# RUN-102 — C10, the canonical-section audit: fifteen choices listed with what each depends on, its tag, how it was quotiented and whether it survives — none survives; each of R0's four arrows canonical in the reduced category at a named layer (C1.10's unique `ω_f`, A6.1's forced exponent, C1.13 lifted by Prop C1.1's injectivity, Kato exact for one Betti class); target and calibrator by the same natural construction with hypotheses instantiated; C9's one flat component used — a section transports path-independently along three paths; so the reduced comparison `Ψ_red` is itself the preferred flat section: **C10 PASS / canonical, T0, ρ = 1** — the PC-001 normalization certificate C1–C10 is closed **as architecture**, BSD stays **open**, and Track B's anchor rules are enforced and drilled though not needed

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the web line's instruction after [RUN-101](../RUN-101/) — C10, T0 or T1: Track A (a canonical natural section from the reduced comparison) before any anchor; every choice one row; the layer of each arrow's canonicity; naturality across partners; C9 used; Track B only on failure and never from `q_fin = 1`, a BSD identity, the target or an arbitrary vector; ten refusals; three controls; external obligations kept out; normalization closed ≠ BSD proved
**Tools:** `src104_c10_canonical_section.py`, `src11_gate_drill.py`
**Logs:** `src104-c10-canonical-section.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/C10-CANONICAL-SECTION/`](../../packages/C10-CANONICAL-SECTION/)

**Result. Track A decides it. The choice table has fifteen rows — `b_f⁺`, the `X` uniformiser, the Gauss convention, `Ω^±`, `q`, the lattice indices, the partner arithmetic, `C_d`, the `L_p` factors, `η_pdR`, `κ_det`, the PC-001 dictionary, `C_π`, the two named extractions, `n` — each with what it depends on, its ledger tag, how it was quotiented (C1.10's unique `ω_f`; the leading coefficient at `X = 0`; RUN-099's cancellation along `Ψ₄`; `q_D`; C8; Round 030 (16); `d`-independence; Eq. 11; RUN-099/100/101's resolved E-factors; `C_π` net 0; C1.8's divisibility) and whether it survives in `Ψ_red`: none does. Arrow by arrow, R0's "canonical" is located: `Ψ₁` canonical after `b_f⁺` (G) which `ω_f` replaces, `Ψ₂` canonical after `X` (G) with A6.1 forcing the lattice exponent, `Ψ₃` a class equality lifted by Prop C1.1 whose scalar `C = c*_f(r)` is determined by `f` and cancels, `Ψ₄` exact for one Betti class whose frame cancels — none merely up to a unit, so `Ψ_red = Ψ₄Ψ₃Ψ₂Ψ₁⁻¹` is choice-free; "`ω_f` is canonical, hence the section is" is refused as trivialising `𝒞_aux` only. Naturality: 389.a1 and 19.a1 are the same theorems with the one auxiliary family, hypotheses (noble, ordinary at 11, residually irreducible, weight-2 newforms) instantiated for both from RUN-095. C9 is used: a section at `L0` reaches `L4` along the chain, along `A′` and along the dictionary detour as the same element on exact instances. Hence T0: C10 PASS / canonical, `s_can` = the class of `Ψ_red`, `ρ = 1` — "trivial monodromy ⇒ section" is planted and refused, as is a defaulted common scalar. Track B is not taken, but its rules are enforced: four inadmissible anchor sources each caught when admitted; the gate accepts an anchored T1 control at `ρ = 3` and refuses PASS to a flat unanchored system. BF-COORD, `Π_E/Π_0`, `c_ES/Kum`, `v_11(q)`, `n` stay open outside C10. With C1–C10 all PASS the PC-001 normalization certificate is closed as architecture; BSD is open and the work re-enters Round 030's DAG A–G, none of which is claimed.**

---

## 0. The verdict in the requested form

```text
C10 = PASS / canonical
   state                        = T0
   canonical reduced comparison = Ψ_red = Ψ₄Ψ₃Ψ₂Ψ₁⁻¹ in the reduced category (G, L, P and the settled E-factors quotiented)
   preferred flat section       = s_can = the class of Ψ_red
   ρ                            = 1
   reason: all noncanonical choices live in G/L/P and were quotiented (§1: fifteen rows, none survives);
           η_pdR = 1 (RUN-099); κ_det = 1 (RUN-100); the residual graph is connected and flat (RUN-101);
           Ψ_red is a theoremically natural transformation — C1.9 + C1.10, C1.6–C1.8 + A6.1, C1.13 + Prop C1.1, Kato —
           natural in the partner (§3), and a section transports path-independently (§4).
   Track B not taken. Its rules are enforced anyway (§5), and the gate accepts a T1 anchor at ρ = 3 and refuses a flat unanchored system.

   ⇒ PC-001 normalization certificate C1–C10: CLOSED — as normalization ARCHITECTURE.   BSD: OPEN.
```

## 1. Every choice, one row each — surviving arbitrary choices = 0

| choice / factor | depends on | tag | quotiented how | survives in `Ψ_red`? |
| --- | --- | --- | --- | --- |
| `b_f⁺` | the auxiliary framing | G | replaced by C1.10's **unique** `ω_f` (RUN-096 gauge law; RUN-097: one free auxiliary frame) | no |
| `X` uniformiser | the weight coordinate on `V₁` | G | inside `c_lead/q_f`; the leading coefficient at `X = 0` is canonical whatever `n` (RUN-097/098) | no |
| Gauss convention | `Ω_F⁻ = Ω_E⁻/G(χ₈)` | G | a convention slot, `G(χ₈)² = 8` | no |
| `Ω^±` / Betti basis `γ_g^±` | B2.1's chosen bases | G | cancels along `Ψ₄`: `κ` and `L_p` scale together (RUN-099) | no |
| `q = [γ_Kato : γ_tree]` | two Z-structures on one Q-line | L | bucket `q_D`; a generator change, not a comparison scalar (RUN-099 + erratum) | no |
| lattice indices | the tree's line in the Manin lattice | L | lengths, C8 (RUN-096) | no |
| partner arithmetic | `E_adj_i, Π_i, λ_i(0), κ_E, ℓ_0` | P | removed first, Round 030 (16); net 0 in `δ` (RUN-097) | no |
| `C_d` | the auxiliary `d`, `(d, 6S) = 1` | cancels | `C_d⁻¹ ⊗ κ_d` is `d`-independent; `C_d = 26` cancels between `Ψ₁` and `Ψ₃` (RUN-098) | no |
| `L_p` factors | `L_p(g×ψ), L_p(g×τ), L_p(Ad g)` | P / cancel | Eq. 11 is the cancellation (RUN-098) | no |
| `η_pdR` | the period–de Rham pairing | E | `= 1`, one comparison pairing (RUN-099) | no |
| `κ_det` | the determinant functor | E | `= 1`, one route, functoriality (RUN-100) | no |
| PC-001 / LR dictionary | PC-001 §1 | E | `= 1`, a definition (RUN-101) | no |
| `C_π = c*_f(r)` | `f` alone (C1.9, C1.11) | E, exact, invariant | net exponent 0: cancels against `Ψ₁`'s `c_f` once `X^n` is counted once (RUN-097/098) | no |
| `log^{[r+1]}`, `X^n` | the two named extractions | extraction | canonical operations: C1.8's divisibility; the leading coefficient (RUN-098) | no |
| `n` (the jet order) | C1.10 | jet order | which coefficient is leading is decided by the class, not chosen | no |

A row left alive while Track A is claimed (`b_f⁺` in the drill) is caught by `c10-choices`; an unknown common scalar defaulted to `1` is caught by `c10-verdict`. The ledger it rests on has 23 records, none unassigned, `C_π` net 0.

## 2. Which layer R0's "canonical" is, arrow by arrow

| arrow | canonical? | up to a unit only? | choice tagged | in the reduced category |
| --- | --- | --- | --- | --- |
| `Ψ₁` | after choosing `b_f⁺` (G), which C1.10's unique `ω_f` replaces | no | G | **canonical** (C1.9 + C1.10) |
| `Ψ₂` | after choosing `X` (G); Theorem A6.1 forces the lattice exponent 1 | no | G | **canonical**: the leading coefficient at `X = 0` |
| `Ψ₃` | yes — a class equality (C1.13) lifted by Prop C1.1's injectivity; its scalar `C = c*_f(r)` is determined by `f` | no | none (`C_π`: E, exact, invariant, cancels) | **canonical** |
| `Ψ₄` | after fixing the Betti class `γ` (frame): Kato's reciprocity is exact for `z_γ ↔ L_p^{(γ)}`; the frame cancels (RUN-099) | no | G (frame), L (`q`) | **canonical** |

So the four layers the instruction named — `ω_f` unique (C1.10), the class equality lifted by injectivity (Prop C1.1), Kato exact after fixing the Betti class, every modular-symbol basis choice quotiented as G/L — are each present, and `Ψ_red` is independent of every basis, frame and generator choice. "`ω_f` is canonical, therefore the residual section is canonical" is refused: `ω_f` trivialises `𝒞_aux` only; the gate that stops there is caught by `c10-choices`.

## 3. Naturality — target and calibrator by the same construction

Both `Ψ_red,E` (389.a1) and `Ψ_red,0` (19.a1) are the same theorems — LR C1.9–C1.13 and Kato's reciprocity — with the one auxiliary family `f = E₂(1, χ₈)`'s critical refinement, instantiated at two partners. The hypotheses per partner, from RUN-095: noble (B1.1), ordinary at 11 (unit roots 7 and 3), residually irreducible at 11 (semistable + Mazur), newforms of weight 2 with trivial nebentypus — all hold for both. Naturality in `g` is the theorems' own functoriality: they are stated for any partner under B1.1 / §B2. A calibrator built by an ad hoc choice while a global section is claimed is caught by `c10-naturality`.

## 4. C9 used — the section transports path-independently

With RUN-101's representation II, a section at `L0` reaches `L4` along the chain `Ψ₄Ψ₃Ψ₂Ψ₁⁻¹`, along the shortcut `A′Ψ₂Ψ₁⁻¹` (`A′ = Ψ₄Ψ₃`, RUN-100) and along the dictionary detour `Ψ₄Ψ₃^PC·id·Ψ₂Ψ₁⁻¹` — the same element on 20 exact instances. One component, both representations flat (RUN-101). An `h = 2` on the `A′` path while C10 is still claimed is caught by `c10-transport`.

## 5. Track B — not taken, still enforced

Track A succeeded, so no 19.a1 anchor is constructed. The admissibility rules are enforced regardless: admissible sources are the 19.a1 LR/Kato exact class equality's residual after G/L/P and `η_pdR = κ_det = 1`, and an exact rank-zero reciprocity theorem; inadmissible are an arbitrary nonzero vector, `q_fin = 1`, a numerical BSD-shaped identity, and the target's result read back onto the calibrator — each admitted in the drill and caught by `c10-anchor`. An anchor need not be 1: control B (a connected flat system, `Mon = 1`, no preferred section, `s₀ = 3e`) is accepted at **T1, ρ = 3**; control C (the same system, no anchor) is refused PASS — **UNMEASURED**; control A (a natural isomorphism with no residual choice) is accepted at T0, ρ = 1.

## 6. The instruction's ten refusals, and where each is caught

| refusal | check |
| --- | --- |
| `ω_f` canonical ⇒ residual section canonical, arithmetic side unhandled | `c10-choices` |
| trivial monodromy ⇒ canonical section | `c10-verdict` |
| an arbitrary nonzero vector as "exact anchor" | `c10-anchor` |
| `q_fin = 1` as anchor | `c10-anchor` |
| a numerical BSD identity as anchor | `c10-anchor` |
| the target's result defining the calibrator anchor | `c10-anchor` |
| a G/L/P choice surviving yet Track A canonical | `c10-choices` |
| two partners' reduced comparisons not one natural construction yet a global section | `c10-naturality` |
| a path-dependent anchor yet C10 PASS | `c10-transport` |
| an unknown common scalar defaulted to 1 | `c10-verdict` |

Plus the two control refusals (a canonical-only gate; a gate that passes the unanchored system). Three controls must stay green: another exact anchor (`7e`) for control B, the seed, fewer trials.

## 7. External obligations — open, and not in C10

BF-COORD, `Π_E/Π_0`, `c_ES/Kum`, `v_11(q)`, `n` stay open. None survives into the residual torsor (§1), so none pollutes C10.

## 8. What a PASS means — and what this round does not claim

`C1–C10` all PASS: the **PC-001 normalization certificate is CLOSED, as normalization architecture**. It is not a proof of anything about `L(E, s)`: BSD stays **OPEN**, and the work re-enters Round 030's post-normalization DAG — A actual BF/Kato classes, B the arithmetic fundamental-line identification, C the regulator / Mordell–Weil determinant bridge, D reciprocity and non-annihilation, E the `p`-adic leading-line equality, F the complex leading-line comparison, G the finite arithmetic certificates. This round does not claim any of A–G, nor BSD, nor the values of `C_π`, `v_11(q)`, `n`, `Π_i`.

## 9. Drill

The whole-tree mutation drill: **590 planted defects, 590 caught by the check named for them, 0 uncaught, 0 caught by the wrong check, 156 controls undisturbed, state restored, over 182 checks** (first run, as 8 shards). Twelve of the defects and three of the controls are this round's. The report and package were written before the drill was launched and nothing was touched while it ran.

## 10. Status

```text
C1-C9  PASS       C10  PASS / canonical (T0, s_can = [Ψ_red], ρ = 1)       R0  PASS (category C)
PC-001 NORMALIZATION CERTIFICATE: CLOSED (architecture)
BSD  OPEN — next: Round 030's post-normalization DAG A-G
```