# RUN-098 — R0.5, the localization audit: the symbolic line's objection — a nonzero divisor is not a unit — was right — `Ψ₁` is injective but **not** surjective for the target because `L_E` has a double zero — verdict **C**: R0 lives over `ℋ_Γ`-lines localized at `X^n` and at the divisor of `L_p^g`, the composite is integral after the two named extractions `log^{[r+1]}` and `X^n`, RUN-097's headline is corrected, and C5 is kept with no new L-factor

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the web line's instruction after [RUN-097](../RUN-097/) — R0.5, distinguish isomorphism / injection by a non-zero-divisor / fractional-line isomorphism / localization-only inverse, arrow by arrow; four verdicts; five mutations; a positive control
**Tools:** `src100_r05_localization_audit.py`, `src11_gate_drill.py`
**Logs:** `src100-r05-localization-audit.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/R05-LOCALIZATION-AUDIT/`](../../packages/R05-LOCALIZATION-AUDIT/)

**Result. RUN-097 called the comparison route "over `ℋ_Γ`-lines" because `Ψ₁`'s scalar `c_f·C_d·L_p^g` is a product of non-zero-divisors and so `Ψ₁` is injective. The web line objected that on a free rank-1 module a non-zero-divisor gives an injection, not an isomorphism, unless it is a unit — and the tree's own data decide it: `L_p(g_E⊗ψ) = L_E = 0 + 0t + 2t² + …` has a double zero at `t = 0` (the rank-2 curve), so it is not a unit of `Λ`, and `Ψ₁` is not surjective for the target whatever `n` and `Π_E` are. `C_d = 26` is a unit mod 11; `L_0`, `λ_E`, `λ_0` are `Λ`-units by their constant terms; `c_f = X^n c_f*` is a unit iff `n = 0`, and `n` is unmeasured; `log^{[r+1]}` is a non-unit of `ℋ_Γ`. Arrow by arrow: `Ψ₁` injective, not surjective, inverse only over `ℋ_Γ[1/L_E][1/X^n]`; `Ψ₂` the one `X^n` extraction; `Ψ₃` fractional as written; `Ψ₄` a non-unit for the target. The composite's scalar, factor by factor: every `L`-function and `C_d` cancel exactly (LR Eq. 11), `X^n` counted once — `C = c*_f(r)` is `c_f` with its `X^n` removed and that removal is `Ψ₂`'s extraction; a first draft counted it twice and the gate's own composite check refused it — leaving `log^{[r+1]}/X^n`, exactly the two extractions LR names. Verdict **C**: localization at explicit divisors, integral after the named extractions and nowhere else. Not A, not D. C5 is kept: every divisor has net exponent 0 in RUN-097's slot table or is a named extraction, and `X^n` is the weight-jet order of Round 030 (14.1), not a δ-factor — new L-factors 0. The five mutations and the positive control are all planted. The stop rule's document-level proxy is left to read this round as "nothing new", which at document level is true, and is treated as the workflow heuristic the web line said it should be.**

---

## 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
```