{"id": "c309", "title": "Independent constraints and claim boundaries", "filename": "490d_C309_Independent_Constraints_and_Claim_Boundaries_20260924.md", "role": "Research action", "step": 309, "sha256": "1fc8b416bde753309e828d75c351dc03ef808224351a0ff5ef0492a9ab4c6475", "bytes": 3998, "origins": ["C01–C479/packets/C309/490d_C309_Independent_Constraints_and_Claim_Boundaries_20260924.md"], "download": "/sources/1fc8b416bde753309e828d75c351dc03ef808224351a0ff5ef0492a9ab4c6475.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "C309 — Independent constraints and claim boundaries\n\n24 September 2026. Dependency audit before C310; no extension of the event register.\n\n Result\n\nThe strongest new529 result is a finite, source-qualified four-node structure, accompanied by explicit domain and phase boundaries. Its six pairwise duration equations have", "content": "# C309 — Independent constraints and claim boundaries\n\n24 September 2026. Dependency audit before C310; no extension of the event register.\n\n## Result\n\nThe strongest new529 result is a finite, source-qualified four-node structure, accompanied by explicit domain and phase boundaries. Its six pairwise duration equations have **rank3**, not six independent scalar constraints. At fixedK, one common translation remains free. The batch strengthens structural understanding without deriving absolute placement or historical intention.\n\nFor the complete coordinate rows i=2,3,4,5, the centers are\n\n    (S,F,R,N) = (−12K,−10K,5K,7K)+i, K=529.\n\nTheir source roles are cumulative LXX_ON Shem, cumulative LXX_OFF Flood, restored regular MT Flood and regular SP Noah; primary SP Noah coordinates also carry qualified LXX_OFF aliases. The six directed widths are2K,17K,19K,15K,17K,2K.\n\n## Count the right objects\n\n| Object | Count or rank | Interpretation |\n|---|---:|---|\n| C286 all-K paths | 12 labels /10 coordinate paths | Source aliases remain distinct labels |\n| Complete four-node family | 4 coordinate rows /6 Noah bindings | First two formal rows lack an admitted Noah endpoint |\n| Six fixedK duration equations | Rank3 | A three-edge spanning chain determines the other widths |\n| UnknownK five-variable system | Rank3, two free parameters | Scale and common translation are not fixed by the pattern alone |\n| FixedK plus one absolute source pin | Rank5 in the five-variable formulation | Absolute placement enters through a source pin |\n\nThe coefficient pattern is extracted from the retained numerical results. Its homogeneous rank is not a derivation of529 from no assumptions. The supplied widths and their source placements calibrate the pattern. Source membership, context and phase are separate obligations that scalar rank does not eliminate.\n\n## Claims that remain separate\n\nThe all-K family and the exact fixed8K reflection arrangement are individually valid but do not coincide in the tested complete source domain. C295 found four all-K configurations, one fixed-reflection configuration and zero joint configurations. The exact reflected arrangement has gaps1060,1058,1060, preserving the two annual offsets instead of silently replacing them by1058.\n\nC301–C302 give28 labeled point matches and12 labeled complete600 segments for the Conquest comparison. C308 gives11 point matches but no complete segment for the supplied Exodus comparison. Full annual point coverage is not evidence for a full biography correspondence.\n\nN01 and N02 are existing graph routes. Their newly identified overlap with the fixed2832 comparison is useful, but their598 relation and1/3 interior differences follow from the same typed affine equations. C307 explicitly separates those consequences from additional constraints. C306 records the admitted600 alternatives and the one-year equation change each would require.\n\nAbsolute map placement remains a premise, as already shown by C276 and C289. The batch does not establish a global z-law, a unique chronological interpretation, statistical rarity or historical intention. No probability model was constructed, and the examined contexts share source data.\n\n## Evidence and next decision\n\nSixteen mathematical/domain checks and fourteen authentication checks passed. The exact incidence-rank calculation, source counts and claim-support bindings replay byte identically. The claim register identifies author interpretation separately from computed facts. Author review only; canonical and graph controls unchanged.\n\nThe final bounded step, C310, will test whether the seven-index LXX600-segment coverage can be supplied by one fixed regular context, or requires a composite of contexts and event roles. It will compute the minimal cover from the existing segment ledger, then close the batch with a source-qualified synthesis and restart package. This resolves a remaining interpretive ambiguity without adding endpoints or searching new anchors.\n"}