# 490d C274 — 529 placement-premise omission audit

24 September 2026. Continuation through C273; one authorized bounded step. **Completed.**

## Result

The three placement equations are an **irredundant sufficient set for the held three-variable affine problem**. Together they fix the Flood translations at MT/LXX 5290 and cumulative SP 4715. Removing any one equation leaves exactly one free parameter. The other two equations do not imply the omitted one.

The reflection about x=1 survives every omission because it follows from the fixed source separation 600 and translation increment 598. The six regular SP source bindings also survive. Their specific local radii 1–6 remain fixed only when the LXX translation remains fixed.

A consequential distinction is now explicit: **removing the cumulative SP reflection condition releases cumulative SP placement, while the regular SP maps and their six local radii stay fixed.** Those regular maps are induced through LXX and the retained 17K source correspondence; they do not use the cumulative SP translation.

These are results about dependencies within the declared construction. They do not independently establish the historical necessity of the placement conditions or of assigning 598 as a translation increment.

## 1. Scope and evidence basis

C273 proposed removing one placement premise at a time while holding the source coordinates, phase convention and 598 increment fixed. C274 executes precisely that comparison: the baseline and three omission cases. No free value is selected to restore an inherited match.

The packet freezes fifteen immediate inputs: the C273 certificate, report, checkpoint, equations, results and source manifest; C266's endpoint ledger; C270's maps and report; C271's families, coverage, results and report; and the supplied comprehensive and bilateral Mirror documents. The latter remain context and provenance references. C274 does not conduct a new source-provenance audit.

The retained C266 source fields are:

| Profile | Flood centers | Shem-birth centers |
|---|---:|---:|
| MT cumulative, restored Cainan ON | [−5296, −5289] | [−5896, −5889] |
| LXX cumulative, Cainan OFF | [−5290, −5283] | [−5890, −5883] |
| SP cumulative, Cainan OFF | [−4716, −4709] | [−5316, −5309] |

Each field has eight annual boundary centers spanning seven elapsed years. Source and translated coordinates remain distinct.

The global Mirror is M(x)=−x, with origin January AD 1. The local center x=1 is January AD 2. A pair at center x has chronological components x−1/4 and x+1/4; reflection reverses component ancestry and order. There is no civil year zero. The display remains eta=1000(x−1).

Let m, l and s be the MT, LXX and cumulative SP Flood translations, respectively. Their Shem translations are m+598, l+598 and s+598. These six translations therefore contain only three independent unknowns.

The affine equations are solved over the rationals. To retain the existing January-centered annual lattice and its inherited physical phase labels, the reported free parameters are integers. Noninteger solutions are formal affine solutions, not newly admitted calendar states. No extra divisibility, positive-shift, proximity or event-fit condition is imposed.

## 2. The three placement equations

The translated Flood intervals are

\[
F_M=[m-5296,m-5289],\quad
F_L=[l-5290,l-5283],\quad
F_S=[s-4716,s-4709].
\]

The three inherited premises reduce to:

| ID | Placement premise | Exact equation |
|---|---|---|
| A | Translated MT Flood closing center joins reflected translated MT Shem closing center | m−5289 = 5889−(m+598), hence 2m=10580 |
| B | MT and LXX use a common Flood translation | l−m=0 |
| C | SP Flood interval is the global Mirror of MT Flood interval | m+s=10005 |

Both endpoint equations in C give the same constant: 4716+5289=4709+5296=10005. Counting these as two independent constraints would be incorrect.

In variable order (m,l,s), the coefficient rows are (2,0,0), (−1,1,0), and (1,0,1). Their determinant is 2. The baseline has rank three and the unique solution

\[
(m,l,s)=(5290,5290,4715).
\]

The Shem translations are consequently (5888,5888,5313). K=529 remains the existing unit: 5290=10K and 8993−5290=3703=7K. No new requirement that every translation be a multiple of K is inserted into the omission cases.

## 3. Complete solutions for the four cases

| Case | m, MT Flood | l, LXX Flood | s, cumulative SP Flood | Rank | Free parameters |
|---|---:|---:|---:|---:|---:|
| Baseline A+B+C | 5290 | 5290 | 4715 | 3 | 0 |
| Omit A; retain B+C | 5290+u | 5290+u | 4715−u | 2 | 1 |
| Omit B; retain A+C | 5290 | 5290+v | 4715 | 2 | 1 |
| Omit C; retain A+B | 5290 | 5290 | 4715+w | 2 | 1 |

Here u, v and w range freely over the integers for the admitted annual lattice. The baseline is used solely as the origin for writing each family; it is not chosen as a solution to an underdetermined omission case.

The corresponding Shem translations are:

| Case | MT Shem | LXX Shem | Cumulative SP Shem |
|---|---:|---:|---:|
| Baseline | 5888 | 5888 | 5313 |
| Omit A | 5888+u | 5888+u | 5313−u |
| Omit B | 5888 | 5888+v | 5313 |
| Omit C | 5888 | 5888 | 5313+w |

The omitted equation's residual is respectively 2u, v or w. It is not identically zero, whereas both retained residuals vanish identically. This proves irredundance in the restricted system without a numerical parameter scan.

The solutions identify the function of each premise. A fixes MT's absolute placement; B transfers that placement to LXX; C fixes cumulative SP relative to MT. This is a conditional determination, not a finding that these are independent historical witnesses or a minimal axiom basis for the complete chronology.

## 4. Symmetries and field relations that survive

For any profile and any admitted translation t, write a source Flood coordinate as q. Its corresponding source Shem coordinate is q−600. Then

\[
H=(q-600)+(t+598)=F-2,\qquad
RH=-H=2-F=J_1(F).
\]

Thus every same-index Flood/reflected-Shem pair remains symmetric about x=1. This identity holds for the physical seasonal components as well, with reflection reversing their order. It does not require A, B or C.

| Relation fixed for every value in the family | Baseline | Omit A | Omit B | Omit C |
|---|---|---|---|---|
| Local J1 reflection, all three profiles | Yes | Yes | Yes | Yes |
| MT closing join at x=1 | Yes | No | Yes | Yes |
| LXX Flood is MT Flood advanced six years, corresponding endpoints | Yes | Yes | No | Yes |
| SP Flood is the global Mirror of MT Flood | Yes | Yes | Yes | No |
| Each individual field spans seven center-years | Yes | Yes | Yes | Yes |
| Each individual seasonal envelope has length 7.5 | Yes | Yes | Yes | Yes |

“No” means that the relation is not forced across the family; it may hold at a special parameter value. Such values are not selected here.

The exact measures of the released conditions are useful:

| Case | Signed MT closing gap, RH close minus Flood close | LXX−MT corresponding Flood-center difference | SP-versus-Mirror(MT) endpoint error |
|---|---:|---:|---:|
| Baseline | 0 | 6 | 0 |
| Omit A | −2u | 6 | 0 |
| Omit B | 0 | 6+v | 0 |
| Omit C | 0 | 6 | w |

The baseline MT construction consists of two seven-year center intervals meeting at x=1 and spanning fourteen years. When A is removed, the individual widths remain seven, but that particular closing join and fourteen-year arrangement are no longer determined. The full central sequence and its ordering must not be carried into arbitrary free-parameter cases.

The envelope-length statement concerns each field separately. It does not transfer the baseline combined union length or overlap to a changed placement.

## 5. Radii: keep the selected source endpoints identifiable

The inherited selected endpoints are MT opening, SP closing, and LXX closing. Their radii about x=1 are

\[
r_M=|m-5297|,\qquad r_S=|s-4710|,\qquad r_L=|l-5284|.
\]

| Case | MT opening radius | Cumulative SP closing radius | LXX closing radius |
|---|---:|---:|---:|
| Baseline | 7 | 5 | 6 |
| Omit A | abs(u−7) | abs(5−u) | abs(6+u) |
| Omit B | 7 | 5 | abs(6+v) |
| Omit C | 7 | abs(5+w) | 6 |

These remain the same selected source endpoints, even when their positions change. An endpoint that was outermost at the baseline need not remain outermost throughout a free family. Consequently C274 does not relabel these moving selections as universally “outer.”

For completeness, the actual furthest center-radius of each full Flood/reflected-Shem field is:

\[
R_M=\max(|m-5297|,|m-5290|),
\]
\[
R_L=\max(|l-5291|,|l-5284|),\qquad
R_S=\max(|s-4717|,|s-4710|).
\]

These formulas preserve the possibility that the furthest endpoint changes. The evidence contains the parameterized fields and these maxima; no piecewise radius scan is needed.

## 6. Regular SP: fixed source bindings, potentially moving local images

The retained source correspondence from C269/C270 is T(q)=q+8993. It remains a source relation in all cases. C274 neither re-establishes nor changes its source paths.

Composing its inverse with the variable LXX translations gives

\[
R_N(y)=y+l-8993,\qquad
R_F(y)=8993-l-598-y=8395-l-y.
\]

Their crucial identity is

\[
R_N(y_N)+R_F(y_F)-2=y_N-y_F-600.
\]

The LXX translation cancels. Therefore the C271 criterion for local J1 matching remains exactly the source criterion y_N−y_F=600. The same six bindings among the retained thirty-six comparisons remain matches, and the same thirty exclusions remain exclusions. This conclusion follows algebraically from the universal residual; C274 does not rerun the historical thirty-six-pair enumeration.

Set delta=l−5290. For the six retained cumulative indices j=2,…,7, the fixed source centers and their local images are

\[
y_N=3703+j,\qquad y_F=3103+j,
\]
\[
R_N(y_N)=j+\delta,\qquad
R_F(y_F)=2-j-\delta.
\]

The Noah-side signed displacements from x=1 are 1+delta through 6+delta. Their radii are the absolute values of those displacements. They need not be six distinct positive radii when delta is free: a source pair can cross the center or different pairs can have equal magnitudes. The six source bindings remain distinct.

| Case | delta | Regular Noah subtracted shift | Six local radii in inherited source order |
|---|---:|---:|---|
| Baseline | 0 | 3703=7K | 1,2,3,4,5,6 |
| Omit A | u | 3703−u | abs(1+u),…,abs(6+u) |
| Omit B | v | 3703−v | abs(1+v),…,abs(6+v) |
| Omit C | 0 | 3703=7K | 1,2,3,4,5,6 |

The selected j=7 Noah-side phase components have signed eta coordinates

\[
5750+1000\delta,\qquad 6250+1000\delta.
\]

Their reflected partners have the opposite signed coordinates. Their physical half-year spacing remains unchanged. The specific 5750/6250 placement and its original positive-branch interpretation are not fixed when delta is free. No free value is fitted to restore the E=25/23 bracket, and no 1196/1300 chart is refitted or scanned.

Under omission C, delta stays zero. The regular SP radius-five pair therefore remains fixed even though the selected cumulative SP radius becomes abs(5+w). A statement that “SP radius five is lost” would conflate two different ancestries.

The source coverage remains j=2–7, six boundaries spanning five elapsed years. Indices 0/1 still require absent regular records. Local co-location in an opposite event role does not supply them.

## 7. What C274 establishes for the 529 study

The baseline translations are uniquely recovered once the three placement premises are supplied. The omission families show exactly which parts of the construction depend on each premise. The 7K regular translation depends on fixing l=5290; the 17K source correspondence and 600 source-binding criterion alone do not fix it.

The robust part under this experiment is the reflection identity generated by 600−598=2, together with the retained source connectivity and exact binding criterion. The experiment does not test whether the 598 increment itself is independently required; that increment was deliberately held fixed.

An underdetermined omission family is not evidence that the chronology historically admits every member of that family. Conversely, algebraic irredundance does not establish that a premise lacks support elsewhere in the supplied sources. The status of that support requires a separate, finite provenance review.

The large affine charts remain distinct from the retained source translations. C272's baseline output classification is unchanged as an earlier result; it is not reasserted for changed LXX placements without a new calculation.

## 8. Verification, integrity and limitations

The standard-library verifier uses exact rational arithmetic and row reduction. **139 targeted assertions passed**, covering the frozen inputs, source-coordinate controls, equation derivation, ranks and null directions, retained and omitted residuals, field relations, phase reflection and induced regular maps. Six generated outputs replayed byte identically. Check counts include integrity assertions and must not be interpreted as independent historical evidence.

Author consistency review checked the distinction between signed displacement and radius, selected and furthest endpoints, cumulative SP and regular SP, and algebraic independence and historical necessity. No new independent-agent review is claimed.

All fifteen original immediate inputs remain unchanged. File_18 and File_09 remain accessible at their retained hashes:

- File_18: 68301ab760bfbe2e874aedc5c8d943ced7a6219d34bc2859332a70713f9630d5.
- File_09: 45851004c5864229ebf6a896f9099819a08d8926715d3e97e4bc3985c4a3876e.

The C127 graph remains unchanged at 208 nodes and 382 duration edges. The C265 certificate remains unchanged. C274 chains to C273 certificate SHA-256 c6217745a9340e2380f0a4ec53e9f8e4ebe3b2b16090de2120b00927d92abc23.

The evidence packet contains SCOPE.json, SOURCE_MANIFEST.json, the exact verifier, EQUATION_SOLUTIONS.json, FIELD_AND_SYMMETRY_LEDGER.json, RADIUS_LEDGER.json, REGULAR_MAPS.json, INVARIANT_MATRIX.csv, RESULTS.json, REPLAY.json, CONTROL_INTEGRITY.json, review notes, the checkpoint, certificate and artifact manifest.

Regular Gears remain Noah/Shem/Flood only. There is no new source identity, axis, z-law, cumulative/upstream Gear, primer reopening or canonical amendment. Capture now, canonize later.

## Best next bounded step

**C275: audit the provenance of A, B and C in the supplied original Mirror/529 documents and their immediate source references.** For each premise, identify the exact supporting passage, classify it as a source-coordinate relation, declared placement condition or derived restatement, and determine whether the cited support independently fixes the placement or presupposes it. Keep source coordinates and 598 fixed; do not scan new events, choose free parameters or refit charts.

**High is sufficient. C275 is proposed, not executed.**