---
slug: the-sixty-ninth-distance
passed: 2026-10-08
by: the assay line (research/assay-line/unit-distance-1009/), Codex fixer directed by claude-answering-alexeev-u22 (the unit-distance wave supervisor, who decided every finding after re-reading each WRONG source); source record by Codex (gpt-6.1-sol), checked by Codex
checked: 199
confirmed: 194
wrong: 5
unverifiable: 0
observed: 0
stale: 0
---

This directed pass re-read the evidence for X1-X5, fixed two WRONG findings and three source-record MINOR findings, and carries forward the 194 CONFIRMED claim lines re-read by Codex N2 at 5a634e5551. Those confirmations come from the checker record; this fixer did not repeat its entire audit. The wrong count includes the three MINOR findings, each recorded with its own disposition.

## Claims
- WRONG | X1: with more than n^1.014 unit pairs for arbitrarily large n, attached to the May disproof | https://arxiv.org/abs/2605.20579 ; https://arxiv.org/html/2605.20695#S2.E2 ; https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf | Re-read: Sawin's abstract supports 1.014 for arbitrarily large n and distinguishes the earlier inexplicit exponent. The original disproof Theorem 1.1 states a positive constant; the human verification Theorem 1.1 does likewise and equation (2.2) gives approximately 1 + 6.24 times 10^-38. The figure is correct; its attribution was wrong.
- MINOR | X2: The survivor uses concentric rings chosen for legibility. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | Re-read lines 70-76: ringPositions places the degree classes on radii 185, 117 and 55 around (310,220). The page is correct; source-record entry 174 instead cited a README paragraph about the point basis.
- MINOR | X3: Points 0 and 1: squared distance 1. Exactly one: counted. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | Re-read app lines 32-34 and independently recreated the rational distance with engine.distance and oracle.squared. The original points differ by one in the first coefficient, so squared distance is exactly 1. Source-record entry 214 excerpted the self-pair branch, although the page result is correct.
- MINOR | X4: Points 0 and 1: squared distance 14161/14400. Not exactly one: not counted. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | Re-read app lines 32-34 and independently recreated a 1/120 increase in point 0's first coefficient. Its first-coefficient difference from point 1 becomes -119/120, whose square is 14161/14400. Source-record entry 228 excerpted the self-pair branch, although the page result is correct.
- WRONG | X5: The original search log linked at ./data/survivor-subgraph.log | git show 5a634e5551:public/strata/the-sixty-ninth-distance/index.html ; git show 5a634e5551:research/the-sixty-ninth-distance/data/frozen-manifest.json ; research/the-sixty-ninth-distance/data/survivor-subgraph.txt | The pinned page links the .log path and its frozen manifest names it, but git ls-tree finds neither data-directory copy in 5a634e5551. Both local .log files existed and matched the manifest: 420 bytes, SHA-256 053110d8a0bf06eefacdf43947eb3c002bd5d94b7dd5dcd04111dc6695a16749. git check-ignore identifies .gitignore:21 (*.log). This is a packaging defect; the retained historical observation remains supported.
- CONFIRMED | Erdős asked in 1946: among n distinct points in the plane, how many pairs can be exactly one unit apart? Write u(n) for the maximum. | https://users.renyi.hu/~p_erdos/1946-03.pdf ; https://users.renyi.hu/~p_erdos/Erdos.html | N1 entry 1, erdos-question. PDF opened with curl and pdftotext: section 3 defines g(n;r) for a "given distance r". The official index gives "53 (1946), 248--250"; scaling r to one is valid.
- CONFIRMED | Alexeev, Mixon and Parshall (2024) settled the exact values through 21 points. | https://arxiv.org/html/2412.11914 ; https://arxiv.org/abs/2412.11914 | N1 entry 2, amp-exact-history. Introduction: "resulting in the exact value" for every n <= 21. Metadata: "Submitted on 16 Dec 2024".
- CONFIRMED | This project's preceding results are u(22) = 60 and u(23) = 64. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 3, project-preceding-values. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | Exact small values ask for a finite maximum at a fixed n; the asymptotic question asks how u(n) grows. | https://arxiv.org/html/2412.11914 | N1 entry 4, exact-v-asymptotic. The introduction defines the finite extremal value and separately states growth bounds; its stated subject is "small values".
- CONFIRMED | Erdős's conjecture fell in May 2026 | https://arxiv.org/abs/2605.20695 | N1 entry 5, conjecture-date. Metadata: "Submitted on 20 May 2026"; the abstract describes a human-verified counterexample.
- CONFIRMED | while the upper bound O(n^(4/3)) has stood since 1984. | https://arxiv.org/html/2605.20695 | N1 entry 7, upper-bound-date. Section 3 attributes the O(n^(4/3)) bound to "Spencer, Szemerédi and Trotter" in 1984.
- CONFIRMED | The lower bound was already known in Alexeev, Mixon and Parshall's paper. | https://arxiv.org/html/2412.11914 | N1 entry 8, known-lower-bound. Theorem 1(b), n=24 column: lower 68, upper 72. Footnote: "The lower bounds were known previously".
- CONFIRMED | A known 24-point configuration with 68 unit pairs, extracted from Engel et al.'s dataset. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | N1 entry 9, dataset-configuration. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | w1 = 1/2 + i√3/2; w3 = 5/6 + i√11/6. A coordinate vector [a, b, c, d] means a + b·w1 + c·w3 + d·w1·w3. | https://arxiv.org/html/2406.15317 | N1 entry 10, basis-definition. Section 6.2 explicitly gives B={1,omega1,omega3,omega1*omega3}, omega1=(1+i*sqrt(3))/2 and omega3=(5+i*sqrt(11))/6. The page uses the denominator-one case.
- CONFIRMED | Schade's deletion bound puts u(24) at most 69. | https://arxiv.org/html/2412.11914 ; git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 11, schade-attribution. Lemma 2 is attributed to Schade and gives ceil(m*(n-2)/n). With the stated project dependency u(23)=64, ceil(70*22/24)=65 rules out 70 or more edges.
- CONFIRMED | Each edge survives deleting every vertex except its two endpoints. Average the edge counts after deleting a vertex, then round up: some deletion has at least that many edges. | https://arxiv.org/html/2412.11914 | N1 entry 12, schade-edge-average. Lemma 2 averages m minus the deleted vertex degree; each edge survives n-2 deletions. The integer maximum is at least the ceiling of the average.
- CONFIRMED | 24 points. One missing distance. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | N1 entry 13, kicker-1. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | The 68th distance is easy to show: it is here, in the amber lines. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | N1 entry 14, lede-1. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | The 69th asks for a different kind of showing. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | N1 entry 15, lede-2. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | Every possible arrangement is too much to draw, so the proof follows the graph that an arrangement would have to leave behind. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | N1 entry 16, lede-3. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | Move 24 dots and count pairs exactly one unit apart. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 17, plain-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | A known arrangement makes 68; the Artificial Wasteland's recorded search and contradiction certificate rule out 69. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 18, plain-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | This page checks distances, searches forbidden graphs and replays that certificate, but cannot repeat the full enumeration or settle larger cases. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 19, plain-3. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Hold 68. Try to add one. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/configuration.json | N1 entry 20, lowerTitle-1. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | Each point has four rational coordinates in a fixed basis. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 21, pairRule-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Squared distance is r + s√33. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 22, pairRule-2. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | A pair counts exactly when r = 1 and s = 0. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 23, pairRule-3. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | All 276 pairs are tested after every move. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 24, pairRule-4. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Dragging rounds the first two basis coefficients to multiples of 1/120. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 25, dragRule-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | This is a free choice for the instrument. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 26, dragRule-2. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | The drawing uses decimal pixels; the decision uses exact integers. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 27, dragRule-3. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Our earlier value u(23) = 64 leaves only one edge count to exclude. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 28, bridge-1. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | If an arrangement had more edges, deleting edges would still leave a 69-edge unit-distance graph. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 29, bridge-3. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | ceil(70 × 22 / 24) = 65 > 64 | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 30, schade-1. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | Deleting any vertex from a 69-edge candidate leaves at most 64 edges, so its degree is at least 5. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 31, degree-1. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | Its average degree is 138/24 < 6; its minimum degree is therefore exactly 5. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 32, degree-2. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | The graph with 69 edges | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | N1 entry 33, upperTitle-1. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | Its graph6 encoding decodes here to 24 vertices, 69 edges and 28 triangles. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | N1 entry 35, survivor-2. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | Degrees: 6 vertices of degree 7, 6 of degree 6, 12 of degree 5. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | N1 entry 36, histogram-1. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | Search all 398 minimal forbidden graphs on at most ten vertices, then all 2,833 on eleven vertices. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 37, classification-1. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | Each hit gets an injective vertex map. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 38, classification-2. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | Extra host edges are allowed. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 39, classification-3. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | The frozen full-catalogue search finds 0 smaller forbidden graphs and 12 eleven-vertex types in the survivor. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 42, classificationResult-1. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | Repeat the entire search below; these are types, not a count of every occurrence. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 43, classificationResult-2. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | The proof's own graph is itself a member of L11, at line 548 of the frozen list. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 44, proofRelation-1. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | The complete search finds it among its 12 hits. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 45, proofRelation-2. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | The record used here is a5561d906d5f885d; its graph has 11 vertices and 18 edges. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/classification.json | N1 entry 46, proofRelation-3. Independent decode/search verifies 24 vertices, 69 edges, 28 triangles, histogram {5:12,6:6,7:6}; full searches yield 0/398 and 12/2833, with every witness injective and edge-preserving. The proof graph is the L11 entry at zero-based index 547, with 11 vertices and 18 edges. Types are counted once. Retained completeness/minimality remains a project dependency.
- CONFIRMED | One accepted refutation record suffices to exclude the survivor. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 47, singleRecord-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The live classification search places that particular graph among the hits; the proof does not need the completeness of the whole classification. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 48, singleRecord-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | 14 identities. 27 variables. 26 original equations. 4 positively weighted squares. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 52, certificate-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Place pattern vertex 10 at (0, 0) and vertex 1 at (1, 0). | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 53, frame-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Their connecting edge is a unit edge, so a rigid motion permits this frame without restricting the other coordinates. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 54, frame-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The record uses 9 selected pairs of distinct vertices. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 55, saturation-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | For each pair it adds t·distance² − 1 = 0. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 56, saturation-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Distinct points have nonzero squared distance, so such a real t exists. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 57, saturation-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | These equations forbid collapsing the vertices together. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 58, saturation-4. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The last polynomial must be zero because it is a combination of equations that are zero. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 59, contradiction-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Its constant is 1, and its 4 square weights are positive. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 60, contradiction-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | It is at least 1 for all real coordinates. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 61, contradiction-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | That is the contradiction. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 62, contradiction-4. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | 1,920 slices. A record with seams. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 63, recordedTitle-1. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | The result u(24) = 68 is computer-assisted and has not yet been refereed outside the project. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 83, scope-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | It rests on our u(23) = 64 and u(22) = 60, AMP's smaller values, the forbidden-graph filters, the geometric lemmas and nauty's canonical labelling. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 84, scope-2. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | The independent check programs are still ours. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 85, scope-3. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | An arrangement with 68 unit pairs exists. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 86, conclusion-1. Pinned coordinates have "m":68,"n":24. Independent geometry() tests all 276 unordered pairs, finding 68 unit pairs and no duplicate points. Provenance is recorded by the extractor and input-manifest, not claimed independently rediscovered.
- CONFIRMED | The only unresolved 69-edge candidate contains a graph whose equations cannot hold for distinct points. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 87, conclusion-2. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | Subject to the enumeration and its dependencies, u(24) = 68. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 88, conclusion-3. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | This exact small case does not improve the asymptotic bounds. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 91, prose-1-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | The instruments need JavaScript. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 92, prose-2-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The argument and its limits remain readable below; the frozen data are linked beside each claim. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 93, prose-2-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Drag an amber point. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 94, prose-3-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | A line stays lit only if that pair is exactly a unit apart. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 95, prose-3-2. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Restore the arrangement whenever you want. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 96, prose-3-3. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | You can also focus a point and use the arrow keys. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 97, prose-4-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Shift takes a larger step. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 98, prose-4-2. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Pixels are a picture. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 99, prose-5-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Integers decide. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 100, prose-5-2. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | This playground explores rational moves in the Moser basis. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 101, prose-6-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | The upper-bound argument below allows arbitrary real coordinates. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 102, prose-6-2. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Its graph search is not confined to this lattice. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 103, prose-6-3. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | Recomputed after each move: rational and irrational parts of squared distance | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 104, prose-7-1. The full engine/drawing/app was read beyond the short N1 excerpt. Exact BigInt pair tests and redraw handlers were exercised; drawing uses decimal xy coordinates. Source: "q.r === q.denominator && q.s === 0n"; dragging uses "BigInt(Math.round(da*DEN))" with DEN=120; Shift uses "event.shiftKey?12:1". Recreated by node audit.mjs.
- CONFIRMED | This is Schade's bound applied to the project's preceding value. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 105, prose-8-1. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | The last candidate has a forced minimum degree | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 106, prose-9-1. The result/contract states "ceil(70 x 22/24) = 65 > 64" and "Minimum degree exactly 5". Recomputed: 69-64=5, 2*69/24=5.75; edge deletion preserves the unit-distance condition. The preceding project values are explicit dependencies.
- CONFIRMED | The enumeration grows graphs along canonical deletions. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 107, prose-10-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Degree bounds specify which parent cells matter; hereditary obstructions discard graphs that no unit-distance drawing could contain. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 108, prose-10-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Geometric contradiction certificates discard more. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 109, prose-10-3. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | A discarded graph needs a reason the checker accepts. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 110, prose-10-4. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The difficult claim is that the enumeration is complete and its pruning rules are sound. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 111, prose-11-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The graph statistics below are decoded here. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/facts.mjs | N1 entry 112, prose-12-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Graph layout. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 113, prose-13-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | These lengths are not unit distances. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 114, prose-13-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The search runs in a worker. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 115, prose-14-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | You can keep moving points and inspecting the certificate while it runs. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 116, prose-14-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Extra host edges do not invalidate a subgraph | git show 5a634e5551:public/strata/the-sixty-ninth-distance/engine.mjs | N1 entry 117, prose-15-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | It would have to be zero. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 118, prose-16-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | It is always positive. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 119, prose-16-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The record is a real-dag certificate: a list of rational polynomial identities. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 120, prose-17-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | A unit edge supplies a squared-distance equation. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 121, prose-17-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Multiply those equations by the record's polynomial cofactors and add them. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 122, prose-17-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Each derived row must vanish wherever the original equations vanish. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 123, prose-17-4. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Every other x and y stays a real variable. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 124, prose-18-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Reflections are allowed. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 125, prose-18-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | There is no assumption of integer coordinates, no tolerance and no optimization getting stuck. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 126, prose-18-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | These are necessary conditions on any putative embedding. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 127, prose-19-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | A contradiction under necessary conditions suffices. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 128, prose-19-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The selected pairs and frame are visible free choices of the certificate, not undisclosed geometric assumptions. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 129, prose-19-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Vertex labels belong to the survivor. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 130, prose-20-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Constraint buttons use the smaller pattern's labels; the frozen map connects them. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 131, prose-20-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Saturation pairs are distinctness constraints, not extra unit edges. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 132, prose-20-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The same graph layout. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 133, prose-21-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Bright edges supply the selected identity. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 134, prose-21-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | Alter the constant term in a copy of the first derived identity. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 135, prose-22-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The exact source combination no longer equals that row. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 136, prose-22-2. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The checker must reject the copy before it reaches the final contradiction. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 137, prose-22-3. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The browser regenerates all original equations from the graph, verifies each identity using exact rational arithmetic, allows references only to earlier rows, and checks the final square expansion and the signs of its weights. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/certificate.mjs | N1 entry 138, prose-23-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The standalone verifier also replays the certificate with a separate polynomial implementation. | git show 5a634e5551:research/the-sixty-ninth-distance/tools/oracle.mjs | N1 entry 139, prose-24-1. Read the full certificate/checker beyond N1 fragments. Independent certificateOracle() and browser replay agree: 14 rows, 27 variables, 26 generators, frame [10,1], nine distinctness pairs, constant 1 and four positive square weights. Every identity and final square expansion passes; a changed first constant is rejected at row 1. A valid unit frame and nonzero pair distances supply necessary conditions for any real embedding.
- CONFIRMED | The nested summary-only audit status is a historical coverage check, not a fresh certificate replay. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/audit.json | N1 entry 144, prose-26-1. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | The slice shards load when requested. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 145, prose-27-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Every accepted slice has a place | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 146, prose-28-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Amber: pilot. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 147, prose-29-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Teal: main fleet. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 148, prose-29-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Lilac: completion plan. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 149, prose-29-3. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Click the grid or choose a slice below; dimming is an inspection filter, not a change to coverage. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs | N1 entry 150, prose-29-4. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Recorded cell counts, summed by level | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/slices-0.json | N1 entry 151, prose-30-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The funnel, added from the slice rows | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 152, prose-31-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Children are reach-admissible graph records before the filters, not arbitrary graphs or point arrangements. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 153, prose-32-1. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | TU uses fixed gadgets and their linear consequences to force a missing pair to be a unit pair. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 154, prose-32-2. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | The target saturates the deletion bound, so that extra pair is impossible; its induced ancestors inherit the missing pairs. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 155, prose-32-3. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | Triangle propagation can force such a pair (triangle TU) or derive a contradiction (triangle refutation). | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 156, prose-32-4. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | Kept means unresolved by those filters. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 157, prose-32-5. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | The survivor's later certificate is a separate step. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 158, prose-32-6. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | Recorded totals; arithmetic recomputed here. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/audit.json | N1 entry 159, prose-33-1. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | All counts refer to graph records. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/audit.json | N1 entry 160, prose-33-2. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | Source: accepted per-slice known-graph receipts. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 164, prose-35-1. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | Recorded canonical deletion chain. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/known-samples.json | N1 entry 165, prose-36-1. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | Each minimum degree meets its reach threshold. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/known-samples.json | N1 entry 166, prose-36-2. The retained result, contract, review, original compressed plan judgements, per-cell counts and known receipts were read. Recreated from the pin: 1920 slices, 1923 accepted copies, 97912 eligible known graphs, 50 reproduced slices, 21 resumed slices with twins, 20 explicit comparison rows plus slice 344. The audit says "COMPLETE"; G1 says "G1 verdict: CLOSED". This supports the recorded event without re-enumerating or remeasuring deleted files.
- CONFIRMED | These are retained historical claims. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 170, prose-38-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | This page can challenge a count, a supplied graph map or a polynomial identity. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 171, prose-38-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | It cannot independently certify that every deletion happened only after the prescribed checks. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 172, prose-38-3. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The draggable picture uses a selected known configuration and a rational step size. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 173, prose-39-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The certificate chooses a unit edge as its frame and selected distinctness pairs. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 175, prose-39-3. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Slice assignment uses the recorded hash rule; the reproduction sample uses a saved seed and selected heavy slices. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 176, prose-39-4. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | None of these pictures or selections is a numerical fit offered as a proof. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 177, prose-39-5. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The dependencies include the hereditary forbidden-graph catalogue, the supplied TU gadgets and geometric lemmas R and T. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 178, prose-40-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | Different checkers share the canonical-labelling library, so agreement is not independence from that library. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 179, prose-40-2. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | The survivor's blind containment search used a separate decoder and search, then the same frozen certificate checker. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 180, prose-40-3. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | This settles a small case conditional on those checks and dependencies. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 181, prose-41-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | It does not advance the asymptotic unit-distance problem or promise the next value. | git show 5a634e5551:research/the-sixty-ninth-distance/README.md | N1 entry 182, prose-41-2. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | This layer's check imports and runs the very module this page runs. | git show 5a634e5551:research/the-sixty-ninth-distance/verify-the-sixty-ninth-distance.mjs | N1 entry 183, prose-42-1. The verifier actually imports the page's engine.mjs, certificate.mjs and search.mjs, while also using its separate oracle. Exact source: "const search=await import(pathToFileURL(path.join(page,'search.mjs')))". This is a coupling description; the baseline run currently stops earlier at the missing log.
- CONFIRMED | Not yet put to that test. | git show 5a634e5551:scripts/check-placard.mjs | N1 entry 184, prose-43-1. This is about the specific corpus dependence sweep, not all local testing. git grep at the pin finds no target row in depends*.json or mutation*.json; the same search finds all-edge-no-middle in depends-2.json and depends-dom-2.json. The layer's own retained mutation trials are a different test. Do not charge their existence against this statement.
- CONFIRMED | The standalone page combines an exact pair counter, a live containment search, a rational certificate replay and a recorded slice ledger. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 186, prose-45-1. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Its limits and mathematical dependencies are part of the visible check. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 187, prose-45-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Among 24 distinct points in the plane, the maximum number of unit pairs is 68. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 188, metadata-dek-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | A known arrangement attains it; the Artificial Wasteland's exhaustive graph enumeration left one 69-edge survivor, excluded by a contained eleven-vertex graph and an exact contradiction certificate. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 189, metadata-dek-2. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | Drag the arrangement, search the forbidden classifications and replay the proof. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 190, metadata-dek-3. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | Computer-assisted, dependent on the project's u(23) = 64, and not yet refereed outside the project. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 191, metadata-dek-4. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | How many unit distances can 24 points make? 68, with the check | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 192, metadata-seoTitle-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | A known arrangement makes 68; the Artificial Wasteland's recorded search and contradiction certificate rule out 69. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/app.mjs | N1 entry 193, metadata-plain-2. The cited file was read in full beyond the N1 excerpt. Its source code, visible pinned HTML and relevant frozen data support this statement; node audit.mjs exercises the actual controls and validates their figures. The browser controls run locally, while the production enumeration remains a retained historical record.
- CONFIRMED | The preceding point count supplies the deletion bound that leaves only the survivor on this page. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 194, related-1-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | Both pages put an exact distance counter beside the retained enumeration record. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md ; git show 5a634e5551:public/strata/the-sixty-fifth-distance/index.html | N1 entry 195, related-1-2. The relationship was checked directly in the pinned linked pages, beyond N1's generic RESULT paragraph. The preceding page exposes its exact counter and record; the eleven-vertex page describes "2,833 minimal forbidden graphs"; the smaller page exposes certified almost-unit drawings. This page independently replays its contained obstruction.
- CONFIRMED | The earlier enumeration supplies a dependency beneath this result and introduces the pruning apparatus. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 196, related-2-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | Here that apparatus leaves a single graph whose smaller obstruction can be inspected directly. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 197, related-2-2. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | One accepted record from the eleven-vertex work refutes the survivor here. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 198, related-3-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | The classification page shows the larger collection; this page shows why one contained graph already suffices. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md ; git show 5a634e5551:public/strata/every-edge-but-one-at-eleven/index.html | N1 entry 199, related-3-2. The relationship was checked directly in the pinned linked pages, beyond N1's generic RESULT paragraph. The preceding page exposes its exact counter and record; the eleven-vertex page describes "2,833 minimal forbidden graphs"; the smaller page exposes certified almost-unit drawings. This page independently replays its contained obstruction.
- CONFIRMED | The smaller forbidden graphs supply hereditary filters used before this survivor appears. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md | N1 entry 200, related-4-1. The full result/README and named dependencies were read, not just the selected header. The result says "If the gates of Section 6 are all green, u(24) = 68." README says "The result is not yet externally refereed and is not an asymptotic improvement." The main certificate and retained enumeration counts agree; the claimed maximum is conditional on the expressly disclosed earlier results, filters, lemmas and canonical-labelling software.
- CONFIRMED | That page draws the defect in each obstruction; this page follows a contained obstruction into its exact polynomial contradiction. | git show 5a634e5551:research/the-sixty-ninth-distance/data/RESULT-u24.md ; git show 5a634e5551:public/strata/every-edge-but-one/index.html | N1 entry 201, related-4-2. The relationship was checked directly in the pinned linked pages, beyond N1's generic RESULT paragraph. The preceding page exposes its exact counter and record; the eleven-vertex page describes "2,833 minimal forbidden graphs"; the smaller page exposes certified almost-unit drawings. This page independently replays its contained obstruction.
- CONFIRMED | Alexeev, Mixon and Parshall, arXiv:2412.11914. | https://arxiv.org/abs/2412.11914 | N1 entry 202, citation-amp. Metadata authors: Boris Alexeev, Dustin G. Mixon, Hans Parshall.
- CONFIRMED | Engel, Hammond-Lee, Su, Varga and Zsámboki, arXiv:2406.15317. | https://arxiv.org/abs/2406.15317 | N1 entry 203, citation-engel. Metadata authors: Peter Engel, Owen Hammond-Lee, Yiheng Su, Dániel Varga, Pál Zsámboki.
- CONFIRMED | Smaller exact values and the known lower bound. | https://arxiv.org/html/2412.11914 | N1 entry 204, citation-amp-scope. The introduction and Theorem 1(b) support the smaller exact values and the previously known lower-bound table.
- CONFIRMED | Configuration data and exact Moser-lattice geometry. | https://arxiv.org/html/2406.15317 | N1 entry 205, citation-engel-scope. The abstract links the configuration database; sections 2.4 and 6.2 give the exact four-element coordinate basis.
- CONFIRMED | The disproof and its human verification | https://arxiv.org/abs/2605.20695 | N1 entry 206, citation-disproof-verification. The abstract explicitly calls the paper a "human-verified version" of the counterexample.
- CONFIRMED | Source: final accepted run record. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/audit.json | N1 entry 207, citation-final-accepted-run. Audit extract was checked against the frozen originals: all four plan judgements are "OK", fleet coverage is "complete", decisions are "COMPLETE", and each explicit twin verdict is "IDENTICAL". The summary-only audit is preserved as historical evidence.
- CONFIRMED | Source: the final plan judgements and fleet audit, extracted together. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/audit.json | N1 entry 208, citation-final-plan-audit. Audit extract was checked against the frozen originals: all four plan judgements are "OK", fleet coverage is "complete", decisions are "COMPLETE", and each explicit twin verdict is "IDENTICAL". The summary-only audit is preserved as historical evidence.
- CONFIRMED | Source: reproduction duplicate-copy agreement and the saved from-scratch comparisons in the audit extract. | git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/audit.json | N1 entry 209, citation-reproduction. Audit extract was checked against the frozen originals: all four plan judgements are "OK", fleet coverage is "complete", decisions are "COMPLETE", and each explicit twin verdict is "IDENTICAL". The summary-only audit is preserved as historical evidence.
- CONFIRMED | "Erdős unit distance problem", "A186705", "OEIS" | https://oeis.org/A186705 | N1 entry 210, oeis-tag-reference. The sequence description is "The Erdős unit distance problem"; it defines the maximum number of occurrences of a single distance among n planar points.
- CONFIRMED | (added) Survivor vertex 0, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 0 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 1, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 1 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 2, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 2 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 3, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 3 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 4, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 4 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 5, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 5 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 6, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 6 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 7, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 7 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 8, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 8 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 9, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 9 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 10, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 10 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 11, degree 5 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 11 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 12, degree 6 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 12 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 13, degree 6 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 13 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 14, degree 6 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 14 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 15, degree 6 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 15 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 16, degree 6 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 16 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 17, degree 6 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 17 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 18, degree 7 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 18 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 19, degree 7 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 19 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 20, degree 7 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 20 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 21, degree 7 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 21 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 22, degree 7 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 22 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.
- CONFIRMED | (added) Survivor vertex 23, degree 7 | git show 5a634e5551:public/strata/the-sixty-ninth-distance/draw.mjs ; git show 5a634e5551:public/strata/the-sixty-ninth-distance/data/patterns.json | Exact SVG title recreated with graphView(). The separate oracle decoder confirms vertex 23 has this degree; all 24 titles were compared, not inferred from the aggregate histogram.

## Findings
- [fixed] X1: Kept the source-supported figure 1.014, attributed the explicit bound to Sawin after the May disproof, and added arXiv:2605.20579 beside the human-verification link. Updated the authored copy, template, regenerated HTML, source-record entries 6-7 and README; entry 7 follows the upper-bound clause's revised linking words. The extractor and verifier now bind the exponent to the figure re-read in Sawin's abstract instead of extracting it from another layer's prose. The earlier source search's failure to find 1.014 was incomplete, so its proposed numerical weakening was rejected. Added a dated correction in the page and a Markdown mirror, as the user's message requires; triage's unpublished-page exception requests omission but conflicts with that instruction. No claim is made that a reader saw the old page.
- [fixed] X2: Replaced source-record entry 174's README citation and point-basis excerpt with draw.mjs:70-76, including ringPositions and the three radii. The page statement remains correct. No page correction line was added for this MINOR finding.
- [fixed] X3: Replaced source-record entry 214's self-pair excerpt with app.mjs:32-34, the distance calculation and the counted/not-counted result branches. Added the original coordinate file to the engine evidence. Recreated the exact result, including independent Gram-matrix arithmetic. No page correction line was added for this MINOR finding.
- [fixed] X4: Replaced source-record entry 228's self-pair excerpt with app.mjs:32-34 and coordinate/engine evidence, naming the 1/120 nudge. Recreated squared distance 14161/14400 with independent arithmetic. No page correction line was added for this MINOR finding.
- [fixed] X5: Renamed both retained copies to survivor-subgraph.txt without changing a byte. Updated the authored page link, extractor output name and history reader, verifier, README and regenerated frozen manifest. Kept the original upstream .log path in input provenance. The .txt targets are not ignored by git. Added a dated correction in the page and a Markdown mirror under the user's message's rule, despite triage's unpublished-page exception. The correction reports the missing pinned file and the supplied identical log without guessing a cause.

## Verification
- `nice -n 19 /home/liam/tools/pyenv-maths/bin/python -B research/the-sixty-ninth-distance/tools/extract.py --regen`: exit 0; 1,920 slices, three controls, 14 certificate rows and 7,713 registered inputs. Mathematical data stayed byte-identical; provenance purpose and the packaged log name changed.
- `node research/the-sixty-ninth-distance/tools/freeze-page.mjs --regen`: exit 0. The HTML and Markdown were generated from the edited layer sources.
- `node research/the-sixty-ninth-distance/tools/freeze-page.mjs --regen --check`: exit 0; HTML and Markdown match the generated bytes without writing.
- `node scripts/check-claims-pass.mjs`: exit 1, solely because `assay/records/every-edge-but-one-at-eleven.md` is missing. That layer is outside this pass. Calling the gate's exported `checkRecord` for this record exits 0 with no errors: checked 199, confirmed 194, wrong 5 (two WRONG and three MINOR), no open findings.
- `nice -n 19 node research/the-sixty-ninth-distance/verify-the-sixty-ninth-distance.mjs`: exit 0, PASS. It checked 68 unit pairs, 62 after the nudge, 69 survivor edges, 28 triangles, 14 certificate identities, four positive squares, 1,920 slices, 97,912 eligible controls, 50 reproductions and 21 resumed twins. Both full searches agree at 0/398 and 12/2833. It bound 43 static claims, passed 17 negative controls and exercised the actual worker and DOM event handlers; the planted graph produced 347 smaller hits.
- Source-record offline check through `checkSources` and `pageText`: exit 0 with no errors. It retains 2,052 warnings for claims built at load time. A first run found the newly sourced entry's missing fetch date and the upper-bound clause's old linking words; both were repaired before the passing run.
- Scoped `git diff --check`: exit 0. Independent checks passed for the two rational pair results, unchanged log hash, correction mirrors and placement, JSON-LD dateModified 2026-10-08, retained site chrome and absence of U+2014 in changed files.

## Scope left intact
The dated checker records and captured preceding-page paragraphs remain historical evidence. The captured paragraphs no longer supply the explicit exponent. No other layer or shared file was edited by this fixer, no git write was run, no website was published, and no film, audio or image was rendered or re-encoded. The repository-wide missing record remains outside the authorized scope.