490d C353 — Source600 repairs and partial operator structure synthesis

24 September 2026. Step 10 of 10 in the C344–C353 authorization.

C344–C353 are complete. The principal result is a source-qualified way to retain the529 rectangle while resolving its primary598 interiors to coherent600 pairs: both sides can use458+600=1058 with their outer2K and17K endpoints unchanged. This preserves the structure through different interior source selections; it does not prescribe those selections.

The two regular repairs and their tradeoff

File_02 §6.3 explicitly gives598×300/299=600 as a whole-year gap conversion. Its worked example holds the Flood and changes the Noah state. The retained primary comparison has two existing endpoint realizations:

Both outputs are coherent same-Gear MT pairs. The first repair directly parallels File_02’s held-Flood example; the second draws on the separately admitted Flood Gear register.

All three same-context MT600 pairs were checked against each primary SP path. With SP Noah N and its matched death D held, the identity N−R=(N−M)+600 settles the tradeoff:

• Keeping2K=1058 forces 458+492+108 through MT Noah and SP death.
• Keeping the 460/490 partition gives 1060, with the death-to-Flood remainder110.

No admitted600 pair preserves both conditions. The original460+490+108=1058 remains valid in its adjacent-Gear598 configuration; the source alternatives are not corrections of one another.

The retained companion comparison uses actual MT Noah against Flood close and has gap599. Its product is601+1/299, not the admitted601. Neither endpoint selection nor any existing seasonal component pairing repairs that mismatch. This finding is specific to those MT endpoint comparisons; the SP companion source states remain retained.

Cumulative counterpart and the unchanged outer structure

For the retained cumulative triple, native-ON Shem S and OFF-Shem P share annual indexi+2; OFF-Flood q has indexi. Thus P−S=460 and q−P=598.

Holding q and selecting the existing OFF-Shem member at i gives 458+600=1058. Holding P and selecting OFF-Flood at i+2 instead gives 460+600=1060. Both replacements exist at every complete C287 indexi2…5, including the inherited matching components.

The458 leg combines a Cainan-state difference with a two-year annual-index difference. Same-index cumulative Cainan insertion remains460. No cumulative Gear, altered lifespan or new annual member is introduced.

At the two primary indicesi3/i5, the cumulative interior shift and regular MT-Noah shift can both be made while S,q,R,N remain fixed. Consequently, both2K sides and both17K cross-spans survive. The cumulative repair also exists at the companion indices, while their regular300/299 counterpart fails as described above. Source domains remain asymmetric and explicitly labelled.

A connected650 source image

The original primary598 interval also has an E=25/23 image under the already established Noah-held comparison:

This combines C325’s MT-Noah→SP-Shem endpoint match with C338’s Flood→base-MT-Shem match. C01 already recorded598/650 scalar and source gains in another placement. The joint interval reading adds a connection between retained results, not a newly discovered scalar or independent witness.

The two outputs600 and650 have ratio12:13 because (300/299)/(25/23)=12/13. No new12/13 source operator is adopted. In the502/98 control, the actual Shem endpoints remain650 apart but are displaced from the selected E images: equality of widths does not identify the endpoint maps.

Reduced endpoint structure and its limits

Within the declared eleven-node primary field, exact center-coordinate matches form these partial relations:

Every listed move has an exact inverse inside the field. The other tested inputs leave that field. This is a bounded collection of partial invertible relations; no total source-state group action is established.

Different fixed endpoints also make composition order matter. The three formal order differences are92/299,40/299 and−2/299. No input in the field supports both complete composition orders. These are diagnostic coordinate identities, not calendar residues, new source points or independent witnesses.

The source biography control is equally sharp. Actual MT Noah birth/death motion is+2/+2 and preserves950. A whole affine300/299 extension scales950 to285000/299 and misses the new source death by−950/299. Source-state translation and gap conversion therefore remain distinct.

Across64 inherited component controls, eight primary repairs pass and no companion repair passes. A single physical-time300/299 dilation changes half-year separation by1/598. Component-preserving comparisons retain the established center/phase distinction. File_02’s exclusion of Residue Protocol arithmetic remains explicit.

What remains unselected

In C319’s held a,b menu, the internal gaps are600−a on the cumulative side and600+e−b on the regular side, where e distinguishes the companion comparison.

Requiring both inputs to be598 selects22 on primaryi3/i5. This is equivalent to the already selected two2K sides plus the primary qualification. If the question instead permits already-valid600 states to remain unchanged, nine admitted primary controls across00,02,20,22 support both600 outcomes.

The shared Shem ambiguity, regular gap conversion and cumulative propagation remain positive source facts. No reviewed instruction requires both sides to start at598 or prescribes the exact coupled endpoint selection. The original selection gap therefore remains open despite the new source-qualified correspondence.

Completion and retained controls

Each preceding step was published with a report, checkpoint and evidence ZIP before the next step. The pre-C353 audit verified nine sealed packets,229 manifested payloads,108 certificate bindings, nine predecessor links and112 frozen source payloads. All18 isolated main output comparisons matched and27 publication receipts passed.

C353 binds those results and their sequential next-step decisions. A separately written root-agent implementation uses integer cross-multiplication, affine coefficients and direct source-record joins to check the repairs, phases, partial relations, commutators and selectors. It is a second method, not an independent-agent review. The final completion record covers all ten sealed packets,30 individual deliverables and the recovery packages.

Actual File_18/File_09 bytes match their retained hashes; File_02 matches its C344 source snapshot. The C265 checkpoint is readable and its certificate hash matches the control. Standing canonical graph and source controls remain unchanged. C343’s primary1058→1150→1250 ladder, its companion1149 and502-mode1152 controls, and all source qualifications remain intact.

Regular Gears remain Noah/Shem/Flood only. SP primary death retains its inclusive/harmonic qualification. The corrected LXX8464 observation remains awareness only; AD3746/2300 stays superseded. Original unprefixed reports, Engine reports and C-series remain distinct. The earlier incomplete September17 ZIP remains distinct from the later verified intact original.

Verification and limits

68 main checks and 57 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.

Finite source comparisons and arithmetic identities do not establish authorial intention, independent witnesses, a prescribed paired selector or a global chronology action. C354 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

C354, proposed and unexecuted: Audit the same598→600 comparison on existing adjacent-Gear Shem birth/death pairs in the frozen regular Noah/Shem/Flood register, keeping contexts and matched SP rails explicit. Determine whether any pair connects to the retained529/E endpoint relations; add no source endpoints, phases, anchors or broader Gear scope.

The Noah/Flood gap rule and its cumulative counterpart are now bounded. Shem’s separately sourced600 lifespan offers the next role-qualified comparison, allowing a direct test of structural reuse without repeating the completed Noah/Flood census. This is proposed only; authorization ends at C353.