# RUN-107 — DAG-D, explicit reciprocity and nonannihilation: **DAG-D PASS**. All three layers land — the first nonzero projective class in `Q = H/L` is Loeffler–Rivero's own `X^n κ̂_d mod X`, the map is not merely nonzero but **injective** (that is literally Proposition C1.1), and the image is nonzero because `L_p(g_E × ψ) ≠ 0` in `Λ`, computed here from `μ = 0`. **D-rational = ISOMORPHISM** and **D-integral = index 11**, compatible, with the local Bloch–Kato factor entered once and its owner named. Schneider's conjecture is **not** in D's dependency set. And one sharpening that cuts *against* us: from mod-11 data `λ = 2` and `μ = 0` are determined and **the exact order of vanishing at `t = 0` over `Z₁₁` is not** — so even the series' order is partly E's obligation, and this round refuses to emit it **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** the symbolic line's instruction after [RUN-106](../../runs/RUN-106/) — hit D only: reclassify the existing LR/Kato chain into "the reciprocity map exists, typed" plus "it is nonzero on the actual transverse quotient", never E's leading-order equality; take the route out of RUN-097/098 rather than rebuilding it; give D its real domain in Round 030's `Q = H/L` language; separate the three nonannihilation layers; never read a positive vanishing order as annihilation; audit the directional question; place RUN-106's local index 11 exactly, once, with an owner; keep C6 frozen; keep Schneider out; keep the normalisation certificate closed; verify target and calibrator separately; at least ten mutations and two positive controls. Do not judge E, F or G **Tools:** `src109_dag_d_reciprocity.py`, `src11_gate_drill.py` **Logs:** `src109-dag-d-reciprocity.json`, `src11-gate-drill.json` **Deliverable:** [`handoff/DAG-D-RECIPROCITY/`](../DAG-D-RECIPROCITY/) · **re-read this round for substance rather than for normalisation:** Loeffler–Rivero `C1.1`, `C1.6`, `C1.7`, `C1.9`, `C1.13` and Kato's explicit reciprocity law, statements in `EXTRACTS.md` **Result. D is a PASS, and the instruction's guess as to why was right: the four theorems D needs were located in RUN-094/095/097/098 and used there only for *normalisation* — read for their substantive content they give D almost everything directly. The route is not rebuilt; it is read off RUN-097's realised chain and RUN-098's localisation audit as three arrows. `Ψ₂` is `C1.6` (the `τ`-quotient image of `κ_d` is zero) plus `C1.7` (the lift is unique and lands in the `ψ`-quotient) — and that is what *defines* D's domain, so the domain is not a new construction either. `Ψ₃` is `C1.13`, whose proof shape matters: equal regulator images first, then equality **of classes** by the injectivity of Proposition C1.1 — a class identity with its lifting theorem named, not a scalar identity in disguise. `Ψ₄` is Kato's explicit reciprocity itself, with the Betti-period convention already audited at RUN-080/096 and frozen by C6. In Round 030's own language: the ambient `H` is the Beilinson–Flach family module, the reference subobject `L` is where `C1.6` puts the class, `Q = H/L` is the `ψ`-quotient line — free of rank one over `ℋ_Γ` by `C1.1`'s proof — and the first nonzero projective class is LR's own `X^n κ̂_d mod X`, which the paper asserts is well-defined and nonzero mod `X`. That is **D-NV1**, cited. **D-NV2** is Proposition C1.1 read for content: at each classical specialisation the relevant global `H¹` is *free of rank one over `ℋ_Γ`*, contains Kato's Euler system as a canonical element, and its localisation at `p` maps under the Perrin-Riou regulator to the `p`-adic `L`-function, which is *not a zero-divisor* — so the map is nonzero, and because the Greenberg Selmer group (its kernel) vanishes, **injective**. Never surjective: that mutation is planted. **D-NV3** is computed here. The reduction of `L_p(g_E × ψ)` mod 11 is `2t² + 2t³ + 2t⁴ + 0t⁵ + 5t⁶ + 6t⁷ + 10t⁸ + 7t⁹`, which is not the zero series — so `μ = 0` and `L_E ≠ 0` in `Λ`; and the same reduction vanishes modulo `(11, t)`, so `L_E ∉ Λ^×`. The Weierstrass split `L̄_E = t²·ū` is exhibited with `ū⁻¹` actually computed (`ū·ū⁻¹ = 1` to the full truncation), and `ord(L̄_E·ḡ) = 2 + ord(ḡ)` on every one of 40 random nonzero witnesses — so multiplication by `L_E` **annihilates nothing**, while its value at the central point is a non-unit. Those are two different statements and this round keeps them apart: her §5's `t^m u(t)`, on the actual object. The directional counterexample is the tree's own series: it is nonzero while both its value and its first derivative at `0` vanish — one variable removes the *choice* of direction, it does not make a derivative nonzero. RUN-106's local index is placed with two verdicts rather than one: **D-rational = ISOMORPHISM**, because `log_ω : H¹_f(Q₁₁,V) → D_dR(V)/Fil⁰ ≅ Q₁₁` is a nonzero map of one-dimensional `Q₁₁`-spaces; **D-integral = index 11**, because `log_ω(H¹_f(Q₁₁,T)) = 11·Z₁₁`. Compatible, and D passes with the integral index `≠ 1`. The factor is entered once, owner named, and E may read it and not re-enter it. Schneider is absent from D's dependency list by construction — no arrow of D's route is a height pairing — and "Schneider unproved ⇒ D FAIL" is planted and caught. Twenty-five mutations (her twelve among them) and both positive controls; `ε` untouched; E, F, G not judged. One finding runs against us and is recorded as such: `λ = 2` and `μ = 0` are what mod-11 data determines, and the exact order of vanishing at `t = 0` over `Z₁₁` is **not** determined by it — so this gate declines to state the order, and hands that to E as an explicit UNMEASURED.** --- ## 0. The verdict in the requested form ```text DAG-D RESULT map source Q = H/L = H^1(Q, V(g)*(psi^-1) (x) H_Gamma(-j)), free of rank 1 over H_Gamma target A_p = the H_Gamma-line spanned by L_p(g_E x psi) theorem Kato's explicit reciprocity law, with LR C1.13 and Proposition C1.1 typed injective YES / surjective NO / status: non-zero-divisor, non-unit target 389.a1 global class nonzero CITED (C1.1: Kato's element is a canonical generator) localisation nonzero CITED (C1.1's proof: nonzero regulator image at p) reciprocity map nonzero PASS (C1.1, and injective) analytic image nonzero PASS (COMPUTED HERE: mu = 0) analytic image a unit NO (COMPUTED HERE: zero mod (11, t)) integral index 11 (RUN-106, local Bloch-Kato line) calibrator 19.a1 global class nonzero CITED localisation nonzero CITED reciprocity map nonzero PASS analytic image nonzero PASS (COMPUTED HERE) analytic image a unit YES (constant term 9 mod 11 -> Lambda-unit) integral index not computed this round one-variable audit relevant tangent directions = 1 (PC-001 fixes one weight/cyclotomic direction) no direction to choose; "one variable therefore a nonzero derivative" REFUSED, with the counterexample taken from the tree's own series local Bloch-Kato factor index 11 owner p-adic reciprocity / the leading-line integral realization counted exactly once; E may read it, E may not enter it again Schneider dependency NONE D verdict PASS D-rational = ISOMORPHISM (nonzero map of one-dimensional Q_11-spaces) D-integral = INDEX 11 (image of the lattice T is 11 Z_11) E input the one tangent direction, with no directional choice left the local factor 11, owner named, already counted once lambda(L_E) = 2 and mu(L_E) = 0 from mod-11 data; the EXACT order of vanishing at t = 0 over Z_11 is UNMEASURED and is E's to establish C = c*_f(r) in L^x; C_d = 26 a unit; L_p(g x tau) a unit; log^{[r+1]} a non-zero-divisor and not a unit D-rational and D-integral are different verdicts; do not conflate them the calibrator's series is a Lambda-unit and the target's is not ``` --- ## 1. The boundary, stated before anything is done D asks three questions and only three: is the reciprocity map **actual** (a map of honest objects), is it **typed** (source, target, ring, integral or rational, named theorem), and is it **nonzero on the actual transverse quotient**. E asks the other three: the order, the leading tensor, the exact leading coefficient and its equality with the arithmetic side. The series fact gives D exactly two things and gives E nothing for free: | from the series | D may use it | E may use it | | --- | --- | --- | | `L_E ≠ 0` in `Λ` (`μ = 0`) | yes — D-NV3 | — | | `L_E ∉ Λ^×` (zero mod `(11,t)`) | yes — the non-unit status | — | | `λ(L_E) = 2` | recorded, not used as an order | an input, not a conclusion | | the exact order of vanishing at `t = 0` over `Z₁₁` | **no — undetermined by mod-11 data** | E's obligation | | the leading coefficient | no | E's obligation | Treating `L_E(t) = 2t² + ⋯` as E already proved is planted as a defect (`dag-d-boundary`), as is emitting any verdict on E. ## 2. The route, read not rebuilt Three arrows, each lifted out of `src99` (RUN-097, the realised chain) and `src100` (RUN-098, the localisation audit). `Ψ₁` — the Coleman/Perrin-Riou computation on the *family* line — is listed for completeness, because it is the source of the scalars and not of `λ_rec`. | arrow | source → target | ring / level | theorem | inj | surj | localised at | | --- | --- | --- | --- | --- | --- | --- | | `Ψ₂` | BF family line `H` → the `ψ`-quotient line `Q` | `ℋ_Γ` after the weight specialisation; integral after the named extractions | `C1.6` (the `τ`-quotient image is zero) + `C1.7` (unique lift, image in the `ψ`-quotient); the leading term `X^n κ̂_d mod X` | ✔ | ✘ | `X^n` | | `Ψ₃` | `Q` → the Kato line (the *same* free rank-1 `ℋ_Γ`-module, `C1.1`) | rational as written; integral after clearing `L_p(Ad g)` | `C1.13`, proved scalar-first and lifted **to classes** by Proposition C1.1 | ✔ | ✘ | `log^{[r+1]}`, `L_p(Ad g)` | | `Ψ₄` | the Kato line → `𝒜_p` | `ℋ_Γ` | Kato's explicit reciprocity law; period convention audited RUN-080/096, frozen by C6 | ✔ | ✘ (target) / ✔ (calibrator) | `L_p(g_E × ψ)` | `Ψ₂` is the one place the passage family → leading term happens, and RUN-098 already named it as a localisation at `X`. In the composite every `L`-function cancels exactly — that is what LR's Equation 11 *is* — and the residual is precisely the two named extractions `log^{[r+1]}/X^n`. Nothing new is localised this round. The mutation here: calling `C1.13` a class identity without the `C1.1` lift. Caught at `dag-d-route`. ## 3. The theorem inventory, read literally | theorem | what D takes | what D may **not** take | | --- | --- | --- | | **LR `C1.9`** | the factorisation, and that `c_f(k)` is a function of `V₁` **alone** — partner-independence is a theorem, not an assumption | the non-vanishing at our point. `k = r` is *exactly* the weight the clause excepts. The nonvanishing comes from `C1.13` instead | | **LR `C1.13`** | `C = c*_f(r) ∈ L^×` — the paper's own assertion in the construction — and `X^n κ̂_d` well-defined, nonzero mod `X` | any leading-coefficient equality | | **LR Prop `C1.1`** | the relevant global `H¹` is **free of rank one over `ℋ_Γ`**, Kato's element is canonical in it, its regulator image is the `p`-adic `L`-function and *not a zero-divisor*; the Greenberg Selmer group — the kernel — vanishes. Hence **injective** | surjectivity, or an integral isomorphism | | **Kato reciprocity** | the arrow `Ψ₄` exists and is injective; `Kato's L_p` = the Mazur–Tate–Teitelbaum `L_p` up to the period frame | reopening the period frame (C6 is frozen) | Every hypothesis points back to a round that actually discharged it, not to a fresh assumption: `κ_d` exists with `(d, 6S) = 1` and `f_β` non-critical at RUN-094; `g` noble, ordinary, `p`-distinguished, prime-to-`p` level, residually irreducible (semistable + Mazur) at RUN-095; and the hypothesis `C1.1` places on the partner is **"ordinary" and nothing else** — adding residual surjectivity would be over-strong and is recorded as a defect, as RUN-094 already noted. The `C1.9` caveat deserves its own line, because it is the kind of thing that is easy to get backwards and expensive if you do: RUN-094 reported "`C` nonzero, `C1.13`" and that was right, but the reason matters — `C1.9`'s non-vanishing clause *excepts* `k = r`, and `k = r` is our point. Stretching `C1.9` over it is planted and caught at `dag-d-theorems`. ## 4. D's real domain: `Q = H/L`, with the parts named **Two different quotients are in play, and they do opposite jobs.** The `τ`-quotient is where `C1.6` says the class **dies** — which is precisely how it descends into the ambient. The `ψ⁻¹`-quotient is where the leading term **survives**, and that one is D's domain. Getting them the right way round is the whole content of this section: ```text enlarged H^1(Q, (1/X)V_c(f,g)*), where kappa_d is constructed ring O(V_1 x V_2 x W), rank 1 tau-quot (1/X)V_c(f,g)*/V(f,g)* = Q_p(tau^-1)(1+r) (x) V(g)* (x) H_Gamma(-j) C1.6: the image of kappa_d here is ZERO <- the direction the class DIES in H ambient H^1(Q, V(f,g)*), the Beilinson-Flach FAMILY module itself ring O(V_1 x V_2 x W), rank 1 <- NOT the transverse quotient the class is here because C1.6 kills the tau-quotient image, and C1.7 says that lift is UNIQUE L reference ker( H -> the psi^{-1}-quotient ) = the image of H^1(Q, V(g)* (x) H_Gamma(-j)) under the inclusion 1 -> V(f_beta)* why the 1-part is the SUB: dualising RUN-093's 0 -> psi -> V(f_beta) -> 1 -> 0 gives 0 -> 1 -> V(f_beta)* -> psi^-1 -> 0 Q quotient H/L = the psi^{-1}-quotient line H^1(Q, V(g)*(psi^-1) (x) H_Gamma(-j)) free of rank 1 over H_Gamma (C1.1's proof) C1.7: the unique lift's image in the psi^{-1}-quotient is what is taken <- D's DOMAIN z first nonzero projective class X^n kappa-hat_d(f,g) mod X, asserted well-defined and NONZERO mod X (LR) which n: the weight-jet order of C1.10 = Round 030's m = RUN-098's Psi_2 localisation ``` Two variables, kept apart on purpose: the **weight** direction `X` is where the leading-term extraction happens, to order `n`; the **cyclotomic** direction `t` is where the target lives, and its vanishing order is E's. Calling the whole family line the transverse quotient, claiming an image while the class in `Q` is zero, or **swapping the two quotients** — calling the class a survivor in the direction `C1.6` kills — is planted and caught at `dag-d-domain`. ## 5. The three layers ```text D-NV1 the class is nonzero X^n kappa-hat_d mod X in Q PASS CITED (C1.13's construction) D-NV2 the map is nonzero lambda_rec : Q -> A_p PASS CITED (Prop C1.1 — INJECTIVE) D-NV3 the image is nonzero (nonzero scalar) * L_p(g_E x psi) PASS COMPUTED HERE ``` D-NV2 is the layer the instruction guessed could come straight from the literature, and it does — but it takes **both** halves of Proposition C1.1, and the two halves are not the same kernel: | | what it gives | which factor of `λ_rec` | | --- | --- | --- | | the **statement** of `C1.1` | the Greenberg Selmer group — *by definition* the kernel of `loc_p` into `H¹(Q_p, F⁻V(g)*(χ)(n) ⊗ ℋ_Γ(−j))` — vanishes | `loc₁₁` is **injective** on `Q` | | the **proof** of `C1.1` | `Q` is free of rank one over `ℋ_Γ`, and the regulator carries Kato's canonical element to `L_p(g × ψ)`, which is **not a zero-divisor** | the Perrin-Riou regulator is **nonzero** on `Q` | | `ℋ_Γ` is a domain (for `Γ ≅ Z₁₁` a ring of functions analytic on the open unit disc) | multiplication by a nonzero element is injective | an injection followed by that is injective | So `λ_rec` is injective, and it is worth saying why the two steps are needed separately: the vanishing Greenberg Selmer group is the kernel of `loc_p`, **not** of the composite. Reading it as the composite's kernel would be getting the right answer from the wrong half of the theorem. D-NV3's scalars, each with its status from RUN-098 and none of them required to be a unit: | scalar | status | | --- | --- | | `C = c*_f(r)` | in `L^×` (`C1.13`) | | `C_d = 26` | unit — `26 ≡ 4 (mod 11)`, computed here | | `L_p(g × τ)` | unit | | `1/L_p(Ad g)` | fractional, cleared | | `log^{[r+1]}` | **non-zero-divisor, not a unit** | | `L_p(g_E × ψ)` | **non-zero-divisor, not a unit** — computed here | RUN-098's rule continues to hold and is planted again: a nonzero divisor is not a unit, and relabelling one as the other is caught at `dag-d-independence`. ## 6. Vanishing order is not annihilation — on the actual series The computation, in `F₁₁[[t]]/t¹¹`, on the tree's own four series (RUN-081, re-read at RUN-090): | series | reduction nonzero | `μ` | `λ` | unit in `Λ` | | --- | --- | --- | --- | --- | | `L_p(g_E × ψ)` — target | ✔ | 0 | **2** | **no** | | twisted, target | ✔ | 0 | 0 | yes | | `L_p(g_0 × ψ)` — calibrator | ✔ | 0 | 0 | **yes** | | twisted, calibrator | ✔ | 0 | 0 | yes | - **the split**: `L̄_E = t²·ū` with `ū = 2 + 2t + 2t² + 0t³ + 5t⁴ + ⋯`; `ū⁻¹` is computed and `ū·ū⁻¹ = 1 + 0t + 0t² + 0t³` to the full truncation, so `ū` is a unit, and reassembling `t²ū` reproduces `L̄_E` coefficient for coefficient. - **nothing is annihilated**: for 40 random nonzero `ḡ`, `ord(L̄_E·ḡ) = 2 + ord(ḡ)` *exactly*, every time — so the kernel of multiplication by `L_E` is `0`. - **what does vanish** is the *value* at the central point: `L_E ≡ 0 (mod 11, t)`. That makes it a non-unit. It does not make it zero, and it does not make the map annihilating. So the correct form is her §5's: `L_E = t^m·u` with `u(0) ≠ 0` — a nonzero map vanishing to positive order. D needs only `u ≠ 0`. The exact `m` belongs to E. **And here is the sharpening that cuts against us.** What mod-11 data determines is `μ = 0` and `λ = 2`. The *order of vanishing at `t = 0` over `Z₁₁`* is **not** determined by it: all we know of the first two coefficients is `v₁₁(c₀) ≥ 1` and `v₁₁(c₁) ≥ 1`, and `c₀ ≠ 0` would already make the order `0`. The tree has carried "order 2" since RUN-081; read strictly that is the order of the *reduction*, i.e. the `λ`-invariant, which bounds the order at `t = 0` above by 2 and no more. This round therefore refuses to state the order, and hands it to E as an explicit UNMEASURED. It costs D nothing — D wanted `≠ 0` and has it — but it means E's "`ord = rank`" has more to prove than the tree's series record looks like it is offering. ## 7. The directional audit PC-001 fixes one weight/cyclotomic direction, so the number of relevant tangent directions is **1**, and Round 011 `(17.1)` applies: in one variable the order is `e + m` exactly, with no direction to choose. Round 030's multivariable requirement `τ_m(ξ^{⊗m}) ≠ 0` needs a direction; with one direction there is nothing to choose. What this does **not** license: "one variable, therefore every derivative is nonzero". The counterexample is the tree's own object — ```text L_E mod 11 is nonzero as a series yes its value at 0 0 its first derivative at 0 0 first nonzero Taylor coefficient index 2, value 2 ``` — a nonzero series in one variable with a vanishing first derivative. Planted, caught at `dag-d-nonvanishing`. ## 8. Where RUN-106's 11 goes, and C6 Two verdicts, not one: ```text D-rational log_omega : H^1_f(Q_11, V) -> D_dR(V)/Fil^0 = Q_11, V = T (x) Q_11 source dim 1 (11 does not divide #E~(F_11) = 16, so E(Q_11)^ = E_1(Q_11) = Z_11) target dim 1, map nonzero (v_11(log_omega(g)) = 1 < infinity) => ISOMORPHISM — Round 011 (3.1)-(3.2) on a real object D-integral log_omega on the lattice: H^1_f(Q_11, T) -> Z_11 image 11^1 Z_11, index 11, integral unit isomorphism NO => INDEX 11 ``` These are compatible: an isomorphism of `Q₁₁`-lines restricted to lattices is an injection with finite cokernel, and the cokernel's order is the index. **D passes with the integral index `≠ 1`.** C6 stays frozen. `η_pdR = 1` settled the *global* Betti/de Rham comparison pairing; this is the *local* `p`-adic Hodge identification through the formal logarithm of the minimal model. They share the differential `ω_E` and nothing that any theorem in this tree identifies. The forbidden inference would turn the index `11` into `1` — planted, caught at `dag-d-integral`, and worth exactly one factor of 11. The ledger, with owners: | factor | value | owner | entered by | times | | --- | --- | --- | --- | --- | | local Bloch–Kato index at 11 | 11 | p-adic reciprocity / the leading-line integral realization | D | 1 | | global index `[Λ_E : ZP+ZQ]` | 1 | `C_NT` (RUN-106) | C | 1 | | 11-adic index of `loc₁₁(E(Q)⊗Z₁₁)` | 1 | RUN-104's `H²` torsion | A | 1 | E may read every row and enter none of them again. Counting the local index in both D's and E's ledgers is planted and caught. ## 9. What D does not depend on D's dependency set, written out: Proposition `C1.1`; Theorem `C1.13`; `C1.9`'s factorisation (not its non-vanishing clause); Kato's reciprocity; one computed nonvanishing; and — for the integral verdict only — RUN-106's `v₁₁(log_ω(g)) = 1`. **No height pairing appears in that list.** Explicit reciprocity asks whether a map of lines is nonzero; the `p`-adic height asks whether a pairing is nondegenerate. So `C_p` being PARTIAL at RUN-106, and Schneider's conjecture being open, cannot make D fail — and "Schneider unproved ⇒ D FAIL" is planted and caught at `dag-d-independence`. The normalisation certificate is not reopened: `C_π`, `q_f`, `G_b`, `Π_i`, `κ_det`, `η_pdR` and `ε` are all things D does not ask for. What D *does* record are the actual arithmetic coefficients — `C_d`, `L_p(Ad g)`, the `λ`-series, `log^{[r+1]}` — and nonzero divisors suffice; units are not required. Target and calibrator are verified on separate rows (§0), and the difference between them is the point rather than an inconvenience: the calibrator's reciprocity **is** an integral isomorphism (`L_0(0) ≡ 9 (mod 11)`, a `Λ`-unit) and the target's is injective with a nontrivial cokernel. That is exactly RUN-098's unit / non-unit split, now with the reason attached. Substituting the calibrator's nonvanishing for the target's is planted and caught. ## 10. The mutations and the two controls Her twelve, each caught by the check named for it: | mutation | caught by | | --- | --- | | `L_E(0) = 0` ⇒ the map annihilates the target | `dag-d-nonvanishing` | | a nonzero series ⇒ the exact leading-coefficient equality | `dag-d-boundary` | | the regulator's injectivity ⇒ a surjective integral iso | `dag-d-theorems` | | the local index 11 turned into 1 by C6 | `dag-d-integral` | | the local index counted in both D's and E's ledgers | `dag-d-integral` | | Schneider unproved ⇒ D FAIL | `dag-d-independence` | | one variable ⇒ the first derivative is nonzero | `dag-d-nonvanishing` | | the calibrator's nonvanishing substituted for the target's | `dag-d-independence` | | a non-zero-divisor forced to be labelled a unit | `dag-d-independence` | | a normalisation scalar reintroduced into D | `dag-d-theorems` | | the quotient class actually 0 while the image is still called nonzero | `dag-d-domain` | | a Kato scalar identity called a class identity with no `C1.1` lift | `dag-d-route` | Thirteen more of mine: `C1.9` stretched over `k = r`; a verdict on E emitted from D's round; **the `τ`- and `ψ⁻¹`-quotients swapped**; `λ` declared 0; `μ` declared 1; the tangent-direction count declared 2; `C_d` replaced by 22 (a non-unit at 11); one nonannihilation layer dropped; the local index declared 1; the local index entered with no owner; the calibrator's series declared a non-unit; control A refused; control B's requirement switched off. **25 planted, 25 caught by the named check.** The swap mutation is there because I got that wrong myself while writing this gate, and caught it before the drill: the first draft named `L` as `ker(H → τ-quotient)`, which makes `Q = H/L` the `τ`-direction — the direction `C1.6` *kills*. The correction is above; the gate was rewritten, the probe re-run, and the mistaken reading is now one of the planted defects, so it cannot come back quietly. **Control A** — `R → R` by multiplication by `t²·u` with `u(0) = 2`, on a free rank-one module. Map nonzero **yes**; vanishing order **2**; annihilates anything **no** (40 witnesses); D-style nonannihilation **PASS**; E-style leading coefficient **NOT CERTIFIED** — the gate emits no leading-coefficient equality, by design. That last row is the point of the control: a method that could certify a leading coefficient from these inputs would be proving E, and it cannot. **Control B** — `Z₁₁ → Z₁₁` by multiplication by 11. Rational isomorphism **yes**; integral image `11·Z₁₁`; integral index **11**; integral unit isomorphism **no**. The two controls together are what keep §8's two verdicts from collapsing into each other. Both controls' *requirements* are themselves drilled: switching a requirement off is planted as a defect, because a control that cannot fail measures nothing — the lesson RUN-106 paid for. Five drill controls disturb nothing: control A's unit replaced by another series with `u(0) ≠ 0`; control A's order raised to 3; control B's multiplier replaced by 121 (index 121, still a rational iso, still not a unit); fewer no-zero-divisor witnesses; a different witness seed. ## 11. What this round does not claim - the **exact order of vanishing** of `L_p(g_E × ψ)` at `t = 0` over `Z₁₁` — undetermined by the mod-11 data the tree holds, and E's to establish; - the leading coefficient, the leading tensor, or any leading-term equality; `ord = rank`; - nondegeneracy of any `p`-adic height (Schneider, open); `det(h_p) ≠ 0`; - the complex BSD comparison; `ε`; E, F, G; BSD. ## 12. Status ```text PC-001 normalization certificate C1–C10 CLOSED as ARCHITECTURE (RUN-102), frozen DAG A PASS by cited theorem (RUN-104) B PASS (RUN-105) C PASS (RUN-106) — C_NT PASS, C_p PARTIAL, local line index 11 D PASS (this round) — D-rational ISOMORPHISM, D-integral INDEX 11, Schneider dependency NONE E–G untouched BSD OPEN ``` Whole-tree drill: **686 planted defects, 686 caught by the check named for each** — 0 uncaught by any check, 0 caught by the wrong check; **179 controls, none disturbed a check**; 218 checks; state restored afterwards. Eight shards, 8331 s. First run, no repair needed.