# 490d C423 — A coherent 2K person profile and its Flood, Mirror and harmonic limits

24 September 2026. Step 10 of 10 in the C414–C423 authorization.

The main new result is a **coherent 2K translation of both Noah and Shem's complete person profiles**. It is unique in the frozen finite person-context menu. Its Flood, Mirror and harmonic extensions have explicit limits.

## The new person profile

The upper context is **SP companion, Gear3, CainanON, Terah+60, full430**. The lower context is **MT, Gear1, CainanOFF, Terah0, short sojourn**. Coordinates retain the regular-register convention; no era or calendar relabelling is introduced.

| Person station | SP companion | MT | Difference |
|---|---:|---:|---:|
| Noah birth |3899|2841|1058 = 2K|
| Shem birth |3399|2341|1058 = 2K|
| Noah death |2949|1891|1058 = 2K|
| Shem death |2799|1741|1058 = 2K|

The translation preserves Noah's950-year and Shem's600-year endpoint lifespans. All six pairwise intervals agree, including the500-year birth interval and150-year death interval. One source-context offset explains all four matches:

**650 +130 +60 +215 +4 −1 =1058.**

The terms are the SP-primary/MT baseline difference, Cainan insertion, Terah shift, sojourn shift, Gear3/Gear1 difference, and the local SP person companion. This is one structured comparison, not four independent witnesses.

C417 examined all96 complete person contexts and4560 unordered pairs without preselecting the tradition, rail, variants or Gear. It found exactly one labelled profile at1058. At the chosen upper settings, SP primary and LXX native both give1059 instead. The companion therefore matters and remains local to SP Noah/Shem.

## How the profile was found

C414 enumerated all existing regular-coordinate pairs at2K:26 coordinate pairs, representing46 labelled endpoint pairs. Six pairs survive simultaneous−500 translation; none survives a second consecutive translation. C415 tested their actual source labels. The six have10 continuous role-defined lifts; nine of19 raw coordinate lifts would switch traditions or rails. The additional3395→2895 bridge from C411 adds no paired2K lift.

Four surviving coordinate cases are the earlier Noah-birth/Flood-start comparisons. Two additional cases share the contexts that produced the new profile:3899/2841 births and3299/2241 mixed Flood boundaries. C416 then tested all four person stations before C417 performed the unfiltered census.

## Flood and Mirror limits

The selected **SP Flood close3299 / MT Flood start2241** adds a fifth2K station. Both columns then have the ordered partition:

**Noah birth —500→ Shem birth —100→ selected Flood boundary —350→ Noah death —150→ Shem death.**

The boundary labels are essential. SP start3300 versus MT start2241 differs1059, as does SP close3299 versus MT close2240. All18 same-boundary comparisons across the three available event Gears remain odd and cannot equal1058. Applying−1058 to the full SP Flood interval gives MT Gear2 close2242 and MT Gear1 start2241, not one same-Gear MT Flood year. This preserves the distinction between SP person companions and independent Flood boundaries. Both valid500/100 and502/98 modes remain available.

The five station pairs were tested against all six sourced cumulative Shem/Flood2K pairs. **None of30 joins has both cross-spans equal17K.** All ten role-specific cumulative endpoints required by those equations are absent. Neither inherited corner exchange, C=−2639 or−2635, gives a sourced cumulative image in the20 tested slots. No new axis or endpoint was fitted.

The new profile therefore supplements the2K findings without replacing the original i3/i5 two17K joins. C414 also retains the older−500 result: moving the original regular side preserves1058 but changes its fixed cumulative cross-spans from8993 to8493.

## One harmonic endpoint, with no sourced path

For each of the five established upper stations, C420 held that coordinate and applied E, E², P and J to its paired lower coordinate. **One of20 outputs is sourced**:

**Held SP companion Noah3899: MT Noah2841 → MT Flood close2649 under E²=625/529.**

The input gap1058 expands to1250. The output is MT Gear3, CainanON/Terah+60/full430 Flood close. The scalar1058→1250 identity is prior arithmetic; the source context and endpoint roles are the new comparison. Its E intermediate2749 is absent. No other tested E or E² output is sourced, and all P/J outputs are nonintegral. J remains a gap conversion, not a general calendar operator.

C421 asked whether other sourced E pivots could produce this same complete affine map. The required equation is25p+23q=48×3899. Even allowing all373 existing regular coordinates as possible pivots, its only sourced solution is p=q=3899, with the same missing2749 intermediate. Integer solutions have p=3899+23t and q=3899−25t; the source maximum3900 forces t=0.

Prime7,13 and5 valuations show that every finite positive E/P/J word with this whole-map slope has exactly two E factors and no P or J. Thus a longer positive word cannot repair the missing source path while retaining the same affine map. This does not exclude different-slope endpoint coincidences or inverse operators, which were not part of the test.

## Source and component warrant

C422 retained twelve current excerpt cards and directly reconstructed all23 relevant person, event and rail-control records. The explicit source passages support endpoints, variant rules, qualified lifespans and named bindings. The cross-context2K selection, finite census, mixed-boundary profile and squared-E endpoint remain derived comparisons; the reviewed passages do not prescribe that combined construction.

The translation preserves the two matched inherited quarter-offset components at each of five stations:10 passes among20 assignments. Fixed held3899 E² preserves none of its four input/output component assignments; matched-offset errors are±24/529. Jointly shifting both pivot and input preserves the common offset, but changes the pivot and is a distinct comparison. No physical-time fixed-pivot action or new phase follows from the integer endpoint equality.

## Ten completed steps and recovery

| Step | Bounded result |
|---|---|
| C414 | Complete26-pair2K census; six paired−500 translations, no second consecutive translation |
| C415 | Ten valid record lifts; nine alias substitutions rejected |
| C416 | Four-station, biography-preserving SP companion/MT2K profile |
| C417 | Unique labelled profile among4560 person-context pairs |
| C418 | Explicit fifth mixed-boundary station; full same-boundary Flood-year obstruction |
| C419 | No fixed17K join or inherited corner-exchange closure |
| C420 | One sourced squared-E endpoint; missing intermediate |
| C421 | Same-map source-path exclusion for all finite positive E/P/J words |
| C422 | Direct source reconstruction and component audit |
| C423 | Synthesis, separate root verification, sealed integrity and restart preparation |

Each research decision was reassessed after its predecessor and recorded in NEXT_STEP_DECISION.json. Each report, checkpoint and evidence packet was completed and published before the next research step. C423 freezes all nine predecessor DATA/RESULTS files, certificates and decisions. The preceding nine sealed packets passed CRC, payload hashes, certificate and predecessor checks, and18 isolated replay-output comparisons before this synthesis.

A separate root-agent implementation imports no main calculation modules. Its595 checks reconstruct all528 regular records and independently compare the pair census, record lifts, context histogram, boundary parity, fixed Mirror requirements, integer affine outputs, all-pivot squared-E equation, prime-factor restriction and quarter components. Its result replays identically against the C423 frozen inputs. This is an alternate implementation by the same agent, not an independent historical witness or a new independent-agent review.

Use the C423 compact restart kit as the active packet after publication. The reading index locates every report, checkpoint and evidence ZIP. The complete C414–C423 continuation archive and its integrity record preserve the full batch. Current decisions and corrections override older archived next-step paragraphs. **C424 remains unexecuted.**


## Verification and limits

75 main checks and 63 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.

Finite uniqueness is relative to the96-context source menu. Endpoint, path, phase and boundary conclusions retain their stated domains. No combined source prescription, global529/Mirror action, historical probability or canonical graph edit is established. C424 has not been executed.

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

**C424, proposed and unexecuted:** Audit the16 labelled Flood-start/Noah-death2K pairings already enumerated in C414 as one closed class. Retain their exact source contexts and test whether an additional documented station in the same contexts supports a coherent correspondence, without adding sources, pivots, phases or Gear scope.

The same-role person profile and its immediate extensions are now bounded. The largest remaining mixed-role class in the existing2K census is a concrete next target, with a fixed finite menu and no need for a new anchor. This is proposed only; the C414–C423 authorization ends here.