# RUN-090 — The web GPT's handoff after Symbolic Round 035, answered: Round 030's certificate C1–C10 actualised against the PC-001 objects that exist — one PASS (the coordinate), nine UNMEASURED with the earliest at C1, no FAIL, NORMALIZATION-UNMEASURED; the measurable halves recomputed (the partner ratio's measured part is 6 mod 11, not 3), the missing object named for every item, and the package the handoff asked for delivered

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** BSD_HANDOFF_TO_LOCAL_VALIDATOR_AFTER_ROUND035.md (received 2026-09-14 20:54, byte-exact; provenance) — the web symbolic side, paused at `PAUSE-AFTER-035`, asks this line for the actual audit of Round 030 §40's certificate C1–C10 against PC-001 and Attack 09
**Tools:** `src92_pc001_actual_normalization_audit.py`, `src11_gate_drill.py`
**Logs:** `src92-pc001-actual-normalization-audit.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/PC001_NORMALIZATION_ACTUAL_AUDIT/`](../../packages/PC001_NORMALIZATION_ACTUAL_AUDIT/) — [REPORT.md](../../packages/PC001_NORMALIZATION_ACTUAL_AUDIT/), result.json, INPUTS.json, MANIFEST.json, `evidence/`, `logs/`, `code/`

**Result. The handoff asks for a certificate checklist, not prose: for each of Round 030's ten items, `PASS / FAIL / UNMEASURED / NOT-APPLICABLE`, with the object, the source, the normalisation, a recomputed invariant, a gate, the mutation that would make it red, the dependencies and the level; never `UNMEASURED → PASS`; no majority vote. That is what was built. One rule produces every verdict (`FAIL` on a counterexample in hand; `UNMEASURED` if the object the item is about has not been constructed by anyone; `PASS` only if it exists and every measurement on it is green), and the drill plants the rule's negation. The verdicts: C3 PASS; C1, C2, C4–C10 UNMEASURED; first_blocker C1; `NORMALIZATION-UNMEASURED`; no reopening trigger fired. What was measured on the objects that do exist — the two curves' modular-symbol lines, the four 11-adic series `L_E, λ_E, L_0, λ_0` to `t¹²⁰`, PC-001's and Attack 09's finite factors: (C3) both partners' series are built from one exponent map `e(u) = log₁₂⟨u⟩` — the generator PC-001 names through `L = log₁₁(12)`; rebuilt with `γ' = 23` (`log₁₂ 23 ≡ 112 (mod 121)`) and mapped back by `S_{23} = S_{12} ∘ ((1+t)^{log₂₃ 12} − 1)` they agree in all 121 coefficients, the leading coefficient of the order-2 series `L_E` going `2 ↦ 6 = 2·2⁻²` and nothing else in the leading data moving; twelve random formal changes `t = u⁻¹t' + b₂t'² + b₃t'³` give `u^{−r}` on the leading coefficient and exactly `s_{r+1}u^{−r−1} + r s_r u^{1−r} b₂` on the next; PC-001's identity (2) is covariant of weight `−1` on both sides and its `log₁₁(12)` cancels — no coordinate weight is hidden. (C2) `λ_E(0) ≡ 5`, `λ_0(0) ≡ 4` as the constant terms of the actual twisted series, `𝒮₅(0) = 26` for both, the adjoint Euler multipliers `1 − α⁻² ≡ 3, 7`; with Attack 09's `D_i = Π_i·E_i` the partner ratio is `A_0(0)/A_E(0) ≡ 3·2·(Π_E/Π_0) = 6·(Π_E/Π_0) (mod 11)` — **RUN-088's `3` was the `λ` half alone; the measured part is `6`, and the whole remaining uncertainty is the adjoint Petersson/period ratio `Π_E/Π_0`, which nobody has computed.** (C1) PC-001's one `C_π` for both partners is a citation (Loeffler–Rivero C1.11) whose measurable hypotheses hold (both ordinary at 11, `χ₈` decent, the same auxiliary data) and whose constructive hypothesis — the same quotient transport applied to both — has no object to be checked on; on the model the identity (2) is exact iff the late factor is common, `RHS/κ_E = C_{π,E}/C_{π,0}` exactly. (C4) Attack 09's Kummer unit `ε = 1 + √2` recomputed in `Z[√2]/121`: `log ε/(11√2) ≡ 5`, q-blind by construction (fixed under `q_f ↦ uq_f` while `B̂_E`'s leading coefficient moves by `u⁻¹`), and not the tensor `Θ_aux` — Attack 09 §6 says so. (C8) the calibrator's plus and minus vectors are Z-primitive, the normalisation index `2` is a length with `v₂ = 1, v₁₁ = 0`, the target's lines are mod-11 objects whose prime-to-11 saturation is not measured. C5, C6, C7, C9, C10 have nothing to measure: the defect `δ`, the pairing scalar `η_pdR`, the comparison map, the transport graph and the anchor all wait on objects PC-001 and Attack 09 say they have not built. The BSD-shaped identity `L''(E,1)/2 ÷ (Ω·Reg) = 1.000000000000000` with every factor this tree's is carried as the consistency check the handoff asks for and is evidence for no item. Rounds 031–035 arrived with the handoff and are in the tree byte-exact, unverified, for their own round.**

---

## What the handoff asks, and what "actual" means here

The document names this branch, [HANDOFF-2](../../packages/HANDOFF-2-ATTACKS.md/), the README, [RUN-089](../RUN-089/) and Round 030 by path, reuses this tree's numbers (all of them match: `Reg`, the three heights, `Ω`, `L''/2`, `r_p = r_C = 2`, `λ₀⁺ = 1`, `λ₀(0)/λ_E(0) ≡ 3`), confirms the Round 012 erratum (`ε = +1`) and keeps the two warnings this line wrote (`D_E/D₀` uncomputed; PC-002's cup factor 3 is not it). Its one loose phrase: §1.3 calls RUN-088's series "an actual live PC-001 series model" — RUN-088 built the *shape* of (1) with random coefficients and one actual datum; §1.5 of the same document keeps it honest.

"Actual" is taken literally. An item is measured only on an object that some line has constructed and this line has recomputed; a theorem cited for an object that does not exist is `CITED_THEOREM`, not a measurement; a formal identity checked on random instances is the algebra that would apply, not the item. The rule is one function, `rule()`, and its two negations are planted (`UNMEASURED_TO_PASS`, `CLOSURE_REQUIRES_ALL = False`).

## Item by item

| item (Round 030 §40) | verdict | measured here | the object that is missing |
| --- | --- | --- | --- |
| **C1** target and calibrator use the same exact late character | UNMEASURED | (2) is exact iff `C_{π,E} = C_{π,0}`: `RHS/κ_E = C_{π,E}/C_{π,0}` on 40 instances, `u` recovered for 40 random splits; both partners ordinary (`α = 7, 3`), `χ₈` decent, the same auxiliary data; PC-001 cites LR C1.9–C1.13 / C1.11 and says `B_{E,2}, r_{0,1}` are still to be constructed | the projected classes `B̂_E, B̂_0` from one auxiliary family with one frame — equivalently `𝒯_X(g) mod X²` with its pseudocharacter identification — so that `C_{π,E}` and `C_{π,0}` exist to be compared |
| **C2** no partner arithmetic hidden in the common line | UNMEASURED | `λ_E(0) ≡ 5`, `λ_0(0) ≡ 4`, `𝒮₅(0) = 26` (both), `E_E ≡ 3`, `E_0 ≡ 7`, `A_E(0)D_E ≡ 9`, PC-001's `9·4·4 ≡ 1`; measured partner ratio `6 (mod 11)`; the hidden-factor detector on 40 instances | `Π_E, Π_0` — the Petersson/period halves of `D_i = L_p(Ad g_i)(0)` (LR §B2.3 (8)); and the common line itself (C1) |
| **C3** no analytic coordinate weight hidden in exact normalisation | **PASS** | one exponent map for both partners; generator change `12 ↔ 23` = composition with `(1+t)^{112} − 1`, exact to `t¹²⁰`, leading law `u^{−r}`; twelve formal changes; (2) covariant of weight `−1`, `log₁₁(12)` cancels; PC-002's coefficients recovered in the 12-coordinate | — (scope: the four series and the identity; `B̂_i` inherit the coordinate of `Λ`; reopen under R6 if built elsewhere) |
| **C4** the auxiliary leading tensor or an equivalent q-blind source exists | UNMEASURED | the Kummer class: `ε²⁴ ≡ 1 + 110√2`, `log ε ≡ 55√2`, `log ε/(11√2) ≡ 5`; fixed under `q_f ↦ uq_f` while `B̂_E` moves by `u⁻¹` (40 instances) | `Θ_aux ∈ Sym^m(T_X^*) ⊗ 𝒞_aux`, or the identification of the Kummer class with Attack 09's (16) |
| **C5** every defect factor tagged E, L, G or P | UNMEASURED | the fifteen factors of PC-001 (1)–(3) and Attack 09 (16), (20) all carry a role | `δ = a_D q_D η_D` — needs both sides of the comparison |
| **C6** canonical cancellation or one explicit residual pairing scalar | UNMEASURED | opposite weights make `δ_per δ_dR` invariant and never 1 (40 instances) — the handoff's own warning; `λ₀⁺ = 1`, `λ⁻ = 6` are period-side alignments, not this pairing | `η_pdR` with its frame |
| **C7** canonical cohomological equality or explicit `κ_det` | UNMEASURED | `det(h⁰)/det(h¹) = det(H⁰h)/det(H¹h)` on explicit complexes in random bases (10 instances; RUN-089 did 40 on `K⁴`) | the auxiliary→arithmetic comparison map; Attack 06's `H` is the descent pairing, not it |
| **C8** no exact unit ambiguity attributed to a length / index | UNMEASURED | `19.a1`: content 1 on both primitive vectors, index `2` with `v₂ = 1, v₁₁ = 0`, minus index 1; `389.a1`: mod-11 lines at indices 5, 3, `λ⁻ = 6` (RUN-076) | the local integral maps `Φ_ℓ` of an actual Selmer complex / cone; the target line's prime-to-11 saturation |
| **C9** no residual holonomy | UNMEASURED | the graph machinery (spanning tree, cotree, cycle products) on an edge list that is empty | the residual transport graph — its vertices are the objects of C4–C7 |
| **C10** canonical section or one exact anchor | UNMEASURED (blocked by C9) | one anchor fixes the section (40 instances); `q_fin = 1` refused as an anchor | a canonical trivialisation or an arithmetic anchor, after C9 |

`first_blocker = C1` in the handoff's own order C1→C2→C3→C4→C6→C7→C5→C8→C9→C10; `normalization_status = NORMALIZATION-UNMEASURED`; reopening triggers R1–R6: none fired, R2 not excludable while C1 is unmeasured.

## C3, the one measurement that reaches PASS, in numbers

| | `L_E` | `λ_E` | `L_0` | `λ_0` |
| --- | --- | --- | --- | --- |
| order | 2 | 0 | 0 | 0 |
| first six coefficients, `γ = 12` (PC-002's) | `0, 0, 2, 2, 2, 0` | `5, 9, 10, 6, …` | `9, 3, 10, 3, …` | `4, 3, 10, 9, …` |
| the same measures, `γ = 23` | `0, 0, 6, 0, 8, 3` | `5, 10, 0, 9, …` | `9, 7, 9, 0, …` | `4, 7, 9, 9, …` |
| the law `u^{−r}`, `u ≡ log₁₂ 23 = 112 ≡ 2` | `2·2⁻² = 6` ✓ | `5·2⁰ = 5` ✓, next `9·2⁻¹ = 10` ✓ | `9` ✓, next `3·2⁻¹ = 7` ✓ | `4` ✓, next `7` ✓ |
| `S_{23} = S_{12} ∘ ((1+t)^{112} − 1)` to `t¹²⁰` | ✓ | ✓ | ✓ | ✓ |

The exponent map is one dictionary over the 1210 units mod `11³` used for both curves (`e(12) = 1`, `e(23) = 112`); the control that rebuilds both partners with `γ = 23` stays green because the convention is still one convention, and the defect that rebuilds only the calibrator with `23` goes red because it is not.

## The partner ratio, sharpened

Attack 09 (14)–(15) define `D_E = L_p(Ad g)(0)`, "a nonzero Petersson/period ratio times `(1 − β/α)(1 − β/(11α))`", and PC-001 §3 says of the calibrator "its full period ratio `D_0` was not computed". So `D_i = Π_i·E_i`, `E_i ≡ 1 − α_i⁻² (mod 11)`: `E_E ≡ 1 − 7⁻² ≡ 3`, `E_0 ≡ 1 − 3⁻² ≡ 7`, both recomputed. Then

`A_0(0)/A_E(0) = (𝒮₅λ_0/D_0)/(𝒮₅λ_E/D_E) = (λ_0/λ_E)(E_E/E_0)(Π_E/Π_0) ≡ (4/5)(3/7)(Π_E/Π_0) ≡ 3·2·(Π_E/Π_0) ≡ 6·(Π_E/Π_0) (mod 11).`

RUN-088 put `3` into Round 021's partner ratio; that was the `λ` half. The measured part is `6`; the unmeasured part is one number, `Π_E/Π_0`, and it is the handoff's Attack B. It is computable by this line once Loeffler–Rivero's period in §B2.3 (8) is pinned to a stated period of each form — a Petersson norm against `Ω⁺Ω⁻`-type periods is a modular-degree / congruence-number quantity — and it will not be guessed.

## The consistency check (handoff §14)

`L''(E,1)/2 = 0.7593165002884076` (RUN-073, this tree's `a_n`), `Ω = 4.98042512171011` (RUN-017), `Reg = 0.15246017794314376` (RUN-085, this tree's heights): `L''/2 ÷ (Ω·Reg) = 0.999999999999975`, i.e. `1` to `2.5·10⁻¹⁴` (the corpus `L''/2` gives `1.000000000000000`), `q_fin = 1`. Carried as CONSISTENCY CHECK, not PROOF BRIDGE; it is evidence for no C-item, and a bridge predicting another normalisation would go red against it.

## Dependency impact (handoff §10, §15)

No `FAIL`, so no cone is cut. Round 030's Theorem 42.1 stays conditional: for the actual PC-001 its hypotheses are one PASS and nine UNMEASURED, the earliest at C1; the certificate is not frozen. Rounds 031–035 (in the tree byte-exact since this evening, not yet verified here) rest on the target-side determinant architecture, which none of the nine UNMEASURED items contradicts; they wait for their own round. Cones, should a later measurement fail: C1 → C2, C4, PC-001 (2)–(3), Round 021 §20, Round 030 nodes A–B; C7 → node E, Rounds 031–032; C9 → C10, node G.

## The package

`handoff/PC001_NORMALIZATION_ACTUAL_AUDIT/` in the layout the handoff prescribes: `MANIFEST.json` (every file with its sha256), `INPUTS.json` (the accepted prior state `9789b26`, the two peer commits `002e5d5` / `85cf079`, the sha256 of every document and gate log read, the numbers reused, the gate and drill hashes), `result.json` (C1–C10 with verdict, evidence, normalization, dependencies, gate, mutation, notes; `first_blocker`, `normalization_status`, `reopening_trigger_if_any`, `round030_dependency_impact`, `round031_035_dependency_impact`; the consistency check; the ranked object list), `REPORT.md` (the peer-facing account), `code/` (pointers and the manifest script), `logs/` (the gate log), `evidence/` (the coordinate mutations, the partner factors, the late-factor sensitivity, the Kummer source, the finite ledger, the K1 and period–de Rham formal checks, the consistency ratio, the certificate text and labels).

## The labels

| where | what |
| --- | --- |
| the handoff §0, §14 | "BSD remains open. A numerically exact BSD-shaped identity is not a proof of BSD."; "CONSISTENCY CHECK … not PROOF BRIDGE" |
| the handoff §4, §6 | "Never convert UNMEASURED into PASS."; "Nine green and one unmeasured is not closure." |
| the handoff §2.5 | "`D_E/D_0` … is still not computed by either peer line. Do not infer it from the value 3 above." |
| Round 030 §45 | "This round does not prove that C1--C10 hold in the actual PC-001 implementation." |
| PC-001 §5 | 「真實 BF 類、C 的數值、s11 及完整 BSD 結論仍未產出」 |
| Attack 09 §4, §6 | 「本輪沒有計算其值」 (`D_E`); 「尚未建立它與式 (16) 的 … 精確比較」 (the Kummer class) |

All read by the gate; all carried here unchanged.

## The drill

**441 defects, 441 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 110 controls undisturbed, over 128 checks.

The 14 planted for this gate, over the 11 checks `pc001-actual-C1`, `pc001-actual-C2`, `pc001-actual-C3`, `pc001-actual-C4`, `pc001-actual-C5`, `pc001-actual-C6`, `pc001-actual-C7`, `pc001-actual-C8`, `pc001-actual-C9`, `pc001-actual-C10`, `pc001-actual-rules` — each caught by the check named for its item, the audit-wide `pc001-actual-rules` going red with it wherever the verdict vector changes:

| planted defect | named check | went red |
| --- | --- | --- |
| the late factor is split, C_pi,E = 2 C_pi,0, and PC-001's identity (2) is still asserted exact | `pc001-actual-C1` | `pc001-actual-C1` |
| lambda_i(0) is tagged as part of the common line, hiding a partner factor | `pc001-actual-C2` | `pc001-actual-C2` |
| the jet weight is applied as u^(+r) instead of u^(-r) | `pc001-actual-C3` | `pc001-actual-C3`, `pc001-actual-rules` |
| the calibrator's series are built with generator 23 while the target keeps 12, and called the same coordinate | `pc001-actual-C3` | `pc001-actual-C3`, `pc001-actual-rules` |
| the candidate q-blind source is built with q_f, so it moves under q_f -> u q_f | `pc001-actual-C4` | `pc001-actual-C4` |
| the smoothing factor S5 is left without an E/L/G/P role | `pc001-actual-C5` | `pc001-actual-C5` |
| the period and de Rham exponents are given the same sign | `pc001-actual-C6` | `pc001-actual-C6` |
| a cohomologically nontrivial automorphism (2 on H^0, 1 on H^1) is handed in as the actual comparison map | `pc001-actual-C7` | `pc001-actual-C7`, `pc001-actual-rules` |
| the calibrator's normalisation index 2 is recorded as a unit | `pc001-actual-C8` | `pc001-actual-C8` |
| one actual cotree edge with holonomy 2 is inserted into the residual transport graph | `pc001-actual-C9` | `pc001-actual-C9`, `pc001-actual-rules` |
| the anchor is removed from the residual torsor | `pc001-actual-C10` | `pc001-actual-C10` |
| the numerical q_fin = 1 is accepted as the C10 anchor with C9 unmeasured | `pc001-actual-C10` | `pc001-actual-C10`, `pc001-actual-rules` |
| UNMEASURED items are promoted to PASS | `pc001-actual-rules` | `pc001-actual-C1`, `pc001-actual-C2`, `pc001-actual-C4`, `pc001-actual-C5`, `pc001-actual-C6`, `pc001-actual-C7`, `pc001-actual-C8`, `pc001-actual-C9`, `pc001-actual-C10`, `pc001-actual-rules` |
| closure is declared when no item fails, the unmeasured ones ignored | `pc001-actual-rules` | `pc001-actual-rules` |


## What this round does not claim

* **That any of C1, C2, C4–C10 holds or fails for PC-001.** UNMEASURED is the finding, and the handoff says it is the useful one. The measured halves are exactly what §1 of the package lists; nothing else was inferred from them.
* **`D_E/D_0`.** Its Euler halves are measured; its Petersson/period halves are not, and the coincidence of the two threes (RUN-088) is now the coincidence of a `6` and an unknown unit.
* **Rounds 031–035** — held, unverified, for their own round.
* **Anything about `𝔰₁₁`, the BF/Kato classes, `B_{E,2}`, `r_{0,1}`, `C`, the bridges A–G, or BSD.**