# R0.5 — the localization audit: verdict **C**, RUN-097's category corrected, C5 kept

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the symbolic line's instruction after RUN-097 — R0.5, the localization audit: a nonzero divisor is not a unit; every arrow typed with the ring its inverse exists over; four verdicts only; C5 stays PASS only once every non-unit divisor is accounted for exactly once
**Gate:** `code/src100_r05_localization_audit.py` · checks `r05-data`, `r05-arrows`, `r05-composite`, `r05-verdict`
**Machine-readable:** [`result.json`](./files/result.json) · **this line's report:** [`reports/RUN-098-R05-LOCALIZATION-AUDIT.md`](../../runs/RUN-098/)

---

## 0. The objection was right, and here is the verdict in the requested form

```text
R0.5 = C
   R0 requires localization at explicit divisors:
      per arrow:      X^n  (C1.10's n, unknown)
                      L_p(g_E ⊗ ψ) = L_E  (order 2 at t = 0: COMPUTED, the rank-2 curve)
                      L_p(Ad g)  (inverted in Ψ₁ and Ψ₃; carries Π_i)
                      log^{[r+1]}  (Ψ₃; a non-unit of ℋ_Γ)
      the composite:  only  log^{[r+1]} / X^n  — and those are exactly the two NAMED extractions
                      (C1.8 divides by log^{[r+1]}; the leading term at X = 0 divides by X^n)

   not A:  Ψ₁ is injective but NOT surjective for the target — its scalar contains L_E, a non-unit of Λ
   not D:  no inverse used in RUN-097 is illegitimate once the localization is named

RUN-097's headline "R0 PASS over ℋ_Γ-lines" is corrected to
   "R0 PASS over ℋ_Γ-lines localized at X^n and at the divisor of L_p^g; the composite is integral
    after the two named leading-term extractions and nowhere else needs a localization."

C5 PASS kept:  every divisor has net exponent 0 in RUN-097's slot table or is a named extraction;
               X^n is not a δ-factor but the weight-jet order m = n of Round 030 (14.1).  New L-factors: 0.
```

I wrote the invertibility of the leading-term map back onto the family map. "Each factor a non-zero-divisor, hence injective" was true and was not enough. The correction is not cosmetic: it is the difference between an `ℋ_Γ`-module isomorphism and an isomorphism after inverting a `p`-adic `L`-function that has a double zero.

## 1. The data — unit or not, from the tree

| scalar | in | first coefficients (mod 11) / value | status | source |
| --- | --- | --- | --- | --- |
| `C_d` | `Z₁₁` | 26 | **unit** (`11 ∤ 26`) | COMPUTED (RUN-095) |
| `L_p(g_E⊗ψ) = L_E` | `Λ = Z₁₁[[t]]` | `0, 0, 2, 2, 2, 0, 5, …` — order **2** at `t = 0` | **non-zero-divisor, non-unit** | COMPUTED (RUN-081/090 series) |
| `L_p(g_0⊗ψ) = L_0` | `Λ` | `9, 3, 10, …` — unit constant term | **unit** of `Λ` | COMPUTED |
| `L_p(g_E⊗τ) = λ_E` | `Λ` | `5, 9, 10, …` | unit | COMPUTED |
| `L_p(g_0⊗τ) = λ_0` | `Λ` | `4, 3, 10, …` | unit | COMPUTED |
| `E_i` (adjoint Euler) | `Z₁₁` | 3, 7 | units | COMPUTED (RUN-090) |
| `Π_i` | `Z₁₁` | unknown value | unit (RUN-091, Attack B) | DERIVED |
| `c_f = X^n · c_f*` | `O(V₁)` | `c_f*(r) ∈ L^×` (C1.13); `n ⩾ 0` unknown (C1.10) | unit **iff `n = 0`** — unmeasured | CITED |
| `log^{[r+1]}` | `ℋ_Γ` | zeros at the characters of degree `⩽ r` | non-unit | CITED (C1.8) |

The one that decides the target's case is `L_E`: an element of `Λ` with zero constant term is not a unit, whatever `n` and `Π` turn out to be. For the calibrator, `L_0` *is* a `Λ`-unit (`μ = λ = 0` at 11 — consistent with rank 0), so `Ψ₁`'s unit status there rests only on `n` and `Π_0`.

## 2. The arrows, in the four states

| arrow | scalar | target 389.a1 | calibrator 19.a1 | inverse exists | divisors |
| --- | --- | --- | --- | --- | --- |
| `Ψ₁` | `c_f · C_d · L_p^g`, `L_p^g = L_p(g⊗ψ)L_p(g⊗τ)/L_p(Ad g)` (Eq. 11) | **non-zero-divisor, non-unit** — injective, not surjective, cokernel `⊇ ℋ_Γ/(L_E)` | non-unit if `n > 0`; unit if `n = 0` and `Π_0` a unit | on the image only; over `ℋ_Γ[1/L_E][1/X^n]` | `X^n`, `L_E`, `L_p(Ad g)` |
| `Ψ₂` | the projection (surjective), then **division by `X^n`** | the one place family ⇝ leading term happens | same | a localization at `X`, named | `X^n` |
| `Ψ₃` | `C · C_d · log^{[r+1]} · L_p(g⊗τ) / L_p(Ad g)` | **fractional** as written (`1/L_p(Ad g)`); non-unit through `log` | same | over the total quotient ring | `log^{[r+1]}`, `L_p(Ad g)` |
| `Ψ₄` | `L_p(g⊗ψ)` | non-unit (`L_E`) | `Λ`-unit (`L_0`) | over `ℋ_Γ[1/L_E]` | `L_E` |

The instruction's five mutations, and where each is caught: a non-zero-divisor forced to "unit" → `r05-arrows`; a base-ring inverse claimed for a non-surjective map → `r05-arrows`; a cokernel divisor forgotten → `r05-composite`; the `X^n` extraction moved to the family map → `r05-arrows`; `Ψ₃`'s total-quotient-ring identity downgraded to an integral one → `r05-arrows`. The positive control — a genuine unit on a free rank-1 module — is classified `isomorphism, inverse over the base ring` and accepted; the drill hands the gate that same control mislabelled and the gate must refuse.

## 3. The composite

```text
Ψ = Ψ₄ Ψ₃ Ψ₂ Ψ₁⁻¹ :   [L_p(g⊗ψ)] · [C·C_d·log·L_p(g⊗τ)/L_p(Ad g)] · [1/X^n] / [c_f·C_d·L_p^g]

   L_p(g⊗ψ)   +1 (Ψ₄)  −1 (Ψ₁)   → 0
   L_p(g⊗τ)   +1 (Ψ₃)  −1 (Ψ₁)   → 0        LR Eq. 11 is the cancellation
   L_p(Ad g)   −1 (Ψ₃)  +1 (Ψ₁)   → 0
   C_d         +1 (Ψ₃)  −1 (Ψ₁)   → 0        κ = C_d⁻¹ ⊗ κ_d is d-independent
   C / c_f     = X^{−n}: the SAME operation as Ψ₂'s extraction, counted once
   log^{[r+1]} +1 (Ψ₃)                        residual
   X^n         −1 (Ψ₂)                        residual

   ⇒  Ψ's scalar = log^{[r+1]} / X^n · (unit)
```

Every `L`-function cancels exactly, `C_d` cancels, and what remains is the cyclotomic logarithm over the weight order — precisely the two extractions LR names (C1.8; C1.7/C1.10). So the composite is integral over `ℋ_Γ` **after** those two named extractions, and nowhere else needs a localization. That is case **C**, with the divisors listed, rather than B: the category is not "fractional in general" but "integral except at named places".

A first draft of this gate counted the `X^n` twice — once as `Ψ₂`'s division, once as `C/c_f` — and its own composite check refused the result. `C = c*_f(r)` is `c_f` *with the `X^n` removed*, and that removal *is* `Ψ₂`'s extraction; one `X^n`, not two. Recorded because it is exactly the kind of slip you warned about ("不要把 Ψ₁* 的可逆性回寫到原始 Ψ₁"), in the opposite direction.

## 4. C5, re-checked against the divisors

| divisor | accounted where | net |
| --- | --- | --- |
| `L_p(g⊗ψ)`, `L_p(g⊗τ)`, `L_p(Ad g)` | RUN-097's slot table: `λ_i`, `Π_i`, `E_i` all net 0 | cancel |
| `C_d` | slot table: `S5` net 0 | cancel |
| `log^{[r+1]}` | slot table: `j` net 0 (`Ψ₃` +1, `Ψ₂` −1) — C1.8's division **is** the accounting | cancel |
| `X^n` | not a δ-factor: the order `m = n` of the weight jet `Sym^m(T_X*)` in Round 030 (14.1) | jet order |

No cokernel produces a new L-factor: **new L-factors = 0**, C5 PASS is kept, and the reason is now the divisor accounting rather than the slot table alone.

## 5. What R0.5 does not claim

* the values of `n`, `Π_i`, `η_pdR`, `κ_det`;
* anything about the calibrator's `Ψ₁` being an isomorphism — that rests on `n = 0`, unmeasured;
* anything past the chain: the leading coefficient in `t` (PC-001 (2)) is downstream.

## 6. On the stop rule

Agreed: the claim-ladder's document-level rule is a workflow heuristic, not an epistemic gate. This round compiled no new document; its `source` field says so plainly ("RUN-097's own chain, re-audited"), and the rule will read that as "nothing new" — which at the document level is true. What advanced is your fifth criterion, the frontier: a category error that would have let C6/C7 "prove a beautiful 1 in the wrong category" is closed. That is the sort of stop condition worth keeping.

## 8. Drill

The whole-tree mutation drill, first run: **541 planted defects, 541 caught by the check named for them, 0 uncaught, 0 caught by the wrong check, 145 controls undisturbed, state restored, over 164 checks** (5460 s as 8 shards). Eight of the defects and two of the controls are this round's — the instruction's five mutations, the cyclotomic log declared a unit, `C_d` misread as 22, and the positive control handed in mislabelled. This time the report and package were written *before* the drill was launched and nothing was touched while it ran.

## 7. Status

```text
C1-C4  PASS       R0  PASS (category C: localized at X^n, L_p^g; composite integral after the two named extractions)
C5     PASS (kept: no new L-factor)          C8  PASS
C6     UNMEASURED — η_pdR: is the same canonical perfect pairing used on both sides of Ψ₄? (opposite weights alone are not enough)
C7     UNMEASURED — κ_det: one route; (18.1) applies iff the same determinant functor on both sides
C9, C10  UNMEASURED
NORMALIZATION-UNMEASURED        BSD  OPEN
```