Conditional κ = 0.001662039077758742 — U10 (two new triangle-hybrid slots) on PR942's T37 owners

research/complex-u10-t37/ALL-T-PROOF.md

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+# A signed-dual upper bound for every replica count
+
+This is the package's independent relaxation bound. The separate construction
+in [PROOF.md](PROOF.md) attains its kappa grid value. For the unchanged U8 local profile, whole-helper widths,
+idle-capacity inventory and seven-step outer recipe, every positive integer
+replica count has a kappa grid upper bound of **0.001661960547713947**.
+The maximum possible improvement over PR890 within this family is therefore
+**0.000000002773066918**. This removes the T <= 22 restriction from the
+*upper-bound* argument. Feasibility and finite precision remain separate.
+
+`independent/all_t_bound.py` reproduces the calculation with Python 3.11+ and the standard
+library. It pins all consumed code and claims by SHA256. Run from the repository
+root, with a fresh output filename:
+
+```sh
+python -B research/complex-t37-owners/independent/all_t_bound.py --out /fresh/ALL-T-BOUND.json
+```
+
+The exploratory arithmetic passed on Windows / CPython 3.14.7; repeated
+fresh-output executions reproduced its complete receipt exactly. Relocation
+changes only path resolution, the fail-closed optimization guard and explanatory
+scope text; the numerical inputs and arithmetic are unchanged. The receipt
+uses both retained exact rational moment engines, independently structured bin
+enumerations, and all 47 retained outer constraints. It does not reproduce the
+full supplier, owner map, hosted invoice, compiler or multiplication theorem.
+
+## Fixed family and necessary inequalities
+
+The U8 local paid profile per call is
+
+```
+{1:2548, 2:338, 3:446, 4:162, 5:72, 8:8}.
+```
+
+There are nine calls and 896 live wires per replica. The whole-helper widths
+per call are `{4:80,5:32,7:54,8:122,9:216}`. The idle stock per replica contains
+672 cap-8 bins and 448 bins of each capacity 16, 32 and 48. These are PR890's
+unchanged data. All geometric and execution-window restrictions are relaxed.
+
+Use the PR795 signed width prices `(-1,-20,-8,-2,4)` at widths `(4,5,7,8,9)`
+and bonuses `(2,4,8,-13)` at capacities `(8,16,32,48)`, as retained by PR890.
+For an attainable residual rank r, let u(r) be the maximum bonus plus price
+over every legal count vector in one bin leaving r. Cartesian enumeration of
+all 1,163 legal vectors and a separate unbounded-fill recurrence agree on u.
+In particular, u(0)=0, u(1)=19 and u(48)=-13.
+
+For T replicas, let n_r count the bins with residual rank r. Summing actual
+rank and signed price gives the necessary inequalities
+
+```
+sum r*n_r = 14382*T,
+sum u(r)*n_r >= -3892*T.
+```
+
+The second direction matters: u is an upper bound on each actual bin's signed
+price, while the sum of actual bin prices is fixed. Eliminating n_48 yields
+
+```
+sum_(0<r<48) v(r)*n_r >= 150*T,
+v(r) = 48*u(r)+13*r,
+v(1) = 925.
+```
+
+Rank zero contributes neither rank, credit, nor moment, so it may be omitted.
+
+## One supporting inequality replaces replica enumeration
+
+At the exact endpoint
+
+```
+z = 0.001664727258740678855773,
+phi(r) = r*(73/r)^z,
+c(r) = 48*phi(r)-r*phi(48),
+lambda = c(1)/925,
+```
+
+both pinned rational engines independently certify lambda > 0 and
+
+```
+c(r) > lambda*v(r)  for every attainable 0<r<48 with r != 1.
+c(1) = lambda*v(1).
+```
+
+The smallest certified strict margin is approximately 0.2913731504478641;
+the receipt retains exact rational lower bounds for every rank. The comparison
+also covers zero and negative v(r), with interval multiplication performed in
+the correct direction for negative coefficients. No floating-point rank or
+sign decision is used.
+
+Consequently every nonnegative count vector satisfying the necessary credit
+inequality satisfies
+
+```
+(sum c(r)*n_r)/T >= 150*c(1)/925 = (6/37)*c(1).
+```
+
+This holds for **every positive integer T**, with no replica cutoff. The idle
+moment numerator is
+
+```
+sum phi(r)*n_r
+  = (14382*T/48)*phi(48) + (sum_(r<48) c(r)*n_r)/48.
+```
+
+All other normalized moment contributions are identical across T. Thus the
+normalized formal remainder profile
+
+```
+n_1/T = 6/37,  n_48/T = 11086/37
+```
+
+is a lower bound on every feasible owner's moment at z. Its smallest integral
+representative is T=37, with `{48:11086,1:6}`. Exact equality in this relaxation
+requires all nonzero non-48 residuals to have rank one and credit equality;
+therefore T must be a positive multiple of 37. This argument alone does not
+establish realization by actual bin patterns, owner time windows, or a complete
+controller. The separate T37 construction and checks supply those facts.
+
+## Root and outer transfer
+
+Adding nine copies per replica of the U8 local profile to the T37 residual
+profile gives live stock 33,152 and rank deficit 11,840. Both exact rational
+engines certify moment < 1 at
+
+```
+0.001664727258740678855772
+```
+
+and moment > 1 at its next 1e-24 tick z. At z every physically feasible profile
+has moment at least this formal T37 profile, hence also rejects z. Every
+proper-child moment is increasing in its saving exponent, so its admitted
+complex saving must be below z.
+
+The unchanged outer map is increasing in this range: each update is
+`a_next=s*(1-s+a_old)`, whose two partial derivatives are positive when
+0<s<0.002 and a_old>=0; the final rational map is increasing in a. The bit
+endpoint 0.001733427039589844234749 remains above the complex endpoint.
+Applying the retained seven-step map to either T37 endpoint produces the same
+1e-18 grid value, **0.001661960547713947**. All 47 strict constraints pass for
+the formal arithmetic, and the next-grid control is rejected. This makes the
+stated grid value an all-T upper bound under the same outer recipe.
+
+The upper-bound use of the retained 512-bit bootstrap *driver* does not admit
+a larger-T physical supplier. It evaluates the same numerical outer map for
+the upper relaxation. A construction using a different payload must separately
+recompute its finite bootstrap cutoffs and complete invoice. This package's
+construction does so with a charged 1024-bit complex allowance.
+
+## Attribution and scope
+
+The signed-dual method and integer prices are inherited from Chafik Boukhalfa's
+(chafreaky) PR795 and PR890.
+PR890's `complex-t21-owner-bound/PROOF.md` supplies the fixed-family equations;
+this note adds the supporting ratio inequality and removes the replica cutoff
+from the relaxation. The U8 construction and T16/T21 controller lineages remain
+credited to their original contributors. Exact arithmetic engines and outer
+driver are imported unchanged and credited through their retained notices.
+
+Prepared for Jonathan with substantial OpenAI Codex assistance, including
+separately performed arithmetic cross-checks. No independent human peer review
+is claimed. The all-size/compiler, common-label/mutable-source realization,
+whole-call hosting, precision/tape, analytic transfer and bit H1-H5 assumptions
+are unchanged. See [NOTICE.md](NOTICE.md) for retained attribution.

research/complex-u10-t37/CLAIM.json

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+{
+  "status": "conditional_finite_construction",
+  "kappa": "831019538879371/500000000000000000",
+  "kappa_decimal": "0.001662039077758742",
+  "delta": "0.000000078530044795",
+  "complex_lower": "0.00166480605047151514435",
+  "bit_lower": "0.001733427039589844234749",
+  "replicas": 37,
+  "calls": 333,
+  "roles": 167832,
+  "live": 33152,
+  "physical": 33666,
+  "rank_mass": 2408256,
+  "deficit": 11840,
+  "prefix_bits": 779,
+  "charged_prefix_bits": 1024,
+  "physical_row_coefficient": 17495,
+  "profile": {
+    "1": 848490,
+    "2": 112554,
+    "3": 149850,
+    "4": 51282,
+    "5": 25308,
+    "8": 2664,
+    "48": 11086
+  },
+  "idle": {
+    "1": 6,
+    "48": 11086
+  },
+  "scope": "Inherits PR942/PR890's all-size/compiler, common-label/physical-hosting, precision/tape, analytic and bit H1-H5 assumptions. No unconditional or formal full-theorem claim. The owner map, widths, capacities and outer recipe are PR942's, unchanged; only the local word changes (U8 -> U10).",
+  "baseline_pr942_kappa": "0.001661960547713947",
+  "local_profile": {
+    "1": 2548,
+    "2": 338,
+    "3": 450,
+    "4": 154,
+    "5": 76,
+    "8": 8
+  },
+  "local_word": "U10 = U8 + two triangle-hybrid slots (construction/u10/RECIPE-U10.json), content sha256 776de990f3e1e9eebb075a3d280d1aa029cb5a171d30945128a713f78dd126d1"
+}