# Sequence provenance and join limits — bounded C265 branch intake

Prepared for the parent task, 24 September 2026. These notes recover exact antecedents for the proposed 5+1+1+6(+1) ledger. They do not change a canonical source or the C265 evidence graph.

## Sources and retrieval

Exact extracted source bytes and archive-member paths are recorded in `extraction_manifest.json`. Only the named S7 antecedents and two direct branch inputs were retrieved; no historical campaign was reread or exhaustively searched.

- S7 comprehensive: `490d_Mirror_Cumulative_Regular_MT_LXX_SP_Comprehensive_Handoff_20260920(1).md`; SHA256 `8a857e112e4a5b774768f92f9872fc75446b202414baa9e9ab788214a6f71eed`.
- S7 bilateral: `490d_Mirror_Bilateral_Integration_and_8395_Correspondence_20260920.md`; SHA256 `152ceaf2b3641ecf9dc1f5f5049640a2b644a881a7c9e674ede099a89f52b0d5`. Recovered from the nested review-backup ZIP, not from an unprefixed numbered-report campaign. This is the exact available source, not a claim of byte identity with a separately suffixed `(1)` attachment named in the new report.
- BRANCH_2116_4232.md and the 3plus7x529 handoff were preserved as direct antecedents. The new integrated report controls wherever its post-C113 corrections supersede them; neither is necessary to fabricate any missing chain segment.

## The literal central sequence is fully recoverable

The comprehensive S7 §§8–9, 14–15, together with bilateral §6, specify these transformed intervals. Pair-center coordinate x uses the existing January AD1 origin; P_b=((b+1)t/bn BC) has center −b and Q_b=(AD bt/(b+1)n) has center +b. Center0 is the pair 1t BC / AD1n, never a civil year0.

| Interval | Source field (BC Nisan bases) | Operation | Final centers |
|---|---|---|---|
| MT cumulative Cainan ON Flood field | 5296→5289 | T_5290 | [−6,1] |
| SP cumulative Cainan OFF Flood field; official Terah145 stack | 4716→4709 | T_4715 | [−1,6] |
| LXX cumulative Cainan OFF Flood field | 5290→5283 | T_5290 | [0,7] |
| MT cumulative Cainan ON Shem-birth field | 5896→5889 | M composed after T_5888 | [1,8] in chronological order |

No regular Gear is applied to these cumulative fields. Full source Shem/Flood matching remains600; the source+600 and translation+598 produce a residual2. Bilateral §6 derives junction x=(600−598)/2=1, January AD2; the fixed Mirror axis is still January AD1.

| Central chronological point | Pair | Provenance | Elapsed from first point |
|---:|---|---|---:|
| −6 | 7t/6n BC | translated MT ON Flood opening | 0 |
| −1 | 2t/1n BC | translated SP OFF Flood opening | 5 |
| 0 | 1t BC / AD1n | translated LXX OFF Flood opening | 6 |
| 1 | AD1t/2n | translated MT ON Flood close; reflected translated MT ON Shem opening | 7 |
| 6 | AD6t/7n | translated SP OFF Flood close | 12 |
| 7 | AD7t/8n | translated LXX OFF Flood close | 13 |
| 8 | AD8t/9n | reflected translated MT ON Shem close | 14 |

The six consecutive coordinate differences are **5+1+1+5+1+1=14**. Dropping only the internal SP closing point at x=6 produces **5+1+1+6+1=14**. Thus the formerly implied final+1 has a supplied endpoint: LXX translated Flood close Q7→reflected translated MT Shem close Q8. The report's full 7+7 completion is T_5290(MT Flood) followed by M(T_5888(MT Shem)).

This is a continuous interval subdivision in the common transformed coordinate line. Its entries retain multiple source roles; shared transformed coordinates do not identify original source events. The construction assigns Creation/Fall/restoration roles typologically. It contains no cumulative Creation event at one of those seven selected boundaries.

Exact supporting passages:

- Comprehensive §9: “This supplies the previously implied final boundary, giving **7+7 = 14**, or the earlier partition **6+1+6+1**.” It identifies the shared paired boundary as AD1t/2n and explicitly warns that it is January AD2 rather than the fixed Mirror axis.
- Comprehensive §14.2: “Mirror(translated SP Flood week) = translated MT Flood week.” Its table supplies all three transformed Flood weeks.
- Comprehensive §15 supplies elapsed5/6 and12/13 as ordinal/cardinal Creation/Fall interpretations, and says cumulative Mirror landings do not acquire the regular SP personal−1 rule.
- Bilateral §6 reproduces signed intervals [−6,1], [−1,6], [0,7] and reflected MT Shem [1,8], derives their junction, and states7.5+7.5 envelopes overlap by0.5 with14.5 union.

## 529 dependency reduction retained from the parent investigation

The parent supplied the present algebraic reduction: T_5290−T_4715=575=529×25/23. The MT-ON versus SP-OFF opening difference580, less translation difference575, exposes5. LXX-OFF versus SP-OFF difference574, against the same575 translation difference, exposes1. These are dependency explanations of the selected source-and-translation construction, not independent corroborating witnesses or a claim of historical novelty.

## Where the stronger cumulative-Flood→Creation path remains open

The new report §§15.2–15.3 specifies the mixed converted-radius construction0→2500→5000→7000 and inserts independently obtained6000 to split the last2000. In its own words, the final2000 is a difference of converted boundaries, “not the P conversion of a2300 segment.” All of this is allowed and exact; it does not assign original source event identities to converted radius5000/6000/7000.

The cumulative Creation layer in new §13 has distinct events and anchoring. Its MT OFF/ON Creation endpoints translate by12696 to1308/1768 BC and then receive the Exodus-centered322+138→350+140 mixed completion. No source passage identifies these transformed Creation events with the literal central Flood-field boundaries above or with a complete sequence of y-radius subinterval endpoints.

The separate S7 Creation support chain **598+3312+5520+9890=19320** is fully specified in comprehensive §13 (and checked in bilateral §7), at six selected internal week positions i=2,…,7. It starts at SP cumulative Shem birth, passes through SP cumulative Flood and ordinary Conquest, then reflected regular MT Creation and reflected cumulative MT Creation. Its component lengths and selection domain differ from the fourteen-year central pattern. It must not be relabeled as5+1+1+6(+1).

The 7000/7500 comparisons supply two further anchor choices:

- S7 §10 fixed3t/2n BC landing gives7000+4+7000=14004 or7500+4+7500=15004 full outer widths, with14000/15000 summing bridges only.
- New report §15 radius diagram doubles aroundzero with no central gap. These are two declared anchoring constructions. Neither warrants silently removing the other's4-year gap.

## Orientation constraint

The literal central points, normalized from their first endpoint, are [0,5,6,7,12,13,14]. The repeated chronological partitions are5,1,1,5,1,1.

The new radial interpretation reads outward on both sides. A symmetric boundary collection ±5000,±6000,±7000 aroundzero has sorted left-to-right segment lengths1000,1000,5000,5000,1000,1000, i.e.1,1,5,5,1,1. Reading the left side inward reverses its outward ordering. The literal sequence centered at its join x=1 becomes[−7,−2,−1,0,5,6,7], not[−7,−6,−5,0,5,6,7].

Therefore one cannot assert a single orientation-preserving affine pointwise identification of every listed boundary in these two constructions. The existing theological correspondence is retained; an embedding needs declared side-dependent orientation, endpoint choices, and scaling. This is a precise remaining question, rather than an objection to mixed conversion.

## Decisive bounded conclusion

The literal central MT/SP/LXX Flood/Shem sequence and its terminal+1 are recoverable as one complete, phase-qualified coordinate subdivision. The converted-radius and cumulative-Creation constructions are exact qualified components; a common event-identified larger path remains unestablished. Preserve both the Year14 and Year15 theological readings, label the overlapping7.5-year envelopes correctly, and record any proposed subsequent connection as an explicit map between these frames.