C13 — Two calibrations control the extended 529 alignment

22 September 2026 · Bounded symbolic continuation of C12.

The extended packet has two independent calibration directions. Releasing the measured 868 attachment and the cumulative 460 insertion shows which equations persist by construction and which equalities require their source values. The 11/2-family alignment fixes the insertion; the 17-family and mixed 20-fold midpoint then fix the attachment. Several prominent checks are therefore different expressions of the same two conditions.

The total mixed 20×529 width alone does not fix its midpoint. Those two requirements together recover both source calibrations in this formal family. A further diagnostic matters: the persistence of 5+5 depends on retaining the equal-interval relation used in C12's compact basis. Holding the directly documented regular +190 profile instead exposes a half-span mismatch away from the 460 calibration.

This is a sensitivity test of the existing equations. No source lifespan, Gear, profile, event date or canonical file is changed.

The two released quantities. In C12's principal N02 packet set

Only the right-hand sides of R09 and R11 are released. The other C12 coefficients, endpoint assignments, reflection rules and 3/2 and 5 allocation ratios are retained, together with a=−4444. Thus the source packet is x=y=0. The symbols x and y do not introduce authorized chronological variants.

The held Creation/regular core is unchanged: b=−4419, g=2458, n=3056, h=3108 and s=3706. Exact solution of the eighteen-variable system gives the moving coordinates below.

The actual BC Noah coordinate X=−3706 is held. All entries are formal signed coordinates; off-calibration values do not inherit the named source labels automatically. The paired Mirror axis remains January AD1, with no civil year zero.

The selected ordinary span remains L=598, while

K is a useful algebraic unit, but it equals the project's fixed 23²=529 only when y=0. The two must be kept distinct during the diagnostic.

Persistent relations retain the structure that was held. Throughout the compact symbolic extension,

The 3+2+5 refinement and the 5+5 equality therefore persist in this system. They preserve the supplied allocations and equal-interval correspondence; their persistence is not a new independent prediction of a particular historical span.

The original held Creation chain also remains 6877→7475→8125, with its two 25/23 steps. Those source endpoints were not varied. The interval-image displacement v−q remains 1248, by the retained correspondence equations.

Releasing the attachment necessarily releases other numerical calibrations that C06's basis derived from it:

Thus the 868 release does not simultaneously hold the original 8395 Flood Mirror width fixed. Adding h−c=8395 as a further fixed constraint would immediately impose x=0. This is a change of which equations are held for the diagnostic, not a new transport rule.

Exact 529 multiples require specific calibrations. The following table uses the fixed unit 529, not the moving K.

The literal 3-, 5- and 7-fold lengths likewise require y=0. All departures in this table lie in a two-dimensional linear space. For example, −y from the 11-fold width and x from the 17-fold width form an independent pair; E supplies the negative of the first departure rather than a third independent condition.

The mixed 20-fold midpoint supplies a separate condition from total width. Its two actual halves are

Their difference is

Hence the landing Z is the midpoint precisely when 2x−17y=0. Requiring both the original total width and that midpoint gives

Within this selected affine family, the total and center jointly recover α=868, κ=460. This is an inverse characterization of the known packet, not new independent historical evidence for its inputs.

The distinction is visible in exact formal controls:

Only the first row is the published source calibration. In the fourth, the total is still 20×529 but the landing is not the midpoint. In the last, the Noah and mixed widths equal 17K and 20K for K=530; the held Creation→MT Flood width remains 6877, whereas 13K would be 6890.

Using a moving unit exposes the shared residual. Define

Then

These four conditions are equivalent in the compact system. The 17-fold alignment, mixed 20-fold alignment, second 10-fold half and Christ-period midpoint do not supply four independent tests.

The other independent residual is

Together these two residuals again force y=0 and x=0. A change of unit can retain some proportional relations, while the independently held Creation chain supplies the remaining calibration condition.

C12's two equivalent source bases give different off-source tests. At x=y=0, C12 showed that either of the following may be used as a basis relation:

• The equal-interval correspondence s−v=c−u.
• The directly documented regular displacement v−g=190=130+60.

Their equivalence used the actual cumulative insertion 460 and the retained MT/SP difference 650. If κ is released, they cannot both remain fixed unless y=0.

In the second calculation, the equal-interval equation has residual y and the interval-image displacement becomes 1248+y. The first and second 5-fold halves are 2645+3y/2 and 2645+7y/2. They coincide at the source calibration.

This qualifies the robustness claim: 5+5 persists because the compact extension keeps its supporting correspondence. It is not invariant under every way of varying the insertion while holding source information. The mixed 20-fold total and its Christ-period midpoint condition are unchanged by this choice, since they do not use the intermediate regular Flood point v/Y. The source's MT ON/+60 profile remains 190 throughout the actual research packet.

Divisibility alone has an infinite formal lattice; the held source fields select its origin. Exact named multipliers are stronger than merely being divisible by 529. For whole-year Nisan endpoints, x is integral and y must be even because A and Z have half-integer coefficients of y. Divisibility of E and the Noah Mirror width requires y and x to be multiples of 529. Consequently the complete locus for the listed base-stage 529 widths is

The multipliers become:

Mixed centering additionally requires m=17n. This concerns the base 529 carrier; the held 7475 and 8125 widths retain their separate 575 and 625 carrier roles.

The source fields sharply limit these formal alternatives. The cumulative OFF later member is 5287+x, and the held field 5283–5290 requires −4≤x≤3. The reflected Conquest candidate is 1400−y, and its held field 1399–1406 requires −6≤y≤1. These intervals contain no nonzero multiples of 529 or 1058, respectively. Thus the lattice intersects these two source bounds only at m=n=0, recovering the published calibration.

Membership without the divisibility or specified-member conditions is weaker. At y=0, all three cumulative endpoints c/c₋/u remain inside their fields for integer −2≤x≤3—six boundary selections, α=866 through 871. Their Noah Mirror widths are 8991 through 8996, with 8993 the sole multiple of 529. These are a bounded membership control, not six new admitted joins. Only x=0 retains N02's specified cumulative members. The original source certificate and the fixed Cainan lifespan already identify the actual values; the modular calculation does not replace that provenance.

What this establishes. The fixed Creation chain, the selected interval correspondences and the chosen partition ratios account for the identities that persist in the compact algebra. The exact square-of-23 lengths and the mixed midpoint require two calibrated inputs. Several family checks share one residual, so their number should not be treated as a count of independent constraints. The conditional full-week 300→299 compression is not used or established by this calculation.

All source restrictions remain active: regular Gears only at Noah/Shem/Flood; SP companions only at Noah/Shem; independent Flood start/close; fixed Creation fields; distinct regular and cumulative Cainan operations; no promotion of generated points to historical events. C12's N04 apparent-boundary and companion-expansion exclusions remain intact. The symbolic calculation does not revise the five sourced mixed cumulative intervals at x=y=0.

Evidence and checkpoint. evidence/check_c13.py passes 114 exact checks. It solves the frozen C12 matrix with affine right-hand sides, verifies polynomial coefficients rather than inferring identities from samples, compares the two basis choices, derives the residual ranks and divisibility lattice, and checks the finite source bounds. C13_SYMBOLIC_CERTIFICATE.json records every endpoint, width and residual coefficient. C13_RESULTS.json contains the formal controls and bounded membership diagnostic. Eight frozen inputs include controlling File_18/File_09 and the relevant C11/C12 material and original field register. Both result files reproduce byte for byte in an isolated replay; SHA256 manifests accompany the packet.

No canonical source or historical graph was changed, and no completed endpoint campaign was rerun. The completed continuation is C01–C13, alongside recovered original Reports 01–10. Interrupted original Reports 11–25 remain unestablished as completed; C13 is a continuation supplement. The previously recorded seasonal interpretation and 5292 typo confirmation remain unresolved.

Best next bounded step: test whether the 12×529 family can attach through C10/C12's 2 BC landing, supplying a contiguous source path for 2+10=12 and clarifying its relation to the companion branch. Keep its later 6900/7500 realizations separately qualified. High is sufficient.