490d C372 — Primitive integer P/J grammar and529
24 September 2026. Step 9 of 10 in the C364–C373 authorization.
Equal integer gains under the retained P and J ratios have a primitive form that contains 529 exactly. The result is conditional on comparing the two maps for agreement.
Let m be their common positive gain, and require integer input and output gaps. Since J adds x/299 to its input x, integer endpoints force x=299m with integer m. Since P adds z/69 to its input z, equal gain gives z=69m. Conversely, these formulas work for every positive integer m.
| Quantity | Complete positive integer family | Actual sourced scale m=2 |
|---|---|---|
| P input → output | 69m → 70m | 138 → 140 |
| J input → output | 299m → 300m | 598 → 600 |
| Pivot separation | 230m | 460 |
| Whole side, when the first source segment equals that separation | 529m | 1058 = 2K |
| Repaired source partition at this scale | — | 458 + 600 = 1058 |
Thus 2×299−69=529. The existing 460-year source separation forces m=2. C370/C371 supply the actual endpoint support; the arithmetic alone does not.
The extra source condition in the whole-side row is checked explicitly. In all four complete cumulative rows, the native-to-OFF Shem segment equals the ON/OFF Flood pivot separation, both 460. In the two primary regular rows, the SP-to-restored-MT Noah segment equals the matched Flood pivot separation, again 460. Adding that first segment to the 598 input gives 1058. The same sourced repair changes the partition to 458+600 while retaining the outer side. These six applications reuse the same source constants and are not six independent witnesses.
Integer inputs alone would be weaker. The rational agreement family is outer input 13t, inner input 3t, separation 10t, with common gain t/23. Integer outputs require t to be divisible by 23. For example, the purely diagnostic separation 10 gives integer inputs but output 300/23. No such diagnostic number is adopted as a source node.
The regular companion controls retain first segment 459, pivot separation 460, outer input 599 and inner input 139. Their whole side remains 1058, but their P/J gains differ. This shows why the 529 side by itself does not establish the paired conversion.
The primitive 230 is a scalar normal form, not a proposed chronology endpoint or a reason to reopen a primer. P, J and 529 are inherited quantities. What is established here is their exact relation under the tested equal-gain condition, together with the already verified source realization at scale two.
Verification and limits
26 main checks and 19 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.
This is a conditional arithmetic explanation, not a new source prescription, independent witness, date anchor or global operator. The separate source first-segment condition is essential for the529 side.
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
C373, proposed and unexecuted: Consolidate C364–C372 into C373; verify the principal results with a separately written root-agent calculation, replay the ten sealed packets, and publish the synthesis, checkpoint and recovery packages. Leave C374 unexecuted.
The new source-supported relation and its finite exclusions are ready for consolidation. A portable, verified handoff is the best use of the tenth authorized step.