C22 — Ratio and scale give the two factors of 23

22 September 2026 · Second of the two authorized steps after C20.

The retained construction separates the two factors in 529:

\frac{598}{2}=299=13\times23, \qquad \frac{460}{2}=230=10\times23,

\boxed{K=(13+10)\times23=23\times23=529.}

The reduced 13:10 ratio supplies the sum 13+10=23. The retained source values fix the common half-scale at another 23. These are different algebraic roles. The observed family multipliers alone do not force the scale to be 23 or the unit to be 529.

C22 also recovers the original 8395=365×23 bridge coefficient from the completed balances. This is a synthesis of the retained source relations, not a new independent numerical witness.

The ratio comes from the local lengths already audited. C17 recorded

\chi=10\ell-13\kappa=0, \qquad\ell=598,\quad\kappa=460.

Their greatest common divisor is 46:

598=13(46),\qquad460=10(46).

In general, integer lengths satisfying χ=0 have ℓ=13t and κ=10t. Then K=(ℓ+κ)/2=23t/2. For the retained integer scalar system, K is integral, so t is even. Write t=2z:

\boxed{\ell=26z,\qquad\kappa=20z,\qquad K=23z.}

The source fixes

\boxed{z=\frac{600-2}{26}=\frac{460}{20}=23.}

The first evaluation uses the retained primary cardinal Noah/Flood span and adjacent-Gear displacement. The second uses the named cumulative Cainan lifespan. Once χ=0 is imposed these evaluations are equivalent; they must not be counted as two new independent witnesses. Their provenance still matters: the age 600, selected offset 2 and Cainan lifespan 460 are supplied by the controlling packet, rather than chosen as free scale parameters.

Four source conditions reduce five scalar quantities to one scale. Retain ℓ, κ, the cumulative OFF opening O, the same-side gap X from C21, and the regular source combination Kreg from C20. The four conditions are:

Condition	Equation	Source role
Local ratio	χ=10ℓ−13κ=0	Retained 598/460 commensurability; C17
Opening calibration	ω=O−5(ℓ+κ)=0	Named cumulative sum and common anchor; C17/C19
Named cumulative/SP balance	C=2X−5ℓ−3κ=0	C21's 1910+269+6 decomposition
Regular/local unit agreement	Ψ=2Kreg−ℓ−κ=0	Primitive source compatibility; C20

Here C is the balance residual, not an earlier Creation endpoint label. In the ordered variables (ℓ,κ,O,X,Kreg), the coefficient matrix has rank four. Its primitive integral null vector is

\boxed{(\ell,\kappa,O,X,K_{\rm reg})=z(26,20,230,95,23).}

All integer solutions of this reduced homogeneous system are integer multiples of that vector. The equations fix the shape; one scale remains. Supplying either source κ=460 or source ℓ=600−2=598 makes the reduced system unique and returns z=23, O=5290, X=2185 and Kreg=529.

This inverse statement is conditional on the previously selected family equations and source settings. It does not derive the original choice of the 10, 13 and 17 multipliers from an unexamined text, predict an unknown Cainan reading, or establish an absolute historical date independently of the earlier anchor work.

The old widths now have one coefficient table. Let H be the retained SP G1 Flood closing coordinate. C21 gives X=O−H, while the original paired Flood bridge is O+H. The source identities are

H=O-X,\qquad M_{\rm Flood}=2O-X,

B=M_{\rm Flood}+\ell=2O-X+\ell,

W=2B-2O-K_{\rm reg}=2O-2X+2\ell-K_{\rm reg}.

W is the Creation-to-reflected-Flood width and B the retained cumulative/regular Noah width. With all four conditions satisfied:

Existing quantity	Coefficient form	At z=23	Role
Regular selected length ℓ	26z	598	Qualified Noah/Flood comparison
Cumulative Cainan κ	20z	460	Named lifespan insertion/removal
Common unit K	23z	529	Half of ℓ+κ
Mixed cumulative difference E	46z	1058	2K
C21 same-side gap X	95z	2185	Opening minus SP G1 close
Cumulative OFF opening O	230z	5290	BC Nisan numeral / paired opening radius, 10K
SP G1 Flood close H	135z	3105	BC Nisan numeral
Original paired Flood bridge	365z	8395	Cross-Mirror width
Creation Mirror width W	299z	6877	13K
Noah Mirror width B	391z	8993	17K
Native cumulative ON opening	250z	5750	O+κ, BC Nisan numeral
Opening paired self-Mirror	460z	10580	2O=20K

The role column is essential: points, radii, same-side differences and cross-Mirror widths do not become identical node classes because they share this scale. In particular, a coefficient 460 in the last row is not an additional occurrence of Cainan's 460-year lifespan.

The old bridge coefficient is now explicit:

\boxed{365=2(230)-95,}

8395=2(5290)-2185=365\times23.

The other two width coefficients follow:

365+26=391=17\times23,

2(391)-2(230)-23=299=13\times23.

Thus 6877=299×23=13×529 and 8993=391×23=17×529 occupy the same reduced system as the original 8395 bridge. Deriving these coefficients from the retained equations does not multiply the evidence count.

The integer family conditions are distinct from the 13:10 ratio. Before setting any residual to zero, define

r_{10}=O-10K,\quad r_{13}=W-13K,\quad r_{17}=B-17K.

The primitive coefficient identities are

\boxed{r_{10}=\omega,}

\boxed{r_{13}=2\omega-\mathcal C-\tfrac12\Psi,}

\boxed{r_{17}=2\omega-\tfrac12\mathcal C.}

Consequently all three family calibrations hold if and only if ω=C=Ψ=0. χ is not required for that statement. The ratio condition supplies the further reduction to the common integer coefficients in z.

C17's native-ON opening dilation supplies the complementary relation:

\boxed{23\left[(O+\kappa)-\frac{25}{23}O\right]=-\chi-2\omega.}

With ω=0 retained, the 25/23 opening dilation is equivalent to χ=0. It is not an additional independent condition. At the source the normalized opening ratio is 250/230=25/23.

The exact coefficient maps in the evidence also retain all error terms. For example, using z=κ/20 even when χ is released:

K=23z+\frac{\chi}{20},

M_{\rm Flood}=365z+\frac{3\chi}{4}+2\omega-\frac{\mathcal C}{2}.

These identities show precisely which assumptions support the displayed source coefficients.

Bounded countermodels distinguish ratio from scale. The following are formal scalar systems, not dated chronology variants or new endpoint matches:

Scalar system	ℓ	κ	O	X	Kreg=K	χ	10/13/17 residuals
Retained source	598	460	5290	2185	529	0	0, 0, 0
Family calibrations held; ratio released	598	462	5300	2188	530	−26	0, 0, 0
All four reduced conditions held at z=24	624	480	5520	2280	552	0	0, 0, 0

The middle row preserves the family multipliers but loses the 25/23 opening dilation, with residual 26/23. The last row preserves the reduced ratio and all family equations while changing the scale. It does not retain Cainan 460; with the actual Gear offset two it would also require a cardinal Noah span of 626 rather than the source 600. These examples establish what the equations alone do and do not fix.

Three further reduced controls alter only ω, C or Ψ. They verify the different effects on the three family residuals. No altered duration is adopted into the source chronology.

A small source-qualified local check locates the actual selected length. Hold MT G1 Noah at 3056 BC and Cainan at 460. Compare only the three existing MT Flood starts:

Flood start	Noah-to-Flood comparison	Local half-sum unit	χ
G1: 2456 BC	600	530	+20
G2: 2458 BC	598	529	0
G3: 2460 BC	596	528	−20

The retained adjacent-Gear comparison is the one that agrees with the 13:10 ratio and gives 529. The other rows are permissible local endpoint comparisons, not newly admitted complete 529-family joins. This is a three-row diagnostic of the retained source relation; no earlier selector campaign was rerun and no frequency or chance claim is made.

The rank counts answer different questions. C19 froze the relative source data and varied Q alone, giving rank one, or Q and the unit, giving rank two. C22 releases five reduced scalar quantities and obtains four homogeneous constraints with one free scale. These statements are compatible. None is a count of independent textual traditions or historical witnesses.

The C20 selected-length match between the regular 598 and cumulative Shem-member 598 remains a source binding. The coefficient table does not grant additional endpoint permission. The completed N02/N04 full constructions, partial branches, SP companion restrictions and fixed Creation fields retain their previous admissions and exclusions.

Validation and checkpoint. The checker passes 78 exact checks. It verifies the current controller hashes, derives the reduced identities from the frozen primitive forms, computes the rank and primitive integral null vector, solves the two source-scale constraints, checks the twelve normalized quantities with their full residual terms, and evaluates six reduced controls plus three local source comparisons. An isolated replay reproduces all three JSON outputs byte for byte. All ten original dependencies remain unchanged.

This report completes the second authorized step. No canonical file was changed and no previous campaign was executed. Regular Gears remain confined to Noah/Shem/Flood; SP person companions remain confined to Noah/Shem. LXX native ON, derived OFF, independent Flood start/close and the paired no-year-zero Mirror convention remain intact. The two primers, Rounded chronology and upstream Gear transport remain outside this synthesis. C21/C22 are new continuation reports, not recovery claims for the interrupted historical Reports 11–25.

Source locators. File_18 §§2.1.3 and 3.1.4 supply the current MT Noah/Flood relations; §§4.2 and 6D supply Cainan 460. File_09 §3.1 controls the paired Mirror convention. C17 supplies the previously proved χ/ω and 25/23 identity; C20 supplies the primitive unit condition and typed inputs; C21 supplies the named gap balance and its equivalence to the remaining family constraint. The inference concerns the supplied project model and does not by itself establish literary intention or independent historical dating.

Best next bounded step — High sufficient. Map the already established 529 families onto this reduced ratio/scale system and their required endpoint bindings, reusing the completed certificates. Distinguish arithmetic consequences from additional source-qualified constructions without repeating the endpoint searches.

C22 — Ratio and scale give the two factors of 23

C22 — Ratio and scale give the two factors of 23

Linked sources and evidence