C05 — The 12-fold and 13-fold families meet in a sourced 6900/7500 comparison

21 September 2026 · Bounded continuation of C04.

The common 6900/7500 lengths have a four-interval realization with retained Creation, Flood and Noah endpoints. One dilation by 25/23, anchored at the SP apparent Creation closing boundary, carries 6877→7475 and 6900→7500. Extending each interval leftward by 300/299 gives the other two sides. Both orders reach the same final 7500 interval from the same initial 6877 interval.

This adds an endpoint-qualified relation between the two numerical families. The older cumulative 12-fold construction retains its own anchoring and generated outer coordinates; the comparison does not turn all its expanded coordinates into historical events.

Family	Two successive 25/23 expansions	Gains
12×529	6348 → 6900 → 7500	552, 600
13×529	6877 → 7475 → 8125	598, 650

The differences between corresponding entries are 529, 575 and 625. The shared lengths arise through a second ratio as well:

6877\times\frac{300}{299}=6900, \qquad 7475\times\frac{300}{299}=7500.

All ordinary regular settings remain full-430, Terah-shift-zero and MT/SP Cainan OFF. Creation stays Normal/G2. The previously declared +30 apparent field is used explicitly: ordinary SP Creation is 4421–4414; its apparent comparison field is 4451–4444. Both fields remain fixed. Three-Gear variation remains confined to Noah/Shem/Flood.

Four retained intervals supply the endpoint realization. The letters below name intervals, not new historical graph nodes.

Interval	SP Creation boundary	Reflected MT target	Width
A	Ordinary opening, 4422t/4421n BC	G1 Flood start, AD 2456t/2457n	6877
B	Ordinary offset 2, 4420t/4419n BC	G1 Noah birth, AD 3056t/3057n	7475
C	Apparent closing, 4445t/4444n BC	G1 Flood start, AD 2456t/2457n	6900
D	Apparent closing, 4445t/4444n BC	G1 Noah birth, AD 3056t/3057n	7500

The two routes are

A\;(6877)\xrightarrow{25/23}B\;(7475) \xrightarrow{300/299}D\;(7500),

A\;(6877)\xrightarrow{300/299}C\;(6900) \xrightarrow{25/23}D\;(7500).

They agree at the level of complete paired endpoints. Both same-season civil calculations check; for example D gives 4445+3056−1=4444+3057−1=7500.

One declared anchor explains the 25/23 sides. Work with the signed paired centers already used in C03: BC Nisan numeral C has center −C; reflected regular numeral R has center +R. Let the fixed apparent center be

\alpha=-4444, \qquad P_\alpha(x)=\alpha+\frac{25}{23}(x-\alpha).

Then

P_\alpha(-4421)=-4419, \qquad P_\alpha(2456)=3056, \qquad P_\alpha(\alpha)=\alpha.

The ordinary Creation opening lies 23 years from the apparent anchor; its image lies 25 years from that anchor and is another admitted ordinary member. The Flood lies 6900 years from the anchor; its image lies 7500 years from it and is the same-Gear Noah-birth reflection. Hence the same affine map sends A to B and C to D.

The calculation acts on pair centers; the ordinary half-year seasonal pair display is retained. The fixed January AD 1 Mirror axis is unchanged. Selecting ordinary Creation offset 2 is an internal member selection, not a Gear applied to Creation.

For the other sides, put e=300/299 and hold each interval’s right endpoint:

E(x,y)=\bigl(y-e(y-x),\ y\bigr).

This sends A to C and B to D. The left extensions are respectively 23 and 25. Applying a common affine map to both endpoints commutes with this relative interval construction, so

E\bigl(P_\alpha(A)\bigr)=P_\alpha\bigl(E(A)\bigr)=D.

The algebra has a general form; source availability is established here on these four intervals. No new universal or indefinitely iterable chronological action is inferred.

The endpoint accounting makes the differing gains transparent:

Comparison	Change of left signed center	Change of right center	Width gain
A → B, 6877→7475	+2	+600	598
A → C, 6877→6900	−23	0	23
B → D, 7475→7500	−25	0	25
C → D, 6900→7500	0	+600	600

This realizes the 598 gain through a two-year ordinary Creation member change alongside the same-Gear 600-year Noah/Flood interval. C01 also retains its different, fixed-Creation realization: 4419 + reflected G2 Flood 2458 = 6877, then the same Creation member to G1 Noah birth gives 7475. Its starting interval is A translated by two years; its final interval is B. Both realizations are qualified and their anchoring choices remain explicit.

The derived 13/12 quotient now has a sourced comparison. Comparing C with B gives

\frac{7475}{6900}=\frac{13}{12}, \qquad \frac{13}{12}\times\frac{300}{299}=\frac{25}{23}.

From C to B, the left endpoint moves 25 years forward while the right endpoint moves 600 years forward. The net gain is 575. B to D restores those 25 years on the left. Thus the direct 600 gain has a sourced decomposition:

\boxed{6900\xrightarrow{+575}7475\xrightarrow{+25}7500, \qquad600=575+25.}

Here 575 is the intermediate square carrier, 23×25. The ratio 13/12 is a derived comparison between these retained intervals, not a newly canonized operator.

The analogous first-stage arithmetic is

6348\xrightarrow{+529}6877\xrightarrow{+23}6900, \qquad552=529+23.

It requires a separate endpoint qualification. The 6348 interval does not share the source anchoring of A–D.

The older 12-fold construction has opposite anchoring. The comprehensive Mirror handoff §10 starts with cumulative native LXX Shem opening 6351t/6350n BC and holds 3t/2n BC fixed. It explicitly generates:

Width	Left BC pair	Held right BC pair	Left-boundary status
6348	6351t/6350n	3t/2n	Retained cumulative Shem opening
6900	6903t/6902n	3t/2n	Generated by expansion
7500	7503t/7502n	3t/2n	Generated by expansion

The original 552 gain moves the left boundary from 6350 to 6902; the next 600 moves it to 7502. In the source intervals C and D, the left apparent boundary is held while the right event advances by 600.

A single exact orientation-reversing alignment expresses the relationship. For signed centers define

H(x)=-4446-x.

It maps the old held endpoint −2 to the apparent endpoint −4444. After reversing endpoint order, it maps the old 6900 and 7500 intervals exactly to C and D. It also carries dilation about the old held landing into dilation about the apparent anchor.

This is a formal interval alignment, not a sourced transport of a cumulative person or a replacement Mirror axis. Its general form maps every width W from [-(W+2),−2] to [-4444,W−4444]; the alignment itself is therefore not an independent numerical witness.

At W=6348, its right image is 1904, giving a calculated interval from 4445t/4444n BC to AD 1904t/1905n. No event at 1904 exists in the tested 66-record full-430 MT/SP/native-LXX Noah/Shem/Flood register. Consequently, the complete three-stage 12-fold sequence is not promoted to a wholly sourced regular Creation chain by this alignment. The common 6900/7500 lengths do have the retained source realization C→D.

An orientation-preserving translation is less uniform: mapping the ordered 6900 endpoints requires +2458, while mapping the 7500 endpoints requires +3058. No single translation identifies both intervals. These are different endpoint constructions connected by the explicit comparison above.

The older MT contact gaps 2556→3108 and 2558→3110 also equal 552: reflected MT Shem birth to reflected SP Flood start. Those source-qualified gaps numerically match the first 12-fold gain, but retain distinct endpoints. The 650 gain remains C01’s same-Gear MT→SP Noah-birth comparison, 3056→3706, giving 7475→8125 with ordinary Creation 4419 held fixed. Thus 552, 598, 600 and 650 have recorded roles without collapsing their different constructions.

The source domain has a sharp boundary. Repeating the four-interval construction with each same-Gear MT Flood/Noah pair requires:

MT Gear	Ordinary source for 6877	Ordinary source for 7475	Required apparent source for both 6900 and 7500	Inside held apparent field?
G1	4421	4419	4444	Yes
G2	4419	4417	4442	No
G3	4417	4415	4440	No

Only G1 supplies the complete four-interval source realization in this domain. All three ordinary member pairs remain retained; the missing apparent candidates are not added to the field. Iteration also requires fresh source checking: applying the same apparent-anchored P to the Noah center 3056 gives 3708+4/23, not a retained integer source station. C04’s distinction between formal continuation and sourced endpoints still applies.

Evidence: evidence/check_c05.py passes 111 exact checks, with a byte-identical isolated replay. Eight frozen dependencies retain the controlling tables, both handoffs, and the relevant C01/C03/C04 records. The checker reconstructs the regular endpoints, verifies both seasonal spans, both routes from A to D, the factor identity and 575+25 triangle, the cumulative alignment, its unsupported 1904 image, and all three Gear candidates. These are verification assertions, not independent witness counts. Canonical files and historical graph counts remain unchanged; the earlier campaigns were not rerun.

Best next bounded step: join this source-qualified 6900/7500 square to C01’s 8125/8993 interface, and identify the smallest set of independent endpoint relations needed to describe the combined construction. High is sufficient.