490d C456 — E-Square Integrality and Source Admission Gates
25 September 2026 UTC. Step 12 of 25, C445–C469.
For integer input and effective pivot, a two-E output is integral exactly when the input gap is divisible by529. The13 sourced-pivot words have17 integer-output cases,15 sourced outputs and11 complete paths. Among nonzero gaps the gates narrow4→2→1; only2148→2020→1956 survives.
The proof is direct: y−H=625(x−H)/529 and gcd(625,529)=1. Conversely, for integer endpoints of any two-E word, H−x=529(x−y)/96. Since gcd(529,96)=1, H is integer exactly when96 divides the endpoint displacement. Thus the529 numerators attached to the four rational effective pivots in C455 are algebraically forced.
| Sourced H | Input | Middle | Output | Admission |
|---|---|---|---|---|
| 3206 | 2148 | 2020 | 1956 | Complete2K path |
| 3208 | 2150 | 2022 | 1958 | Middle absent |
| 3266 | 2737 | 2655 | 2641 | Middle and output absent |
| 3268 | 2739 | 2657 | 2643 | Middle and output absent |
The distinction is now exact: divisibility supplies an arithmetic candidate; source admission supplies the evidence. Repeated529 factors generated by the same E² identity are not extra historical witnesses.
The remaining positive question can be stated without an effective pivot. A common affine map expands the gap between any two inputs by its slope. Applying the same sourced word to both members of an actual529 pair therefore asks whether a complete interval ladder exists, with both endpoint histories retained. This differs from asking whether one endpoint is reachable from the other.
Verification and scope
4866 calculation checks and 19 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.
The absence claims are restricted to the frozen source register. No missing2022,2655,2641,2657 or2643 record is generated or admitted.
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
C457, proposed and unexecuted: Starting from all60 actual529/1058 source pairs, compute full pair reachability when the same admitted E or reflection is applied to both endpoints, preserving endpoint ancestry and every intermediate source pair.
Single-vertex reachability excludes529 endpoint connections, while E still transports intervals. A full shared-word pair closure tests the meaningful interval question directly and has a finite stopping condition.