# C272 — Fixed charts and the outer radius selection

24 September 2026. Bounded continuation of C271 in the existing 490d Unified Chronology research. Capture now, canonize later.

**Only the outer radius 6 produces source matches under C268’s fixed charts.** The complete test has 72 image instances: ten match the retained regular register, representing eight distinct source records, and all ten reproduce C268. The inner five radius families add no target matches. Their source ancestry from C271 remains valid.

The geometry has an exact explanation: the 1196-scale outer endpoint and the 1300-scale inner endpoint coincide only at radius 6. At smaller radii their intervals overlap. This qualifies how the 17K-backed local register connects to the retained 13K/598/650 construction; it does not create additional independent witnesses.

## 1. The finite test

C271 supplied six positive local pair centers x=2 through 7, with radii ρ=1 through 6 about x=1. Each row has an existing cumulative LXX OFF Flood member and a reflected regular SP Noah-birth counterpart. Keep those source identities and use the three local points

\[
x=1+\rho+\delta,\qquad \delta\in\{-1/4,0,+1/4\}.
\]

The outer two points are actual local seasonal components; the middle point is their pair center. C268’s charts remain

\[
\Phi_{s,a}(x)=a+s(x-1),
\]

with s=1196 or 1300 and fixed anchors a=−4419 for N02 or a=−4417 for N04. These anchors are retained ordinary SP Creation-field members, not Creation Gears. No coefficient is refitted.

There are 6×3×2×2=72 images. The target domain is exactly the frozen C01 register: 48 source records at 48 distinct positive reflected Noah/Shem/Flood pair centers, ranging from 1956 to 3710. A match requires the exact signed center and retains the stored tradition, event role, Gear and rail. Negative outputs are not replaced by absolute values or matched against original BC counterparts.

All target images are integers because both scales are divisible by four. This does not make all images source records: it makes them eligible numerical pair centers, whose source status must still be checked.

## 2. Complete image classification

For N02, the signed target triples are shown below in inner / center / outer order. **Every N04 entry is the corresponding N02 entry plus 2.** Thus the table and that exact anchor relation specify all 72 image instances.

| Radius ρ | N02 at scale 1196 | N02 at scale 1300 |
|---:|---|---|
| 1 | −3522 / −3223 / −2924 | −3444 / −3119 / −2794 |
| 2 | −2326 / −2027 / −1728 | −2144 / −1819 / −1494 |
| 3 | −1130 / −831 / −532 | −844 / −519 / −194 |
| 4 | 66 / 365 / 664 | 456 / 781 / 1106 |
| 5 | 1262 / 1561 / 1860 | 1756 / 2081 / 2406 |
| 6 | 2458 / 2757 / 3056 | 3056 / 3381 / 3706 |

The counts distinguish calculations from source identities:

| Counted object | Result |
|---|---:|
| Image instances | 72 |
| Distinct signed image centers | 70 |
| Matching image instances | 10 |
| Distinct matched source records | 8 |
| Generated image instances | 62 |
| Distinct generated centers | 62 |
| Source matches from radii 1–5 | 0 |

The only repeated image centers are 3056 and 3058. Each is the outer endpoint of the 1196 chart and the inner endpoint of the 1300 chart at radius 6. Their repetition supplies the shared stage boundary, not an additional event identity.

“Generated” here means absent from the declared reflected target register. It is not a claim that the coordinate cannot occur in another chronology, event list or project construction. No broader historical lookup was performed.

## 3. The retained outer matches and midpoint distinction

All source matches belong to the already established outer triple x=27/4, 7, 29/4:

| Anchor and scale | Inner image | Center image | Outer image |
|---|---|---|---|
| N02; 1196 | 2458 — MT G2 Flood start | 2757 — SP G2 companion Noah death | 3056 — MT G1 Noah birth |
| N04; 1196 | 2460 — MT G3 Flood start | 2759 — SP G3 companion Noah death | 3058 — MT G2 Noah birth |
| N02; 1300 | 3056 — MT G1 Noah birth | **3381 — generated midpoint** | 3706 — SP G1 primary Noah birth |
| N04; 1300 | 3058 — MT G2 Noah birth | **3383 — generated midpoint** | 3708 — SP G2 primary Noah birth |

The first chart has three sourced target centers in each anchor case. The second has sourced endpoints but an unsourced midpoint in this register. The established 299+299 and 325+325 distinction is therefore unchanged.

Measured from either Creation anchor, the first outer-row triple is 6877 / 7176 / 7475 and the second is 7475 / 7800 / 8125. Their shared endpoint retains the chain

\[
6877\longrightarrow7475\longrightarrow8125
=13(529\longrightarrow575\longrightarrow625),
\]

with gains 598 and 650. C272 adds the finite comparison with the five inner families, not a new discovery of these source targets.

The input at radius 6 is cumulative LXX OFF Flood closing base 5283 and, through C270/C271’s induced map, reflected SP G3 primary Noah birth 3710. The chart’s target SP companion Noah deaths are different source events. Their identities do not replace the input’s primary-Noah identity.

## 4. Why only radius 6 joins the two stages

At any one radius and anchor, the two image intervals have endpoints

\[
I_{1196}=[a+1196\rho-299,\ a+1196\rho+299],
\]

\[
I_{1300}=[a+1300\rho-325,\ a+1300\rho+325].
\]

Their widths remain 598 and 650 in every row. Their center separation is 104ρ. The signed gap from the first interval’s outer endpoint to the second interval’s inner endpoint is

\[
\begin{aligned}
G(\rho)
&=\Phi_{1300,a}(1+\rho-1/4)
-\Phi_{1196,a}(1+\rho+1/4)\\
&=104\rho-624\\
&=\boxed{104(\rho-6)}.
\end{aligned}
\]

Hence the endpoint equality occurs only at ρ=6. For radii 1–5, the intervals overlap by 520, 416, 312, 208 and 104 respectively. They do not form the same adjacent two-stage chain.

The equality is equivalent to

\[
\frac{\rho+1/4}{\rho-1/4}
=\frac{1300}{1196}
=\frac{25}{23},
\]

which again selects ρ=6. Thus C271’s outer seasonal-ratio condition and C272’s shared-stage-endpoint condition express the same algebra under the fixed charts.

For every radius, the general anchored expansion identity still holds:

\[
\Phi_{1300,a}(x)=a+\frac{25}{23}\bigl(\Phi_{1196,a}(x)-a\bigr).
\]

What fails at the inner radii is the particular outer-to-inner endpoint joining and the target-register matches, not the affine identity itself. Endpoint joining alone would not exclude an isolated source match elsewhere; the full image classification checks that separate question.

## 5. Why the inner rows have no target matches

The target register’s smallest reflected center is 1956. Radii 1–3 give 36 negative outputs; radius 4 remains below 1956. At radius 5, every 1196-scale image is at most 1862, and the two 1300-scale inner images are 1756 and 1758. All of those lie below the register.

Only four inner-family images fall inside its numerical range: 2081, 2083, 2406 and 2408. None is one of its 48 reflected source centers. The outer generated midpoints 3381 and 3383 bring the total of within-range misses to six.

Across the whole test, 56 image instances are below the register’s minimum, and 16 are within its range. Ten of those sixteen match; six do not. This is an exact finite exclusion, not a probability calculation or a claim of global historical absence.

## 6. Source and phase limits

All six input families retain their C271 ancestry. Failure of a rescaled output to match the target register does not invalidate the input’s cumulative or regular source membership. The unit-magnitude C270/C271 maps and the C268/C272 rescalings serve different purposes.

In these charts, the seasonal input components map to target **pair centers**. A local half-year separation becomes 598 or 650 years. The standard half-year display of a matched target comes from its own stored source record; it is not raw transport of the input season through the large scale factor. Local pair centers have center-to-center images, with generated status where appropriate.

The scales were originally calibrated to the radius-6 bracket in C268. Retaining their radius-6 matches is therefore expected. The new finite test establishes that those same fixed charts do not acquire additional source targets on the other five admitted families. It is not independent evidence for the original calibration, a statistical discovery, or a universal uniqueness theorem.

The result creates no source-event identification or stored directed graph edge. C269’s retained closing-source paths and C271’s source membership remain separate from these affine target comparisons. The generated midpoints are not promoted to Shem or another historical event.

## 7. Verification and retained controls

Fourteen immediate inputs are frozen. **326 exact assertions pass**, and six generated evidence outputs reproduce byte for byte in separate main and independent-review directories. Independent mathematical and source-role reviews confirm the counts, all target identities, the midpoint statuses and the overlap formula. Final review found no substantive mathematical or provenance errors. The authored report and checkpoint are not computational replay outputs.

The certificate continues C271 and binds the report, checkpoint, source manifest, results, replay and control-integrity record. File_18, File_09, the C265 certificate and the C127 graph remain unchanged. No old campaign was rerun, no target register was widened, and no new historical event search was performed.

Keep K=529, E=25/23, P=70/69 and ρ=radius. Regular Gears remain Noah/Shem/Flood only; SP companions remain local to Noah/Shem, while Flood closes remain independent event boundaries. No cumulative or upstream Gear, new global axis, z-law or canonical amendment is asserted. Both primers remain quarantined.

Retain C267’s 2500/7500 distinctions and the complementary 7000/7500 interpretations. Original unprefixed Reports 01–10 and interrupted, unestablished original Reports 11–25 remain distinct from the C-series. The superseded AD 1905–1911 search remains unexecuted.

## 8. Best next bounded step

**C273 — consolidate the dependency structure of C266–C272.** Produce one source-qualified account of how the 17K correspondence, 7K Noah shift, six-radius register, outer 25/23 bracket and 13K/598/650 chain depend on one another. Identify the inherited assumptions, derived consequences and strongest remaining unsupported link. Confine the work to these seven completed steps and their immediate dependencies; introduce no new numerical search. **High is sufficient. C273 has not been executed.**