Prove same-rho Schur closure for the completed parity-triangular caller

research/schur-majorant-closure/CROSS-RESULT.json

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+{
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+  "endpoint_digest": "5b774faf86a7bd5a9070aaa0710f3437f0e7861954780c8984d78aab0023591d",
+  "families": {
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+    "dense_modulus_3": 8,
+    "dense_modulus_4": 4,
+    "dense_mu_3": 16,
+    "dense_mu_4": 8,
+    "toeplitz_4": 4
+  },
+  "input_sha256": {
+    "independent_oracle.py": "716bb5e5c7ffb0d2ef44951a13fec49bb875a138e07503de8b6e652276e1f8ab",
+    "primary_verify.py": "c5c9de309ae227eae0c9cead43b7bb0ec724e5d7b80cc68fddb3c80d1bf1c040"
+  },
+  "matrices": 2441,
+  "oracle_invariants_checked_on_every_partition": true,
+  "principal_E_K_D_partitions": 13501,
+  "scope": "Exact agreement of existing primary Gaussian elimination and frozen independent bordered-minor oracle, including modulus-only oracle fixtures. Reusing their code is integration, not a third independent oracle or a universal proof. No source gate compiler, precision or physical program executed.",
+  "status": "PASS_SCHUR_ALGORITHM_AGREEMENT_INTEGRATION_NO_NEW_KAPPA",
+  "wrong_sign_control": {
+    "inverse_identity_rejected": true,
+    "norm_bound_still_holds": true
+  }
+}

research/schur-majorant-closure/INDEPENDENT-REVIEW.json

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+{
+  "checked_utc": "2026-10-20T15:57:39.533877+00:00",
+  "files": {
+    "PROOF.md": {
+      "bytes": 14113,
+      "sha256": "45c02c2f5c158b58b88114cf39ee43b681f4274b7bee37c27cc58ea1d355a8fc"
+    },
+    "RESULT.json": {
+      "bytes": 2440,
+      "sha256": "a55a2c28e1356835d3d1f26e305cb2408ab97e7e346cffe6a56ae78577298f34"
+    },
+    "canonical_original_check_circuits.py": {
+      "bytes": 10774,
+      "sha256": "06d1861bfe8af543908e26557ffda7e451ff176bceb9bacfaf0f04f234b336d5"
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+      "sha256": "716bb5e5c7ffb0d2ef44951a13fec49bb875a138e07503de8b6e652276e1f8ab"
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+      "sha256": "54bd0aca6e7108e0be6559e3968db1e4c4a38e1a0c18a06901b3a966302176ec"
+    },
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+      "sha256": "abea26e3c86a9970225b070be034c47e7970e10fe1b5686f405e0b301acf3159"
+    },
+    "source/PR4205-triangular.py.txt": {
+      "bytes": 9884,
+      "sha256": "322e5b821e0db6c97d39ffb65e5e108755a91d91255ef47f7a63ad30c3ef8413"
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+      "bytes": 7685,
+      "sha256": "270044fc9fd1203a7eee45ec68a7ebb6b6fab668a109ec956434b8a6cbb483ff"
+    }
+  },
+  "hypotheses": [
+    "finite square complex A=I+H",
+    "nonnegative majorant of modulus or submultiplicative component envelope",
+    "both row and column budgets<=rho<1 for both conclusions",
+    "principal restrictions/elimination; simultaneous coordinate permutations only",
+    "source parity commutation established in truncated lower-Toeplitz algebra",
+    "component local constants use actual executable component admission"
+  ],
+  "no_frontier_change": true,
+  "no_new_kappa": true,
+  "proof_assessment": "Same-rho majorant closure is valid under separate row and column budgets with rho<1, for modulus or component mu. Nesting/principal restriction/simultaneous permutation closure and local parity consequence are valid.",
+  "remaining": [
+    "Source-bound integration and final package/source-history/cold/public/hosted binding are coordinator follow-up",
+    "No revised global prefix/grid/rounding/online-offline width allocator",
+    "No active compensated-generator program audit",
+    "No complete setup/coefficient-delivery/physical caller or multiplication theorem",
+    "No human or formal-kernel acceptance"
+  ],
+  "review_prompted_clarifications": [
+    "Explicit source E-OE^-1(ZO)=E-ZOK equality uses commuting Toeplitz algebra",
+    "Modulus-only versus component-admission boundary control z=(3+3i)/10"
+  ],
+  "reviewer": "separate OpenAI Codex agent; not human or formal-kernel acceptance",
+  "source_head": "203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20",
+  "status": "PASS_TWO_SIDED_SCHUR_CLOSURE_AND_LOCAL_SOURCE_REVIEW",
+  "substantial_ai_assistance": true,
+  "unresolved_findings": [],
+  "verification": {
+    "canonical_original_source_matches_acquired_raw_bytes": true,
+    "compiled_source_or_global_precision_rerun": false,
+    "exact_primary_matrices": 64,
+    "five_raw_acquired_sources_hashes_checked": true,
+    "four_retained_source_exports_checked": true,
+    "nested_quotient_identities": 656,
+    "nonleaf_shears": 33,
+    "oracle_method_review": "code inspection; cofactor determinants distinct from primary Gaussian elimination",
+    "primary_normal_exit": 0,
+    "primary_receipts_byte_identical": true,
+    "primary_receipts_match_final_author_RESULT": true,
+    "primary_result_sha256": "a55a2c28e1356835d3d1f26e305cb2408ab97e7e346cffe6a56ae78577298f34",
+    "primary_strict_optimized_exit": 0,
+    "principal_schur_cases": 1440,
+    "recursive_nodes": 78,
+    "restriction_identities": 1440,
+    "toeplitz_roots": 12
+  }
+}

research/schur-majorant-closure/INDEPENDENT-REVIEW.md

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+# Independent Schur-majorant and local-caller review
+
+Status: PASS for the stated two-sided near-identity envelope theorem and its local parity-caller consequences. Separate OpenAI Codex agent review is not human acceptance or a formal-kernel proof.
+
+I checked the proof for a counterexample or a missing hypothesis before considering finite receipts. With A=I+H, alpha equal to complex modulus or mu(z)=|Re z|+|Im z|, and P=alpha(H) having every row and column sum at most rho<1, the positive comparison matrix is Q=P_RR+P_RE(I-P_EE)^(-1)P_ER. Principal restriction discards only nonnegative budget terms. The eliminated rows imply P_ER*1<=rho*(I-P_EE)*1; the inverse is entrywise nonnegative, so its product with P_ER*1 is at most rho*1. Therefore Q*1<=rho*1-(1-rho)P_RE*1. Transposing gives the distinct column statement. The component scalar envelope is submultiplicative and dominates modulus, so entrywise Neumann majorization applies to it as well. No positivity, symmetry or Toeplitz property of A itself is assumed.
+
+The strict rho<1 hypothesis ensures every principal pivot and resulting complement is invertible. Empty elimination reduces to principal restriction; the empty retained case has no nonempty output norm. Reapplying the same-rho conclusion proves arbitrary finite nesting. The quotient identity retains previously eliminated indices inside the appropriate original principal restriction. Simultaneous coordinate permutations preserve the identity and row/column budgets. Nonprincipal selections and arbitrary nonisometric similarities are correctly excluded.
+
+The Euclidean conclusion uses both usual modulus bounds through ||J||2<=sqrt(||J||1*||J||infinity); one side alone would not suffice. Disjoint coordinate blocks combine by the maximum operator norm. This statement requires actual disjoint supports and does not claim a new compiled scheduling audit. Direct and inverse bounds 1+rho and 1/(1-rho), and the unchanged rho defect, are attained by the written diagonal examples with a nonempty eliminated set.
+
+At the retained source boundary, triangular.py admits the component coefficient defect sum at most1/2. A full lower triangular Toeplitz matrix has that same component row and column defect envelope. In the parity decomposition T=[[E,ZO],[O,E]], the source S=E-ZOK is the true Schur complement because E,O,E^-1,K and the lower shift Z are in the commuting truncated lower-Toeplitz algebra. This reordering would not hold for arbitrary noncommuting blocks. E is a principal restriction and S a principal Schur complement; the bound therefore propagates through every source E/S descendant, including simultaneous wire relabeling. The local consequences M<=3/2, M^-1<=2, O<=1/2, K/ZK<=1 and completed shears/inverses<=2 are valid. They do not establish revised global prefix, grid, coefficient-width, rounding or physical-caller allocations.
+
+A suggested scope control was added before freeze: z=(3+3i)/10 has |z|^2=9/50<1/4 but mu(z)=3/5>1/2. Thus modulus-only admission cannot be silently promoted to the stronger component admission used by the executable source. Both variants of the theorem are valid with their corresponding hypotheses and conclusions. The proof also now spells out the source-specific commuting-block equality explicitly.
+
+The final primary checker was independently replayed in ordinary and strict optimized Python3.12.14. Both exit zero, produce byte-identical receipts, and match the author's final RESULT.json SHA256 a55a2c28e1356835d3d1f26e305cb2408ab97e7e346cffe6a56ae78577298f34. It checks64 exact Gaussian-rational matrices,1440 principal restriction/Schur cases,656 nested quotient identities and1440 restriction identities, plus12 Toeplitz roots/78 recursive nodes/33 nonleaf shear checks. Explicit exceptions retain checks under optimized mode. These finite component-envelope tests corroborate the written proof; they are not its arbitrary-size proof or a compiled-source execution.
+
+The independently frozen oracle was inspected. Its bordered-minor determinant formula is code-disjoint from the primary Gauss-Jordan inverse routine; rational Pythagorean complex entries exercise modulus majorants separately from component majorants. It includes distinct row-only and column-only failures, rho=1 singularity and nonprincipal controls. Its sign-fault witness is correctly tested by an exact block-of-inverse identity: both correction signs can satisfy the same norm bound, so a norm-only test would not detect that fault. The oracle's full replay and source-bound integration are separate coordinator evidence, not a repeated execution claimed by this review.
+
+Source binding inspected here: five acquired source documents/modules match their recorded raw byte lengths, SHA256 and Git blob values; the original check_circuits.py is exactly the canonical capped-phase-fusion/evidence/inplace_algebra_ultra baseline, not a reconstructed script. Four minimal retained exports in SOURCE-PINS match those raw source values. This review does not certify the source's complete rounded circuit or its active compensated-generator extension.
+
+Credit Jose Manuel/jmbarrancoml for PR4205's retained rational parity-triangular execution/precision proof, Vlado Vojdanovski/PR1673 for the parity recursion and descendant argument, Chafik Boukhalfa/PR628 for phases, Cinna Davis for the later caller, and the retained NOTICE lineage. The earlier inverse bound and descendant description remain predecessor work. The new result is the invariant-rho majorant closure and these local consequences. Complete bounded-generator/caller composition, source setup, coefficient delivery, physical tape lowering, global precision/width savings, human/formal acceptance and the multiplication theorem remain open. No new kappa or frontier change follows.

research/schur-majorant-closure/ORACLE-NOTES.txt

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+Independent principal-Schur majorant oracle
+
+Source read: Jose Manuel (jmbarrancoml), completed PR4205 recursive head
+203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20, TRIANGLE-PROOF.md. Its parity
+factorization/recursion and Schur-descendant lineage credit Vlado Vojdanovski
+PR1673; its phase lineage credits Chafik Boukhalfa PR628, and later caller
+context credits Cinna Davis. No new primary implementation was read/imported
+before this oracle and receipt were frozen. Substantial OpenAI Codex assistance;
+Apache-2.0 repository contribution.
+
+New candidate statement corroborated here: for A=I+H, a nonnegative entry
+majorant P with every row and column sum <=rho<1 retains that SAME bound
+after arbitrary principal restriction and principal Schur elimination.
+An entry majorant may dominate either complex modulus or
+mu(z)=abs(Re z)+abs(Im z), with the corresponding output interpretation.
+This sharpens the predecessor's loose rho+rho^2/(1-rho) bound, which is 1
+at rho=1/2. It does not change a precision allocation automatically.
+
+Written argument, separate from the finite computation: put B=retained,
+E=eliminated after first dropping other indices. The entry majorant is
+Q=P_BB+P_BE(I-P_EE)^(-1)P_EB. The inverse is nonnegative by the convergent
+Neumann series. Since P_EB*1 <= rho*1-P_EE*1,
+(I-P_EE)^(-1)P_EB*1 <= 1-(1-rho)(I-P_EE)^(-1)*1 <= rho*1.
+Consequently Q*1 <= P_BB*1+rho*P_BE*1 <= rho*1. Transpose the argument for
+columns. Each side separately needs its corresponding original budget;
+neither side alone implies the other. Entrywise majorization of the true
+Schur correction follows from submultiplicativity and the inverse's
+Neumann series. Gaussian component norm mu and complex modulus both have
+the needed triangle/product inequalities. Principal restrictions reduce
+nonnegative row/column budgets. Iterated restrictions/eliminations retain
+the original eliminated set in the principal-Schur description.
+
+Independent method: each complex Schur entry is the ratio of its bordered
+minor determinant to the eliminated principal determinant. There is no
+Gaussian-elimination routine or primary-source import. Fractions and pairs
+of Fractions are exact; no floating comparisons are used. Majorant Q is
+computed independently as I minus the corresponding bordered-minor Schur
+complement of I-P. All 2401 two-by-two matrices over seven small Gaussian
+rationals are checked, plus 36 bounded dense matrices of dimension3/4 and
+four lower triangular Toeplitz dimension4 matrices. Every retained,
+eliminated and dropped partition is included. Selected dense cases explore
+all one-index elimination/drop histories until one index remains.
+
+Controls: a row-only rho=3/4 three-by-three input gives a Schur correction
+with maximum column sum9/8; transpose gives the opposite one-sided failure.
+rho=1 admits a singular eliminated block. A nonprincipal submatrix of I2
+can be [0], violating near-identity conclusions. A wrong Schur correction
+sign fails the exact block-of-inverse identity although both signs satisfy
+the norm bound; the finite checker does not pretend that norm testing
+alone detects this fault. Diagonal examples with nonempty elimination
+attain rho exactly, proving no smaller uniform scalar bound is possible.
+
+Finite evidence is not a formal-kernel proof of the arbitrary-size lemma.
+The four Toeplitz examples are generic exact matrices, not a fresh replay
+of the predecessor's triangle_plan, phase compiler or rounded precision
+argument. Full bounded-generator/caller composition, tape delivery, setup,
+online/offline coefficient widths and multiplication/outer theorems remain
+outside this oracle. No new kappa or physical speedup is claimed.
+
+Replay: python3 -I -B oracle.py --output ORACLE.json
+Strict replay: python3 -I -O -W error -B oracle.py --output ORACLE-O.json
+API for crosscheck: matrix_cases(), G, mu(), schur(), principal(), determinant().

research/schur-majorant-closure/ORACLE-PROVENANCE.json

@@ -0,0 +1,20 @@
+{
+  "commands": [
+    "python3 -I -B oracle.py --output ORACLE.json",
+    "python3 -I -O -W error -B oracle.py --output ORACLE-O.json"
+  ],
+  "files": {
+    "NOTES.txt": "38f87bf4f79ffb15c8afacaa8bff32e0dc58c0f367355847372132d64a677db2",
+    "ORACLE-O.json": "785e4c8f93a4fb38353e28def436d6ec4639fb293a393caf2759baba180b8f93",
+    "ORACLE.json": "785e4c8f93a4fb38353e28def436d6ec4639fb293a393caf2759baba180b8f93",
+    "oracle.py": "716bb5e5c7ffb0d2ef44951a13fec49bb875a138e07503de8b6e652276e1f8ab"
+  },
+  "frozen_at": "2026-10-20T15:51:57.203798+00:00",
+  "normal_optimized_results_identical": true,
+  "python": "3.12.14",
+  "source_head": "203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20",
+  "source_main": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "source_pr": 4205,
+  "source_proof_sha256": "abea26e3c86a9970225b070be034c47e7970e10fe1b5686f405e0b301acf3159",
+  "status": "FROZEN_BEFORE_PRIMARY_IMPLEMENTATION_VIEW"
+}

research/schur-majorant-closure/ORACLE-RESULT.json

@@ -0,0 +1,47 @@
+{
+  "controls": {
+    "column_only_output_max_row": "9/8",
+    "nonprincipal_restriction_I2_rows0_cols1_is_zero": true,
+    "rho_1_singular_eliminated_block": true,
+    "row_only_input_rho": "3/4",
+    "row_only_output_max_column": "9/8",
+    "sharpness_diagonal_nonempty_elimination_rhos": [
+      "0",
+      "1/4",
+      "1/2",
+      "3/4",
+      "7/8"
+    ],
+    "wrong_schur_sign_also_satisfies_norm_bound": true,
+    "wrong_schur_sign_rejected_by_inverse_identity": true
+  },
+  "endpoint_digest": "5b774faf86a7bd5a9070aaa0710f3437f0e7861954780c8984d78aab0023591d",
+  "families": {
+    "all_2x2": 2401,
+    "dense_modulus_3": 8,
+    "dense_modulus_4": 4,
+    "dense_mu_3": 16,
+    "dense_mu_4": 8,
+    "toeplitz_4": 4
+  },
+  "finite_cap": {
+    "all_2x2_alphabet_size": 7,
+    "all_2x2_rho": "1/2",
+    "max_matrix_dimension": 4,
+    "other_rhos": [
+      "1/4",
+      "1/2",
+      "3/4",
+      "7/8"
+    ]
+  },
+  "limitations": [
+    "Finite corroboration; arbitrary-size lemma needs the separate written proof.",
+    "No physical tape, full caller/generator precision, multiplication theorem, or new kappa."
+  ],
+  "matrices": 2441,
+  "method": "exact Gaussian-rational bordered-minor ratios; no Gaussian-elimination routine",
+  "nested_history_states": 1244,
+  "principal_E_K_D_partitions": 13501,
+  "status": "PASS"
+}