Prove Clifford tensor faithfulness and a 104-return paid-repair floor

research/clifford-pointwise-faithfulness/APPLICATION-RESULT.json

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+{
+  "affected_independent_inputs": 104,
+  "ambient_axes": 17,
+  "finite_gram_lower_at_72": "15012035871336244158376742625882962056980917290227300430718349610187586298795724967750288241818388072331510355393216731687558989143138428798507710704884164192105304698649639726569/129672361527531029953512745740348785969138944757576153124864291552832900356653379574990845279596993571506183956603149661949848471106617978371464838566061365220661931356297172615168",
+  "finite_gram_lower_at_84": "3307688356600097832824604549253505021711762694675820630380988647763450661166142843805702790691122208883654326597829149758145507314212369741999146206603406894760888959824077189445396581768015141945681474091631148125685268535301461615363915903/3334007216439927137039925895360628898572379161157954080198128905882018618908816035760716100435777145371464955296716620222944400827059682540181678026165415023047578789757007279231539142955907012364482508067943300990845374018738230645581938688",
+  "first_proved_width": 72,
+  "late_grade7_words": 5408,
+  "later_squared_error_lower_bound": "125/126",
+  "later_width": 84,
+  "new_kappa": null,
+  "normalized_squared_error_lower_bound": "1/9",
+  "operator_norm_error_lower_bound": "1/3",
+  "proof_scope": "One frozen passive columnwise Clifford contract; see APPLICATION.md",
+  "source_replay": "PASS_PASSIVE_SAVED_QUERY_CEILING",
+  "status": "PASS_PASSIVE_CEILING_FROM_WIDTH_72_NO_NEW_KAPPA"
+}

research/clifford-pointwise-faithfulness/APPLICATION.md

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+# The retained passive ceiling holds from width 72
+
+This applies the new pointwise theorem to the retained
+[passive saved-query ceiling](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/passive-saved-query-ceiling/README.md).
+Chris R supplies that proof and its independent source/grade audit in
+[PR #4978](https://github.com/CrocSwap/integer-mult-bounds/pull/4978).
+The actual native source is Dugongue's
+[PR #4948](https://github.com/CrocSwap/integer-mult-bounds/pull/4948), head
+`86e474038e05f99a76d41f71d81946ad2327e4a4`.
+The retained source's result holds for all sufficiently large widths.
+The new conclusion supplies an explicit width and a nonzero error bound.
+
+## Contract and source facts
+
+Read the frozen
+[proof](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/passive-saved-query-ceiling/evidence/frozen/pr4948-saved-query-macro-gate/PROOF.md)
+with its mandatory
+[uniform-alphabet addendum](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/passive-saved-query-ceiling/evidence/frozen/pr4948-saved-query-macro-gate/UNIFORM-ALPHABET-ADDENDUM.md)
+and
+[owned-reset clarification](https://github.com/CrocSwap/integer-mult-bounds/blob/32890bc5de034ab8fc93ac88ead7279a6a416d04/research/passive-saved-query-ceiling/evidence/frozen/pr4948-saved-query-macro-gate/OWNED-RESET-CLARIFICATION.md).
+All three are byte-pinned replay inputs. The following facts are inherited:
+
+- Four original main banks have 112 independent input fields each. Nine
+  macro cuts and three paid responses supply 39 observed bank fields.
+- For 104 zero-diagonal source ports in original bank 0, the allowed
+  coefficient words have Fourier grades 0, 1 or 7. The exact audit checks
+  all 5,408 nonzero late grade-7 words. Their target `D1 Q^-1` has grade 9.
+- Every allowed left code factor is a scalar sum of monomial Clifford
+  words. Left multiplication by each such word preserves Fourier grade.
+- Every complete one-column word belongs to the same 17-axis Clifford
+  group. The finite universe is independent of width. A word containing
+  a Gaussian need not be monomial; the full Clifford theorem is used.
+
+Scalar coefficients, choices of controls and finite numbers of decoder
+operations may depend on the width. The paid word and each paid input
+expression stay fixed. Source scalar coefficients remain outside tensor
+powers. Only a scalar inside a native operator acquires its f-th power.
+
+## New quantitative corollary
+
+Fix any allowed decoder at any integer width f>=72=4*17+4. Expand one
+affected input-to-target coefficient into its finite Clifford words.
+Combine projective copies, with their phases raised to f. Call the result
+`sum_g b_g U_g^(tensor f)`, where each U_g is unitary. Fourier grade is
+unchanged by scalar phase, so the grade-9 target T is not one of these
+projective classes. Equation (4) of the [new proof](PROOF.md) gives
+
+    2^(-17f) ||T^(tensor f)-sum_g b_g U_g^(tensor f)||_HS^2
+        >= (1/9)(1+sum_g |b_g|^2) >= 1/9.
+
+The operator norm error in this coefficient is at least 1/3. Set all other
+independent inputs to zero and choose a unit vector which attains this
+coefficient's operator norm. The corresponding complete endpoint also has
+error at least 1/3 on that input, when the endpoint uses the direct-sum
+Euclidean norm. This does not assert error on every input or add 104 errors
+on one common input.
+
+At every f>=84=4*17+16 the same normalized squared-error lower bound can
+be replaced by 125/126. The norm error is then at least sqrt(125/126).
+These constants hold uniformly over all allowed width-dependent choices.
+No enumeration of the 17-axis group or dense tensor matrix is needed.
+
+Thus this passive contract cannot provide even a uniformly small-error
+version of the missing endpoint at these widths. This corollary adds no
+new lower bound on reset cost. Storage, COPY, finite code work and owned
+CLEAR/reset may all be granted for free, as the retained clarification says.
+
+## Verification and limits
+
+[verify_application.py](verify_application.py) first authenticates every
+retained package file. It then runs the unchanged retained verifier, which
+replays the author bank/grade calculation, the independent trace/grade
+audit and the five wrapper rejection controls. Its complete output must
+match [PASSIVE-REPLAY.json](PASSIVE-REPLAY.json). It computes both new exact
+finite Gram bounds and compares [APPLICATION-RESULT.json](APPLICATION-RESULT.json).
+
+The source replay establishes the finite source facts. The written proofs
+establish the uniform group premise, grade preservation and the universal
+quantifiers. This is not an independent human review or a formal proof
+kernel. No actual new decoder, physical emission, selected cold replay,
+complete pricing, supplier admission or kappa improvement is claimed.
+
+Active feedback into later paid gates, new paid calls, changed words or
+targets, other observations, native internal prefixes and noncolumnwise
+payload routing change the contract. They are outside this corollary.
+No active-feed task is reserved or rejected by this package.

research/clifford-pointwise-faithfulness/MUTATIONS.json

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+{
+  "checks": [
+    {
+      "diagnostic": "Exact first Clifford overlap level",
+      "mutation": "first Clifford overlap",
+      "optimized": false
+    },
+    {
+      "diagnostic": "Exact first Clifford overlap level",
+      "mutation": "first Clifford overlap",
+      "optimized": true
+    },
+    {
+      "diagnostic": "Native/scalar phase residue",
+      "mutation": "native phase outside tensor",
+      "optimized": false
+    },
+    {
+      "diagnostic": "Native/scalar phase residue",
+      "mutation": "native phase outside tensor",
+      "optimized": true
+    },
+    {
+      "diagnostic": "Stabilizer pointwise Gram estimate",
+      "mutation": "projection scale outside tensor",
+      "optimized": false
+    },
+    {
+      "diagnostic": "Stabilizer pointwise Gram estimate",
+      "mutation": "projection scale outside tensor",
+      "optimized": true
+    },
+    {
+      "diagnostic": "Nonzero-trace Pauli fiber count",
+      "mutation": "unsigned fixed Pauli count",
+      "optimized": false
+    },
+    {
+      "diagnostic": "Nonzero-trace Pauli fiber count",
+      "mutation": "unsigned fixed Pauli count",
+      "optimized": true
+    },
+    {
+      "diagnostic": "Tensor phase covariance",
+      "mutation": "repair phase outside tensor",
+      "optimized": false
+    },
+    {
+      "diagnostic": "Tensor phase covariance",
+      "mutation": "repair phase outside tensor",
+      "optimized": true
+    },
+    {
+      "diagnostic": "Complete repair invoice hinges",
+      "mutation": "missing added repair charge",
+      "optimized": false
+    },
+    {
+      "diagnostic": "Complete repair invoice hinges",
+      "mutation": "missing added repair charge",
+      "optimized": true
+    }
+  ],
+  "source_sha256": {
+    "audit_mutations.py": "ace28d7781cda16d2e1e4de015112f3d3a6bcccd67b23955e736ca4ac25e9b4d",
+    "bounds.py": "e2a742d1cd1a76d0f49cef4b233ca74a5c8c8b3bee5c954c0e003bc065878af2",
+    "census.py": "145fe9e335aa57c39e1c1411da351049d9b52fdc58fb02f994066e166a8235d4",
+    "native_pointwise.py": "7dea28f405137881de8048010951e7e42857a250d98972faa14cb4d3ed6228b2",
+    "source.py": "1ef7dada5daba9d0e89f8117285c944a401b0ba74c00c3a70673a9926bfe45ea",
+    "stabilizer.py": "7011f435441657d5168b23e32f32ed2a263cf608dd5c139cf86a40e88427a738",
+    "verify.py": "47fcf94881aee98843d10194560c41bc3dc2db13a3956caeba1a9e1f7e3a953e",
+    "verify_application.py": "f783fae54da7d8c6d2377d12c2013fce9ced55ebbbd27bb002ebd0a68e58b325",
+    "verify_repair.py": "013fb6f9743bc296e541b71a2e6b95f45b46d846ad26e64b23002bf189c8c073"
+  },
+  "status": "PASS_TWELVE_SCIENTIFIC_MUTATION_REJECTIONS"
+}

research/clifford-pointwise-faithfulness/PASSIVE-REPLAY.json

@@ -0,0 +1,38 @@
+{
+  "author_exact_stdout_receipts": [
+    {
+      "bytes": 4031,
+      "receipt": "BANK-SPAN.json",
+      "sha256": "c92a574492a18d501ae6acd8cb792aaaa28070312a7558990923d7a7e810073c"
+    },
+    {
+      "bytes": 702206,
+      "receipt": "GRADE-GATE.json",
+      "sha256": "8267ff814fd7fa86638b6ce6384f0ac717a341d871bd302742de6aaf8ba00b6d"
+    }
+  ],
+  "independent_exact_stdout_receipt": {
+    "bytes": 5574,
+    "receipt": "INDEPENDENT.json",
+    "sha256": "bfbfeb0a835e11ef9a4638f57e9179758e7bda560c58993e26c6ec9cde8a47d9"
+  },
+  "late_grade7_words": 5408,
+  "macro_cuts": 9,
+  "native_source": "86e474038e05f99a76d41f71d81946ad2327e4a4",
+  "new_kappa": null,
+  "observed_banks": 39,
+  "passive_only": true,
+  "phase": "all",
+  "physical_emission": false,
+  "status": "PASS_PASSIVE_SAVED_QUERY_CEILING",
+  "sufficiently_large_width": true,
+  "target_grade": 9,
+  "wrapper_mutation_rejections": [
+    "unretained Git source request",
+    "undeclared dependency dispatch",
+    "altered mandatory reset clarification",
+    "altered scientific input bytes",
+    "wrong scientific receipt"
+  ],
+  "zero_diagonal_inputs": 104
+}