490d C343 — Noah-held double expansion and source-binding synthesis

24 September2026. Step10 of10 in the C334–C343 authorization.

C334–C343 are complete. The main addition is a source-qualified Noah-held realization of1058→1150→1250 under E=25/23, connected to the existing950+108 Noah-death partition. Its exact scope is primary, Noah-bound and phase-matched. The double-E outer endpoint relation survives wider controls; the complete intermediate does not.

The positive source realization

Hold the existing reflected SP Noah endpoint N. Let R be the next-Gear restored-MT Flood start, H the same-Gear base-MT Shem birth, and F the same-Gear base-MT Flood start. SP is OFF/Terah0/full430; R is MT ON/Terah60/full430; H and F are MT OFF/Terah0/full430.

Both E steps are exact, and their gains are92 and100, with92×E=100. These are admitted source endpoints under their stated contexts; no endpoint was fitted to a desired interval. The two rows are translated Gear realizations of the same interval structure.

Insert M, the same-Gear restored MT Noah, and D, the matched SP Noah death. The primary chain N→M→D→R→H→F gives

460 +490 +108 +92 +100 =1250.

Thus the successive source distances are1058=950+108,1150=950+200 and1250=950+300. Primary death retains File_18's inclusive/harmonic qualification. This is a cross-context comparison chain, not a single historical biography.

Controls that limit the result

The companion cases would require a missing one-year MT person companion to supply1150. That node is not admitted. Flood close remains a separate event boundary. The502/98 mode remains valid and is not corrected to500/100; it simply does not realize this particular intermediate E step. The modal test uses the already named G2/G1 comparison and creates no Gear0.

The original C287 complete SP domain remains i2…5. Primary full ladders are i3/i5; both companion rows remain valid double-E endpoint comparisons. Co-located LXX labels do not acquire a new death-ladder identity through coordinate equality.

What the source-choice audit establishes

C334 positively finds a shared Shem±2 origin and cumulative propagation rule. The sources distinguish that rule from localized Noah/Shem binding and ordinary coherent regular Flood Gear transport. The two C319 displayed +2 changes also have opposite underlying BC directions. No reviewed instruction binds exactly ON-Shem i+2 and next-Gear MT Flood while the other comparison endpoints are held.

The source choices factor cleanly. Cumulative1058 requires ON-Shem indexi+2 with OFF-Flood indexi, leaving six left pairs. Regular1058 requires restored MT CainanON/Terah60, the next Gear, and primary/start or companion/close matching, leaving four right pairs. Their24 combinations have two1058 sides; exactly the four equal-index joins recover both17K cross-spans and C287's full source paths.

Coherent whole-pair shifts preserve the native intervals600, or601 to Flood close. The selected1058 and598/599 readings use endpoints held across different profiles or Gears. Those comparisons remain valid under their own labels; they do not redefine a source lifespan or age.

The regular E ladder derives from650,500,600 and192=130+60+2. In C319's frozen control menu it has the same selector as C332's death criteria: 02 and22 both remain at i3/i5. Adding the17K condition imposes a=b and selects22. The new regular ladder therefore does not close the cumulative coupling gap or supply an independent source prescription.

Phases, maps and prior work

The complete ladder passes on the two inherited matching components of each primary path. All80 component assignments across the five controlled chains were tested; four complete component ladders remain. Mismatched components do not repair the companion or502/98 failures.

On one selected component, the interval matches can be written as an affine expression fixing that existing Noah component. A single affine E dilation on physical time cannot preserve both seasonal components: it enlarges their half-year separation by1/23. Nor does it preserve the source Noah lifespan:950 maps to23750/23. The diagnostic death image is unadopted. Center-based expansion with suffix phases retained and whole-physical-time dilation must remain distinct.

The scalar ladder was already recorded in C01/C09. Locator-selected C89, verified against its recorded hash, contains a different complete endpoint realization. Its final base-MT Flood/SP-Noah1250 interval is shared here; its earlier stages use other events. File_46 explicitly licenses two integral E expansions of a23²-multiple. Those positive precedents are preserved without transferring their endpoint identities or prescribed scope to this particular construction. No archive-wide novelty or absence claim is made.

Completion and recovery

Each of C334–C342 was sealed and saved with its report, checkpoint and evidence ZIP before the next step. The pre-C343 audit verified all nine packets,237 manifested payloads,108 certificate bindings, nine predecessor links and120 frozen source payloads;18/18 isolated main output comparisons matched, with27 successful publication receipts.

C343 binds the nine predecessor results and reassessments. A separately written root-agent check uses integer cross-multiplication and direct source-record joins to verify the factorization, ladder, phase counts, death chains and selector result. It is a second method, not an independent-agent review. After C343 is sealed, the final completion record covers all ten packets and30 individual deliverables, followed by the reading index and both recovery archives.

Actual File_18 and File_09 bytes match their expected hashes. The C265 checkpoint is readable and its certificate hash matches the retained control. Canonical graph and other standing controls remain unchanged. Original unprefixed reports, Engine reports and C-series remain distinct. The earlier incomplete September17 ZIP is not the later verified intact original. The corrected LXX8464 supplement remains awareness only:1564→1700,2484→2700,920→1000; AD3746/2300 is superseded.

Verification and limits

68 main checks and 56 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. These checks and the source interpretation were performed by the root agent; no new independent-agent review is claimed. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

Conditional arithmetic and source membership do not establish authorial intention, a source-prescribed coupled selector, independent witnesses or a chronology-wide map. C344 is unexecuted.

Mirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.

Decision after this result

C344, proposed and unexecuted: Audit File_02 §6.3’s explicitly prescribed gap conversion598×300/299=600 against the retained adjacent-Gear MT Noah/Flood endpoints. Test the existing held-endpoint realizations and their effect on the Noah-held2K/E ladder, keeping companion and modal states separate. No new endpoints, Gears, phases or anchors.

The present batch separates source choice, coherent Gear movement and E interval matching. The explicit598→600 source rule offers a focused next comparison of those mechanisms without another broad endpoint search. It is proposed only; the ten-step authorization ends at C343.