# Independent integration review: C901–C910

**Arithmetic and source-family consistency: PASS.** One interpretive claim requires a narrower final wording, and one taxonomy cell benefits from clarification. No source or journal edits were made by the reviewer.

## Scope and method

Read all ten journal records, their ten generated models, relevant calculation scripts, the six-head carrier model, the original inverse-input packet and gain field, and the native/restored NT–MT models. Verified every model artifact hash bound directly by these ten records.

Independently checked the endpoint basis as a polynomial identity in λ, all eight endpoint recoveries, all64 original span invariances, the nineteen distinct already-derived inverse gaps, every one of the18 festival grid classifications, the complete22-node Cainan response, both exact retained/uniform partitions, and the general internal-cut difference. Seventeen checks passed; see `evidence/INDEPENDENT_C901_C910.json`. The review did not rerun the entire historical or current cycle suite.

## Main findings

### Endpoint basis and recovery

For `D=C−R` and `Wλ=λC+(1−λ)R`, the coefficient matrix on `(C,R)` is `[[1,−1],[λ,1−λ]]`, with determinant1. Its inverse gives `C=Wλ+(1−λ)D` and `R=Wλ−λD`. This is valid for every fixed λ; the independent check verified the symbolic polynomial identity, beyond the representativeλ12 checks in the step script.

All six macro source pairs and both inherited weighted Creation pairs recover exactly. The equal12026 weighted coordinate paired with9890 versus9900 retains the Actual/Rounded distinction. The integration does not identify the source coefficients12 and4/5 or promote any generated macro head to a primary event.

### Frame change and anchor change

The original64 durations remain identical under common source/anchor translation. The nineteen different paired inverse gaps are consistent with changing only the anchor while retaining the source date. The distinction is correctly structural: whole-frame translation preserves the input digits because it preserves the original duration, whereas a source-fixed anchor change can alter digit length, borrowing and placeholder state.

This is covariance in the declared coordinate chart, not permission to translate every source chronology as a historical state. The frozen NT+30 pair and the source macro paired shifts provide actual admitted instances; a general translation of all32 inverse rows remains a formal diagnostic.

### Festival grids

All18 prefix classifications satisfy `7c∈49Z ⇔ 7|c`, `7c∈70Z ⇔ 10|c` and `7c∈490Z ⇔ 70|c`. Both source walks meet the common490 duration grid exactly at counts0 and140. Their other shared counts59 and199 do not meet it. This is complete-domain classification, not selected-hit counting, and does not equate dates belonging to different anchors.

### Cainan topology and measures

All twelve common nodes Adam through Arphaxad change their NT-minus-MT gap by−130 under the admitted MT insertion; all ten nodes Shelah through Jacob remain unchanged. The model retains130 regular,460 cumulative,70 Luke display and57 BJ secondary birth-interval weights with their separate source roles. A shared name alone does not supply a universal numerical conversion. The inserted Cainan node itself is separate from this common-node response list.

### Retained subdivision

For `S=144900`, `l=147`, `k=70/69`, the uniform partition is `(3377570/23,3430/23)` and the retained partition is `(146853,147)`. Both total147000; their internal cuts differ by49/23, exactly `(k−1)l`. The equality condition `k=1 or l=0` is correct over these ordinary numerical domains.

The source-appointed terminal147 is retained after changing the outer span. It is not the uniformP image of the old terminal147. This provides a sound new instance of the inherited distinction between shared outer measurement and identical complete path.

## Required interpretive refinement: digit information versus independent source input

C909's broad sentence that every retained information type has a witness showing what its removal loses should be narrowed for the digit register. The nonaffine paired-field result proves that a single affine column rule is insufficient. It does **not** prove that the decimal digits, core length or trailing-zero count are independent additional primitive source data.

With an original nonnegative integer span and the globally fixed base-ten/placeholder rule, these digit features are recoverable algorithmically from the span. They must remain **operationally available**, either computed when needed or stored for explanation. The representation policy is a necessary part of the rule; the derived digits need not increase the per-row primitive input count.

Recommended final wording:

> Each tested family requires certain source distinctions or a specified representation rule. Some information must be supplied independently; other information, such as the digits of an original span under the fixed decimal convention, can be computed from the retained input.

This refinement strengthens the Strategy's compression claim. It does not change the32-row arithmetic, its six cases, or any source choice. C910's phrase “digit register where used” is acceptable with this qualification.

## Minor taxonomy clarification

In C907's retained-subdivision row, “held: outer endpoints and source biography” can suggest both compared routes retain the original biography. Only the retained branch does. For the final human-facing table, use:

> Common: outer endpoints. Branch choice: retain the source biography or uniformly transform its internal cut.

The underlying C898/C906 mathematics already makes the distinction correctly, so no numerical repair is needed.

## Constraint ledger, necessity and schema

The six-domain constraint ledger correctly separates forced calculations from particular source inputs. Its strongest examples are the extra off-diagonal50=50 balance in Genesis32 and the difference between automatic NT grid reflection and source-appointed BJ internal boundaries. The homogeneous grid's490 coarsening presupposes the declared70 and49 measures; it does not independently derive them.

The finite necessity examples establish useful limits on overly compressed representations. Their conclusions remain relative to the tested tasks, not a proof that one schema is uniquely or globally minimal. The five operation groups in C910 are a practical explanatory grouping, not five mathematically irreducible primitives. The explicit disclaimer of historical origin and global minimality is appropriate.

The integration preserves the one-pass decimal restriction, the selected coordinate charts, generated/source roles, the quarantined primer boundaries and the current Gear scope. It does not introduce new target searches or a new statistical tally.

## Conclusion

The new integration is coherent and mathematically correct. It explains how complete families can share endpoint algebra, duration grids, insertion support or outer measurements while differing in source roles, representation rules and retained internal structure. The final synthesis should incorporate the digit-information refinement and the small retained-branch taxonomy clarification above.