Construct complementary B/C cancellation and prove a serial-consumer cost ceiling (no new κ)

research/complementary-consumer-feasibility/.gitattributes

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+inputs/* -text

research/complementary-consumer-feasibility/OVERLAP.json

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+{
+  "base": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "pr_searches": [
+    "4703",
+    "complementary OR mixed controller OR endpoint gauge"
+  ],
+  "related": [
+    4703,
+    4712,
+    4674,
+    4660,
+    4839
+  ],
+  "discussion_pages": [
+    1163,
+    1166,
+    1167
+  ],
+  "discussion_coverage": "Bounded rendered first-page read and term search; pagination not exhaustive.",
+  "local_helper": "find_overlap complementary --refresh failed remote gh unavailable; local3614/6319 note paths searched,2705 absent.",
+  "novelty": "No exhaustive novelty claim. Direct-price method credited4674; paired serial contract and refreshedC profiles are distinct scope.",
+  "source_heads": {
+    "4703": "4cc96829fad977aecf0018b26beb067cfd2689a5",
+    "4712": "b75a27bf07f1584b20babfddf887493981cd5949",
+    "4674": "84cf25a580db9ed2b4b1371c40689a035711364e"
+  }
+}

research/complementary-consumer-feasibility/PROOF.md

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+# Complementary cancellation and its complete serial budget
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.
+
+## Sources and obligation
+
+PR4703 at 4cc96829fad977aecf0018b26beb067cfd2689a5 supplies actual complementary B/C native interfaces. PR4712 at b75a27bf07f1584b20babfddf887493981cd5949 independently audits the endpoints, frames, phases, helper ownership and refreshed local profiles. The unchanged audit and relevant inputs are retained byte for byte from canonical main 624de9dc66858af0945e5f8a0942488d62280d43. This package consumes those conditional interface facts; it does not replay the native scientific suite.
+
+The older PR4674 at 84cf25a580db9ed2b4b1371c40689a035711364e supplies the direct-price normalization method for the earlier B supplier. Its method is credited, not claimed as new. The new scope is an explicit complementary two-call cancellation on the refreshed B/C suppliers, and a screen of their serial fixed-owner consumers. PR4712's 120-head query and dirty-helper failures are different models and are not repeated.
+
+## Actual two-call word
+
+Use a shared central Gaussian Q and two disjoint seven-axis Gaussian blocks D1,D2. They commute because they act on disjoint address axes; the field matrices below are not tensor-powered. Let L(c)=[[1,0],[c,1]]. At stage s:
+
+    A_s = D_s [[1,0],[Q^-1,1]],
+    B_s = D_s [[1,0],[1,Q]],
+    C_s = D_s [[Q,0],[1,1]].
+
+The whole first invocation is the C provider and the whole second is the B provider. Use the same 56 exceptional ports in both. Each call has its own 408 independent helper inputs. Every main input remains arbitrary. On exceptional pairs the actual chronological word C1, B2, L(-2) gives
+
+    L(-2) B2 C1 = diag(D2 D1 Q,D2 D1 Q).
+
+On the 56 ordinary pairs the chronological word A1,A2,L(-2Q^-1) gives
+
+    L(-2Q^-1) A2 A1 = diag(D2 D1,D2 D1).
+
+The second terminal is a genuine rank-one query on each pair, implemented with the source contract's owned temporary, Q^-1 child, scalar multiply/add and owned clear. It cannot be replaced by L(-2). All monomial source wrappers and actual inverse-Gaussian normalizers are inherited. The helper fields are untouched by these main-only terminals and each returns its original independent Gaussian job exactly once. No helper is credited as a recovered main job.
+
+For any matrix W=T B2 C1, the actual raw dual is
+
+    W^-T=T^-T B2^-T C1^-T.
+
+The execution order is C1^-T, B2^-T, T^-T, the same chronological order with every gate replaced by its inverse transpose. Explicitly
+
+    B_s^-T=D_s^-1 [[1,-Q^-1],[0,Q^-1]],
+    C_s^-T=D_s^-1 [[Q^-1,-Q^-1],[0,1]],
+    A_s^-T=D_s^-1 [[1,-Q^-1],[0,1]],
+    L(c)^-T=[[1,-c],[0,1]].
+
+These expressions use symmetric canonical D_s,Q and retain their inverse powers; they do not identify Q with Q^-1. The verifier computes the inverse transpose algebraically and checks both complete endpoints over rational Laurent polynomials in D1,D2,Q without finite-order assumptions.
+
+This is a fully specified typed consumer with paid query children. It is not a newly emitted 1040-field native trace. Physical validity remains conditional on the audited supplier/normalizer/owned-temporary interface; no claim of new absolute-phase native replay or compiler admission follows.
+
+## Complete recursive bill and finite work
+
+The audited net local rank masses are B=1688 and C=1680. Add 56 rank-one terminal jobs. The new main jobs are 112 rank14 and112 rank15, rank mass3248. All816 helpers occur once in both the moment and ownership census. Thus the complete rank-mass deficit is
+
+    1688+1680+56-3248=176.
+
+The full paid and target histograms appear in RESULT.json. There are1040 independent persistent inputs:224 main and816 helpers. One owned query temporary can be reused serially; its clean entry/exit is not productive credit. The whole word inherits two local finite allowances each less than10^8 V under PR4703's native model, plus112 terminal additions,56 owned query COPY/clear sequences, their coefficient work and monomial normalization. Address permutations, storage, precision and the outer compiler remain separate finite obligations. Those costs are granted free only in the optimistic rejection screen below; their exact finite native constants are not needed to reject a negative recursive moment. No recursive price is invented for finite scalar work.
+
+## Refreshed direct-price lemma
+
+Let t=1-C and P_B,H_B,P_C,H_C be the authenticated current profiles. Define
+
+    g_J(C)=P_J(t)-H_J(t)-224*7^t-112.
+
+The direct primitives of either actual A/B or A/C interface cost224 rank-seven jobs and112 rank-one jobs, with helper jobs paid separately. B's exceptional output scalar shear and C's exceptional input scalar shear give their direct implementations. This is the existing PR4674 comparison extended to the current C profile.
+
+At C=0, g_B=8 and g_C=0. Normalize g_J/7^t. In its derivative, every signed-rank term is nonnegative except the negative rank-six helper term. The rank-seven term is constant. The rank-one term alone contributes at least (P_J[1]-112)/7 * log7; the rank-six term has magnitude at most H_J[6]*log(7/6). Since log7>1 and log(7/6)<1, the derivative is strictly greater than
+
+    B: (1956-112)/7-7 =1795/7 >0,
+    C: (2011-112)/7-6 =1857/7 >0.
+
+All other signs are checked against the authenticated profiles. Therefore for0<=C<1:
+
+    g_B(C)>=8*7^-C,
+    g_C(C)>=0,
+
+and both are strict relative to their starting normalized value for C>0. This proves the domain, not a numerical sample or monotonicity extrapolation. It also explains why the C local improvement does not supply a positive primitive discount at positive saving.
+
+## Scoped serial ceiling
+
+Fix n>=2 whole invocations on the same224 main owners, one shared Q, and n fresh disjoint seven-axis blocks. The only other productive fields are the independent408 helpers per invocation, preserved as their audited outputs. Require every completed main Gaussian job to act within these7n+1 axes. Allow arbitrary main permutations and scalar boundary programs, but no extra productive inputs, helper repurposing or uncharged rank outside that domain. Any additional recursive cleanup is charged; grant all cleanup and finite work free for this upper bound.
+
+Each main target has rank at most7n+1. Thus with n_B B invocations and n_C C invocations, n_B+n_C=n, the complete target-minus-paid moment has upper bound
+
+    M(C)<=224*(7n+1)^t -n*(224*7^t+112)-8*n_B*7^-C.
+
+For0<t<1, strict subadditivity and n^t<=n give
+
+    (7n+1)^t < (7n)^t+1 <= n*7^t+1.
+
+Consequently M(C)<224-112n-8*n_B*7^-C<=0. At C=0 the exact same optimistic mass bound is
+
+    M(0)<=224-112n-8*n_B.
+
+The best case n=2, n_B=0 only ties zero at C=0 and is strictly negative at every positive saving. A B/C pair loses at least8 rank units even if every main field is optimistically awarded rank15 and all cleanup is free. The actual cancellation word loses176 rank units. All B/B, B/C, C/C serial two-call choices are therefore screened without a search.
+
+This statement is confined to the declared serial whole-call fixed224-owner domain. It does not exclude a controller with more independent main owners, interleaved extra productive jobs, changed helper ownership, different common/transverse geometry, or genuinely shared producer work. A new controller must identify which premise changes and price that change. More digits of the #4610 fixed root or more serial calls do not reopen this screen.
+
+## Expansion decision and next question
+
+Stop this serial consumer family. The next useful constructive question must alter the ownership or incidence contract enough to exceed the target cap, or change the native paid profile to restore a local primitive discount. A feasible first test is one actual shared producer/cancellation word across a scalar contact with a source-disjoint target census. Its complete optimistic bill must be favorable before native emission, compiler reserve and outer transfer. No such word is established here. No new kappa is claimed.

research/complementary-consumer-feasibility/README.md

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+# Complementary B/C cancellation loses its complete serial bill
+
+A two-call typed consumer of PR4703 is exact on every arbitrary main input: match its56 exceptional ports, run C then B on disjoint seven-axis blocks sharing Q, and subtract twice X from Y. This gives two rank15 jobs per exceptional pair. The remaining56 ordinary pairs need an actual rank-one query each and give rank14 jobs. With all816 independent helper outputs retained once, the complete recursive rank-mass deficit is176.
+
+The optimistic screen is stronger than this witness. Every serial n>=2 whole-call consumer on the same224 main owners, shared Q and n disjoint seven-axis blocks, with main target ranks at most7n+1 and only the source helpers as additional productive fields, has nonpositive moment at C=0 and strictly negative moment throughout0<C<1, even granting free cleanup and finite work. This is a scoped stop result, not a universal controller obstruction or a new kappa.
+
+[PROOF.md](PROOF.md) specifies the chronological word, actual raw inverse transpose, complete bill, all-exponent proof and family-expansion decision. [RESULT.json](RESULT.json) contains exact profiles and checks. The refreshed native facts are inherited from PR4712's independent audit of PR4703; this package does not repeat that native/phase audit. PR4674's direct-price proof method is credited. This is a typed-interface construction and analytic screen, not a freshly emitted complete native trace or admitted multiplier.
+
+Run from any exported copy, using standard-library Python3.12 or later:
+
+```sh
+python3 -I -O -B research/complementary-consumer-feasibility/verify.py
+```
+
+The new evaluator authenticates source bytes, checks exact Laurent endpoints and gatewise inverse transposes, counts every helper once, and checks symbolic derivative sign premises for the entire declared interval. It rejects a wrong cancellation coefficient, omitted ordinary Q correction, forward word substituted for raw dual, source corruption and type-sensitive receipt corruption. No Lean, new full native replay, precision execution, compiler or outer admission is claimed. The selected record and #4610 comparison remain unchanged.
+
+## Open next questions
+
+Stop enlarging the losing serial model. Reopen only with a concrete change to target ownership, shared producer work or native paid profile; identify an actual scalar contact where a paid job disappears and verify that it does not return in restoration. Price the optimistic whole bill first. A separate physically valid witness, positive complete moment and funded kappa transfer remain distinct milestones.
+
+Credit Dugongue (#4703), Cao Yifan / intqwq (#4610/#4712), Cinna Davis (#4674 and retained controller lineage), zxv, Jacob Alan Sussman, DaysSky and every retained source notice. Prepared by Chris R / Clear1nsight with substantial OpenAI Codex assistance, as a delegated construction study. Apache-2.0. No independent human review is claimed.

research/complementary-consumer-feasibility/RESULT.json

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+{
+  "complete_pair_paid": {
+    "1": 4023,
+    "2": 895,
+    "3": 830,
+    "4": 239,
+    "5": 40,
+    "7": 8,
+    "8": 8
+  },
+  "complete_pair_rank_deficit": 176,
+  "complete_pair_targets": {
+    "3": 28,
+    "4": 28,
+    "6": 13,
+    "7": 95,
+    "8": 652,
+    "14": 112,
+    "15": 112
+  },
+  "new_kappa": false,
+  "normalized_direct_gap_derivative_lower_bounds": {
+    "B": "1795/7",
+    "C": "1857/7"
+  },
+  "persistent_independent_inputs": 1040,
+  "profiles": {
+    "B": {
+      "direct_gap_mass": 8,
+      "helpers": {
+        "3": 2,
+        "4": 26,
+        "6": 7,
+        "7": 47,
+        "8": 326
+      },
+      "net_mass": 1688,
+      "paid": {
+        "1": 1956,
+        "2": 450,
+        "3": 403,
+        "4": 127,
+        "5": 28,
+        "8": 8
+      }
+    },
+    "C": {
+      "direct_gap_mass": 0,
+      "helpers": {
+        "3": 26,
+        "4": 2,
+        "6": 6,
+        "7": 48,
+        "8": 326
+      },
+      "net_mass": 1680,
+      "paid": {
+        "1": 2011,
+        "2": 445,
+        "3": 427,
+        "4": 112,
+        "5": 12,
+        "7": 8
+      }
+    }
+  },
+  "serial_domain": "n>=2 complete whole calls; same224 main owners; independent408 helpers per call; one shared Q; n disjoint seven-axis transverse blocks; each main target rank<=7n+1; no extra productive inputs or ranks",
+  "status": "PASS_SCOPED_COMPLEMENTARY_SERIAL_SCREEN",
+  "two_call_screens": {
+    "0B+2C": {
+      "maximum_main_count": 224,
+      "maximum_main_rank": 15,
+      "optimistic_mass_deficit": 0
+    },
+    "1B+1C": {
+      "maximum_main_count": 224,
+      "maximum_main_rank": 15,
+      "optimistic_mass_deficit": 8
+    },
+    "2B+0C": {
+      "maximum_main_count": 224,
+      "maximum_main_rank": 15,
+      "optimistic_mass_deficit": 16
+    }
+  },
+  "uniform_optimistic_moment_bound": "M(C)<=224*(7n+1)^(1-C)-n*(224*7^(1-C)+112)-8*n_B*7^(-C); strict negative for0<C<1, n>=2",
+  "verification_scope": "typed-interface Laurent identities and source receipt arithmetic only; source native/phase audit inherited from4712, not freshly replayed"
+}

research/complementary-consumer-feasibility/SNAPSHOTS.json

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+  },
+  {
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research/complementary-consumer-feasibility/SOURCES.json

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+  "base_commit": "624de9dc66858af0945e5f8a0942488d62280d43",
+  "files": [
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