# C19 — The 10-, 13-, and 17-fold equations recover one shared anchor

22 September 2026 · Bounded continuation of C18.

**All three audited calibrations return the Conquest coordinate Q=1406 BC. With the relative source inputs and K=529 fixed, they impose one independent absolute-anchor equation.** Their agreement contains source compatibility information, but it is not three independent historical datings.

The equations are

\[
\boxed{
Q+3884=10K,\qquad
2Q+4065=13K,\qquad
2Q+6181=17K.
}
\]

A stronger diagnostic leaves the unit unknown as well: every pair recovers **Q=1406 and K=529**, with coefficient rank **2** and one remaining consistency check. The 10/13/17 multipliers and chronology settings were already selected in this research. This is an inverse consistency result within that packet.

**Use relative inputs before reading the comparison anchor.** Q denotes the positive Nisan numeral of the paired Conquest boundary. C19’s equation constructor receives relative durations, not the known 1406 coordinate. In this pass Exodus becomes Q+40, and the old Flood bridge becomes a function of Q. The existing absolute labels are compared with the solved result afterward.

The new source subtotal is the MT Arphaxad-to-Abraham begetting chain in File_18 §2.2, using Terah 70 in the retained Terah-zero state:

\[
35+30+34+30+32+30+29+70=290.
\]

The corresponding SP chain is

\[
135+130+134+130+132+130+79+70=940.
\]

Their relative difference is **650**, obtained here from the begetting inputs rather than from absolute event dates. The shared Abraham-to-Conquest suffix already verified in C18 is

\[
A=100+60+130+430+40=760.
\]

The 430 is the retained full-430 comparison; the last 40 is the retained Exodus-to-Conquest duration. These are accepted relative inputs of the existing source system. Some prior source tables express such durations through date differences, so withholding the absolute anchor from this calculation does not make the source history or the selected multipliers independent of earlier work.

The required quantities are:

| Relative source quantity | Construction | Value |
|---|---|---:|
| LXX cumulative OFF Arphaxad–Moses subtotal T | C17 lifespan sum, excluding second Cainan | 3879 |
| Shared regular suffix A | 100+60+130+430+40 | 760 |
| SP fixed normal head → Arphaxad P | 654+52+500+102 | 1308 |
| MT G1 Flood-start offset f | 290+A; G1 start co-registers with fixed MT Arphaxad | 1050 |
| SP G1 Flood-start offset s | 940+A | 1700 |
| SP Creation endpoint offset R | P+s | 3008 |
| MT/SP same-boundary displacement d | s−f | 650 |
| Cumulative opening offset wL | C17 source envelope | 5 |
| Regular Creation-week width wC | Fixed File_18 field | 7 |
| Flood start-to-close duration τ | Independent event boundary | 1 |
| Selected Noah/Flood comparison ℓ | Retained 600−2 | 598 |
| Cumulative Cainan κ | Native LXX lifespan insertion/removal | 460 |

The regular Three Gears remain confined to Noah/Shem/Flood. P uses the fixed Normal source path; it does not introduce upstream Gear transport. LXX OFF remains derived from native ON, and the SP person companion remains separate from τ.

**The three width functions now follow without a fixed Q.** Let O be the LXX cumulative OFF opening magnitude, C₀ the fixed SP Creation opening, F₀ the MT G1 Flood start, and H₀ the SP G1 Flood start. Then

\[
\begin{aligned}
O(Q)&=Q+T+w_L=Q+3884,\\
C_0(Q)&=Q+R+w_C=Q+3015,\\
F_0(Q)&=Q+1050,\\
H_0(Q)&=Q+1700.
\end{aligned}
\]

The original close-anchored Flood bridge is

\[
M(Q)=O(Q)+H_0(Q)-\tau=2Q+5583.
\]

The Creation and Noah widths are therefore

\[
\boxed{W(Q)=C_0(Q)+F_0(Q)=2Q+4065,}
\]
\[
\boxed{B(Q)=M(Q)+\ell=2Q+6181.}
\]

O is the opening radius whose paired self-Mirror is 2O. W and B are cross-Mirror widths, so each contains two copies of the common BC anchor. The paired-center convention and ordinary same-season civil count agree; no civil year zero is introduced.

The expression B=M+ℓ refers to the retained matched interior construction. It does not assign a Noah birth to the opening pair plus 598. Its two complete source realizations are explicit:

| Branch | Cumulative OFF member D | SP Creation member C | MT Flood F | SP Noah N | Resulting widths |
|---|---|---|---|---|---|
| N02 | Q+3881 | Q+3013 | Q+1052 | Q+2300 | C+F=2Q+4065; D+N=2Q+6181 |
| N04 | Q+3879 | Q+3011 | Q+1054 | Q+2302 | Same two width functions |

At Q=1406 these reproduce the existing **5287/4419/2458/3706** and **5285/4417/2460/3708** source packets. Both full 6877→7475→8125 expansions remain intact. No endpoint search was repeated.

**Solving with the retained unit gives the same anchor three ways.** First obtain

\[
K=\frac{\ell+\kappa}{2}=\frac{598+460}{2}=529.
\]

| Calibration | Inverse equation | Recovered Q |
|---|---|---:|
| O=10K | 5290−3884 | **1406** |
| W=13K | (6877−4065)/2 | **1406** |
| B=17K | (8993−6181)/2 | **1406** |

The constructor does not hold Exodus 1446, the 5290 opening, the 8395 bridge, or the apparent-Creation anchor fixed as additional coordinate inputs. It solves the chosen width conditions and then checks the result against File_18’s Conquest coordinate. This prevents another retained absolute label from silently fixing Q in advance.

With the relative constants and K fixed, the three residual equations are simply

\[
(Q-1406),\qquad2(Q-1406),\qquad2(Q-1406).
\]

Their coefficient rank and augmented rank are both **1**. The equations express one absolute-epoch condition once the source data are frozen. Their agreement depends on the relative constants fitting together.

**The source structure explains the dependence of the 17-fold equation.** The regular chains share the downstream suffix, giving

\[
R=f+P+d.
\]

Consequently the difference between the Creation opening and MT G1 Flood start is anchor-independent:

\[
G=C_0-F_0=P+d+w_C=1308+650+7=1965.
\]

Also define the source-derived scalar

\[
J=d-\tau+\ell=650-1+598=1247.
\]

J is a combination of retained durations; it is not assigned a new Noah event endpoint. The width formulas can be written

\[
W=2F_0+G,\qquad B=O+F_0+J.
\]

Eliminating F₀ gives the exact source-structural identity

\[
\boxed{B=O+\frac{W+K_{\rm reg}}{2},\qquad K_{\rm reg}=2J-G.}
\]

Here

\[
\boxed{K_{\rm reg}=2(1247)-1965=529=K.}
\]

This regular-source compatibility agrees with the independently assembled local/cumulative unit **(598+460)/2**. It is another expression of the source constraints, not a newly identified 529-year event interval. At the retained values it makes the multiplier relation transparent:

\[
\boxed{17=10+\frac{13+1}{2}.}
\]

Thus the seventeenfold calibration follows from the tenfold and thirteenfold calibrations **after** the checked source relation Kreg=K is supplied.

For the precise qualification, define

\[
r_{10}=O-10K,\quad r_{13}=W-13K,\quad r_{17}=B-17K,
\qquad\mu=\frac{K_{\rm reg}-K}{2}.
\]

Then

\[
\boxed{r_{17}=r_{10}+\tfrac12r_{13}+\mu.}
\]

At the retained source inputs μ=0. If the two individual anchor solutions differ while μ remains zero, the 17-fold solution lies at their mean:

\[
Q_{17}=\frac{Q_{10}+Q_{13}}2.
\]

The identity is therefore conditional on a specific relative source compatibility. It is not forced by the bare integers 10, 13, and 17 alone.

**Two anchor-independent checks locate the compatibility burden.** Put a=3884, b=4065, c=6181. Eliminating Q from the fixed-unit equations gives

\[
J_{13}=b-2a+7K=0,
\qquad J_{17}=c-2a+3K=0.
\]

The actual shared regular structure gives

\[
\boxed{J_{17}=\tfrac12J_{13}+\mu.}
\]

The nontrivial source agreement can thus be recorded as the match between the cumulative and regular anchor solutions, together with μ=0. Once both are retained, the third anchor equality adds no independent equation. These are algebraic dependencies within the chosen source packet, not a statistical assessment of historical witnesses.

**Allowing the unit to vary provides a stronger inverse consistency test.** Replace K by an unknown U, while retaining the same source durations and previously identified multipliers:

\[
Q+a=10U,\quad2Q+b=13U,\quad2Q+c=17U.
\]

Every pair uniquely determines Q and U:

| Equation pair | Unit recovered after eliminating Q | U | Q |
|---|---|---:|---:|
| 10 and 13 | (2a−b)/7 = 3703/7 | **529** | **1406** |
| 10 and 17 | (2a−c)/3 = 1587/3 | **529** | **1406** |
| 13 and 17 | (c−b)/4 = 2116/4 | **529** | **1406** |

The numerator relation **3703−1587=2116** connects the familiar 7-, 3-, and 4-fold quantities. The 2116 retains C18’s interpretation as the sum of unequal flanks; this calculation adds no new contiguous fourfold event path.

The coefficient matrix in (Q,U) has rank **2**, and the augmented matrix also has rank **2**. The third equation tests the remaining source consistency condition

\[
\boxed{7c-3b-8a=0.}
\]

Equivalently, the residual rows obey **−8e₁₀−3e₁₃+7e₁₇=0** at these source constants. The unused equation closes exactly for each pair’s solution. This is a useful inverse check; the prior choice of multipliers and chronology settings prevents calling it a blind discovery of either 529 or 1406.

**Bounded controls distinguish a common anchor shift from a changed source relation.** With the relative sources held, a common one-year change of Q changes O by one and W/B by two:

| Formal Q | O | W | B | Residuals from 10K, 13K, 17K |
|---:|---:|---:|---:|---|
| 1405 | 5289 | 6875 | 8991 | −1, −2, −2 |
| **1406** | **5290** | **6877** | **8993** | **0, 0, 0** |
| 1407 | 5291 | 6879 | 8995 | +1, +2, +2 |

The same-side attachment remains **868** throughout this formal translation. It therefore cannot select the absolute anchor on its own.

The following relative-data controls keep K=529. Non-source duration changes are diagnostic releases, not adopted variants; fractional roots are algebraic outputs, not new Nisan chronology states.

| Bounded change | Q from 10 | Q from 13 | Q from 17 | μ |
|---|---:|---:|---:|---:|
| Retained source | 1406 | 1406 | 1406 | 0 |
| Cumulative subtotal T+1 | 1405 | 1406 | 1405½ | 0 |
| Shared regular suffix A+1 | 1406 | 1405 | 1405½ | 0 |
| SP head-to-Arphaxad span P+1 | 1406 | 1405½ | 1406 | −½ |
| Use G1 Flood start in place of the retained close calibration | 1406 | 1406 | 1405½ | +1 |

The first two releases preserve the regular structural relation and expose the mean-root property. Changing P alters Kreg, so μ must be retained. The final row is the actual start/close alternative already distinguished in C18; it loses the common fixed-unit anchor. These controls show which accepted durations and boundary choices support the agreement.

**Validation and checkpoint.** The checker passes **86 exact checks**. It parses the MT/SP begetting subtotals, extracts relative durations from the completed C17/C18 records, constructs and solves the equations before comparing the canonical anchor, proves both rank statements and the residual identities, verifies the two existing complete source chains, and records the bounded controls. An isolated replay reproduces all three generated JSON files byte for byte. All eight original dependencies remain unchanged.

The packet contains this report, the standard-library checker, the relative-input ledger, the anchor/dependency certificate, the result register, eight frozen inputs with provenance and hashes, the replay record, and the artifact manifest. No earlier campaign was rerun or canonical source edited. The 1446 and 1876 primers, Rounded chronology, upstream Gear transport, and interrupted historical Reports 11–25 remain outside this completed step.

**Source locators.** File_18 §2.2 supplies the MT post-Flood begetting chain; §§3.1–3.4 supply the fixed SP chain, primary stations, Flood start/close, and shared suffix; §6D supplies the LXX cumulative Cainan value and the anchor used for final comparison. File_09 §3.1 controls paired Mirror inversion. C17 supplies the verified cumulative lifespan subtotal; C18 supplies the verified regular relative path and boundary attachment. Earlier scripts are not execution dependencies. The result concerns the supplied project model and does not replace its historical anchor argument.

**Best next bounded step — High sufficient.** Audit the primitive lifespan and begetting dependencies behind **Kreg=529** and **K=(598+460)/2=529**. Determine which shared inputs explain their agreement and which relative source relation remains an additional constraint.