paid[8*size] += 208*count
+    for rank, count in {145: 1, 73: 2, 1: 1, 17: 1, 41: 1}.items():
+        targets[rank] += 104*count
+    paid[16] += 104
+    paid[40] += 104
+    return targets, paid
+
+
+def difference(after, before):
+    out = Counter({int(k): v for k, v in after.items()})
+    out.subtract({int(k): v for k, v in before.items()})
+    return {rank: count for rank, count in out.items() if count}
+
+
+def compare(after, before):
+    targets = difference(after['targets'], before['targets'])
+    paid = difference(after['paid'], before['paid'])
+    need(targets == {16: -104, 40: 104, 41: 208, 57: -208},
+         'Complete productive change from PR6545')
+    need(paid == {16: -104, 40: 312, 56: -208},
+         'Complete paid change from PR6545')
+    signed = difference(targets, paid)
+    need(signed == {40: -208, 41: 208, 56: 208, 57: -208},
+         'Exact two-interval concave difference')
+    return {'targets': targets, 'paid': paid, 'signed': signed,
+            'gain_formula': '208*(H41-H40-H57+H56)',
+            'rank_mass_change': sum(r*n for r, n in targets.items())}
diff --git a/research/retirement-rebalanced-query-tree/pricing.py b/research/retirement-rebalanced-query-tree/pricing.py
new file mode 100644
index 0000000000..b659d28e3d
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/pricing.py
@@ -0,0 +1,189 @@
+"""Check the complete mixed-projector-root invoice and outer transfer.
+
+The interval and compiler bounds are by zxv and Cao Yifan. The productive
+recursion is by Jacob Alan Sussman. The source price adapter is by Ali Romman.
+The source lifts and tree invoice are by Ali Romman, with OpenAI Codex
+assistance. This adapter keeps the native word and the retirement identity.
+It includes all nine output contractions and their productive target debits.
+This adaptation changes the last two child trees under the PR6545 retirement.
+The original sources retain their credit. Apache-2.0.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from tree_invoice import formula_invoice, need
+
+KAPPA = Q('0.00329492')
+TESTED = Q('0.003294925')
+PREVIOUS = Q('0.00329394')
+
+
+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 assemble(local, native, word, physical):
+    paid, targets = Counter(), Counter()
+    counts = Counter(e[0] for e in word['events'])
+    need(local['helpers'] == native['helpers'], 'All source helper targets have native maps')
+    idle = Counter()
+    for event in word['events']:
+        if event[0] == 'LOCAL':
+            paid.update(local['paid'])
+            targets.update(local['helpers'])
+        elif event[0] == 'QUERY':
+            paid[1] += 104
+        elif event[0] in ('IDLE', 'IDLE_ONE'):
+            rank = 8*len(event[3] if event[0] == 'IDLE' else event[2])
+            count = 208 if event[0] == 'IDLE' else 104
+            paid[rank] += count
+            idle[rank] += count
+    main = Counter()
+    for term in word['diagonal']:
+        main[int(term[0] % 2)+8*sum(term[1:])] += 104
+    targets.update(main)
+    need(main == physical['main_targets'], 'All main target phases and ranks')
+    need(idle == physical['whole_idle_jobs'], 'All paired and individual whole idle jobs')
+    expected_targets, expected_paid = formula_invoice(local)
+    expected_targets.update({1:104,40:104,41:-104})
+    need((targets, paid) == (expected_targets, expected_paid), 'Independent rebalanced tree, retirement delta and emitted-event counts')
+    need((counts['LOCAL'], counts['QUERY'], counts['IDLE']) == (18, 17, 14) and counts['IDLE_ONE'] == 2 and counts['LAMBDA'] == 34, 'Complete caller census')
+    need((sum(targets.values()), sum(paid.values())) == (13756, 67096), 'All independent fields and paid calls')
+    need(sum(r*n for r, n in targets.items())-sum(r*n for r, n in paid.items()) == 760, 'Complete rank-mass surplus')
+    need((max(targets), max(paid)) == (145, 56), 'Maximum target and paid ranks')
+    return targets, paid, counts
+
+
+def run(local, native, word, physical, L):
+    T, P, counts = assemble(local, native, word, physical)
+    E = L.load(L.ROOT/'research/higher-moment-strictification/verify.py', 'balanced_intervals')
+    other = L.load(L.ROOT/'research/kappa-rebuild/architecture/mixed_target_probe.py', 'balanced_second_intervals')
+    def margin(k):
+        a, b = E.moment(T, 1-k), E.moment(P, 1-k)
+        return a[0]-b[1], a[1]-b[0]
+    from root_interval import construct
+    bracket, _ = construct(E, T, P)
+    lower, upper = Q(bracket['lower'], bracket['grid']), Q(bracket['upper'], bracket['grid'])
+    signed = Counter(T)
+    signed.subtract(P)
+    cap, derivative_upper = Q('.0034'), Q(0)
+    for rank, count in signed.items():
+        logarithm = E.log_bounds(rank)
+        a = E.mul(E.power(rank, 1-cap), logarithm)[0]
+        b = rank*logarithm[1]
+        derivative_upper -= count*(a if count >= 0 else b)
+    need(derivative_upper < 0, 'Unique raw crossing below the BIT cap')
+    need(margin(lower)[0] > 0 and margin(upper)[1] < 0, 'Strict rational crossing enclosure')
+    endpoints = {}
+    for label, k in [('lower', lower), ('upper', upper)]:
+        tl, th = other.moment(T, k)
+        pl, ph = other.moment(P, k)
+        second = (tl-ph, th-pl)
+        first = margin(k)
+        need(max(first[0], second[0]) <= min(first[1], second[1]),
+             'Both rational engines agree at the raw endpoint')
+        need(second[0] > 0 if label == 'lower' else second[1] < 0,
+             'The second engine certifies both raw endpoint signs')
+        endpoints[label] = {'first': E.display(first), 'second': E.display(second)}
+    need(PREVIOUS < KAPPA < TESTED < lower, 'The claim and tested saving are below the raw crossing')
+    t, offset, power, cutoff, floor = 1-TESTED, Q(1, 7), 1920, 12, 13
+    target, paid = E.moment(T, t), E.moment(P, t)
+    tl, th = other.moment(T, TESTED)
+    pl, ph = other.moment(P, TESTED)
+    need(tl > ph and max(tl, target[0]) <= min(th, target[1])
+         and max(pl, paid[0]) <= min(ph, paid[1]), 'Two independent rational moment engines')
+    alpha = E.divide(paid, target)
+    gamma = E.divide(E.moment(P, t+offset), E.mul(E.power(cutoff, offset), target))
+    delta = E.divide((Q(sum(T.values())*floor),)*2, E.mul(E.power(cutoff, TESTED), target))
+    reserve, decay = Q(1, 10**27), Q(97, 100)
+    need(alpha[1] < 1-reserve and gamma[1] < decay and delta[1] < decay, 'All charged tail ratios')
+    need(delta[0] > Q(31, 32), 'The source tail decay does not fund this tree')
+    need(power % 128 == 0 and decay**128 < Q(1, 16), 'The previous block estimate applies')
+    coarse_tail = 4*Q(1, 16)**(power//128)
+    need(1/(1+power*reserve)+coarse_tail > 1, 'The coarse tail bound cannot fund this claim')
+    exact_tail = 4*decay**power
+    normalized = 1/(1+power*reserve)+exact_tail
+    need(normalized < 1 and cutoff**64 < floor**63, 'Completion, splitting and rounding')
+    theta, logarithm_power = Q(63, 64), 2100
+    # Keep each exponential inside the retained engine's proved domain.
+    half = E.power(145, Q(power, 2)*TESTED)
+    factor = E.mul(half, half)
+    eta = E.scale(E.divide((Q(1), Q(1)), factor), t)
+    need(2*(cutoff**power)**64 <= (145**power)**63
+         and eta[0] > theta**logarithm_power, 'Shifted exits and complement capacity')
+    need(factor[1] < 2**logarithm_power, 'Compiler initial range')
+    gap = (TESTED-KAPPA)/7
+    need(gap.numerator == 1, 'Exact compiler integer grid')
+    step = gap.denominator
+    def wins(j):
+        return (step*j+1)**logarithm_power < 2**j
+    left, right = 1, 2
+    while not wins(right):
+        left, right = right, 2*right
+    while right-left > 1:
+        middle = (left+right)//2
+        if wins(middle):
+            right = middle
+        else:
+            left = middle
+    H = step*right
+    need(H > 8*power and wins(right) and not wins(right-1), 'Uniform threshold and adjacent-grid rejection')
+    local_routes = (sum(local['paid'].values())+native['helper_normalizers']
+                    +native['main_adapters']+4*104+4*local['owned_lifetimes'])*1000
+    need(local['real_scalar_allowance']+local_routes < 10**7, 'Every local ordinary operation is funded')
+    raw = 18*10**7 + 17*1664 + 32*35*104*len(word['events']) + 35*104*(26+1000)
+    need(all(abs(Q(e[-1])) <= 16 for e in word['events'] if e[0] in ('ADD', 'SCALE')),
+         'Every emitted scalar coefficient has the declared growth bound')
+    need(all(Q(e[-1]).denominator & (Q(e[-1]).denominator-1) == 0
+             for e in word['events'] if e[0] in ('ADD', 'SCALE')),
+         'Every emitted rational scalar is dyadic')
+    need(raw < 10**9, 'Complete ordinary-work budget')
+    bit = L.read(L.ROOT/'research/query-thinned-controllers/RESULT.json')['ordinary_BIT']
+    atom = Q(bit['atom'])
+    ordinary = (1-atom)*Q(bit['after_cover']) + atom*Q('.0000384599')
+    admitted = Q(bit['admitted'])
+    need(atom > ordinary > admitted > TESTED and atom < 1-ordinary, 'Retained BIT admission and stock')
+    outer_module = L.load(L.ROOT/'research/pr4359-p7-padding/dependencies/compiler-refill/endpoint.py', 'balanced_outer')
+    params = {'kappa': KAPPA, 'tested_saving': TESTED, 'complex_uniform_saving': KAPPA+6*gap,
+              'b': KAPPA+5*gap, 'a': KAPPA+4*gap, 'q': KAPPA+3*gap, 'bit_saving': admitted,
+              'epsilon': 1-gap, 'beta': 1-3*gap, 'r0': gap/10, 'stop': gap, 'layout': gap, 'd': gap}
+    outer = outer_module.ledger(params)
+    outer['functional_checks']['actual_new_BIT_stock_sublinear'] = params['beta']*(1-atom)
+    need(len(outer['strict_checks']) == 54 and len(outer['functional_checks']) == 4, 'Full outer predicate inventory')
+    need(all(x > 0 for key in ('strict_checks', 'functional_checks') for x in outer[key].values()), 'Every outer inequality is strict')
+    old_completion = E.divide((Q(sum(T.values())*17),)*2, E.mul(E.power(16, TESTED), target))
+    need(old_completion[0] > 1, 'The old completion cutoff does not fund the new inventory')
+    bad = outer_module.ledger(dict(params, bit_saving=Q('.0027')))
+    need(any(x <= 0 for x in bad['strict_checks'].values()), 'Insufficient BIT saving rejected')
+    return encode({'kappa': KAPPA, 'kappa_decimal': '0.00329492', 'previous_fixed_root': PREVIOUS,
+                   'refinement_kind': 'Rebalance the last two binary children under the fixed Gaussian retirement.',
+                   'gain': KAPPA-PREVIOUS, 'relative_gain_percent': 100*(KAPPA/PREVIOUS-1),
+                   'raw_root_interval': [lower, upper], 'raw_headroom': [lower-KAPPA, upper-KAPPA],
+                   'raw_endpoint_margins': endpoints,
+                   'derivative_upper': derivative_upper, 'margin_at_previous': E.display(margin(PREVIOUS)),
+                   'targets': T, 'paid': P, 'productive_fields': sum(T.values()), 'paid_calls': sum(P.values()),
+                   'rank_mass_surplus': 760, 'full_forward_calls': 18, 'thin_queries': 17,
+                   'whole_idle_jobs': 3120, 'strictification': {
+                       'tested': TESTED, 'power': power, 'cutoff': cutoff, 'floor': floor,
+                       'tail_offset': offset, 'reserve': reserve, 'tail_decay': decay,
+                       'exact_tail_upper': exact_tail, 'coarse_tail_upper': coarse_tail,
+                       'alpha': E.display(alpha), 'gamma': E.display(gamma), 'delta': E.display(delta),
+                       'normalized_paid_upper': normalized},
+                   'compiler': {'theta': theta, 'logarithm_power': logarithm_power, 'eta': E.display(eta),
+                                'gap': gap, 'H': H, 'N': '2^'+str(H), 'integer_test_exponent': right,
+                                'uniform_saving': params['complex_uniform_saving'], 'word_materialized': False},
+                   'nonrecursive': {'local_scalar': local['real_scalar_allowance'], 'local_routes': local_routes,
+                                    'raw_scalar_upper': raw, 'raw_budget': 10**9,
+                                    'peak_owned_fields': local['peak_owned_fields'],
+                                    'scalar_growth_bound': 'At most 2^(10^9 + R*f), where R is the full paid rank mass.',
+                                    'paid_rank_mass': sum(r*n for r, n in P.items())},
+                   'ordinary_BIT': bit, 'outer': outer,
+                   'negative_controls': {'old_tail_decay': 'The source tail decay 31/32 is below the new completion ratio.',
+                                         'old_completion': 'Cutoff16 and floor17 give a ratio above one.',
+                                         'old_BIT': 'An outer inequality fails at BIT saving0.0027.',
+                                         'coarse_tail': 'The block tail bound cannot fund this near-limit claim.'}})
diff --git a/research/retirement-rebalanced-query-tree/source.tar b/research/retirement-rebalanced-query-tree/source.tar
new file mode 100644
index 0000000000..025b8b9c6d
Binary files /dev/null and b/research/retirement-rebalanced-query-tree/source.tar differ
diff --git a/research/retirement-rebalanced-query-tree/verify.py b/research/retirement-rebalanced-query-tree/verify.py
new file mode 100644
index 0000000000..6c399a5738
--- /dev/null
+++ b/research/retirement-rebalanced-query-tree/verify.py
@@ -0,0 +1,201 @@
+"""Replay the retained source and the rebalanced retirement caller.
+
+The archive checks and explicit extension use the PR6545 verifier. Native,
+caller, phase, interval and compiler sources keep their original credit.
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from collections import Counter
+from copy import deepcopy
+from hashlib import sha1, sha256
+from pathlib import Path, PurePosixPath
+import importlib.util
+import json
+import sys
+import tarfile
+import tempfile
+
+sys.dont_write_bytecode = True
+HERE = Path(__file__).resolve().parent
+STATUS = 'PASS_REBALANCED_RETIREMENT_KAPPA_00329492'
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def read(path):
+    return json.loads(Path(path).read_bytes())
+
+
+def load(path, name):
+    spec = importlib.util.spec_from_file_location(name, path)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[name] = module
+    spec.loader.exec_module(module)
+    return module
+
+
+def reject(action):
+    try:
+        action()
+    except ValueError as error:
+        return str(error)
+    raise ValueError('Invalid candidate accepted')
+
+
+def unpack(record, root):
+    archive = HERE/record['archive']
+    need(sha256(archive.read_bytes()).hexdigest() == record['archive_sha256'],
+         'Exact retained archive')
+    pins = {p['path']: p for p in record['files']}
+    seen = set()
+    with tarfile.open(archive, 'r:') as tar:
+        for member in tar.getmembers():
+            path = PurePosixPath(member.name)
+            need(member.isfile() and not path.is_absolute() and '..' not in path.parts
+                 and member.name in pins and member.name not in seen,
+                 'Declared regular source member')
+            seen.add(member.name)
+            data = tar.extractfile(member).read()
+            pin = pins[member.name]
+            need(sha256(data).hexdigest() == pin['sha256'] and
+                 sha1(b'blob '+str(len(data)).encode()+b'\0'+data).hexdigest() == pin['git_blob'],
+                 'Exact source SHA256 and Git blob')
+            target = root/member.name
+            target.parent.mkdir(parents=True, exist_ok=True)
+            target.write_bytes(data)
+    need(seen == set(pins), 'Complete retained inventory')
+
+
+def adapted(source, kind):
+    text = source.read_text()
+    for before, after in read(HERE/'ADAPTERS.json')[kind]:
+        need(text.count(before) == 1, 'Unique declared '+kind+' edit')
+        text = text.replace(before, after)
+    return text
+
+
+class Extension:
+    def __init__(self, source_candidate, source, temporary, write):
+        self.source_candidate = source_candidate
+        self.source = source
+        self.temporary = temporary
+        self.write = write
+
+    def run(self, native_word, old_caller, local, native, L):
+        # The source runs all its checks and compares its frozen caller receipt.
+        source_result = self.source_candidate.run(native_word, old_caller, local, native, L)
+        controller = sys.modules['controller']
+        original = Path(controller.__file__)
+        constructor_path = self.temporary/'tree_constructor.py'
+        constructor_path.write_text(adapted(original, 'controller'))
+        constructor = load(constructor_path, 'tree_constructor')
+        data = read(HERE/'ROOT.json')
+        original_data = read(original.with_name('ROOT.json'))
+        need({k: v for k, v in data.items() if k not in ('children', 'child_sizes')} ==
+             {k: v for k, v in original_data.items() if k not in ('children', 'child_sizes')},
+             'Only the child trees and their size declaration change')
+        need(data['child_sizes'] == [7, 2, 5, 4], 'Declared child allocation')
+        base = constructor.construct(data)
+        controller.replay(base)
+        need(base['events'][:329] == old_caller['events'][:329],
+             'The complete source prefix through the retirement cut is unchanged')
+        sizes = Counter(len(node['stages']) for node in base['nodes'])
+        need(sizes == {2: 8, 3: 2, 4: 2, 5: 1, 7: 1}, 'Every binary supplier size')
+        make = load(self.source/'retirement.py', 'tree_retirement')
+        checks = load(self.source/'check_caller.py', 'tree_independent')
+        word, witness = make.construct(base)
+        forward = controller.replay(word)
+        reverse = checks.transpose(word)
+        pair = checks.dual_pair(base, make.G, make.J)
+        controls = controller.controls(word)
+        for label, index in [('boundary', 0), ('cut', witness['handoff_after']),
+                             ('return', len(word['events'])-1)]:
+            bad = deepcopy(word)
+            del bad['events'][index]
+            controls['missing_'+label] = reject(lambda bad=bad: controller.replay(bad))
+        bad = deepcopy(word)
+        bad['diagonal'][-1][0] = -1
+        controls['unearned_central_target'] = reject(lambda: checks.transpose(bad))
+        bad = deepcopy(word)
+        bad['events'][witness['handoff_after']][-1] = '-1'
+        controls['wrong_cut_sign'] = reject(lambda: controller.replay(bad))
+
+        def jobs(caller):
+            return [(e[0], e[1] if e[0] == 'LOCAL' else e[3] if e[0] == 'QUERY' else
+                     tuple(e[3] if e[0] == 'IDLE' else e[2]))
+                    for e in caller['events'] if e[0] in ('LOCAL', 'QUERY', 'IDLE', 'IDLE_ONE')]
+
+        need(jobs(base) == jobs(word), 'Retirement keeps every rebalanced paid job')
+        physical = load(self.source/'physical.py', 'tree_physical').physical(word, native_word, L)
+        invoice = load(HERE/'invoice.py', 'tree_invoice')
+        need(adapted(self.source/'pricing.py', 'pricing') == (HERE/'pricing.py').read_text(),
+             'Every price edit is declared')
+        price = load(HERE/'pricing.py', 'tree_price')
+        bound = price.run(local, native, word, physical, L)
+        changes = invoice.compare(bound, source_result['complete_cost'])
+        old_T = {int(k): v for k, v in source_result['complete_cost']['targets'].items()}
+        old_P = {int(k): v for k, v in source_result['complete_cost']['paid'].items()}
+        E = L.load(L.ROOT/'research/higher-moment-strictification/verify.py', 'tree_comparison')
+        other = L.load(L.ROOT/'research/kappa-rebuild/architecture/mixed_target_probe.py', 'tree_second_comparison')
+        first = E.moment(old_T, 1-price.KAPPA), E.moment(old_P, 1-price.KAPPA)
+        tl, th = other.moment(old_T, price.KAPPA)
+        pl, ph = other.moment(old_P, price.KAPPA)
+        need(first[0][1] < first[1][0] and th < pl,
+             'Both rational engines reject PR6545 at the new claim')
+        bad = deepcopy(local)
+        bad['paid'][1] -= 1
+        controls['unpaid_native_job'] = reject(lambda: price.assemble(bad, native, word, physical))
+        bad = deepcopy(physical)
+        bad['main_targets'][1] += 104
+        controls['extra_gaussian_credit'] = reject(lambda: price.assemble(local, native, word, bad))
+        for name, value in [('CONTROLLER.json', word), ('WITNESS.json', witness)]:
+            if self.write:
+                (HERE/name).write_text(json.dumps(value, indent=2)+'\n')
+            else:
+                need(value == read(HERE/name), 'Exact '+name)
+        print('PASS rebalanced caller, independent reverse rows, dual pairs, physical maps and complete price', flush=True)
+        candidate = price.encode(dict(caller=forward, reverse=reverse, dual_pair=pair,
+            child_sizes=data['child_sizes'], internal_node_sizes=dict(sizes),
+            physical=physical, complete_cost=bound, changes=changes, controls=controls,
+            source_margin_first=E.display((first[0][0]-first[1][1], first[0][1]-first[1][0])),
+            source_margin_second=E.display((tl-ph, th-pl))))
+        return {'source': source_result, 'candidate': candidate}
+
+
+def run(write=False):
+    with tempfile.TemporaryDirectory(prefix='rebalanced-retirement-') as directory:
+        root = Path(directory)
+        old_cache = sys.pycache_prefix
+        sys.pycache_prefix = str(root/'bytecode')
+        try:
+            pins = read(HERE/'RETAINED.json')
+            unpack(pins, root)
+            source = root/'research/gaussian-root-retirement'
+            # The two edits add one explicit extension parameter. They do not
+            # remove or alter a source check, price, receipt or archive.
+            driver_path = source/'tree_extension_driver.py'
+            driver_path.write_text(adapted(source/'verify.py', 'driver'))
+            driver = load(driver_path, 'tree_source_driver')
+            result = driver.run(extension=lambda candidate: Extension(candidate, source, root, write))
+            candidate = result['candidate']['candidate']
+            result['candidate'] = result['candidate']['source']
+            need(result == read(source/'RESULT.json'), 'Every PR6545 result and control is unchanged')
+            return {'status': STATUS, 'source_head': pins['source_head'],
+                    'source_result_sha256': sha256((source/'RESULT.json').read_bytes()).hexdigest(),
+                    'candidate': candidate, 'assumptions': result['assumptions'],
+                    'selected_frontier_changed': False}
+        finally:
+            sys.pycache_prefix = old_cache
+
+
+if __name__ == '__main__':
+    need(sys.argv[1:] in ([], ['--write-result']), 'Only --write-result is supported')
+    write = bool(sys.argv[1:])
+    result = run(write)
+    if write:
+        (HERE/'RESULT.json').write_text(json.dumps(result, sort_keys=True, indent=2)+'\n')
+    else:
+        need(result == read(HERE/'RESULT.json'), 'Exact complete scientific receipt')
+    print(STATUS, flush=True)