Three manuscript traditions, eighteen named inverse paths

Research continuation C1433–C1482 · September 29, 2026 UTC

Status: 50 of 50 actions completed and reassessed. Cycle complete.

What the study adds

The six primary inverse spines are MT, LXX and SP in Regular and Cumulative mode. This continuation holds their source states fixed and tests one source-named 600-year interval in each: Noah–Flood in Regular chronology, and the serial Shem lifespan in Cumulative chronology. Each source trunk is compared at the Flood division, at Noah/Shem alone, and with both nodes retained.

MT Regular and SP Cumulative preserve their heads through all three partitions. LXX Regular and MT Cumulative preserve the head when the division moves but add 990 when both internal nodes are retained. The latter agreement comes from cancelling equal local failures of additivity.

The SP cumulative refined upper branch translates all three selected nodes together: 4716/5316/13396 becomes 2736/3336/11416, a shift of −1980. Its inverse Shem node 3336 matches native Rounded LXX Regular Shem. With the Nativity leg included, its inverse Flood is 5436, matching Rounded MT Cumulative Shem. These are precisely tagged coordinate interfaces, not identifications of their source roles.

The pre-existing C1432 Covenant Key and the supplied LXX–SP 1700 relation now form one closed circuit:

The direct Key route and the path through 13306, 11606 and 14306 share source inputs. They connect the families without providing independent statistical evidence.

These findings are integrated into the updated How the chronological families fit together, especially its new chapter on pages 14–16 of the printable edition.

All eighteen outcomes

The heads below are measured from held Conquest 1406 BC. “Both” retains Flood and Noah/Shem, so it has three original intervals. Adding the original Nativity leg and reversing it once raises every head by 2700; it does not reverse any output again.

For original legs a,b, the local differences are:

This one law reproduces the entire comparison. The primary SP Fall construction remains accepted. MT Regular alternatives and the MT Cumulative both-node 13016 branch were already established; the systematic LXX/SP comparison and complementary classification extend those results.

Other useful explanatory results

• MT and SP primary cumulative source heads and both inverse heads preserve a 610-year gap. The original component differences 120+490 become 2100−1490 after reversal.
• The LXX–MT cumulative source upper-leg difference −20 becomes −2000; with their fixed lower difference 910, the source head gap 890 becomes inverse −1090. This explains their change of order.
• The LXX–SP cumulative source head difference 1500 becomes inverse −480 because LXX contracts 1980 more. This is the dependent third side of the preceding cumulative comparisons.
• The mode interaction is 1170−480=690=500−(−190); one manuscript-wide head translation would erase it.
• The existing LXX 386 BC paired-Key join follows from its 500-year head difference and the 40-year Exodus–Conquest target difference. The same fixed comparison gives output gaps 1000/23 for MT and 60 for SP.

Exact limits and corrections retained

The Covenant Key maps the specified LXX Regular head to the SP Cumulative head, while the corresponding internal primary nodes do not match under that same affine operation. Applying it to both two-stage heads with Covenant 1866 unchanged gives a signed residual of −5400/23. An initial positive-sign expectation during this cycle was corrected from the actual calculation; no source input changed.

The Prophetic route to 13306 followed by Priestly expansion about Rounded MT 4106 gives 14106. It reaches the 10000-year pair 14116→Actual MT 4116 only by translating both endpoints by ten. Moving only the held anchor changes the output by −20/23, not +10.

The two-stage/three-stage labels in inherited model keys path2 and path3 identify the 1406-held and Nativity-extended contexts. A refined path retaining both internal nodes has three or four original intervals, respectively. The reader uses “Conquest-held” where this distinction matters.

No second decimal reversal, unconstrained partition scan, new statistical rarity claim, or universal conversion of full genealogies was performed. The research question was the selected source spines and their named refinements.

Source and verification basis

The full File52c copy matches the C1432 frozen edition. The attached shorter edition is retained as an older source record. The supplied September 28 comparative study controls the LXX and accepted SP inputs. The frozen native Rounded LXX model binds Noah 3836 and Shem 3336; Files51a, 47 and53 support the named-node constructions. model/source_bindings.json identifies the exact editions. Existing source files and the sealed C1431/C1432 records remain unchanged.

verify_cycle.py recomputes decimal reversal arithmetically with integer division, independently of the construction code’s string method. It checks the eighteen paths, local laws, Key relations, source hashes, reader integration and sequential journal. The final command is python3 verify_cycle.py --require-50. Check counts describe verification groups, not independent discoveries.

Sequential actions and reassessments

C1433 — Set the comparative spine question

Question: Which source-controlled internal relations extend the six inverse trunks beyond their known head equalities?

Result: Use named Noah/Shem partitions and source-derived connections to test how far the common construction extends.

Reassessment: Resolve the source editions before reconstructing the six objects.

C1434 — Resolve and freeze controlling sources

Question: Does the attached short File 52c supersede the full edition already used, and which document controls LXX/SP?

Result: The full File 52c remains the MT control; the September 28 comparative study controls the native LXX and accepted SP Fall paths.

Reassessment: Register exactly those six source pairs, keeping the source heads and pivots explicit.

C1435 — Register the six declared source pairs

Question: What are the complete source and one-pass paths for each manuscript and mode?

Result: The six objects use one reconstruction rule with distinct original intervals and source roles.

Reassessment: Compare against the inherited results and identify what is already explained.

C1436 — Separate inherited results from the new question

Question: Which relationships are established before this cycle?

Result: Primary heads and C1432 remain inherited results; this cycle tests their internal extent and explains their connections.

Reassessment: Use the shared tail to reduce duplicate work across anchor stages.

C1437 — Derive the universal stage translation

Question: Does adding the original Nativity leg act uniformly on every complete spine?

Result: Adding the reversed 1400 tail translates the reconstructed Conquest-and-earlier suffix by 2700 for all six objects.

Reassessment: Analyze mode differences at one stage; the second stage follows from this common translation.

C1438 — Reconstruct MT convergence at component level

Question: What compensates when MT switches from Regular to Cumulative source measurement?

Result: MT switches between source measurements differing 9900, but its reversed component changes−1580 and+1580 cancel.

Reassessment: Test whether the LXX divergence has the same local compensation structure.

C1439 — Reconstruct LXX residual at component level

Question: Which component prevents LXX Regular/Cumulative convergence?

Result: The LXX lower-leg difference 530 is only offset by 30 in the upper leg, leaving the known 500-year residual.

Reassessment: Test the accepted SP state without changing its Fall choice.

C1440 — Reconstruct SP residual at component level

Question: What causes the accepted SP cumulative head to fall 190 years later than its regular head?

Result: SP cumulative/regular mode changes−5770 and+5580 leave−190, under the same accepted Fall convention in both modes.

Reassessment: Explain why these changes lie in different decimal registers before testing new partitions.

C1441 — Explain gains by decimal register

Question: Can the signed reversal gains be computed from the source digits without treating them as unrelated constants?

Result: For abc 0 spans the gain is 990(c−a); for SP regular ab 00 spans it is 900(b−a). Source digits account for the six signed gains.

Reassessment: Use these register laws to distinguish forced residues from exact cancellations.

C1442 — Locate the obstruction to cross-mode coincidence

Question: Are LXX/SP nonzero head differences avoidable without altering their declared source states?

Result: MT and LXX primary spans preserve gaps modulo 990; LXX residue 500 prevents convergence. Across SP digit widths the safe common modulus 90 leaves residue 80, also preventing convergence.

Reassessment: The next test should change only source-named partitions, not tune inputs toward convergence.

C1443 — Bind the named600-year hinge in both modes

Question: Which additional internal node can be used consistently in all six source paths?

Result: Regular uses Noah birth 600 before Flood; Cumulative uses the Shem lifespan boundary 600 before Flood. These are separate role-tagged source paths.

Reassessment: Test MT Regular Flood-only, Noah-only and both-node partitions first, retaining the same endpoints.

C1444 — Test MT_regular at Noah and Flood

Question: Do the two named coarse partitions and the path retaining both source nodes have the same inverse head?

Result: MT Regular keeps its established head under Flood-only, Noah-only and the path retaining both nodes.

Reassessment: Apply exactly the same 600-year change to native LXX.

C1445 — Test LXX_regular at Noah and Flood

Question: Do the two named coarse partitions and the path retaining both source nodes have the same inverse head?

Result: LXX Regular shares the Flood/Noah coarse endpoint, but retaining both source nodes raises the head 990.

Reassessment: Test SP without modifying its accepted Fall head or Noah 600 interval.

C1446 — Test SP_regular at Noah and Flood

Question: Do the two named coarse partitions and the path retaining both source nodes have the same inverse head?

Result: SP Regular changes by−6300 at the Noah-only partition and by−1800 when both nodes are retained.

Reassessment: Because the Regular responses differ, test the same source-named 600 interval in Cumulative mode next.

C1447 — Test MT_cumulative at Shem and Flood

Question: Do the Flood-only, Shem-only and both-node partitions agree under the same accepted source state?

Result: MT Cumulative repeats the LXX Regular response: the two coarse splits agree, while the retained 600 refinement adds 990. Its 13016 refined head is inherited, not a correction.

Reassessment: Test whether native LXX Cumulative shares that balanced response.

C1448 — Test LXX_cumulative at Shem and Flood

Question: Do the Flood-only, Shem-only and both-node partitions agree under the same accepted source state?

Result: LXX Cumulative adds 990 both when the partition moves to Shem and when both nodes are retained.

Reassessment: Test whether SP Cumulative instead supplies the fully stable counterpart to MT Regular.

C1449 — Test SP_cumulative at Shem and Flood

Question: Do the Flood-only, Shem-only and both-node partitions agree under the same accepted source state?

Result: SP Cumulative preserves its accepted head under all three partitions, matching the MT Regular response.

Reassessment: Collect all retained intermediate nodes, rather than stopping at head totals.

C1450 — Retain the complete named-node comparison

Question: What do the eighteen paths do to their internal 600-year edge and source head?

Result: Every refined path preserves the explicit 600-year edge; its placement and the final head respond differently to the surrounding source digits.

Reassessment: Explain the whole eighteen-path table using the two local merging defects.

C1451 — Derive the two local partition laws

Question: Can two local additivity differences reconstruct every observed change of partition?

Result: Two local defects explain all eighteen results: alternate−primary=dL−dU; refined−primary=−dU.

Reassessment: Classify the exact shared responses without confusing cancellation with local additivity.

C1452 — Identify the complementary manuscript-mode pairings

Question: When is invariance genuine at both merges, and when does it arise from cancellation?

Result: MT Regular and SP Cumulative have two zero defects; LXX Regular and MT Cumulative have two equal−990 defects. The second pair preserves coarse totals by cancellation while refinement adds 990.

Reassessment: Inspect the complete source-node displacement vectors for an actual multi-node translation.

C1453 — Locate translations of complete selected upper branches

Question: Does any refined path translate its Flood, Noah/Shem and head by the same amount?

Result: Both refined SP upper branches translate uniformly: Regular by+5400 and Cumulative by−1980. The completed SP cumulative branch is its source branch+720 after the common 2700 tail change.

Reassessment: Relate SP cumulative preservation to MT cumulative head preservation, keeping intermediate differences visible.

C1454 — Explain the preserved MT–SP cumulative610 gap

Question: Why does the MT–SP cumulative source-head gap survive both primary inverse stages?

Result: The common−1980 contraction preserves the 610 head gap, while a 1980 transfer changes component differences(120,490) to(2100,−1490). It preserves heads, not the full primary path.

Reassessment: Compare the LXX upper-leg change, where the same contraction no longer applies.

C1455 — Explain the LXX–MT cumulative order change

Question: How does a 20-year upper-leg source difference become a 2000-year reversed difference?

Result: The retained lower gap 910 combines with upper−20 before reversal and upper−2000 after it: source 890 becomes inverse−1090. The changed decimal place explains the reordered heads.

Reassessment: Use the MT/SP preserved 610 gap to explain the third cumulative comparison without treating it as independent.

C1456 — Close the cumulative three-manuscript comparison

Question: Does the LXX–SP reversal of head order follow from the two already explained component relations?

Result: The LXX–SP cumulative source gap 1500 becomes inverse−480 because LXX contracts 1980 more. This is the dependent third side of the cumulative comparison.

Reassessment: Check how the Regular/Cumulative switch interacts with the LXX–SP manuscript change.

C1457 — Measure manuscript–mode interaction

Question: Can manuscript change and mode change be represented by one uniform head translation?

Result: LXX→SP shifts Regular heads 1170 but Cumulative heads 480; the 690 interaction is exactly 500−(−190). A manuscript-wide constant translation would erase this source mode dependence.

Reassessment: Look for source-named interior incidences in the stable SP cumulative branch instead.

C1458 — Connect the new SP interior to LXX and MT Shem

Question: Does the refined SP cumulative path reach independently named Shem coordinates in the other traditions?

Result: The SP cumulative refined Shem node 3336 coincides with native Rounded LXX Regular Shem; its completed inverse Flood 5436 coincides with MT Cumulative Shem. Their path roles and anchors remain explicit.

Reassessment: Connect these source-derived branches to the existing Covenant head map without claiming a full genealogical conversion.

C1459 — Determine the scope of the Covenant head map on complete paths

Question: Which corresponding nodes of the declared LXX Regular and SP Cumulative paths are carried by the existing Covenant Key?

Result: The Covenant Key relates the heads; it does not map the other three corresponding primary nodes. The newly proved SP branch translations are different, explicitly local maps.

Reassessment: Close the known head-level connections into a source-defined circuit, preserving this scope.

C1460 — Close the two routes from LXX Regular to SP Cumulative

Question: Does the Covenant Key agree with the supplied Prophetic/SP1700/stage/mode route?

Result: Both routes reach 14116: direct Covenant Key, or 13136→13306→11606→14306→14116. The original SP1700, common 2700 and SP mode−190 now form one connected circuit.

Reassessment: Trace dependencies rather than counting the circuit as an independent numerical confirmation.

C1461 — Link the source490 coefficient with preserved610 and the Covenant circuit

Question: How do the local SP cap reductions participate in the three-head comparison?

Result: The source decomposition 610=120+490 and the Covenant gain 980=2×490 join the MT/SP head relation to the LXX/SP Key relation. Their 1590 total is the same three-head triangle, not extra evidence.

Reassessment: Test an apparently similar two-Key route while retaining its actual held endpoints.

C1462 — Test the adjacent Prophetic–Priestly composition

Question: Does expanding the 13306 head about Rounded MT Creation 4106 reach the accepted SP cumulative head?

Result: The composition gives 14106, not 14116. It produces a 10000-year pair that becomes the SP14116→Actual MT4116 pair only after both endpoints translate+10.

Reassessment: Check whether changing only the anchor could legitimately replace that whole-pair translation.

C1463 — Distinguish anchor change from whole-pair translation

Question: Does moving only 4106 to 4116 close the ten-year two-Key discrepancy?

Result: Changing only the held anchor adds−20/23, not+10; translating source and anchor together adds 10. The source states cannot be exchanged silently.

Reassessment: Use a genuinely shared pair of source-appointed anchors to understand the LXX386 convergence.

C1464 — Explain the LXX paired-Key join by its mode gap

Question: Why do LXX cumulative-to-Exodus and regular-to-Conquest Priestly routes meet at 386?

Result: The 500-year LXX head gap is exactly(25/2)×40, the condition for the two head-held Priestly routes to share a target. The known 386 join is an algebraic expression of that compatibility.

Reassessment: Apply the same fixed Exodus/Conquest comparison to the other two manuscripts, retaining every result.

C1465 — Compare the paired-anchor criterion across all three manuscripts

Question: Does the same head-held Priestly comparison join for MT and SP as well as LXX?

Result: At the same declared anchors, the resulting separations are MT1000/23, LXX0 and SP60. Only the LXX source head gap meets the shared-target condition.

Reassessment: Check how the common stage shift interacts with the Keys before treating the closed circuit as stage-independent.

C1466 — Measure stage–Key interaction

Question: Does translating a spine by 2700 commute with the existing anchored Key relations?

Result: Keeping Covenant 1866 fixed sends the two-stage LXX head 5400/23 later than the SP target (signed residual−5400/23). The LXX paired join remains equal between modes at 14278/23.

Reassessment: Group successful equalities and exact misses into a compact explanatory dependency structure.

C1467 — Compress the findings into dependency families

Question: Which results add structural understanding, and which are consequences of already selected inputs?

Result: Promote the comparative partition law, SP selected-branch maps and closed source-named circuit. Keep residue tests and exact nearby misses in the research record.

Reassessment: Express a compact reconstruction contract sufficient to reproduce the complete declared comparison.

C1468 — State the reconstruction contract

Question: What must remain attached to a spine so that its relations are reproducible?

Result: One operation applied to six source pairs and three named partitions reconstructs all eighteen paths; source identity, partition and anchor are essential inputs.

Reassessment: Review the independently produced algebra and source controls before drafting the new explanation.

C1469 — Reconcile independent algebra and source reviews

Question: Do independently checked partitions, source states and novelty classifications agree with the cycle results?

Result: Independent arithmetic and source reviews support the comparative extension; inherited MT results are marked explicitly and selected branches retain their source roles.

Reassessment: Write the integrated explanation around the strongest results, with exact alternative heads retained in the research evidence.

C1470 — Draft the comparative partition explanation

Question: How can the verified path relationships be explained compactly without burying the main Strategy goal?

Result: Draft leads with the complementary partition law, then the SP branch and named Shem interfaces, and finally incorporates C1432 through the closed Covenant circuit.

Reassessment: Integrate the findings into the existing explanation and update its overview and evidence scope.

C1471 — Integrate findings with the existing family explanation

Question: Can the new chapter strengthen the whole explanation while preserving unrelated source-family sections?

Result: The new findings are integrated into the main explanation, overview and closing account; unrelated genealogy, Keys, Covenant and NT sections are preserved.

Reassessment: Have the actual integrated prose checked against its sources and exact arithmetic before rendering.

C1472 — Render the revised printable explanation

Question: Does the expanded report retain readable equations, tables and chapter flow in its printable form?

Result: The revised report rendered successfully with no renderer warnings.

Reassessment: Inspect rendered pages and equations rather than relying on a successful build.

C1473 — Inspect the printable report layout

Question: Are the new chapter, tables and long circuit equation readable and unclipped in the rendered pages?

Result: The 22-page printable report retains readable tables and equations; the new chapter fits on pages 14–16.

Reassessment: Apply the independent source review to the actual integrated prose before final arithmetic replay.

C1474 — Review the integrated prose against source roles

Question: Does the new reader chapter accurately distinguish inherited results, new comparisons, node roles and accepted states?

Result: The source review found no required correction. The 3336 incidence is bound to the frozen native Rounded LXX model; SP branch and Covenant scope are correctly stated.

Reassessment: Run the portable independent arithmetic verifier against the complete accumulated models.

C1475 — Replay the complete mathematical comparison independently

Question: Does numeric digit reversal and exact rational arithmetic reproduce all recorded model results and their bound sources?

Result: Independent arithmetic replay passes, including all eighteen paths, exact Key relations and signed counterexamples; the source bindings and sequential chain agree.

Reassessment: Verify the reader artifact hashes and formatting separately from the arithmetic models.

C1476 — Verify editorial integrity and retained content

Question: Does the delivered Markdown match the reviewed integration and preserve unrelated sections?

Result: The report matches the reviewed text; all tables have consistent columns and display math is balanced. The portable verifier now checks editorial hashes and preserved sections as well.

Reassessment: Write a readable research record that distinguishes structural gains, inherited comparisons and exact misses.

C1477 — Write the research synthesis and step register

Question: Which concise record will make all fifty reassessed actions reviewable without fifty separate reports?

Result: The research synthesis records every tested partition, substantive connection, exact miss and step reassessment. One anchor-context label was clarified in the main report.

Reassessment: Prepare a self-contained evidence packet and instructions for independent replay.

C1478 — Prepare the portable evidence structure

Question: Can the research and updated explanation be reviewed without the original working directory?

Result: The packet can reproduce the arithmetic and preserve the explanation, source editions and complete reassessment chain.

Reassessment: Test its portability in a separate directory before final sealing.

C1479 — Check detached replay before sealing

Question: Does the verifier depend only on the included relative files?

Result: Detached replay succeeds without original workspace dependencies; all arithmetic and editorial models are accounted for.

Reassessment: Confirm the final printable wording and preserve the independent visual review.

C1480 — Confirm the final reader after the anchor-label clarification

Question: Did the small Conquest-held wording correction preserve layout and the reviewed mathematical content?

Result: The final PDF remains 22 pages with a clean new chapter on pages 14–16. The clarified anchor label introduces no layout problem.

Reassessment: Assess whether the retained findings improve the common explanation and choose the next bounded question.

C1481 — Select the result and next bounded question

Question: Which claims should carry forward as the actual advance toward the Strategy goal?

Result: The advance is a connected account of path behavior, selected multi-node maps and Covenant links. The next question extends the proven local rule to the existing Noah–Shem lifespan block.

Reassessment: Read the full Strategy afresh and close the 50-action cycle against its larger objective.

C1482 — Review the full Strategy and close the cycle

Question: Do all fifty actions serve a compact explanation of the families rather than accumulating unrelated numerical matches?

Result: The complete 50-action cycle meets the Strategy goal through a common local partition law, exact selected-branch maps and a connected Covenant circuit; the source-dependent limits are retained.

Reassessment: Stop this cycle and deliver the revised explanation and reproducible research record. C1483 remains unexecuted.

Strongest next bounded question

Can the same local partition law explain the already source-appointed MT cumulative Noah refinement, and its LXX/SP counterparts, without changing anchors or accepted source states? Freeze the distinct Regular age and Cumulative lifespan meanings first. The present 600-year family supplies the mechanism to test; it should not be counted again as new evidence.