# 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.**