# C324 independent mathematical and method review

24 September 2026. C324 only. The C323 checkpoint/synthesis and held C322 biography DATA were read as inherited evidence. Relevant passages in File_18, File_09, File_46 and AXIS_REPORT were examined for the present dependency question; no C325 calculation was performed.

**Finding:** the supplied sources positively prescribe several local cumulative `460→500` completions. The reviewed passages do not prescribe transporting those endpoint rules to the C322 SP→MT whole-biography correspondence or identify their generated `500` with its regular Noah-bound `500`. The gap is a typed endpoint/dependency link, not the elementary ratio identity. This is a bounded source finding, not mathematical proof that no further source can supply the link.

## Exact arithmetic and operator distinction

Use the batch notation `K=529` and `E=25/23`; AXIS_REPORT calls the ratio itself `K`, so its notation must be translated explicitly.

`460E=500`, `500−460=40`, `529E=575`, and `529E²=625`. File_46's `4232=8×529` gives `4232→4600→5000`. For an integer input, divisibility by `23²` licenses two integral `E` expansions because `25` and `23` are coprime. In particular, `460=20×23` licenses the first integral expansion, but `460E²=12500/23` is not integral. The `4232` double-expansion fact does not make `460` a `529` multiple.

In the held BC-label coordinate, C322's correspondence is the uniform translation `T(b)=b−460`. It applies to all fifteen primary same-Gear role endpoints and thirty same-phase components. Its internal spans stay `500,100,350,150`; Noah's lifespan remains `950`, Shem's `600`.

An expansion around a common anchor would have the form `D_a(b)=a+E(b−a)`. For any two endpoints, `D_a(u)−D_a(v)=E(u−v)`, independently of `a`. The held Noah–Shem source and target gaps are both `500`, whereas an `E` map would give `12500/23`, with nonzero residual `1000/23`. **Therefore no one common anchored `E` map can equal the C322 whole-biography translation.** This conclusion requires no anchor solving or search. It concerns that uniform-map interpretation only; it does not rule out separately sourced local interval constructions.

The held Noah/Shem quadruple also forces `MT Noah−SP Shem=40` and `SP Noah−MT Shem=960=500+460` in each Gear. These cross-role widths follow from the existing `500` sides and `460` context translations. Their numerical compatibility with `460+40=500` is not, by itself, an instruction to apply `E` to a particular pair of historical endpoints.

## Positive source instructions and their bounds

| Supplied source passage | Positive instruction or fact | What it does not supply for C322 |
|---|---|---|
| File_18 §1.3.5, lines755–771 | Both regular Noah/Flood-bound `500→600` and Arphaxad/text-bound Shem100 states are valid; the latter gives the distinct two-year binding. | An `E` derivation of the Noah-bound relation, or a selection rule linking it to the SP→MT context translation. |
| File_18 §1.6.1–1.6.2, lines1123–1146 | Regular Cainan restoration is `+130`; cumulative restoration is `+460`, with explicit upstream/downstream scope. | Authority to replace the regular `+130` insertion by cumulative `+460` merely because C322's full regular context composition has net−460. |
| File_09 §3A.2.3, lines1045–1074 | `460E=500`; a `40` gain may be placed on either side of its declared cumulative field. The named boundaries `14511→14471→14011→13971` yield two500 spans and full outer540. | One universally selected orientation or an extension of those generated endpoints to the C322 regular biography. |
| File_09 §3A.3.3, lines1133–1155 | Restored Flood/Arphaxad `5296–5289→4836–4829` is460; the locally completed target `4796–4789` makes500. | Identification of that cumulative Flood/Arphaxad construction with regular Noah birth→Shem birth. |
| File_09 §3A.3.4, lines1157–1189 | Cainan `4858–4851→Shelah4398–4391` is460; generated completed `4358–4351` makes500. | An independently transmitted regular endpoint or Noah/Shem selector. |
| File_09 machine guards, lines482–500 | Local scope, raw/generated node classes and Key-expanded-span firewall are explicit. Cainan's source lifespan stays460; generated endpoints are not Primary anchors. | Permission to erase role, chronology, context, or source/generated distinctions. |
| File_46 §6A.3 and Appendix A.3, lines363–365 and422–428 | Two integral expansions for a23²-multiple; the appendix example is4232→4600→5000. The500 Noah resonance is explicitly a display motif. | A regular SP→MT endpoint prescription. The777-overlay example remains appendix-dependent. |
| File_46 §5 and §6A scope guard, lines291–303 and347–353 | The lattice-local `460+30` motif is static; the deep-time460 states retain their own load-bearing frames. | Either a universal prohibition on all460 calculations or a transferable targeting rule from the static motif. |
| AXIS_REPORT §§2,7.1,11, lines108–124,401–407,629 | Converted durations are distinguished from raw widths; biography translation preserves its source durations; assigning dates to a converted width requires holding a stated boundary. | The missing C322-specific expansion boundary and dependency. |

File_46 §6A.5 additionally registers the declared Adam-field `920E=1000`, with `920=460+460`. Dividing this scalar identity by two recovers `460E=500`; it does not select an endpoint in a different field. The same restraint applies to reading the display-motif500 in its double-expansion example.

## Claim typing and next bounded question

The ratio identity is an arithmetic fact. The C322 mapping is a verified source-qualified translation between declared contexts. File_09 supplies positive local endpoint constructions with generated completion coordinates. Noah-bound500 is an independently retained regular modal relation. Their unification into one prescribed operation remains **not established in the reviewed passages**. The proof against a common uniform `E` map is narrower and mathematically exact; it must not be substituted for the bounded source-audit conclusion.

The most useful next bounded question is to inspect only the fifteen already admitted same-Gear primary biography endpoints for a source-prescribed, role-typed `460+40=500` endpoint triple and determine whether its40 leg has an explicit local dependency. Record orientation, tradition/context, role, phase ancestry and source/generated status. Do not generate a new endpoint, choose a fitted anchor, apply an additional `E` transformation, or promote a numerical triple to a source instruction. This C325 proposal is unexecuted here.

The companion standard-library audit reads frozen `C322_DATA.json`, verifies the finite arithmetic ingredients, preserves the distinct502 binding, and records the anchor-independent span obstruction. Its checks are verification counts, not independent witnesses or evidence of source absence. The global paired Mirror remains January AD1; local junction remains x=1/January AD2; no civil year zero or new Mirror is introduced.