P112 rank-57 cover with scoped rational decoder cost floors

research/p112-rational-decoder-gate/CANONICAL-SHARP-PROOF.md

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+# Sharp minimum cost across all canonical-span decompositions
+
+For any rank57 fitting matrix on the pinned PR2757 mask, assume both factor
+families lie in that matrix's column and row spaces. If there are m=57+s
+nonzero rational rank-one terms, then
+
+    C >=334+2s.
+
+The saved57-term construction attains334. Thus allowing arbitrary extra terms
+within both spans cannot lower the invoice. The statement excludes exterior
+factor directions, higher-rank fitting matrices, changed covers and changed
+physical contracts. It is a finite decoder-cost result, not supplier admission.
+
+## Common definitions and inherited normal form
+
+Normalize the common diagonal to1. The existing normal-form proof gives
+K=[I,A;B,BA] and factors U=[P;BP], V=[Q;A^T Q], with PQ^T=I57. The57 latent
+coordinates consist of eighteen three-coordinate paired components H_i and
+the three-coordinate component J. Invertible paired maps make each nonzero
+paired component touch both physical triads, so its full port space is H_i.
+The residual coefficients remain general linear functionals of component8.
+
+Call a term pure when either latent factor is supported entirely in J, and
+let p_J count these terms. This is the same definition in both inequalities
+below. Each pure term costs3; each nonpure term costs at least4. Let L4,L5
+count nonpure terms of actual tariff4 and5, and write D=2L4+L5. Since the
+residual diagonal is nonzero, some nonpure term costs at least7. Consequently
+
+    C >=6m-3p_J-D+1.                                      (1)
+
+The extra1 is charged to a term outside both low-price classes. The pure
+terms, the actual-price classes and this residual term are unchanged when
+the identity is subsequently restricted.
+
+## Full and restricted acyclic budgets
+
+The existing complete40-space closure checks all272 ordered low-rank span
+rectangles. Their maximal component supports are disjoint. A rank4 rectangle
+has size1 by1; rank5 rectangles have size1 byk or k by1, k<=3. Each actual
+term's supports are subsets of these maximal supports. Replacing a maximal
+support by a nonempty subset only increases the probability that all source
+components precede all reader components in a random order. The geometric
+rank may be below the actual price because the closure omits the residual
+line; this also only increases the discount available from the probability.
+
+For a random order of all19 components, select each low-price term when its
+source support entirely precedes its reader support. Selected terms sum to
+a strictly block triangular N. Therefore I57-N remains invertible, so at
+most m-57 terms were selected. The exact census probabilities give
+
+    D <=4(m-57).                                          (2)
+
+Now remove the p_J pure terms and restrict to the54 ordinary coordinates.
+The remaining m-p_J terms sum to I54, and every one has nonzero projections
+on both sides. Delete J from each low-price support rectangle. The supports
+are still nonempty and disjoint, and selection probabilities cannot decrease.
+Repeating the same argument for a random order of these18 components yields
+
+    D <=4(m-p_J-54).                                      (3)
+
+This second argument requires no minimality of the remaining factorization:
+the only rank assertion is that I54-N is invertible and a sum of its remaining
+rank-one terms. The first argument has the identical property for I57.
+
+## Combine without discarding the pure-term count
+
+Substituting(2) and(3) separately into(1) gives
+
+    C >=2m-3p_J+229,
+    C >=2m+ p_J+217.
+
+One quarter of the first lower bound plus three quarters of the second
+cancels p_J, proving
+
+    C >=2m+220 =334+2s.
+
+In particular m=58 costs at least336. The apparent333 possibility came from
+retaining p_J in only the restricted inequality while discarding it in the
+full inequality; it is not an unresolved construction after the two bounds
+are combined.
+
+## Exact replay and review scope
+
+`check_canonical_sharp.py` authenticates the candidate, selected archive and
+payload, reconstructs the40 spaces using explicit vector sets and
+annihilators, and checks both selection probabilities on all272 rectangles.
+It also verifies the exact affine combination, rejects missing closure
+members and duplicate JSON keys, and prints a receipt without changing files.
+The old checker uses an independent echelon representation. The proof was
+separately audited by another AI agent; no human review or formal proof
+assistant verification is claimed.
+
+    python -I -O -B check_canonical_sharp.py --repo PATH_TO_REPOSITORY
+
+Receipt: `CANONICAL-SHARP-RESULT.json`. Its low-rectangle table SHA256 is
+`12858a508b318f62a568fe5205d16a399fc4e3e7fdc9cb7e8ea413f0a444832e`.

research/p112-rational-decoder-gate/CANONICAL-SHARP-RESULT.json

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+{
+  "archive_sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6",
+  "candidate_lf_sha256": "dc5fbe51f9cd3a14361256ac136736bffb18971d0247b891accf0d2ee060d06b",
+  "combined_bound_coefficients": [
+    2,
+    0,
+    220
+  ],
+  "controls": [
+    "removed paired generator",
+    "removed generated span",
+    "duplicate JSON key"
+  ],
+  "full_bound_coefficients": [
+    2,
+    -3,
+    229
+  ],
+  "full_low_rectangles": {
+    "(1, 1, 4)": 20,
+    "(1, 1, 5)": 178,
+    "(1, 2, 5)": 33,
+    "(1, 3, 5)": 4,
+    "(2, 1, 5)": 33,
+    "(3, 1, 5)": 4
+  },
+  "generated_spaces": 40,
+  "ordinary_bound_coefficients": [
+    2,
+    1,
+    217
+  ],
+  "ordinary_low_rectangles": {
+    "(1, 1, 4)": 20,
+    "(1, 1, 5)": 222,
+    "(1, 2, 5)": 13,
+    "(1, 3, 5)": 2,
+    "(2, 1, 5)": 13,
+    "(3, 1, 5)": 2
+  },
+  "payload_sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+  "rectangle_count": 272,
+  "rectangle_table_sha256": "12858a508b318f62a568fe5205d16a399fc4e3e7fdc9cb7e8ea413f0a444832e",
+  "result": "C >= 2m+220 = 334+2s, m=57+s",
+  "scope": "Any rank57 fit on the fixed mask with BOTH factor families in its column/row spaces; arbitrary rational coefficients and term count. Exterior factors and larger fitting rank excluded.",
+  "space_dimension_histogram": {
+    "0": 1,
+    "3": 1,
+    "6": 18,
+    "7": 2,
+    "8": 17,
+    "9": 1
+  }
+}