# 490d C371 — Paired-map uniqueness, selectors and phase controls

24 September 2026. Step 8 of 10 in the C364–C373 authorization.

**Agreement is unique for the specified pivots, and it selects the same two primary controls already retained.** Use an oriented coordinate with the J pivot at 0 and the P pivot at D. For an input x:

`J image = (300/299)x`

`P image = D+(70/69)(x−D)`

Their difference is `(10/897)x−D/69`. Consequently they agree only at `x=13D/10`, with common output `30D/23`. At the sourced separation D=460 this is exactly 598→600, with the inner gap 138→140. The two affine maps remain different operators.

| Test in the fixed twenty-control C319 menu | Source-admitted survivors |
|---|---|
| Cumulative agreement: a=2 |20 and 22, each at i=3,4,5|
| Regular agreement: b=2, primary rail |02 and 22, each at i=3,5|
| Joint agreement |22 at i=3,5|

The cumulative input gap is 600−a; the regular gap is 600+e−b, where e=1 for the companion rail and 0 for primary. These formulas account for the selection directly. Formal agreement outside the frozen source ranges is not promoted to a source hit. The joint result equals C368’s intersection obtained from fixed-19K Noah transport and the joined regular comparison. These are related consequences of the retained constants, not independent historical witnesses or a source instruction choosing the controls.

The alternative companion boundary is informative. Holding the MT Flood close but using the SP Flood start makes the pivot separation 461. P alone then maps the actual 138 gap to 140 and reaches next-Gear MT Noah birth. J still misses that source by 1/299, so the two maps do not agree. A standalone source hit cannot replace the matched-boundary joint criterion. For separation 461, agreement would require the unsourced input gap 5993/10.

All twelve inherited same-offset components pass when each input, pivot and output carries its matching ±1/4 component. With the two pivots instead fixed at their centers, changing the input by u produces a P−J discrepancy `(10/897)u`, equal to ±5/1794 at the inherited offsets. Neither fixed-pivot image equals the target’s same-offset component. Component-preserving comparisons therefore do not establish one physical-time affine map across the phase pair.

## Verification and limits

93 main checks and 17 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.

The agreement condition is imposed for this test, not established as a textual prescription. Boundary and phase alternatives retain their identities. No new pivots, integer rounding, source endpoints or chronological anchors are adopted.

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

**C372, proposed and unexecuted:** Derive the primitive integer form of equal gains under P=70/69 and J=300/299, then bind its scale to the sourced 460-year separation and the existing 529 rectangle. Keep the derivation conditional on the tested agreement criterion.

The unique intersection exposes a compact arithmetic structure. Proving its integer form can show exactly which part explains 529 and which part still depends on source and comparison choices.