# 490d Mirror checkpoint after C413

24 September 2026. **C413 complete; C414 unexecuted.** 0 steps remain in the current authorization. Read this checkpoint and report first; frozen predecessor records retain earlier findings. Current decisions override stale archive instructions. Do not repeat completed research.

## Current result

C404–C413 complete. The existing Noah-to-Shem E charts retain only the two sourced J arrows; required Shem E pivots3016/3018 and P pivots2853/2855 are absent. The fixed source500 correspondence succeeds: all12 labelled/six coordinate Noah E/J loops translate to Shem births/deaths, including both complete E/P/J graphs. The complete target graph has MT Shem2531,2533,2535, E pivots2556/2558, J death pivots1933/1935 and P death pivots2393/2395. Every path is conjugated byT(t)=t−500 with unchanged slope and circuit multipliers1750/(3K),7500/(13K). A direct2048-case Shem-family census recovers exactly those loops. The valid502/98 binding maps only originalG2–G3 to targetG1–G2; the other six labelled cases would require unavailableGear0. The older translated squared-E endpoints were already established inC361/C363; the new return paths cannot replace their missing2056/2058 intermediate. Three slope-compatible EE paths fail all six whole-map comparisons and48 phase comparisons; the closest discrepancy is−4/529. The existing Mirror corner exchanges admit none of104 reflected loop-record slots or208 inherited components. Translating the original regular2K side−500 preserves1058 but changes both17K cross-spans to8493. The full fixed−500 source-domain census has169 coordinate arrows,392 labelled pairings and141 maximal coordinate chains. A single extra coordinate bridge3395→2895 produces3895→3395→2895→2395→1895. Its formula100+405−4−1=500 forces all three variant differences, SP companionG1 and FloodG3. It has two continuous record lifts, choosing SP or LXX CainanOFF for the Flood/death pair; the sequence is not one biography. All528 regular records reconstruct from twelve current source excerpts. The new graphs and bridge remain derived comparisons; no new source endpoint, protocol prescription, canonical edit or global529/Mirror action is established.

Latest completed C413; C414 is unexecuted and outside this completed authorization. Use the C413 compact restart kit as the active packet; read synthesis/checkpoint then locator. C404 separates failed E-chart closure from C405 positive−500 role correspondence. C406 gives the independent target-family census; C407 the500/502 Gear boundary; C408 the all-positive-word obstruction to substituting a new return path for C361 squared-E map; C409 the fixed Mirror source obstruction; C410 the complete−500 coordinate domain; C411 the bridge formula and record-path lifts; C412 the direct528-record source reconstruction and warrant matrix. The complete Shem graphs are2531↔2533↔2535, with E return pivots2556/2558, J forward death pivots1933/1935 and P forward death pivots2393/2395. Their source correspondence with the C400 Noah graph isT_-500, not the earlier E charts toSP/LXX Shem2991/2993/2995. Preserve source labels even when affine maps agree. The four-step500 chain3895→3395→2895→2395→1895 begins with SP companionG1 CainanON/Terah60/full430 Noah and Shem births, crosses toSP or LXX OFF G3/base-Terah/half430 Flood start and Shem death, then ends atMT Noah death. Its opening own deaths remain2945 and2795, giving950/600. The bridge alone is a derived cross-context comparison; no2000-year biography is adopted. C361/C363 already recorded the−500 squared-E endpoints; newC408 is the replacement-path obstruction, not a new discovery of that translation. Retain C403/C393/C383 finite source normal forms, conditional529 equivalences and missing Mirror/pivot members. No global2K/17K-preserving action is established. K529,E25/23,P70/69; File02 J300/299 is gap conversion, not Residue Protocol or permission to transform dates globally. Global paired MirrorJanuaryAD1; local junctionJanuaryAD2; no civil yearzero; signed cumulative coordinates are not civil dates. Regular Gears Noah/Shem/Flood only. SP person companions remain local, Flood close independent, primary Noah death inclusive/harmonic; both500/100 and502/98 valid. LXX nativeCainanON is not given a second insertion; OFF is theoretical removal. No new source endpoint, phase, fitted anchor, global axis, upstream/cumulative Gear, primer reopening or canonical graph edit. Corrected LXX8464 supplement awareness only:1564→1700,2484→2700,920→1000; AD3746/2300 superseded. C-series, original unprefixed reports and Engine reports remain distinct. The earlier incompleteSeptember17 ZIP differs from the later verified intact original. Counts, aliases, repeated paths and second implementations are not independent historical witnesses. Proposed C414 tests existing2K regular source pairs under one and two applications of the fixed−500 comparison; no such new pair census has been executed.

## Retained context

C394–C403 complete. The two primitive source E23→25 Noah comparisons have exact J return arrows at existing Flood pivots, but not P returns. A4096-trial coherent-family census, without529 or prior-PJ preselection, finds12 labelled E/J loops at six coordinate configurations. Two MT configurations also admit P returns and coincide with the lower-full430 common P/J repairs. Their shared G2 Noah source joins the two graphs into three vertices3031,3033,3035 and six operator arrows. Every admitted path fromu tov has affine formF(t)=v+E^e P^p J^j(t−u), with(v−u)/2=p+j−e. The402 paths through six arrows yield78 classes with endpoints retained as part of the classification; an algebraic proof gives the normal form for every finite admitted path. At each source vertex, complete P–E and J–E circuits commute with independent multipliers1750/(3K) and7500/(13K), K529. Individual E/P/J operators do not commute. The original17K branch obtained by+215 translation lacks required E pivots3271/3273. All ten common P/J source edges were checked: exactly the two lower regular edges admit an E return; no cumulative edge does. Across2238 circuit-generator/input trials in373 regular coordinates, only the fixed source point survives each circuit. Reduced denominators prove this for every nonempty positive circuit, excluding a global529-preserving chronology action. Source-table and variant reconstruction confirms all52 labelled record slots across12 loops. Endpoints, variants and scalar ratios are sourced; the new affine path grammar is a derived comparison, not an established prescription in the reviewed source passages. Whole950 biographies and a single physical seasonal action are not preserved. No new source endpoint, pivot, Gear scope or canonical graph edit was introduced. C393 is an earlier synthesis. The sourced E/J square connects MT Noah two-year repair to SP primary/LXX Cainan-OFF Shem births; E_(N+2) after J_A equals J_(A-40) after E_N with the transported pivot at next-Gear Shem death. Four coordinate squares have eight labelled contexts and form two two-edge strips. The required P pivots are absent; the two original17K reflections lack three cumulative image points each. Primary500/100 passes; companion/502 modes remain valid but fail the vertical E comparison. A4096-case coherent-family census without529 prefilter recovers the same cases. Source500/600 gives460+598=1058=2K; target1100 differs by42. The1492-evaluation held-Noah E census has29 nontrivial edges and25 maximal paths; only the four old double-E ladders have two edges. Two extra source E23→25 edges are3683→3681 about3706 and3685→3683 about3708, with explicit−215+130+60 context displacement. No global529-preserving E/P/J action is established. C383 is an earlier synthesis. The complete frozen Flood-pivot common-P/J census has six cumulative signed-minus-two and four regular signed-plus-two source quadruples. Two regular lower-full430 contexts are minus215 translations; only original full430 i3/i5 joins retain17K. P/J moving-pivot squares commute exactly when pivot separationD=230h. Finite paths have at most three cumulative or two regular edges. In translated frames the maps share the existing source input as center; operator counts determine the word map, and distinct counts retain distinct slopes. Paired iteration raises cross-spans by4, so it does not preserve17K. Fixed-pivot order defect is±20/897. J300/299 is File02 gap conversion, not calendar Residue Protocol. C373 is an earlier synthesis. At source Flood pivots 460 apart, P=70/69 on 138→140 and J=300/299 on 598→600 reach the same existing two-year repair. All four complete cumulative rows and regular primary i3/i5 pass; companion regular rows fail. Twenty C319 controls leave 22:i3/i5 conditionally. Equal integer gains have family69m→70m,299m→300m,pivot230m, source-qualified side529m; actual460 fixesm2. The pairing is not a reviewed source prescription or a global map. Same-offset comparisons pass; fixed-pivot phase transport fails. Native cumulative Noah950 translates by existing19K to reflectedSP950 with birth/death roles exchanged. The full source bridge matrix has primary69/138 and companion70/140 remainders; the companion459 includes an explicit annual-index difference. The cumulative joined500/1100 comparison has no source support in its frozen menu. C373 retains all earlier qualifications. C363 is an earlier synthesis. Joined460/598→500/600 reaches sourced SP Shem birth/death and changes one original17K span by42;550/529 is diagnostic only. The source Shem600+50+600 rectangle gives1250; Shem-held squaredE maps1058→1250 but has no admitted2056/2058 intermediate. Both new constructions leave02/22:i3/i5, with cumulativea unselected. Phase ancestry and source membership remain explicit. C353 adds the primary458+600=1058 paired source realization with original2K/17K outer endpoints unchanged, the598→650 SP/MT Shem image under E, and bounded partial relations.600 output alone leaves nine primary controls; both598 inputs select22 conditionally. Whole-affine gap conversion is not biography or seasonal-component transport. C343 is the earlier source-ladder synthesis. The primary Noah-held source ladder1058→1150→1250 under E=25/23 has refinement460+490+108+92+100. Companion actual MT Shem gives1149 and the valid502/98 mode gives1152; both retain double-E outer endpoints. C341 distinguishes component-preserving comparisons from one physical-time affine dilation or a biography-preserving map. C334 shared Shem±2 origin is explicit, but exact cumulative/regular endpoint coupling is unestablished in reviewed passages. C342 full regular ladder leaves C319 controls02 and22 at i3/i5; separate17K selects22. C287 full SP domaini2…5 remains; C319 is onlyi3…7. C314 reversal requiresi+j=10 with opposite cumulative ancestry phase; C323 conditionali=j5 uses its extra support requirements. SP primary death retains inclusive/harmonic qualification; matched lifespan950 remains. Source500/100 and502/98 bindings are valid alternatives. Scalar/C89 prior art remains distinct from the Noah-held endpoint realization. Corrected LXX8464 supplement is awareness only:1564→1700,2484→2700,920→1000; AD3746/2300 superseded. Earlier incompleteSeptember17 ZIP differs from the later verified intact original. Original unprefixed reports, Engine reports and C-series remain distinct. No independent witness or source-prescribed selector is inferred from repeated coordinates.

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.

## Verification

75 main checks and 68 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.

## Best next bounded step

C414: Within the fixed C410 regular source menu, enumerate existing endpoint pairs exactly2K=1058 apart and test whether both ends remain sourced after one and two applications of the fixed−500 comparison. Preserve source roles and contexts; compare surviving pairs with the established529 rectangle while keeping its cumulative corners fixed. Search no new intervals, shifts, phases or axes.

The new500 graph has a complete finite source domain, but its precise support for paired529 endpoints has not been classified. This brings the new regular structure back to the Mirror/529 question without treating the2000-year chain as a new research direction. C414 is proposed only and requires a new instruction.

The current authorization ends here. C414 requires a new instruction.