J return loops, three-point path grammar and limits of 529 transport

24 September 2026. Step 10 of 10 in the C394–C403 authorization.

C394–C403 is complete. Six sourced E/J endpoint loops have been identified, and two also admit the existing P return. Those two share the same middle Noah record and form a three-point path structure. Its affine rules are exact, but its source domain is narrow: no nonempty circuit preserves another coordinate in the frozen regular register. C404 has not been executed.

The result extends the 529 investigation with a bounded algebraic structure. It does not establish a general chronology transformation or a prescribed historical use of these comparisons.

What the ten steps established

Step	Bounded result and resulting decision
C394	The two E23→25 Noah comparisons have sourced J returns at Flood pivots3083/3085. The six other required P/J pivot slots are absent; both original17K reflections fail complete cumulative source membership. This led to an affine and biography check.
C395	The two endpoint loops are dilations about their respective source endpoints, with multiplier7500/6877. They are not inverse maps. Matched source components pass, but whole950-year biographies and one physical seasonal action fail. This led to an unfiltered family census.
C396	All4096 coherent family/adjacent-Gear/boundary trials give12 labelled E/J loops at six coordinate configurations. Two MT configurations also admit P returns. This led to analysis of their complete circuits.
C397	Complete P–E and J–E circuits about the same source point commute and have independent multipliers1750/(3K) and7500/(13K). Their constituent operators remain noncommuting. This led to a check against the original17K branch.
C398	Translating the lower common-P/J branch by215 preserves the known P/J comparisons but requires absent E pivots3271/3273. Phase choices do not repair the absence. This led to complete return-pivot closure of the existing P/J census.
C399	Only two of all ten common-P/J coordinate edges admit an E return at an existing source point: the same two lower regular edges. Six cumulative edges and the two upper regular edges fail. This led to joining the two surviving graphs.
C400	The identical MT G2 Noah record joins the graphs into three vertices and six arrows. Every admitted path has a normal form determined by endpoints and operator counts;402 paths through six arrows give78 classes with endpoints retained. This led to a test of the wider source domain.
C401	In2238 circuit-generator/input trials, each map retains only its fixed source point. A denominator proof extends this to every nonempty positive circuit and excludes preservation of the529 lattice. This led to a direct source-warrant audit.
C402	Ten excerpt cards reconstruct all52 labelled record slots from the current tables and variant rules. Endpoints and scalar ratios are sourced; the assembled affine circuit grammar is derived research, with no prescription established in the reviewed passages. This led to consolidation.
C403	Consolidates the results, authenticates the predecessor chain, and adds266 alternate-method checks. The next comparison is proposed below and remains unexecuted.

The two complete source graphs

Write E, P and J for the ratios25/23,70/69 and300/299, with the explicitly stated pivot held in each application. J retains File02's gap-conversion meaning.

Segment	Lower MT Noah	Higher MT Noah	Held E Noah	J Flood pivot	P Flood pivot
G1↔G2	3031	3033	3056	2433	2893
G2↔G3	3033	3035	3058	2435	2895

E carries the higher Noah coordinate to the lower; P and J both carry the lower to the higher. The E pivots are MT Noah records in the other specified chronology context. The P pivots retain their SP/LXX source labels; the J pivots retain the moving MT family's Flood identity. A circuit compares these records; it does not combine them into one source biography.

The middle3033 is the same source record in both segments. Thus the union has three vertices, not four. Across the full E/J census,12 source-labelled comparisons reduce to six coordinate loops: two MT, two SP-companion and two primary SP/LXX coordinate configurations. The companion cases use the existing Flood-close event independently of the minus-one Noah person rail. LXX native Cainan-ON and theoretical removal states remain explicit for each participating record.

For a path from source point u to source point v, with e E-arrows, p P-arrows and j J-arrows, the whole affine map is

F(t)=v+\left(\frac{25}{23}\right)^e \left(\frac{70}{69}\right)^p \left(\frac{300}{299}\right)^j(t-u), \qquad \frac{v-u}{2}=p+j-e.

This follows by composing maps that carry each admitted source vertex exactly to the next. Prime valuations at7,13 and5 recover p,j,e from the slope, so the counts are unique. This is an all-finite-path statement; the402-path computation through length six is a finite verification, not its entire justification. Source endpoints are part of the classification, and coincident affine maps do not erase different pivot histories.

For a closed path, e=p+j. Its two elementary circuit multipliers are

\lambda_P=\frac{1750}{1587}=\frac{1750}{3K},\qquad \lambda_J=\frac{7500}{6877}=\frac{7500}{13K},\qquad K=529.

At each source point c, every closed path acts as $c+\lambda_P^p\lambda_J^j(t-c)$ . Complete circuits about c commute; their positive operator counts give distinct maps. Every nonempty circuit has multiplier greater than one and therefore is not an identity. This grants neither commutation of individual E/P/J operators nor an inverse protocol.

The source and Mirror boundaries

For a common P/J arrow u→u+h, the unique pivot that would give an E return is

V=u+\frac{25h}{2}.

All ten previously admitted common-P/J coordinate arrows were checked. The lower regular branch supplies3056/3058. Its215-year translation to the original17K branch would require3271/3273, neither present in the frozen regular register. Keeping the old E pivot leaves error430/23. The six cumulative arrows require−5913 through−5908, absent in the cumulative menu. Exact algebra and a translated P/J pair therefore do not supply a complete reflected or upper-branch E/P/J loop.

The wider-domain check is equally restrictive. For a nonempty positive circuit, the reduced multiplier denominator is

3^{\max(p-j,0)}13^j529^{p+j}.

For one circuit,1587 or6877 already exceeds every displacement from its source center to another frozen regular coordinate. For two or more circuits, the denominator is at least529². Since all register centers are integers, no other input in the373-coordinate menu can have an integer output under that circuit. The six generator maps therefore yield just six retained inputs in2238 trials: their own fixed points. This is a proof for the stated register and positive circuits, not a claim about every imaginable chronology.

Neither2K nor17K is preserved. Their formal images are3500/3 and29750/3 under the P-circuit multiplier, or15000/13 and127500/13 under the J-circuit multiplier. These are diagnostic spans, not new dates. The occurrence of3K and13K in the loop denominators is exact arithmetic; it does not establish a global529-preserving action.

Source warrant and recovery

C402 checks File18's chronology tables and variant conventions, File02's MT Noah and Flood/gap material, File46's scalar expansion protocols, and File11's scoped Gear increments. All52 loop-record slots reconstruct exactly. The reviewed passages support those inputs and ratios; a prescription to build this particular multi-operator graph is not established. File02 explicitly converts a gap, not a date. That distinction governs interpretation of every J-labelled comparison here.

The separate alternate method forces pivots from source endpoint pairs, composes normalized integer affine triples, merges paths by dynamic programming, checks source membership from the output side, and reconstructs table coordinates independently of the primary calculation. Its266 checks pass, including a byte-identical portable replay using the final frozen inputs. This is a second implementation by the root agent, not an independent-agent or independent-historical-witness claim.

All nine predecessor evidence packets have passed sealed ZIP, payload, certificate, source and isolated-replay checks, with27 confirmed incremental deliverables. C403's own two primary replays agree byte for byte. The final ten-packet integrity record and recovery ZIPs accompany this synthesis as separate delivery artifacts.

Actual File18 and File09 bytes match the retained hashes; File02 and File46 match the C393 frozen source records. File11 is bound to the C402 source manifest. The C265 checkpoint is readable and its certificate matches its retained hash. No exhaustive rereading of historical reports was required. Current corrections take precedence over older archive instructions, including the corrected supplemental LXX8464 comparison; it has not redirected this batch.

Verification and limits

75 main checks and 65 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.

Exhaustive claims apply to the stated frozen menus, role/context conditions and positive circuits. The all-path and all-circuit extensions have explicit algebraic proofs. Source labels, repeated coordinates, verification counts and affine equalities are not independent historical witnesses or proof of intended source pairing. No global529-preserving action or source-prescribed circuit grammar is established. C404 remains 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

C404, proposed and unexecuted: Compare the C400 three-vertex Noah return-loop grammar with the already sourced C385–C388 Noah-to-Shem E/J strips. Test whether the existing vertical E comparisons carry the admitted return paths and circuit normal forms to a source-qualified Shem graph, using only existing endpoints, pivots, chronology contexts and phases. Record any missing pivot or order obstruction. Do not extend Gear scope, add generated source points, or execute a general search.

The present batch closes the source census and establishes a precise local path structure, while the preceding batch supplies a distinct source-qualified Noah-to-Shem comparison. Their finite compatibility is the next direct test connecting the two results. C404 is proposed only and requires a new instruction.

Source E/P/J return loops, three-point path grammar and limits of 529 transport