490d C314 — Reversed index and phase correspondence
24 September 2026. Step 1 of the ten-step authorization C314–C323.
The explicitly reversed index and cumulative phase correspondence succeeds in all five selected source biographies and all ten components. For the fixed inherited formulas
N(i,eN)=3703+i+eN, S(j,eS)=3213−j−eS, D(j,eS)=2613−j−eS,
an exact500 requires i+j=10 and eN=−eS. The integer index term cannot absorb a nonzero half-year phase sum. A finite100-comparison check finds exactly one Noah component for each Shem component.
| Shem j | Noah i | SP state | Noah → Shem birth → Shem death, BC | C287 at i / at j |
|---|---|---|---|---|
| 3 | 7 | G3 primary | 3710 →3210 →2610 | No / Yes |
| 4 | 6 | G3 companion | 3709 →3209 →2609 | No / Yes |
| 5 | 5 | G2 primary | 3708 →3208 →2608 | Yes / Yes |
| 6 | 4 | G2 companion | 3707 →3207 →2607 | Yes / No |
| 7 | 3 | G1 primary | 3706 →3206 →2606 | Yes / No |
All states retain Cainan OFF/Terah0/full430 and the documented regular +155 normalization. Matched physical output phases preserve500,600 and1100; inherited cumulative phases are opposite. C313's zero same-ancestry result is unchanged.
The finite label map J(j,e)=(10−j,−e) is an involution on the selected five indices and two phases. It has five component two-cycles and no fixed exact component; j5 is fixed only as an annual center. In signed cumulative coordinates it is R(q)=−10570−q, with derived local center−5285. The original eight-index field is not closed under this map. This local algebra does not establish a new global Mirror axis.
Direct17K admission survives all five pairings. C287 full four-node support survives at the paired Noah index in three cases and at both indices only in the middle case. For paired i6/7 the required ON-Shem indices8/9 remain outside the frozen field. Source membership is checked against C286; exact Noah ancestry is independently checked against C271. File_18 §3.2.1 and File_09's component/coordinate distinctions supply the source and typing controls.
Verification and limits
60 main assertions and 14 frozen-input/predecessor checks passed. A second main replay produced byte-identical DATA and RESULTS. Separate review scope and agreement are recorded in the evidence. All frozen originals and standing controls remain unchanged. These are verification counts, not independent-witness counts.
A conditional finite correspondence, not a prescribed selector, new global axis, extended source field or repair of earlier phase failures.
Mirror/529 remains the focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction x=1 January AD2; no civil year zero. Regular Gears remain Noah/Shem/Flood only. SP minus-one companion is local to SP Noah/Shem. No cumulative/upstream Gear, new endpoint or phase, fitted anchor, global axis, z-law, canonical or graph edit. Capture now; canonize later. Source state, event identity, generated coordinate and phase ancestry remain distinct. Exact arithmetic and source membership do not establish intention or independent historical witnesses.
Decision after this result
C315, proposed and unexecuted: Can these five reversed pairings extend to the source-labelled Flood and Noah-death parts of each regular SP biography while retaining the existing8395 Flood path?
The500 binding now works conditionally. Testing the remaining admitted biography endpoints asks whether this success composes with the existing Flood interface, instead of repeating the same symmetry.