Research: sharp nine-coordinate threshold for five-subset shared nodes (seed 48)

research/five-subset-shared-node-seed48/CERTIFICATE.json

@@ -0,0 +1,123 @@
+{
+  "format": "five-subset-shared-node-seed48-v1",
+  "base_commit": "ce7ad1b6ab09452c3478723a2e96ed5ff212dc08",
+  "seed": 48,
+  "new_kappa": false,
+  "sharp_ambient_threshold": 9,
+  "canonical_census": {
+    "8_source_intersection_3": {
+      "target_labels": 11,
+      "target_pairs_tested": 55,
+      "rectangles": 0,
+      "labeled_source_pairs": 840,
+      "target_intersection_histogram": {}
+    },
+    "9_source_intersection_1": {
+      "target_labels": 32,
+      "target_pairs_tested": 496,
+      "rectangles": 288,
+      "labeled_source_pairs": 315,
+      "target_intersection_histogram": {
+        "3": 288
+      },
+      "dimension_four_frames_tested": 44640,
+      "legal_frame_count_histogram": {
+        "48": 288
+      }
+    },
+    "9_source_intersection_3": {
+      "target_labels": 30,
+      "target_pairs_tested": 435,
+      "rectangles": 96,
+      "labeled_source_pairs": 3780,
+      "target_intersection_histogram": {
+        "1": 24,
+        "3": 72
+      },
+      "dimension_four_frames_tested": 14880,
+      "legal_frame_count_histogram": {
+        "48": 96
+      }
+    }
+  },
+  "labeled_rectangles": {
+    "8": 0,
+    "9": 453600
+  },
+  "labeled_by_source_intersection": {
+    "8": {
+      "3": 0
+    },
+    "9": {
+      "1": 90720,
+      "3": 362880
+    }
+  },
+  "canonical_frame_records_sha256": "402c740ffe67383768de7439144e917fa4bf8d4ac6e7643be3ed415b13d84417",
+  "fixture": {
+    "h": 9,
+    "sources": [
+      31,
+      481
+    ],
+    "targets": [
+      47,
+      87
+    ],
+    "kernel": [
+      [
+        1,
+        1
+      ],
+      [
+        -1,
+        -1
+      ]
+    ],
+    "frame_basis": [
+      31,
+      481,
+      3,
+      40
+    ],
+    "frame_characteristic": 40,
+    "complement_characteristic": 471,
+    "incoming_residual_bases": [
+      [
+        52,
+        457,
+        458
+      ],
+      [
+        28,
+        55,
+        482
+      ]
+    ],
+    "outgoing_residual_bases": [
+      [
+        59,
+        67,
+        131,
+        259
+      ],
+      [
+        131,
+        279,
+        367,
+        379
+      ]
+    ],
+    "incident_residual_dimensions": [
+      3,
+      3,
+      4,
+      4
+    ]
+  },
+  "controls": {
+    "semantic_controls": 6,
+    "isometry_vector_pairs": 65536,
+    "five_subset_kernel_pairs": 3136
+  }
+}

research/five-subset-shared-node-seed48/CLAIM.json

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+{
+  "track": "structural decoder research",
+  "seed": 48,
+  "randomness": "One fresh Python secrets.randbelow(100) draw, operating-system randomness",
+  "base_commit": "ce7ad1b6ab09452c3478723a2e96ed5ff212dc08",
+  "started_utc": "2026-10-13 19:15:16 UTC",
+  "claimed_utc": "2026-10-13 19:20:01 UTC",
+  "expected_finish_utc": "2026-10-13 19:45:16 UTC",
+  "verification_reserve_utc": "2026-10-13 19:37:46 UTC",
+  "coordination": "No pinned sprint coordination issue found: open non-PR issues 650 and 913. Claim recorded locally.",
+  "candidates": [
+    "Five-subset norm-one labels: change the local shared-node geometry and test a nine-coordinate threshold",
+    "Bit E8 decoder beyond 25 punctures: deferred due adjacent PR1084/1020 and potentially larger classification work",
+    "P104 delivery-stage mixing: deferred due recent PR1125's comprehensive fixed-term carrier bound and expensive full invoice"
+  ],
+  "object": "One two-source/two-target shared node, five-subset labels, both endpoint pairs degenerate, signed rank-one kernel, disjoint binary spans and historical orthonormal-residual frame interface",
+  "method": "Prove impossibility below nine via the general frame floor and the eight-coordinate complement isometry to triples; exhaust canonical source intersections 1 and 3 at h=9, test every legal dimension-four completion and give explicit residual bases",
+  "domain": "Canonical h=8/9 five-subset rectangles; all quotient two-subspaces for h=9 dimension-four frames, plus an independent labeled rectangle census",
+  "stopping_rule": "Complete this fixed local class and certificate; do not construct or price a global decoder",
+  "optimistic_budget": "The mechanism changes only local geometry. No paid saving is claimed; even zero-cost local sharing leaves global label coverage, source access, helper lifecycle, copies, restoration, transfers, recursive children, hosting and analytic obligations unresolved.",
+  "overlap": "PR920 supplies the general norm-one floor and old fixtures. PR1139 proves the triple-only ten-coordinate theorem; this tests a different odd-weight label alphabet. PR968/1077/1084/1125 concern punctured E8 pairing or delivery classes, not this local five-subset frame geometry. Refreshed open PRs through1154; targeted quintuple/five-subset/shared searches found no matching scope.",
+  "expected_deliverable": "One structural research note with a sharp scoped threshold, exact finite census, witness and semantic controls; no new kappa",
+  "attribution": "Prepared for ToikY with substantial OpenAI Codex assistance. Credit ArnavSwiggy PR920, woklerx PR1139, Douglas Colkitt historical parity decoder and prior frame-completion work. Apache-2.0.",
+  "result": {
+    "status": "completed structural research note",
+    "threshold": 9,
+    "labeled_rectangles_h9": 453600,
+    "canonical_frames_tested": 59520,
+    "residual_dimensions": [
+      3,
+      3,
+      4,
+      4
+    ],
+    "new_kappa": false,
+    "completed_utc": "2026-10-13 19:35:00 UTC",
+    "link": "https://github.com/ToikY/integer-mult-bounds/tree/codex/decoder-seed48/research/five-subset-shared-node-seed48",
+    "overlap_refresh": "Open PRs refreshed through1171; PR1139 unchanged at82da1e827c51d94691134c6aec346357d89622bc; distinct constant-weight-five class."
+  }
+}

research/five-subset-shared-node-seed48/PROOF.md

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+# Sharp threshold in the weight-five class
+
+Use the class defined in README.md, over the standard binary dot product.
+The proof combines PR920's residual dimension floor with PR1139's triple
+support classification. The eight-coordinate complement step and the
+nine-coordinate weight-five fixture are this package's additions.
+
+## The four-plus-four floor
+
+For any nondegenerate binary space W, its characteristic w_W uniquely satisfies
+w_W dot x = x dot x for all x in W. A norm-one vector s splits W as its line
+orthogonally summed with the nondegenerate residual W intersect s-perp.
+That residual is alternating exactly when w_W=s: its characteristic vanishes
+exactly when the characteristic of W lies in the line spanned by s.
+This is the historical criterion used in PR920.
+
+Distinct norm-one sources with s1 dot s2=1 span a degenerate plane E with
+radical r=s1+s2. A nondegenerate U containing E cannot have dimension two.
+If dim U=3, then E=r-perp inside U. Since r has norm zero, w_U lies in E.
+Pairing w_U with both sources must give one; the only two possibilities
+inside E are s1 and s2. At least one incoming residual is therefore an
+alternating plane and has no orthonormal basis. Hence dim U>=4. Apply the
+same argument to the targets inside U-perp to obtain dim U-perp>=4.
+Thus h>=8 for every legal doubly degenerate norm-one rectangle.
+
+## Eight-coordinate complements preserve the interface
+
+In F2^8 let w be the all-ones vector and define J(x)=x+(x dot w)w.
+The map is linear, w dot w=0, and J is its own inverse. Expanding the two
+cross terms shows J(x) dot J(y)=x dot y. It is therefore an isometry and
+preserves every containment, orthogonal complement, nondegeneracy and
+orthonormal basis condition in the local interface. It also preserves
+disjointness of E and T, since it is an invertible linear map.
+
+Every weight-five label has odd parity, so J sends it to its three-element
+complement. For two weight-five supports with intersection size k,
+their complements intersect in k-2. Consequently
+K(J(s),J(t))=-K(s,t); the entire 2x2 matrix changes by one common sign and
+rank one is preserved. Degenerate pairs remain degenerate. A legal weight-five
+rectangle at h=8 would therefore give a legal triple rectangle there.
+
+## Why a doubly degenerate triple rectangle needs ten coordinates
+
+The following is PR1139's small support classification, included to make the
+dependency inspectable without importing pending code. Write source triples
+as {c,a,b} and {c,d,e}, and let O be the other coordinates. Every triple
+even-orthogonal to both sources has one of four forms:
+
+| Type | Support | Signed kernel column |
+| --- | --- | --- |
+| A | three elements of O | (+1,+1) |
+| B | {a,b,o}, o in O | (-1,+1) |
+| C | {d,e,o}, o in O | (+1,-1) |
+| D | {c,x,y}, x in {a,b}, y in {d,e} | (-1,-1) |
+
+If c is present, parity forces one element from each displayed pair. If c
+is absent, each pair contributes zero or two elements. This proves the list
+complete. Rank one groups the two targets into A/D or B/C. A/D targets are
+disjoint, and two distinct B or two distinct C targets intersect twice,
+so those pairs are not degenerate. Two D targets intersect once only when
+their x and y both differ; their sum is the nonzero source radical
+{a,b,d,e}. Likewise a B/C pair intersects once only with a common o, and
+then has the same sum. These possibilities violate E intersect T={0}.
+Only A/A remains, requiring five coordinates outside the five source
+coordinates. Thus h>=10. In particular no such triple rectangle exists
+at h=8, even before checking incident residuals.
+
+The isometry excludes weight-five rectangles at h=8. The general floor
+excludes h<8. Together these prove the lower bound h>=9 in our local class.
+
+## Nine coordinates attain the bound
+
+Take sources (31,481), targets (47,87), and U=span(31,481,3,40) in F2^9,
+with the masks defined in README.md. The endpoint intersections are one and
+three, every cross intersection is even, and the kernel is
+[[1,1],[-1,-1]]. Direct span inspection gives E intersect T={0}.
+U is nondegenerate, contains E, and is orthogonal to T. The four explicit
+residual bases in README.md each span their full residual and have identity
+Gram matrix. These exact finite vector identities certify this witness;
+both implementations check them. Therefore the minimum ambient dimension
+for this weight-five class is exactly nine.
+
+For h>9 embed these supports and U by zero extension. The complement becomes
+the old U-perp orthogonally summed with standard coordinate lines; each
+outgoing residual gains those same orthonormal lines. Thus existence holds
+for every h>=9. This proves a threshold, not a classification or count of
+all weight-five rectangles in larger dimensions. The h=9 exhaustive census
+is separate finite evidence. No complete supplier, global simultaneous
+frame assignment, cost recurrence or new multiplication exponent follows.

research/five-subset-shared-node-seed48/README.md

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+# Five-subset shared nodes: sharp nine-coordinate threshold
+
+**Structural geometry research note; no new kappa or paid supplier.** In the
+historical orthonormal-residual interface, a doubly degenerate two-source,
+two-target shared node with constant-weight-five labels first exists at
+ambient dimension **nine**. This changes the label alphabet in the triple-only
+ten-coordinate classification of [woklerx's PR1139](https://github.com/CrocSwap/integer-mult-bounds/pull/1139).
+It does not lower the eight-coordinate floor for unrestricted norm-one labels
+already attained in [ArnavSwiggy's PR920](https://github.com/CrocSwap/integer-mult-bounds/pull/920).
+
+From the repository root, Python 3.9 or later, standard library only:
+
+```sh
+python3 -I -B research/five-subset-shared-node-seed48/verify.py
+python3 -I -B research/five-subset-shared-node-seed48/independent.py
+```
+
+The first command prints
+`PASS_FIVE_SUBSET_SHARED_NODE threshold=9 rectangles_h9=453600 canonical_frames=59520 residual_bases=4 no_new_kappa`.
+The second independently exhausts all labeled rectangles at h=8,9 and checks
+the four residual bases using coordinate sets. Its final lines are
+`PASS_INDEPENDENT_FIVE_SUBSET_RECTANGLES h8=0 h9=453600` and
+`PASS_INDEPENDENT_RESIDUAL_BASES dimensions=3,3,4,4`.
+Both commands are offline, read-only and retain checks under optimized Python.
+The optional `verify.py --write-certificate` regenerates only this package's
+certificate. See [VALIDATION.md](VALIDATION.md) for completed checks.
+
+## Precise local class
+
+Labels are incidence vectors of five-subsets of a labeled h-coordinate ground
+set, in the standard dot-product space F2^h. Each side is an unordered pair
+of distinct labels; source and target sides are distinguished. Require:
+
+1. Both endpoint spans E=span(s1,s2) and T=span(t1,t2) are degenerate, so
+   both pair intersections have odd size (one or three).
+2. Every cross dot product is zero, and the 2x2 signed scalar matrix
+   K(s,t)=Re(i^|s intersection t|) has rank one over Q.
+3. E intersects T only in zero.
+4. A nondegenerate U satisfies E subset U subset T-perp, and all four
+   incident residuals U intersect si-perp and U-perp intersect tj-perp
+   have orthonormal bases.
+
+The historical side kernel has a common factor 1/2, irrelevant to its rank.
+The label lines and target hyperplanes are fixed; no copied carriers, phases,
+multiple shared nodes or relaxed residual contract enter this class.
+The lower bound is proved in [PROOF.md](PROOF.md), independently of enumeration.
+
+## Exact nine-coordinate witness
+
+Masks use zero-based coordinates, with bit i representing coordinate i:
+
+```text
+sources = (31,481) = ({0,1,2,3,4}, {0,5,6,7,8})
+targets = (47,87)  = ({0,1,2,3,5}, {0,1,2,4,6})
+K = [[1,1],[-1,-1]]
+U = span(31,481,3,40), dimension 4
+w_U = 40, w_(U-perp) = 471
+```
+
+U and U-perp have orthonormal bases `(28,55,481,482)` and
+`(47,59,67,131,259)`, respectively. The exact incident residual bases are:
+
+| Endpoint | Orthonormal basis | Dimension |
+| --- | --- | ---: |
+| Source 31 | 52,457,458 | 3 |
+| Source 481 | 28,55,482 | 3 |
+| Target 47 | 59,67,131,259 | 4 |
+| Target 87 | 131,279,367,379 | 4 |
+
+Both implementations compare each basis's complete generated vector set with
+the actual residual and check every Gram-matrix entry. These dimensions are
+local geometry, not a paid recursive profile.
+
+## Completed finite coverage
+
+Coordinate permutations reduce source pairs to intersection sizes one and
+three. At h=8 only intersection three is possible. At h=9, every valid
+rectangle has a dimension-four frame, and each has exactly **48** such legal
+frames. The complete canonical frame census tests all 155 two-dimensional
+subspaces of T-perp/E for every rectangle: **59,520** candidates total.
+
+| h | Source intersection | Compatible target labels | Canonical valid rectangles | Labeled source pairs | Labeled rectangles |
+| ---: | ---: | ---: | ---: | ---: | ---: |
+| 8 | 3 | 11 | 0 | 840 | 0 |
+| 9 | 1 | 32 | 288 | 315 | 90,720 |
+| 9 | 3 | 30 | 96 | 3,780 | 362,880 |
+
+The independent implementation visits every labeled source pair and every
+cross-compatible target pair, using set intersections, literal integer kernel
+values, determinants and explicit nonzero spans. It imports no primary code
+and uses no symmetry multiplier. It also checks source/target swap symmetry.
+At h=8 all 4,200 degenerate signed-rank-one candidates fail common-span
+disjointness. At h=9 there are 511,560 such candidates, of which 57,960 fail
+that condition. The survivors total **453,600**.
+
+The seed is **48**, one fresh OS-random draw. It selects structural decoder
+research and the odd-weight-five extension, rather than another triple or
+current-frontier search. The fixed domain, three candidate tasks, scope,
+stopping rule and local claim are in [CLAIM.json](CLAIM.json). This is complete
+finite coverage at h=8,9, not an all-dimension census or a formal proof of the
+full multiplication theorem.
+
+## Cost decision, scope and attribution
+
+This mechanism changes copied-information geometry by changing the label
+alphabet. No complete cost improvement follows. Even setting local operations
+optimistically to zero leaves global label coverage, source access, scalar
+gates, helper stock, frame changes, copies, compensation, restoration, recursive
+children, whole-call hosting, precision/tape and analytic transfer obligations
+unresolved. No global decoder is implemented or fed to assembly. The experiment
+stops after the sharp local threshold, witness and exact census; it does not
+reopen the rejected fixed h=26 parity-DAG repair with more computation.
+
+Exact base: **ce7ad1b6ab09452c3478723a2e96ed5ff212dc08**. The reviewed selected
+result at this base is PR1094 with conditional
+kappa=836330098538459/500000000000000000. Its manifest, actual rare-pattern
+construction, assembly certificate and maintainer review were followed.
+Neither the selected frontier nor upstream/ is changed. [SOURCES.json](SOURCES.json)
+records exact source commits and hashes, including pending PR1139's proof.
+Our lower-bound proof restates the small triple classification with attribution,
+so reproducing the new certificate requires no pending code or network access.
+
+Credit Douglas Colkitt's historical parity-side kernel and frame obstruction;
+ArnavSwiggy's PR920 for the residual criterion, four-plus-four floor and general
+norm-one fixtures; woklerx's PR1139 for the triple support classification; and
+the seed47 frame-completion note for the distinction between separating frames
+and legal incident transfers. No predecessor implementation is imported.
+
+Prepared for GitHub contributor **ToikY**, with substantial **OpenAI Codex**
+assistance in mathematics, implementation, algorithmically independent checks,
+review and writing. This is not independent human peer review. New files are
+under Apache-2.0, matching the repository.