C273 — The 529 dependency synthesis through C272

24 September 2026. A bounded synthesis of C266–C272 and their immediate retained dependencies in the existing 490d Unified Chronology research. Capture now, canonize later.

The seven steps establish a connected, source-qualified comparison with a compact set of dependencies. The 17K source correspondence induces the 7K Noah shift within the declared local placement. The regular SP bindings then give six radii, and the outer radius 6 supplies the 25/23 seasonal bracket that C268’s charts connect to the retained 13K/598/650 chains.

Several apparent confirmations are consequences of the same equations. The strongest unresolved question is whether the chronology independently requires the placement and calibration conditions used to connect these constructions. The selected source paths are established, and the conditional arithmetic is exact. A common rule selecting the entire construction remains unestablished.

1. What this synthesis establishes

C273 introduces no new event search, source register, fitted anchor or scale. It brings seven completed steps into one dependency account, verifies their report/result certificate chain, and checks the equations needed to distinguish inputs from consequences. It also makes the conditional determination of the central translations explicit.

This table records kinds of progress, not independent witness counts. The machine-readable claim ledger has seven input groups, ten derived groups and one assessment group. Those groups are an explanatory organization; C273 does not prove that they are a minimal independent axiom system.

2. The dependency structure

Two operations connect the source data at different levels: source translations and reflections preserve actual seasonal spacing; the large affine charts rescale it. The chart construction also needs the retained 13K chains as an input. Those chains are not derived from 17K alone.

flowchart TD
    P["Sources and declared placements"] --> L["Local endpoint and phase geometry"]
    T["17K paths and regular SP register"] --> R["Induced maps and six radii"]
    L --> R
    C["Retained 13K chains"] --> A["Calibrated 1196 / 1300 charts"]
    L --> A
    R --> V["Bounded 72-image comparison"]
    A --> V

The arrows show dependencies of claims. They are not chronological paths or stored graph edges. The two retained source-path families share material, so separate boxes do not imply statistical independence.

The input groups are:

“Held input” describes its role in this synthesis. It does not mean that all seven groups are independent of one another or that each source member is separately attested as a historical date.

3. The local placement is conditionally determined

The cumulative Shem source lies 600 before its corresponding Flood source. The declared Shem translation is 598 greater than the Flood translation. Therefore the translated Shem field is two earlier:

The common local reflection is consequently about x=1, January AD 2. This result follows from the residual 600−598=2. The global Mirror remains at x=0.

The translations can be recovered from the declared placement conditions. Let f be the MT Flood translation and h=f+598 the MT Shem translation. The MT ON closing source centers are −5289 and −5889. Requiring the translated Flood close to coincide with the reflected translated Shem close gives

Hence

and the shared center is 1. Equivalently, the inherited general condition for two reflected seven-year fields to meet, with residual 2, fixes the MT image to [−6,1] and its partner to [1,8]. The bilateral handoff had already supplied this conditional geometry; C273 states its translation consequence explicitly.

Using the same Flood translation for LXX places its field at [0,7]. Requiring the translated SP Flood interval to be the global reflection of the MT Flood interval places SP at [−1,6]. Its opening source is −4716, so its shift is 4715; adding the declared 598 gives its Shem shift 5313.

The absolute translation values are thus rigid given these conditions. The remaining question concerns the conditions themselves: the choice of the 598 translation increment, the MT closing join, the common MT/LXX shift and the SP/MT interval reflection. A source-supported 598 interval is available in C269; using that interval as the translation increment remains a specified assignment in the construction.

The same placement retains the cross-profile identity

and the central subdivision 580−575=5, 575−574=1. These are related consequences of the selected source differences and translations.

4. From the 17K paths to the six regular radii

C269 established the closing comparison:

Both columns have a 600 separation and the same 598+2 subdivision through −5285 and 3708. The translation T(q)=q+8993 preserves pair centers and actual components.

The Flood closing path was already present in C254:

Its Creation-to-MT-Flood prefix is 6875, not 6877. The unsupported exact-13K Flood candidate 2462 remains unsupported. The frozen C127 graph has the Shem closing node and edge; it lacks the exact Flood closing node −5283. The later C254 route supplies source support without silently changing that graph.

Composing T⁻¹ with the local cumulative maps gives

The first shift is therefore

This is an induced identity from the retained path and local placement. It does not independently verify the choice of that placement.

For a cumulative index j, the local images are j and 2−j. The admitted regular SP register supplies j=2 through 7. With Gear g and selector c=0 for primary/start or c=1 for companion/close,

The complete pairing condition is

Exactly six of the 36 comparisons have Δ=0, identically in both seasonal components. This is the source of the six-radius register. The thirty failures, missing cumulative indices 0/1, and distinct status of Flood closes remain part of the result.

Source membership in those six rows does not automatically certify six new stored graph paths. The additional regular comparisons remain dependent constructions using T and the held register.

5. The outer bracket and the 13K charts

The outer LXX pair has center 7 and components 27/4 and 29/4. Their radii from the local junction are 23/4 and 25/4. Therefore

The scale η=1000(x−1) labels those radii 5750 and 6250, around center-radius 6000. The ratio does not require the factor 1000; the displayed numbers do. The negative partner has components at the opposite radii, with its source-phase order reversed.

C268 separately imports two retained chains with widths

Once the comparison requires Φ(1)=a and maps inner radius 23/4 to the first or second width, the scales are determined:

The 299 midpoints are source-qualified; the 325 midpoints remain generated in the held register. The complete Gear and rail identities remain in the included C268 correspondence ledger.

Three formulations of the outer selection express one condition under these fixed charts:

The first formulation compares seasonal radii. The second is the second chart’s inner endpoint minus the first chart’s outer endpoint. The charts share the retained MT Noah endpoint only at radius 6; the smaller intervals overlap.

C272’s finite source test is a separate classification: 72 image instances yield ten matches, representing eight distinct retained targets, all at radius 6. The remaining 62 instances stay generated in that register. The algebraic endpoint equality would not, on its own, exclude an isolated target match elsewhere; the finite register check supplies that exclusion.

The scales had been calibrated to radius 6. Their successful outer fits, the shared endpoint and the matching ratio cannot be counted as three independent discoveries.

6. Why connected source paths do not identify the two operations

The current evidence establishes source connectivity for the closing members. The remaining gap should not be described as “no source path.” What remains distinct is the operation performed on that source.

The mismatch is visible at the actual source attachment. Writing the cumulative Flood center as q gives the chart

Its source anchor is −5289. The retained 868 attachments instead begin at −5287 or −5285 and end at a=−4419 or −4417. Applying Ψ at those defining source points misses the attachment target by:

Thus the large charts do not implement those source translations. A shared target center or a connected source path does not remove the difference in anchor, scale or geometric role.

7. The broader axis addition remains properly classified

The synthesis preserves the full geometric vocabulary established by C266/C267:

The central matching-season span is 14. Its two full seasonal envelopes have length 7.5 each, total 15, overlap 0.5 and union 14.5. The core’s most distant component radius is 7.25, not 7.5. These distinctions preserve Dean’s complementary Year 14 restoration and Year 15 renewed-age/eighth-day readings on their stated quantities.

The source-based Creation support route retained in C266 and the macro conversion constructions remain useful connected evidence on their own declared source histories. Equal converted durations, local co-location and a matched radius arrangement do not yet assign every macro marker to one continuous historical event path. No physical three-dimensional metric or z-law is supplied by this synthesis.

8. Assessment: the strongest remaining link

The seven steps have reduced the bookkeeping to a small set of explicit operations and conditions. They establish actual source endpoints, selected source paths, exact induced maps, a complete finite binding test, and a precisely bounded chart comparison. The available source labels and negative results are part of that structure.

The strongest unresolved question is now selection of the construction: why should a common chronology rule require these joining conditions and this calibration, rather than merely permit the resulting exact comparison?

Conditional uniqueness is already demonstrated in two places. The central translations follow once their joining conditions are imposed, and 1196/1300 follow once their chart fits are imposed. Neither conditional result alone establishes the necessity of its premises. The useful next test is therefore an audit of which premises actually determine which results, with the source data held fixed.

C273 does not discard the inherited construction or replace its parameters. It identifies the point where further evidence would most improve the unification claim. A proof of literary intention, universal chronology action or complete source-path realization is not among the present results.

9. Evidence, review and restart state

The packet freezes 43 immediate inputs, including all seven reports, results and certificates, the selected supporting ledgers, the two controlling source files and the relevant handoffs. Their certificate chain is checked from C265 through C272. This verifies retained artifacts; it does not rerun the seven historical campaigns or the axis report’s separately mentioned 855-check package.

The C273 verifier passes 156 targeted checks of integrity, dependency references/order and explicit core equations. Six generated evidence files reproduce byte for byte in a separate directory: the claim ledger, dependency graph, core equations, limits register, results and compact CSV claim table. The authored report and checkpoint are not computational replay outputs. Dependency-edge semantics and the assessment remain documented scholarly judgments rather than claims of machine proof.

C273 received an author consistency review. Prior steps’ independent reviews remain prior evidence; no new independent-agent review is claimed for this synthesis. The source files, C265 certificate and C127 graph remain unchanged.

Keep K=529, E=25/23, P=70/69 and ρ=radius. Regular Gears remain Noah/Shem/Flood only. SP Noah/Shem companions remain local; Flood closes retain their independent event-boundary status. Both primers remain quarantined. No canonical amendment, upstream/cumulative Gear, new global axis or historical endpoint search is introduced. Original unprefixed Reports 01–10 and interrupted, unestablished original Reports 11–25 remain distinct from the C-series. The superseded AD 1905–1911 proposal remains unexecuted.

10. Best next bounded step

C274 — perform a finite symbolic audit of the placement conditions. Hold the source coordinates, phase convention and 598 increment fixed. Remove one declared placement condition at a time—the MT closing join, the common MT/LXX Flood shift, or the SP/MT global interval reflection—and solve the remaining affine constraints. Record which translation values, radii and symmetries are still determined and which become free. Do not choose free values to recover the original matches, add source events or reopen the primers. This directly tests the dependency question identified above. High is sufficient. C274 has not been executed.