Exclude the actual negative E ordinary decoder from width 296 with an error floor

research/negative-e-decoder-gram-bound/INPUTS.json

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+[
+  {
+    "path": "source/decoder-proof.md",
+    "repository": "JoyboyBrian/integer-mult-bounds",
+    "commit": "7e6824d560337e2b2e49da8b98727183491f6ecf",
+    "source_path": "research/odd-e8-negative-exit-algebra/PROOF.md",
+    "git_blob": "bc945b4901cc1c20da7d904ad238224aada71634",
+    "sha256": "6109fd631868ca7cf55c1d9178ff0a2ac030afe4353bc701e715757e4a754b06"
+  },
+  {
+    "path": "source/decoder-result.json",
+    "repository": "JoyboyBrian/integer-mult-bounds",
+    "commit": "7e6824d560337e2b2e49da8b98727183491f6ecf",
+    "source_path": "research/odd-e8-negative-exit-algebra/RESULT.json",
+    "git_blob": "0813bbb062c5ab28247d24fadac94d3cd67db137",
+    "sha256": "df10c3db7f0513e4cd43311bb22c9db15af8422c272339fffa933081c01768c7"
+  },
+  {
+    "path": "source/clifford-proof.md",
+    "repository": "aliromman/integer-mult-bounds",
+    "commit": "32e014bec9206f3987857e6153ec0fd8058f04e3",
+    "source_path": "research/clifford-pointwise-faithfulness/PROOF.md",
+    "git_blob": "465009d9d1b6947e80e6f1e868022f47ddab1f3c",
+    "sha256": "45483e4b5f604a5e70602c5caaa4394d4a0d0d1a6d0b6e8a93b836d50b23f764"
+  }
+]

research/negative-e-decoder-gram-bound/README.md

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+# A width-296 exclusion and error floor for the actual negative E decoder
+
+This source-bound application lowers the ordinary-decoder exclusion threshold
+in JoyboyBrian's [PR #5508](https://github.com/CrocSwap/integer-mult-bounds/pull/5508)
+from width 4198 to 296. It also gives a nonzero approximation-error floor.
+It uses the pointwise Clifford Gram theorem in
+[PR #5526](https://github.com/CrocSwap/integer-mult-bounds/pull/5526).
+The complete kappa calculation and selected frontier are unchanged.
+
+## Exact scope and conclusion
+
+Use the fixed source endpoint chart and negative-bank observation contract of
+PR #5508. There are 73 binary address axes. At width f the required postdecoder is
+
+```text
+J_f = Q_f^-1 E_(1,f)^-1 E_(2,f) E_(3,f)^-1 Q_f.
+```
+
+The only available dependent input is the emitted negative bank. Other owned
+or dirty inputs are independent. Ordinary address operations are tensor lifts
+of one-column monomial Cliffords. Arbitrary scalar owner mixing, instruction
+count, independent workspace and f-dependent coefficients are permitted.
+
+For every integer f>=296, no such ordinary program implements J_f. More
+strongly, for every such ordinary endpoint O_f, in the fixed source chart,
+
+```text
+||J_f-O_f||_op >= 1541/1255956.
+```
+
+This is a worst-case unit-input error floor, not an error on every input. It
+also applies to any larger endpoint required to contain the specified decoder
+block, by restriction to its original input and projection onto its output.
+It is not invariant under arbitrary nonunitary rescaling of that chart.
+
+No minimum width is asserted. At smaller widths the new argument gives no
+decision. Paid calls, correlated original-input histories, operations mixing
+tensor columns, nonlinear rounding and changed target contracts remain outside
+scope. This is not a paid-child lower bound or a physical completion invoice.
+
+## Pinned premises and their roles
+
+The package keeps byte-exact copies of three public files. INPUTS.json records
+their repositories, commits, Git objects and SHA256 values. The verifier checks
+those bytes before using the consumed data. Frozen predecessor files are not
+edited and no predecessor executable is imported.
+
+1. [PR #5508 proof](https://github.com/JoyboyBrian/integer-mult-bounds/blob/7e6824d560337e2b2e49da8b98727183491f6ecf/research/odd-e8-negative-exit-algebra/PROOF.md):
+   the actual owner code, negative endpoint, ordered fresh-stage path uniqueness
+   and unique nonzero path class are cited premises.
+2. Its [exact result](https://github.com/JoyboyBrian/integer-mult-bounds/blob/7e6824d560337e2b2e49da8b98727183491f6ecf/research/odd-e8-negative-exit-algebra/RESULT.json):
+   the selected owner endpoints, signed 25-axis map, source coefficient and
+   sparse conjugated column are checked independently below.
+3. [PR #5526 proof](https://github.com/aliromman/integer-mult-bounds/blob/32e014bec9206f3987857e6153ec0fd8058f04e3/research/clifford-pointwise-faithfulness/PROOF.md):
+   projective Clifford tensor independence and its dimension-uniform Gram
+   bounds are cited premises. The consumed rational row bounds are recomputed.
+
+These pending written proofs are not declared independently accepted by this
+application. Source authentication and the checks below do not replace their
+full mathematical review or the source producer's physical admission.
+
+## Verify the full alphabet premise
+
+Each stage source factor is a scalar owner matrix conjugated by block-diagonal
+signed linear address permutations L_i. Expanding the fixed owner block gives
+
+```text
+(J_f)_(i,j) = sum_path c_path (Q^-1 M_path Q)^(tensor f),
+```
+
+where M_path is a product of signed linear binary permutations. The scalar
+owner coefficients c_path are applied once; they are not raised to f.
+A binary invertible linear permutation is a CNOT Clifford, and a linear sign
+is a Pauli Z product. Thus every M_path is unitary Clifford. The source Q is
+the one-central-axis square root of X,
+
+```text
+Q = ((1+i)/2) I + ((1-i)/2) X.
+```
+
+It is unitary, Q squared is X, and it conjugates Paulis to Paulis. Hence every
+conjugated path is a 73-axis unitary Clifford with Gaussian-dyadic entries.
+This checks the actual expanded alphabet; the sum J itself need not be a
+Clifford. The distinction is required for applying PR #5526.
+
+The source's ordered path separation says that distinct paths between the
+fixed owner endpoints have distinct linear maps. Therefore they are not
+projectively equal, before or after conjugation by Q. At each width, folding a
+representative phase changes a path coefficient only by a unit phase. It does
+not cancel the chosen coefficient with another path.
+
+For each original-input/output block, any ordinary program in the stated
+alphabet lies in the scalar span of lifted monomial Cliffords. Composition,
+addition, scalar multiplication, COPY and CLEAR preserve this property, for
+any number of independent workspace fields. This also holds for each specified
+f when the word and its scalars depend on f.
+
+## The actual path is outside the ordinary alphabet
+
+The retained endpoints are j=(0,0,0) and i=(1,0,0). The source root rows have
+squared norm four and mutual dot product two. The inverse-stage cross entry
+is therefore 1/4, while the diagonal inverse entry is one. The middle-stage
+entry is
+
+```text
+39*98/(2744*97) + 1078/(2744*1079) = 1541/104663.
+```
+
+The selected path coefficient is
+
+```text
+c_star = (1/4)*(1541/104663)*1 = 1541/418652.
+```
+
+These rational identities are recomputed from the consumed source data.
+The complete path map reduces to the signed 25-axis map supplied in the
+source result, extended by identity on the other 48 axes. Its first binary
+column is 657934, so it moves the central direction e0.
+
+If a signed monomial M has linear map G with Ge0 different from e0, expansion
+of Q^-1 M Q gives four distinct output addresses on every basis input x:
+Gx, Gx+e0, Gx+Ge0 and Gx+e0+Ge0. Their coefficient magnitudes are 1/2.
+The four destinations are distinct because G is invertible and Ge0 is neither
+zero nor e0. Thus the chosen Clifford class is nonmonomial, even after a global
+phase. At input zero the exact consumed column is
+
+```text
+0       :  1/2
+1       :  i/2
+657934  :  i/2
+657935  : -1/2.
+```
+
+The independent sparse interpreter reconstructs this column and 146 further
+basis columns, checks invertibility of the binary map, and checks Q inverse
+and square identities. The all-input conclusion follows from the four distinct
+addresses, not extrapolation from the finite sample.
+
+## Apply the pointwise Gram theorem
+
+Put D=2^73. For distinct projective unitary Clifford classes, PR #5526 gives,
+at every f>=4*73+4=296,
+
+```text
+D^(-f) ||sum_g b_g U_g^(tensor f)||_HS^2 >= (1/9) sum_g |b_g|^2.
+```
+
+Subtract any ordinary block O from the selected decoder block. Its monomial
+classes cannot change the chosen nonmonomial coefficient. After all projective
+classes are combined, that coefficient still has magnitude c_star. Therefore
+
+```text
+D^(-f) ||(J_f)_(i,j)-O||_HS^2 >= c_star^2/9
+                                    = 2374681/1577425473936.
+```
+
+Since the normalized Hilbert-Schmidt norm is at most the operator norm, this
+proves the error floor and exact nonmembership. Restricting the complete
+endpoint to this input block and projecting onto this output block cannot
+increase operator norm, which gives the stated complete-endpoint consequence.
+
+The later source bound also gives normalized squared error at least
+(125/126)*c_star squared for every f>=308. It is a stronger sufficient constant,
+not a claim of optimal distance. The source's old support argument first
+passes at f=4198 because 2f-216 must exceed 8178. Its ADD-count lower bound and
+broader cross-column exclusions retain their original scopes.
+
+## Verification and limitations
+
+```sh
+python3.12 -B scripts/research_package.py --package research/negative-e-decoder-gram-bound
+```
+
+Use Python 3.12 and the standard library. Registered warning-strict normal and
+optimized runs print `PASS_NEGATIVE_E_DECODER_GRAM_BOUND` and the launcher prints
+`PASS_REGISTERED_RESEARCH_PACKAGE`. Each run authenticates three immutable
+inputs and checks 147 sparse columns, 36 finite rational Gram row bounds,
+complete dimension-uniform geometric tails and nine rejecting or semantic
+controls. It checks changed coefficients, source axes, owner endpoints, signs,
+unproved width, projective phase cancellation and external-scalar placement.
+
+The complete source owner-path family and general Clifford counting theorem
+are credited written premises. Their full scientific suites, independent human
+review, proof-kernel verification, selected cold replay, physical emission,
+precision application and complete paid invoice were not run here. No native
+supplier, new kappa or change to the five selected framework assumptions follows.
+
+## Open next questions
+
+Bind a changed paid decoder or a correlated-history interface to the complete
+caller and its remaining invoice. That change can evade the present observation
+contract. State the new original-input identity first, then check every paid
+call, helper lifetime and precision budget. The present floor must not be
+subtracted as a paid-call count.
+
+## Credit
+
+JoyboyBrian supplies PR #5508, its actual E-code/path binding, signed normalizer
+and support consequence, with the source producer and helper lineage retained
+in the copied proof. Ali Romman (@aliromman) supplies PR #5526 and the earlier
+fresh-stage separation lineage. The retained sources credit their native,
+Clifford, stabilizer and compiler predecessors. The new contribution is this
+source-bound width and error application. Proof and checks by Ali Romman with
+substantial OpenAI Codex assistance; Apache-2.0. No global novelty claim is made.

research/negative-e-decoder-gram-bound/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "fc3d4ba105e1f097d0d7d051a308281e86ab1446",
+      "sha256": "66974efdced4083c00025f97d95d83909a226cdf801edb9301c195c6641f6903"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    }
+  ],
+  "external_provenance": [
+    {
+      "path": "source/decoder-proof.md",
+      "repository": "JoyboyBrian/integer-mult-bounds",
+      "commit": "7e6824d560337e2b2e49da8b98727183491f6ecf",
+      "source_path": "research/odd-e8-negative-exit-algebra/PROOF.md",
+      "git_blob": "bc945b4901cc1c20da7d904ad238224aada71634",
+      "sha256": "6109fd631868ca7cf55c1d9178ff0a2ac030afe4353bc701e715757e4a754b06"
+    },
+    {
+      "path": "source/decoder-result.json",
+      "repository": "JoyboyBrian/integer-mult-bounds",
+      "commit": "7e6824d560337e2b2e49da8b98727183491f6ecf",
+      "source_path": "research/odd-e8-negative-exit-algebra/RESULT.json",
+      "git_blob": "0813bbb062c5ab28247d24fadac94d3cd67db137",
+      "sha256": "df10c3db7f0513e4cd43311bb22c9db15af8422c272339fffa933081c01768c7"
+    },
+    {
+      "path": "source/clifford-proof.md",
+      "repository": "aliromman/integer-mult-bounds",
+      "commit": "32e014bec9206f3987857e6153ec0fd8058f04e3",
+      "source_path": "research/clifford-pointwise-faithfulness/PROOF.md",
+      "git_blob": "465009d9d1b6947e80e6f1e868022f47ddab1f3c",
+      "sha256": "45483e4b5f604a5e70602c5caaa4394d4a0d0d1a6d0b6e8a93b836d50b23f764"
+    }
+  ]
+}

research/negative-e-decoder-gram-bound/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "32890bc5de034ab8fc93ac88ead7279a6a416d04",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Bind the pointwise Clifford Gram theorem to the actual 73-axis negative E postdecoder.",
+  "scope_change_reason": "Take the released PR5508 decoder boundary and the unreserved actual-caller application in PR5526.",
+  "bottleneck_link": "Lower the width of the actual ordinary-decoder obstruction and quantify its worst-case approximation error.",
+  "budget_minutes": 60,
+  "stop_rule": "Check the consumed alphabet, coefficient and sparse native map; do not claim paid-child pricing or correlated-history coverage.",
+  "outcome": "completed",
+  "evidence": "Pinned source proofs, independently checked source coefficient and sparse columns, exact Gram row bounds.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance.",
+  "open_questions": [
+    {
+      "question": "Can a paid decoder or changed correlated-history contract complete the actual caller within its full budget?",
+      "why_it_matters": "Those interfaces can evade the negative-bank-only ordinary span.",
+      "first_step": "State the complete original-input identity and added-paid/lost-target ledger.",
+      "blocked_by": "No physical completion, retained-history ownership or complete precision and cost bill is supplied."
+    }
+  ]
+}

research/negative-e-decoder-gram-bound/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Authenticate source proofs and data; check exact path, sparse native map, rational Gram bounds and controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Authenticate source proofs and data; check exact path, sparse native map, rational Gram bounds and controls."
+    }
+  ]
+}