# 490d Mirror checkpoint after C270

24 September 2026. Latest completed step: **C270 — the regular SP counterpart of the LXX seasonal bracket**. Continues the existing research through C269; no restart.

## Completed transport

C269’s closing source correspondence is T(q)=q+8993:

| Cumulative source | Reflected regular SP source |
|---|---|
| LXX OFF Shem closing −5883 | G3 Flood start 3110 |
| LXX OFF Flood closing −5283 | G3 primary Noah birth 3710 |

C266’s maps remain F(q)=q+5290 and R_H(q)=−(q+5888). Composing each with T⁻¹ gives the unique induced regular maps, conditional on those inherited inputs:

\[
R_N(y)=y-3703,\qquad R_F(y)=3105-y.
\]

Here R_N maps the reflected regular Noah branch and R_F maps the reflected regular Flood branch. The constants are 8993−5290 and 8993−5888; no new scale or fitted endpoint offset is introduced. The Noah shift is 3703=17K−10K=7K, with K=529: a dependent identity from the inherited maps.

| Regular reflected phase | Local phase | η(x)=1000(x−1) |
|---|---|---:|
| SP G3 Noah AD 3710t | AD 7t | +5750 |
| SP G3 Noah AD 3711n | AD 8n | +6250 |
| SP G3 Flood AD 3110t | 5n BC | −5750 |
| SP G3 Flood AD 3111n | 6t BC | −6250 |

The negative branch reverses component order; its chronological display is 6t/5n BC. Both maps preserve actual half-year separation. No phase reattachment is needed.

Original regular BC ancestry remains explicit: Noah 3710n BC → local AD 7t, and Noah 3711t BC → AD 8n; Flood 3111t BC → local 6t BC, and Flood 3110n BC → 5n BC. Original Mirror and final local reflection must be counted separately.

## Structural result and limits

The source 600 relation gives

\[
R_F(y-600)=J_1(R_N(y)),\qquad J_1(x)=2-x.
\]

Thus the regular G3 counterpart reproduces the LXX local centers −5 and 7 and all four components. It retains 6250/5750=25/23. Inner, same-season and outer bilateral display spans are respectively 11.5, 12 and 12.5, giving 11500, 12000 and 12500 at the inherited scale. These are dependent quantities from the same source configuration.

The intermediate cumulative member −5285 corresponds to reflected SP G2 primary Noah 3708 and remains on the F/R_N branch, mapping to center 5. The other branch would give −603; source roles determine the branch.

The two maps are role-dependent. A single affine center fit would have slope 1/50 and would shrink a half-year pair to 1/100, so it cannot replace the two phase-preserving branch maps. C268’s 1196/1300 charts remain distinct rescalings with their own source-to-center interpretation.

This is a completed source and phase transport of the C269 rectangle. It does not independently derive C266’s original 5290/5888 maps, supply an independent witness, merge historical event identities or establish a universal chronology action.

## Evidence and retained controls

Ten frozen inputs; **111 exact assertions**; six generated outputs replayed byte identically; independent mathematical and source-phase reviews; certificate chained to C269; source/artifact integrity manifests. The authored report and this checkpoint are not computational replay outputs.

File_18 SHA-256: `68301ab760bfbe2e874aedc5c8d943ced7a6219d34bc2859332a70713f9630d5`.

File_09 SHA-256: `45851004c5864229ebf6a896f9099819a08d8926715d3e97e4bc3985c4a3876e`.

The C127 graph and C265 certificate remain unchanged. Keep K=529, E=25/23, P=70/69 and ρ=radius. Global Mirror remains January AD 1; local junction x=1 is January AD 2. No civil year zero. Regular Gears remain Noah/Shem/Flood only, with the SP companion rule confined to its admitted persons. Capture now, canonize later; no canonical or graph mutation, cumulative/upstream Gear, primer reopening or z-law.

C267’s 2500/7500 role distinctions, C269’s source-path qualifications, and the complementary 7000/7500 interpretations remain retained. Original unprefixed Reports 01–10 and interrupted, unestablished original Reports 11–25 remain distinct from the C-series. The superseded AD 1905–1911 historical lookup remains unexecuted.

## Best next bounded step

**C271:** apply the same derived maps to the finite three-Gear SP Noah/Flood register, including the permitted Noah companions and independently tagged Flood closes. Test the correctly bound pairs for local J₁ symmetry and cumulative source ancestry while holding all maps fixed. **High is sufficient.**

C271 is proposed, not executed. Publish completed reports and incremental evidence as work proceeds; await Dean’s next instruction.