# PC-001 ACTUAL NORMALIZATION AUDIT — Round 030's certificate C1–C10 against the actual PC-001 objects **From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`), answering `BSD_HANDOFF_TO_LOCAL_VALIDATOR_AFTER_ROUND035.md` (received 2026-09-14 20:54; byte-exact copy at `data/external/gpt-symbolic-rounds/handoffs/`) **Project:** `389.a1`, `p = 11`, `P = (0,0)`, `Q = (1,0)`, `a₁₁ = −4`, `w = +1`; calibrator `19.a1` (`a₁₁ = 3`, rank 0) **Gate:** `code/src92_pc001_actual_normalization_audit.py` · drill `code/src11_gate_drill.py` (checks `pc001-actual-C1` … `pc001-actual-C10`, `pc001-actual-rules`) **Machine-readable verdicts:** [`result.json`](./files/result.json) · **inputs and hashes:** `INPUTS.json`, `MANIFEST.json` · **evidence:** [`evidence/`](./files/evidence/) · **log:** `logs/` **This line's own report:** [`reports/RUN-090-PC001-ACTUAL-NORMALIZATION-AUDIT.md`](../../runs/RUN-090/) --- ## 0. The verdict, in the form asked for ```text PC-001 ACTUAL NORMALIZATION AUDIT: C1 UNMEASURED late common character — C_{π,E}, C_{π,0} do not exist as separate objects C2 UNMEASURED partner factors — λ_i(0), 𝒮₅(0), E_i measured; Π_E/Π_0 (adjoint Petersson/period) not C3 PASS coordinate factors — one cyclotomic coordinate for both partners, jet weights only C4 UNMEASURED pre-leading source — the Kummer-unit class is q-blind but is not yet Θ_aux C5 UNMEASURED defect classified — the existing factors are all tagged; δ itself does not exist C6 UNMEASURED period–de Rham — no pairing scalar η_pdR exists on the auxiliary side C7 UNMEASURED determinant — no auxiliary→arithmetic comparison map exists C8 UNMEASURED lattice / finite data — the calibrator's lattice data measured; Selmer/cone lengths absent C9 UNMEASURED residual flatness — the transport graph has no vertices yet C10 UNMEASURED residual anchor — blocked by C9; q_fin = 1 is a number, not an anchor first_blocker = C1 normalization_status = NORMALIZATION-UNMEASURED reopening_trigger = none fired (R2 cannot be excluded while C1 is unmeasured) ``` No item is `FAIL`. No `UNMEASURED` was promoted. No majority vote was taken. The rule that produced every verdict is one function (`rule()` in the gate): `NOT-APPLICABLE` if the item does not apply; `FAIL` on a counterexample or a nontrivial residual in hand; `UNMEASURED` if the object the item is about has not been constructed by anyone; `PASS` only if the object exists and every measurement on it is green. The drill plants the negation of that rule (`UNMEASURED_TO_PASS`, `CLOSURE_REQUIRES_ALL = False`) and both are caught. BSD remains open. `L''(E,1)/2 = Ω·Reg·q_fin` with `q_fin = 1` (every factor this tree's) is carried as a **CONSISTENCY CHECK** and is evidence for no C-item. --- ## 1. What was actually measured (the halves that exist) Everything below is recomputed in the gate from this tree's own objects — the mod-11 modular-symbol lines of `389.a1` (RUN-068/077), the rational lines of `19.a1` (RUN-080), the layer-`11³` measures and the four series `L_E, λ_E, L_0, λ_0` to `t¹²⁰` (this line's rebuild of RUN-081). The producers' scripts are not run. ### C3 (PASS) — the coordinate * **Same convention for both partners, measured.** Both series are built from one exponent map `e(u) = log_γ⟨u⟩`, `⟨u⟩ = u·ω(u)⁻¹`, `γ = 12` — the generator PC-001 names through `L = log₁₁(12)`. The same series rebuilt with `γ' = 23` (`log₁₂ 23 ≡ 112 (mod 121)`, `12¹¹² ≡ 23 (mod 1331)`) and mapped back by `S_{23} = S_{12} ∘ ((1+t)^{log₂₃ 12} − 1)` agree with the `γ = 12` build **in all 121 coefficients**, for all four series; the first eleven coefficients in the `12`-coordinate are PC-002's (`L_E = [0,0,2,2,2,0,5,6,10,7,0]`, `λ_E = [5,9,…]`, `L_0 = [9,3,…]`, `λ_0 = [4,3,…]`). * **Coordinate changes act only through the declared jet weight.** Under `t' = ut + O(t²)` the leading coefficient of an order-`r` series scales by `u^{−r}` and nothing else in the leading data moves: `L_E` (order 2, leading 2) becomes `[0,0,6,0,8,3,…]` in the `23`-coordinate — `6 = 2·2⁻²` with `u ≡ 112 ≡ 2 (mod 11)`; `λ_E, L_0, λ_0` (order 0) keep their constant terms `5, 9, 4` and their first coefficients scale by `u⁻¹` (`9 ↦ 10`, `3 ↦ 7`, `3 ↦ 7`). Twelve random formal changes `t = u⁻¹t' + b₂t'² + b₃t'³` on all four series: order preserved, `u^{−r}` on the leading coefficient, the next coefficient exactly `s_{r+1}u^{−r−1} + r·s_r·u^{1−r}b₂`, the tail moves. * **PC-001's identity (2) is covariant, not hiding a coordinate weight.** On the series model of (1) with the actual constant terms, under the same coordinate change `B_{E,2} ↦ u⁻²B_{E,2}`, `r_{0,1} ↦ u⁻¹r_{0,1}`, `κ_E ↦ u⁻¹κ_E`, both sides of (2) carry weight `−1`, and the constant `L = log₁₁(12)` cancels (`B_{E,2}` changes when `L` does, the right-hand side of (2) does not). * **Scope.** The PASS is on the objects that exist: the four series and the identity. The unconstructed `B̂_i` inherit the coordinate once they live in the same `Λ = Z₁₁[[Γ]]`; if they are ever built in another coordinate this item reopens under R6. * Level: a convention (Round 026's `G`); of the handoff's three, `E` — exact, neither partner- nor lattice-dependent. ### C2 (UNMEASURED) — the partner ratio, sharpened PC-001 (1): `A_i(t) = 𝒮₅(t)λ_i(t)/D_i`; Attack 09 (14)–(15): `D_i = L_p(Ad g_i)(0) = Π_i · E_i`, a Petersson/period ratio times the adjoint Euler multiplier `(1 − β_i/α_i)(1 − β_i/(11α_i))`. | factor | 389.a1 | 19.a1 | measured here | | --- | --- | --- | --- | | `λ_i(0)` (the `x⁻¹` moment, ω⁻¹ branch) | `5` | `4` | yes — constant terms of the actual twisted series | | `𝒮₅(0) = 5² − χ₈(5)⁻¹` | `26` | `26` | yes — partner-independent, `gcd(5, 6·8·389·19·11) = 1` | | `E_i = 1 − α_i⁻² (mod 11)` | `3` | `7` | yes — from the unit roots `7`, `3` | | `Π_i` (Petersson/period ratio in Loeffler–Rivero §B2.3 (8)) | — | — | **no; by nobody** | So `A_0(0)/A_E(0) = (λ_0/λ_E)(E_E/E_0)(Π_E/Π_0) ≡ 3 · 2 · (Π_E/Π_0) = 6 · (Π_E/Π_0) (mod 11)`. RUN-088 supplied the `3`; with the Euler multipliers the measured part is `6`; **the entire remaining uncertainty of the partner ratio is the adjoint Petersson/period ratio `Π_E/Π_0`** — the handoff's Attack B. PC-002's cup factor `3` is not it (RUN-081). Cross-checks: `A_E(0)D_E = 26·5 ≡ 9` (Attack 09), PC-001's "product of three nonzero factors" `9·4·4 ≡ 1`. The hidden-factor detector (the partner ratio read off the role tags equals the model's `A_0(0)/A_E(0)`, 40 instances) goes red exactly when a partner factor is tagged common. ### C1 (UNMEASURED) — the late common factor PC-001 §1 writes one `C_π` for both partners by citing Loeffler–Rivero C1.9–C1.13 and C1.11's uniqueness of the meromorphic Eichler–Shimura map given the auxiliary frame — `CITED_THEOREM`. The hypotheses measurable here hold: both partners ordinary at 11 (`a₁₁ = −4, 3`; unit roots `7, 3`), the auxiliary character decent (`χ₈` even, nontrivial, `χ₈(11) = −1`), the same auxiliary data named for both (`E₂(1,χ₈)`, `d = 5`). The hypothesis that cannot be instantiated is the construction itself: that the **same** auxiliary quotient transport is applied to both partners — a property of projected classes `B̂_E, B̂_0` that nobody has built (PC-001 §2: 仍要求真正構造 `B_{E,2}` 與 `r_{0,1}`; Attack 09: 未宣稱計算出 Coleman 族). What is measured: the derivation of (2) is exact **iff** the late factor is common — on the model, `RHS(2)/κ_E = C_{π,E}/C_{π,0}` exactly, so a split late factor is precisely the defect (2) would carry (40 instances, and `u` recovered for 40 random splits). ### C4 (UNMEASURED) — the q-blind source The candidate that exists is Attack 09's Kummer unit `ε = 1 + √2`: recomputed in `Z[√2]/121`, `ε²⁴ ≡ 1 + 110√2`, `log₁₁ ε ≡ 55√2`, `log ε/(11√2) ≡ 5 (mod 11)`. It is q-blind by construction (its computation references no quotient transport): under the pure reframe `q_f ↦ u q_f` (Round 021's frame, re-evaluated on the model) the source is fixed while `B̂_E`'s leading coefficient moves by `u⁻¹` (40 instances). It is **not** the tensor `Θ_aux ∈ Sym^m(T_X^*) ⊗ 𝒞_aux`: no weight-direction jet exists, and Attack 09 §6 says its comparison with the Eichler–Shimura leading period (16) is not established. ### C8 (UNMEASURED) — the finite ledger Measured: `19.a1`'s plus and minus vectors are Z-primitive (content 1, global saturation measured, not only 11-adic); PC-001's normalisation index `2` (coordinate 0 of the primitive plus vector) is recorded as a length, `v₂ = 1`, `v₁₁ = 0`; the minus index is `1`; `389.a1`'s lines are mod-11 objects normalised at their first nonzero coordinates (5, 3), the minus line a unit multiple `λ⁻ = 6` of the `Ω⁻`-normalised symbol (RUN-076) — its prime-to-11 saturation is **not** measured; `G(χ₈)` is a convention (`G² = 8`, an 11-unit). What C8 is about in Round 030 — `v_ℓ(det Φ_ℓ)` of the local integral maps of a Selmer complex or mapping cone against Smith / cone Euler lengths — has no complex to be computed on. ### C5, C6, C7, C9, C10 (UNMEASURED) — nothing to measure yet * C5: every factor of PC-001 (1)–(3) and Attack 09 (16), (20) carries a role — `C_π, c_lead, q_f: E`; `j, log₁₁(12), G(χ₈): G`; `𝒮₅, 16 = e₁₁/θ, 416, the index 2: L`; `λ_i(0), E_i, Π_i, κ_E, ℓ_0: P` — but the defect `δ = a_D q_D η_D` needs both sides of a comparison, and the auxiliary side (C4) is not built. * C6: no `η_pdR`; the archimedean alignments `λ_0⁺ = 1`, `λ⁻ = 6` are period-side data, not the period–de Rham pairing. The formal fact the handoff warns about is confirmed on 40 instances: opposite weights make `δ_per δ_dR` invariant, never `1`. * C7: no comparison map. The abstract identity `det(h⁰)/det(h¹) = det(H⁰h)/det(H¹h)` is recomputed here on explicit complexes in random bases (as in RUN-089); Attack 06's `H` (RUN-073) is the descent pairing matrix, not this map. * C9: the residual transport graph has no vertices — its vertices would be the objects of C4–C7. The machinery (spanning tree, cotree, cycle products) is in the gate and goes `FAIL` on one planted nontrivial cotree edge. * C10: blocked by C9. The numerical `q_fin = 1` is refused as an anchor (handoff C10, §14); the formal "one anchor fixes the section" is exercised and goes red when the anchor is removed. --- ## 2. Dependency impact (handoff §10, §15) * **Round 030:** no `FAIL`, so nothing is cut. Theorem 42.1 stays a conditional theorem whose hypotheses, for the actual PC-001, are one `PASS` (C3) and nine `UNMEASURED`, the earliest at C1. The certificate is not frozen; NORMALIZATION-CLOSED may not be declared. * **Rounds 031–035:** no dependency broken by this audit. They are in the tree byte-exact (received 2026-09-14 18:20–18:47) and **not yet verified here**; the target-side determinant architecture they rest on is contradicted by none of the nine `UNMEASURED` items. * **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's determinant comparisons. C9 → C10, node G. ## 3. What would move the frontier — the objects, ranked (handoff §12, §20) 1. **C1 / Attack A** — the projected late classes `B̂_E(t)`, `B̂_0(t)` from one auxiliary Eisenstein family `E₂(1,χ₈)^crit` with one frame and one quotient transport; equivalently the Coleman-family trace jets `𝒯_X(g) mod X²` with a stated pseudocharacter identification. Until then `C_{π,E} = C_{π,0}` is a citation, not a measurement. 2. **C2 / Attack B** — `Π_E, Π_0`: the Petersson/period halves of `D_i = L_p(Ad g_i)(0)` in Loeffler–Rivero's normalisation (§B2.3, eq. (8)). This line can compute a stated period ratio once the definition is pinned to a period; it will not guess the period. 3. **C4 / Attack D** — `Θ_aux`, or the identification of the Kummer class with the Eichler–Shimura leading period of Attack 09 (16). 4. **C7 / Attack E** — the auxiliary→arithmetic comparison map and its `κ_det`. 5. **C6** — `η_pdR` with its frame. 6. **C9–C10 / Attack F** — the transport graph, then the anchor. ## 4. The drill (handoff §17) Fourteen defects planted for this gate, each caught by the check named for its item, four controls undisturbed; the full tally (all gates of this tree) is in this line's report and in `logs/`. The planted ones: the late factor split `C_{π,E} = 2C_{π,0}` (C1); `λ_i` tagged common (C2); the jet weight as `u^{+r}`, and the calibrator built with generator 23 while the target keeps 12 (C3); the candidate source built with `q_f` (C4); `𝒮₅` left untagged (C5); period and de Rham exponents given the same sign (C6); a cohomologically nontrivial automorphism handed in as the comparison map (C7); the index 2 recorded as a unit (C8); one nontrivial cotree edge (C9); the anchor removed, and the numerical `q_fin = 1` accepted as an anchor (C10); `UNMEASURED` promoted to `PASS`, and closure declared with unmeasured items ignored (rules). Controls: another seed, more trials, more coordinate mutations, and both partners rebuilt with generator 23 — the same coordinate for both, PC-002's coefficients recovered in the 12-coordinate. ## 5. Claim labels used (handoff §16) `COMPUTED_HERE`: the four series and their coordinate laws, `λ_i(0)`, `𝒮₅(0)`, `E_i`, the Kummer logarithm, the calibrator's lattice data, the consistency ratio, the K1 identity on explicit complexes. `CITED_THEOREM`: Loeffler–Rivero C1.9–C1.13 / C1.11 (C1), the adjoint interpolation (C2). `DERIVED_FROM_VERIFIED_INPUTS`: the role ledger (C5), the covariance of (2) (C3). `UNMEASURED`: everything in §3. `FAILED`: nothing. ## 6. Not claimed Partner-independence of `C_π`; `D_E/D_0`; the BF/Kato classes; `B_{E,2}`, `r_{0,1}`; `C`; `𝔰₁₁`; any bridge A–G; BSD.