C310 — Minimal context cover and final529 synthesis

24 September 2026. Latest completed research: C310. This closes the authorized C277–C310 batch, comprising34 steps from the C276 checkpoint. The endpoint C310 governs the earlier “33 steps” wording. C311 has not been executed.

C310 result: a composite source cover

The seven supported cumulative LXX_OFF600-segment indices require two regular context families and an event-role change. The unique smallest cover is ordinary MT Noah→Flood plus the alternate SP Shem-life family. One annual position remains unsupported.

Here a context family fixes tradition, Cainan state, Terah setting, sojourn setting and endpoint roles. Existing Gear and rail labels remain attached to each segment; they are not erased by grouping.

The third family adds source labels at existing coordinates. Its three positions are already covered by SP. No single family covers all seven supported positions, and no combination covers the full eight-position field. Index1 requires upper3057 and lower2457; only the lower MT Flood-close endpoint is admitted.

When Gear is also fixed in each group, the minimum cover uses five groups: MT Gears2/1 and SP Gears3/2/1. A second equally small cover substitutes LXX Gear1 for the co-located SP Gear1 primary segment. This preserves the distinction between context coverage and branch identity.

The regular Terah/sojourn change contributes+60−215=−155 within its own tradition. Moving between the two covering families also changes tradition and the event roles. Consequently the combined coverage is a comparison construction assembled from qualified contexts. This step supplies no prior historical rule selecting the switch.

What the batch adds to529

The principal gain is a finite source-qualified structure with explicit boundaries. Starting from the retained598 alignment,

598+8395 = 8993 = 17K
598+460 = 1058 = 2K
8395−460 = 7935 = 15K.

C282–C286 established the mixed-state2K+15K route and exhaustively classified it in the stated domains. The widened Noah/Shem/Flood register has528 qualified records; across135168 source-pair/target comparisons it gives12 labeled all-K paths, representing10 coordinate paths. This is a finite census, not a count of independent historical witnesses.

The resulting normal form is

S,F,R exist at i0…5. N completes four coordinate rows, i2…5; the requirements3703 and3704 at i0/1 are absent. The cross-width matrix is [[17,19],[15,17]]K, with2K on both sides. Its six duration equations have rank3. FixedK leaves a common translation free; absolute placement still needs its retained source premise.

Adding the established16K bridge interlaces complete rows at levels(4,5,6,7)K+i. These translated landings are generated coordinates. They are not additional historical AD events.

Boundaries that now have explicit answers

The exact fixed8K-reflection arrangement and the pure-K rectangle are separately admitted but do not coincide in the tested complete domain. C294’s exact reflected arrangement has gaps1060,1058,1060. C295 found one fixed-reflection configuration, four all-K configurations and zero simultaneous configurations. The two annual offsets cannot be discarded to combine their strongest features.

The25/23 expansion also has to retain its operation type. A matching center does not establish a matching physical phase pair: expansion changes a half-year pair to25/46 years. Cainan translation and dilation therefore cannot be substituted for one another merely because a center lands correctly.

For the fixed16K/24K comparison, regular reflectedAD6 and cumulative reflectedConquestAD1406 give C−B=2832. The held72×528 domain yields28 labeled points and12 labeled complete600 segments, representing18 point coordinates and eight segment coordinates. The LXX subset supplies the seven-index composite cover proved above.

Using the already supplied cumulative reflectedExodusAD1446 changes the condition to2792. It yields11 labeled points at eight coordinates, covering all SP_OFF Shem annual indices, but no complete600 segment. The missing regular lower endpoints1924…1917 explain the failure. Complete point coverage and complete path coverage are different results.

The two overlaps with17K bind to existing C127 routes N01 and N02. Both have direct17K and868+13K+1248 paths; only N02 has the retained299+299+650 sourced expansion. Their fixed2832 MT Flood selections differ from the MT endpoints inside those stored paths. Imposing both interfaces gives same-Gear SP widths599/601, and every admitted600 alternative changes one fixed equation by a year. Exact phase comparisons preserve this boundary.

Status and best next bounded step

C310 adds15 mathematical/domain checks and13 authentication checks, with byte-identical replay. Each step has a report, incremental checkpoint, frozen dependencies and reproducible evidence. Reassessment after every step is recorded in NEXT_STEP_DECISION.json. Author review only; no new independent agent review is claimed. File_18, File_09, C265 and C127 remain unchanged. The graph remains208 nodes and382 duration edges.

The batch establishes exact internal relationships and their finite source limits. It does not derive a global z-law, absolute placement, historical intention or statistical rarity. The corrected supplemental LXX Creation/Jared observation remains in the checkpoint as awareness only; it did not redirect this batch.

Best next bounded step — C311, unexecuted: inspect only the existing File_18, File_09 and axis-source statements for a pre-existing rule that selects the MT Noah/Flood→SP Shem-life context switch. Produce a short premise-and-citation ledger distinguishing an explicitly prescribed switch from a numerically assembled cover. Hold all endpoints, bridges, anchors and Gear limits fixed. If no such rule is documented, retain the composite-cover classification. This is the most useful remaining question after the finite arithmetic and source membership checks.