490d C455 — Two-E Effective Pivots and Sourced 529 Input Gaps
25 September 2026 UTC. Step 11 of 25, C445–C469.
Across23,872 input/word trials,64 ordered E pairs yield15 complete paths. Thirteen words have a sourced effective fixed point;11 complete paths use one. The only nonzero529-multiple input gap is the established1058 gap atH3206, with2148→2020→1956. Four additional paths have rational, unsourced effective pivots.
For first pivot a and second pivot b, the word has slope625/529 and effective fixed point H=(25a+23b)/48. That H is an algebraic parameter; a source-coordinate test is required separately.
| Complete path with unsourced H | H | H−input |
|---|---|---|
| 2735→2733→2731 | 66169/24 | 529/24 |
| 2620→2608→2520 | 76105/24 | 13225/24 |
| 2746→2670→2582 | 87593/24 | 21689/24 |
| 3105→3055→3081 | 12949/4 | 529/4 |
The remaining11 paths are eight pivot self-paths, the forward and reverse station returns, and the nonzero2K path. No rational effective pivot is promoted to a source event or new axis.
The repeated529 numerator in the table deserves an algebraic control before being interpreted. Since y−H=(625/529)(x−H), the effective gap can be expressed directly in terms of the integer endpoint displacement. The next step will distinguish this forced denominator arithmetic from the genuinely restrictive availability of all source points.
Verification and scope
23872 calculation checks and 20 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.
Effective fixed points are comparison parameters. Only13 of64 are sourced, and only one complete nonzero source-qualified529-gap expansion is found in this menu.
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
C456, proposed and unexecuted: Prove the exact E² integrality criterion and stage the13 sourced-effective-pivot words through integer output, sourced output and sourced intermediate gates, separating forced529 factors from source admission.
The finite census has isolated rational effective pivots. An explicit integrality theorem and admission-stage counts prevent automatic529 factors from being treated as additional source evidence.