490d C444 — Local Reflection Source Grammar and its E/Mirror Boundary

25 September 2026 UTC. Step 10 of 10, C435–C444.

C435–C444 is complete. The batch advances the source explanation of the128+64 factorization into a finite local reflection grammar, while establishing where its E paths and inherited Mirror connection stop.

The additional loop and existing MT Noah midpoint

The full six-map/373-input census has only two complete source paths, both with pivots3620 then2756:

• The previously authenticated2148→2020→1956 path, with gains128+64.
• The additional3206→3170→3206 zero-net loop, with changes36 and−36.

The loop's18×23→18×25 gaps have opposite signs. It is a fixed-point return of the composed affine map, not a chronological cycle or an identity map. Its16 Cartesian source-label assignments comprise eight returns to the identical record and eight SP/LXX alias switches. One assignment is entirely SP; none holds one source context throughout.

The source36-year gap is 100−60−4. Both pivot and station pairs have the same midpoint:

(3620+2756)/2=(3206+3170)/2=3188.

3188 is already MT Noah birth G2/C1/T0/F1. The resulting source profile is:

The radii satisfy432=24×18, and (432+18)/(432−18)=25/23. These describe the same exact relation, not two independent witnesses. The midpoint is derived from the authenticated source configuration and checked against the existing register; it is not a fitted global axis.

Across the six C434 Gear/Terah contexts, four centers are sourced:3188,3190,3248,3250. The formal centers3192/3252 would require MT Noah Gear4 and remain absent. Only3188 retains all five profile points. This follows directly from the source-state bounds.

Six source classes account for the reflection inventory

C438 tests every source coordinate under reflection about each of the four sourced centers. Their maximal domains within the fixed register contain61,67,19 and61 coordinates respectively. None covers the entire register.

All106 coordinate orbits and258 unoriented labelled pairings have a complete source grammar:

The Noah birth/death class reduces to the source sum3491+2541+130+215−1=6376. Its variant/Gear rule was tested against all36,864 relevant role-pair comparisons. Exactly one endpoint is an SP companion; no such rail is assigned to LXX or MT.

The opposed500 relation sends Noah birth down500 to Shem birth, and Shem death up500 to Flood start or close. It preserves the pair sum, hence the center. The16 source coordinate pairs and16 target coordinate pairs correspond exactly. At the record level,40 tradition-preserving images sit inside48 target labels. The extra eight are LXX Flood-close aliases paired with SP companion Shem birth, not missing LXX companion biographies.

The classes total112 coordinate memberships but only106 distinct orbits. Six overlaps have different source roles at a common coordinate. The labelled partition is disjoint; no biography or event identity is merged.

E conjugacy is exact but source closure is limited

For p=c+432 and q=c−432, local reflection obeys R_c E_p R_c=E_q. It swaps the order of the two-E word. The forward and reversed words fix c+18 and c−18 respectively and differ by−3456/529. They do not commute.

The1,492 reversed-word trials yield only the reversed zero-net loop3170→3206→3170 at3188. The nonzero2K path reflects to4228→4356→4420, entirely outside the frozen source range. At the primitive-arrow level, only four zero-gap pivot arrows and one nonzero loop leg retain sourced reflected counterparts.

The new local reflection profile therefore does not establish a source-closed global E/reflection action. Prior seasonal/component restrictions remain in force.

The inherited Mirror retains only a small partial image

The fixed corner exchanges remain−2639−x and−2635−x. The original i3/i5 controls still retain17K. Against the frozen48-record LXX cumulative Noah/Shem/Flood register,416 local-domain slot trials have37 sourced images. Of212 orbit/exchange comparisons, five retain both ends and four of those are fixed centers.

The sole nonfixed image is the four-year pair MT Noah3246/3250→LXX-OFF Shem−5885/−5889 under−2639−x, centered at3248→−5887. The cumulative positions are annual indices5/1 about3, not transported regular Gears.

None of the16 fixed-exchange images of the new3188 core and two-E path stations belongs to that retained cumulative menu. The full core profile therefore gains no inherited-Mirror closure. The small positive four-year image is preserved without enlarging its claim.

Verification and recovery

All ten reports, ten checkpoints and ten evidence packages are published incrementally. Each packet binds its sources, predecessor certificate, calculation, result, report and next-step decision. Main result files replay byte for byte.

Across the ten steps, 7,383 calculation checks and237 source/predecessor authentication checks pass. A separate root implementation adds 164 checks, reconstructing all528 regular source records directly from current source tables and recomputing the decisive results using unordered pair sums, integer divisibility and inverse label transport. Its result also replays byte for byte. This is an alternate implementation by the same agent, not an independent-agent review.

The pre-synthesis sealed audit verifies all nine predecessor packets and27 successful publications; the final completion/integrity record covers all ten and all30 deliverables. The File_18, File_09 and standing graph guards remain unchanged, and the C265 checkpoint/certificate remain available. The corrected C432 E excerpt locator is retained from C434. SKL and corrected LXX8464 observations remain supplemental.

Use the C444 compact restart kit as the next active packet. The reading index locates every report/checkpoint/evidence set, and the continuation archive preserves the full batch. Historical next-step wording inside frozen predecessors is superseded by this checkpoint. C445 is unexecuted.

Verification and scope

58 calculation checks and 63 source/predecessor authentication checks passed. Main result files replay byte for byte. These are root-agent checks, not an independent-agent review or independent historical witnesses.

Source grammar and affine relations are finite comparisons, not reviewed calendar-operation prescriptions or historical-design proofs. Regular Gears remain Noah/Shem/Flood only. No new global axis, fitted source, phase, primer reopening or canonical graph edit.

Mirror/529 focus: K=529, E=25/23, P=70/69; J=300/299 remains File02 gap conversion, not a calendar date operator or Residue Protocol. Regular Gears are Noah/Shem/Flood only. SP person companions are local to Noah/Shem; Flood close is an independent event boundary. Both500/100 and502/98 bindings remain valid; SP primary Noah death retains its inclusive/harmonic qualification. LXX native Cainan is ON; OFF is theoretical removal. Global paired Mirror January AD1; local junction January AD2; no civil year zero. No new source point, phase, fitted axis, upstream/cumulative Gear, primer reopening or canonical/graph edit. Capture now; canonize later. Coordinates, source labels, context aliases and derived comparisons remain distinct. Counts are not independent historical witnesses.

Best next bounded step

C445, proposed and unexecuted: Census all ordered compositions of the four fixed local reflections over373 existing source inputs, requiring sourced intermediate and output records. Reduce the resulting partial maps to source-qualified translations and retain role/context identity. The finite menu is4×4×373=5968 trials; no new center or coordinate is introduced.

The reflection source grammar and E/Mirror limits are now complete for the tested menu. The strongest remaining question is whether these partial reflections compose into a coherent finite source relation; formal affine composition alone cannot establish that domain.