# 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

\[
\alpha=b-c=868+x,
\qquad\kappa=c_- -u=460+y.
\]

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.

| Endpoint | Signed paired-center expression |
|---|---|
| c, cumulative LXX OFF later | −5287−x |
| c₋, cumulative LXX OFF earlier | −5885−x |
| u, cumulative LXX ON earlier | −6345−x−y |
| q, reflected Conquest candidate | 1400−y |
| v, reflected alternate MT Flood candidate | 2648−y |
| A, cumulative MT ON Flood candidate | −5293−3y/2 |
| Y, BC reflection of v | −2648+y |
| Z, Christ-period landing candidate | −3+7y/2 |

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

\[
\boxed{E=c-u=1058+y,\qquad K=E/2=529+y/2.}
\]

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,

\[
q-b+E=g-b=6877,
\]
\[
X-A=3K,\qquad Y-X=2K,\qquad Z-Y=5K,
\]
\[
Y-A=Z-Y=5K,\qquad Z-X=7K.
\]

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:

\[
h-c=8395+x,\quad q-u=7745+x,\quad s-c=v-u=8993+x.
\]

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.

| Quantity | Symbolic value | Condition for the displayed source multiplier |
|---|---|---|
| Held Creation→MT Flood, 13×529 | 6877 | Always held |
| Creation→Conquest, 11×529 | 5819−y | y=0 |
| Selected comparison E, 2×529 | 1058+y | y=0 |
| Cumulative→SP Noah, 17×529 | 8993+x | x=0 |
| 3+7 outer interval, 10×529 | 5290+5y | y=0 |
| Mixed cumulative total, 20×529 | 10580+x+3y/2 | 2x+3y=0 |
| Mixed interval's second half, 10×529 | 5290+x−7y/2 | 2x−7y=0 |

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

\[
H_L=Z-A=5290+5y,
\qquad H_R=-c-Z=5290+x-\frac72y.
\]

Their difference is

\[
\boxed{H_R-H_L=x-\frac{17}{2}y.}
\]

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

\[
2x+3y=0,\qquad2x-17y=0
\quad\Longrightarrow\quad\boxed{x=y=0.}
\]

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:

| Diagnostic | α | κ | Creation→Conquest | Noah Mirror width | Mixed total | Mixed halves |
|---|---:|---:|---:|---:|---:|---|
| Source packet | 868 | 460 | 5819 | 8993 | 10580 | 5290 / 5290 |
| Attachment alone +1 | 869 | 460 | 5819 | 8994 | 10581 | 5290 / 5291 |
| Insertion alone +2 | 868 | 462 | 5817 | 8993 | 10583 | 5300 / 5283 |
| Preserve total 10580 only | 865 | 462 | 5817 | 8990 | **10580** | **5300 / 5280** |
| Preserve centering with K=530 | 885 | 462 | 5817 | 9010 | 10600 | 5300 / 5300 |

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

\[
\rho=x-\frac{17}{2}y.
\]

Then

\[
(s-c)-17K=\rho,
\]
\[
(-A-c)-20K=\rho,
\qquad(-c-Z)-10K=\rho,
\qquad H_R-H_L=\rho.
\]

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

\[
(q-b)-11K=(g-b)-13K=-\frac{13}{2}y.
\]

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.

| Diagnostic choice | Alternate MT coordinate v | Regular profile displacement | Difference between the two 5-fold halves |
|---|---|---|---|
| Retain equal-interval R12 | 2648−y | 190−y | 0 |
| Hold documented MT +190 profile | 2648 | 190 | 2y |

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

\[
\boxed{x=529m,\qquad y=1058n,\qquad m,n\in\mathbb Z.}
\]

The multipliers become:

| Width | Multiplier of 529 on this formal lattice |
|---|---|
| Held Creation→MT Flood | 13 |
| Creation→Conquest | 11−2n |
| E | 2+2n |
| Noah Mirror | 17+m |
| 3+7 outer interval | 10+10n |
| Mixed total | 20+m+3n |
| Mixed second half | 10+m−7n |

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.**