490d C383 — Finite P/J source transport and moving-frame synthesis

24 September 2026. Step 10 of 10 in the C374–C383 authorization.

C374–C383 are complete. The strongest addition is an exact moving-pivot composition law for P and J, with a clear source boundary and a clear limit on preserving the full529 rectangle.

The same relation that generated the primitive529 side, D=230h, also makes the two mixed operator orders agree. Here D is the separation of the existing Flood pivots and h is their common shift/gain. The source realizations use D=±460 and h=±2.

The surviving composition law

Let J_A(x)=A+(300/299)(x−A) and P_B(x)=B+(70/69)(x−B). When the next source configuration shifts both pivots by h,

P_(B+h) ∘ J_A − J_(A+h) ∘ P_B = (B−A−230h)/20631.

The slopes of these two compositions are equal, and their difference is the displayed constant. Thus the mixed square commutes exactly when B−A=230h. All six admitted consecutive source-edge pairs pass this test.

A regular example uses actual MT Noah birth records:

flowchart LR
    X["MT Noah 3246"] -->|"J: Flood pivot 2648"| A["MT Noah 3248"]
    X -->|"P: Flood pivot 3108"| B["MT Noah 3248"]
    A -->|"P: next pivot 3110"| Y["MT Noah 3250"]
    B -->|"J: next pivot 2650"| Y

The two middle boxes represent the same source record reached along different operator paths. The diagram illustrates the source transport; the affine coefficient calculation establishes equality of the complete mixed maps, not only of this one output.

With fixed pivots, the order defect is±20/897. Therefore the result is specifically about the source configurations with moving pivots.

A reduced coordinate description explains it. Subtract the accumulated shift th from the input frame and(t+1)h from the output frame. P and J then become dilations about the same existing source point:

x₀=A+299h=B+69h.

No new epoch anchor is introduced. For a finite path of length n, a word containing p P-steps has the form

x ↦ xₙ + P^p J^(n−p)(x−x₀).

All24 available P/J word routes verify this formula. Operator order does not matter when counts are fixed. Different counts still give different slopes, although the one admitted starting source reaches the same ending source. This is a precise local composition structure; it is not a claim that P and J are the same operator.

Complete finite source census

The broader test used every ordered pair of distinct already registered Flood coordinates, with existing source inputs and outputs required at the unique common-image intersection.

Only the two-year gain has complete support: coordinate gain−2 cumulative and+2 regular. Neither menu contains a230-year Flood-pivot pair for unit gain. The regular±920 candidates yield scale four algebraically, but lack their required input and output source records.

The four regular common-image configurations are:

The input/output and J-pivot records share restored-MT Cainan ON/Terah+60 context, with adjacent Noah Gears and a Flood start in the output Gear. The P pivot has native-SP and Cainan-OFF-LXX aliases, with matching preceding-Gear Noah birth aliases. These identities remain distinct even where coordinates coincide.

Every regular configuration has the sourced first segment460 and input gap598, giving1058=2K. The lower pair is a uniform−215 translation of the upper pair. It preserves local geometry but does not preserve the17K cross-spans against the fixed cumulative sources.

All24 pairings of the six cumulative and four regular repairs were tested. Only the original full430 configurations at i=3 and i=5 retain both2K sides and both17K cross-spans. The translated copies and tradition aliases do not provide independent witnesses.

Finite paths and the17K obstruction

At any one fixed pivot pair, the shared source repair has no further nontrivial P/J continuation. Source paths appear only when the next existing pivot pair moves with the input:

These are all ten common coordinate edges. Cumulative paths use the retained eight annual members; their third local edge need not belong to the smaller complete-C287 pairing domain. Regular paths stop at Gear3. Neither an extra Gear nor a cumulative Gear operation is introduced.

Pairing the opposite-direction paths yields eight admitted two-step comparisons. All change the source frames by

(S,q,R,N) → (S−2,q−2,R+2,N+2).

The2K sides and endpoint sum S+N survive, but each cross-field separation rises by4. None of the eight paths preserves17K at both steps. The first original rectangle can continue only into a17K+4 frame; the second stops at Gear3. The original529-coupled configurations are therefore not closed under repeated forward repair.

This distinction already appears inside one fixed rectangle. Repairing its cumulative middle point by−2 and its regular middle point by+2 preserves all four outer corners but changes the central cross-span from17K+138 to17K+142. Applying P to the remainder138 would instead give140. The local138→140 conversion must not be mistaken for an expansion of that complete bridge.

Source and phase qualifications

The existing corner exchange r_C(x)=C−x conjugates each cumulative anchored map to its regular counterpart at the paired Flood pivot. This exact affine relation explains their matching structure. On primary rows it also carries the actual input/output source records. On companion rows it would require MT Noah dates one year below the admitted records, which are not supplied.

The corner exchange reverses phase: r_C(x+u)=r_C(x)−u. It does not carry both inherited same-offset seasonal components under one fixed reflection. The original same-offset component comparisons and the fixed-center affine maps remain distinct.

The fixed-pivot census also found four J-only299→300 edges,2946→2947→2948→2949→2950, at successive MT Flood pivots2647…2650. These belong to one restored-SP Noah-death family with its legitimate local companions. A translation of one year carries the matched950-year birth/death pair; the anchored J map instead misses the required birth target by950/299. These unit edges have neither the shared P conversion at the selected SP pivot nor a source counterpart in the cumulative menu. LXX and Flood aliases at their coordinates do not change that source ancestry.

Ten completed decisions

The nine predecessor packets passed isolated replay, integrity and publication checks before this synthesis. A separately written root calculation reversed the search direction—enumerating source endpoint pairs to recover required pivots—and used integer2×2 affine matrices for the composition checks. Its179 checks agree with the main results. It shares the same frozen sources and analyst; no new independent-agent review is claimed. The final completion record covers all ten sealed packets and all30 individual deliverables.

The outstanding connection is to the already retained Noah-held E=25/23 ladder. The new P/J structure supplies a precise comparison to test against it, using existing source pivots and endpoints. That next test is proposed only.

Verification and limits

75 main checks and 61 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 exhaustive statements concern only the frozen menus. Exact local composition does not establish a source-prescribed pairing, independent historical witnesses, unlimited source closure or a global529-preserving action. C384 is unexecuted.

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

C384, proposed and unexecuted: Compare the moving-frame P/J composition structure with the existing Noah-held E=25/23 source ladder. Test a finite mixed diagram using only its already recorded pivots, endpoints, contexts and phases; determine whether a source-qualified commuting square exists and record any obstruction without changing anchors or Gear scope.

The paired P/J comparisons now have an exact source-supported composition law and a known17K closure limit. Connecting that reduced structure to the retained E ladder is the next direct unification question. C384 is proposed only and requires a new instruction.