# RUN-106 — DAG-C, cut in two as instructed: **C_NT PASS** — `P, Q` are proved here to be a *global primitive* basis (`m = 1`: Hermite's cap, one cited height bound, an exhaustive box of 27 points, and an ℓ-sieve that needs no citation at all), the Néron–Tate convention is fixed and discriminated by the BSD identity, and `det(h♭) = Reg_NT = 0.152460139`; **C_p PARTIAL** — Macias Castillo–Sano's Theorem 4.2 identifies two *p-adic* constructions up to a sign, valued in `Q^k = I^k/I^{k+1}` and not in `Z₁₁`, and says nothing about nondegeneracy, which at `k = 1` is Schneider's conjecture, open; and the local Bloch–Kato line turns out **not** to be integrally trivial: `v₁₁(log_ω(g)) = 1`, so `log_ω` carries `H¹_f(Q₁₁,T)` onto `11·Z₁₁` — index 11, exactly the factor the forbidden "η_pdR = 1 ⇒ local log = 1" inference would have lost: **DAG-C PASS**, controlled by C_NT **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** the symbolic line's instruction after [RUN-105](../RUN-105/) — split C into `C_NT` and `C_p` and never mix them; go back to global for the index; fix the Néron–Tate convention; read MCS §4 as a literal type; realise the local logarithm and audit its relation to C6; p-adic ≠ real; eleven mutations, two positive controls; four outcomes; D, E, F, G untouched **Tools:** `src108_dag_c_regulator_bridge.py`, `src11_gate_drill.py` **Logs:** `src108-dag-c-regulator-bridge.json`, `src11-gate-drill.json` **Deliverable:** [`handoff/DAG-C-REGULATOR-BRIDGE/`](../../packages/DAG-C-REGULATOR-BRIDGE/) · **the paper read this round:** Macias Castillo–Sano §4 (Hypothesis 4.1, Theorem 4.2, (4.1.1)–(4.1.3), (4.2.1), Lemma 4.4, Remark 4.3), statements in `EXTRACTS.md` **Result. The round's own instruction says DAG-C's verdict is controlled by `C_NT`, because Rounds 031/035 define C as Mordell–Weil determinant ⟶ classical Néron–Tate regulator; everything below keeps the two tracks apart. Track A. The canonical heights are recomputed as `4^{-k}h(x(2^k·))` at `k = 10`: `ĥ(P) = 0.327000763`, `ĥ(Q) = 0.476711547`, and the pairing by two independent routes — half the polarisation and the parallelogram — agreeing to `2.4e-9`, giving `Reg = det(h♭) = 0.152460139368` (residual `3.9e-8` against the external value). The convention is not merely declared: a doubled or halved pairing would move `Ω·Reg·∏c_v/#tors²` away from `L''(E,1)/2` to `0.9326` or `1.0168`, while the ½ convention gives `0.99999975` — that identity is what discriminates it, and it is a consistency check, never the bridge. Then the index, which the instruction demanded come from a certificate rather than from 11-adic saturation. Two independent arguments. (i) Hermite in rank 2: a sublattice of index `m` forces a nonzero point of canonical height `≤ (2/√3)√Reg/m`, so `m ≥ 2` forces one of height `≤ 0.2254`; a Silverman-type height-difference bound — the single cited ingredient, and measured here against 120 points, where the shortfall actually needed is none — no point fell short against the `6.479` used — turns that into naive height `≤ 6.704`, i.e. `x = a/d²` with `|a|, d² ≤ 816`; that box is searched exhaustively, yields 27 points, and its smallest canonical height is `0.32696 > 0.2254`. Hence `m = 1`. (ii) With no citation whatsoever: for each prime `ℓ ≤ 19` and each projective class `(a:b)` mod `ℓ`, a prime `q` of good reduction with `ℓ | #E(F_q)`, cyclic `ℓ`-torsion, and `(aP+bQ) mod q ∉ ℓ·E(F_q)` — a complete finite proof that `ℓ ∤ m`, one witness per class. So `D_MW = Z(P∧Q)` is the primitive Mordell–Weil determinant and `m²·Reg_NT = Reg_NT`. Track B. Hypothesis 4.1 is instantiated: (i) `11 ∤ Tam·#Ẽ(F₁₁) = 1·16` computed here; (ii) the image contains a Cartan subgroup because `ρ̄₁₁` is surjective (RUN-069's six witnesses). Theorem 4.2 is then read as a type rather than as a slogan: `⟨,⟩_k : S_p^{(k)} × S_p^{(k)} → Q^k = I^k/I^{k+1}` for `1 ≤ k ≤ p−1 = 10`, with the Bertolini–Darmon filtration of (4.2.1), symmetric, and the theorem's content is `⟨,⟩^BD = −⟨,⟩^Nek` — an equality **with a sign**, between two `p`-adic constructions, valued in a line that only becomes `Z₁₁` after a topological generator of `Γ` is chosen (a gauge of the same species as RUN-103's augmentation generators). It gives no nondegeneracy: at `k = 1` that is Schneider's conjecture, open for a rank-2 curve, so `det(h_p) ≠ 0` is UNMEASURED and `C_p` is PARTIAL. Track C. The local line is realised rather than assumed: the formal group of the minimal model gives `ω = (1 + t² + 2t³ − 3t⁴ + …)dt` and `log(t) = t + t³/3 + t⁴/2 − 3t⁵/5 + …`; since `11 ∤ #Ẽ(F₁₁) = 16`, `H¹_f(Q₁₁,T) = E(Q₁₁)⊗̂Z₁₁ = E₁(Q₁₁)`, and at `t(16P)` the computation gives `v₁₁(t) = 1` and `v₁₁(log_ω) = 1` with `log/11 ≡ 328 (mod 11³)`. **So `log_ω(H¹_f(Q₁₁,T)) = 11·Z₁₁`: the local line carries an index 11 and is not integrally trivial.** C6's `η_pdR = 1` concerns the global Betti/de Rham comparison pairing, not this local `p`-adic Hodge identification; they share the differential `ω_E` and nothing else that any theorem in this tree establishes — and here the forbidden inference would have cost exactly that factor 11. The three indices in play (global `1`, this local `11`, RUN-104's 11-adic `11⁰`) are different quantities and each enters once. Verdict DAG-C PASS, controlled by C_NT, with C_p PARTIAL and the `p`-adic realization handed to D/E with its gap named. Eleven mutations and both controls planted; `ε` neither computed nor demanded; D, E, F, G untouched.** > **Correction, 2026-10-18.** Two figures in this report were wrong and have been corrected > in place. (1) The Hermite cap was printed `0.225432`; the gate's log records > `0.2254330198886279`, which rounds to `0.225433`. (2) The report said the largest measured > shortfall over 224 points was `−0.00005`. That figure came from the gate's docstring, which > described an earlier computation that no longer runs. The code measures **120** points and > starts its running minimum at `0.0`; the logged `0.0` means **no point fell short at all**, and > also that a positive minimum would not have been recorded. Neither change touches a > conclusion: the index argument does not depend on the cap's last digit, and the cited bound's > margin is used even less than the report said. Found by the crystallization standard's > number trace, which requires every load-bearing figure in a paper to appear in a log. > **Correction, 2026-10-18 (second).** The individual canonical heights were also stale: the > regulator line was updated when the height depth was raised to 10, the line of individual > heights was not. The depth-10 log gives `ĥ(P) = 0.327000763`, `ĥ(Q) = 0.476711547`, > `⟨P,Q⟩ = 0.058522646`, and the two pairing routes differ by `2.4e-9`, not `2.8e-9`. The > regulator `0.152460139368` and every conclusion are unchanged. These were invisible to the > first version of the number trace — its filter for commit hashes also ate the digit run after > a decimal point, so every high-precision figure was exempt — and were found once that was > fixed. The old `ĥ(P)` had in fact been "traced", to the 7th iterate of a convergence sequence > in an unrelated gate: a real number, attached to the wrong quantity. --- ## 0. The verdict in the requested form ```text DAG-C = PASS (controlled by C_NT, per the instruction's §0) C_NT = PASS global lattice m = [Λ_E : ZP + ZQ] = 1, PROVED HERE by two independent arguments (§2); 11-adic saturation was not used for it, and LMFDB/Cremona's own saturation statement is recorded as agreement, never as a step NT convention = ½(ĥ(P+Q) − ĥ(P) − ĥ(Q)); two routes agree to 2.4e-9; the BSD identity discriminates it (0.99999975 against 0.9326 / 1.0168) (§1) h♭ and det h♭ : M_E → M_E^∨; det(h♭) = Reg_NT = 0.152460139368 (residual 3.9e-8); m²·Reg_NT = Reg_NT since m = 1 ε not touched, as instructed C_p = PARTIAL Hypothesis 4.1 (i) COMPUTED: 11 ∤ Tam·#Ẽ(F₁₁) = 1·16; (ii) DERIVED: ρ̄₁₁ surjective ⇒ the image contains a Cartan subgroup literal type ⟨,⟩_k : S_p^{(k)} × S_p^{(k)} → Q^k = I^k/I^{k+1}, 1 ≤ k ≤ 10, symmetric; Theorem 4.2: ⟨,⟩^BD = −⟨,⟩^Nek — identification, with a sign, between two p-adic pairings; Remark 4.3: Nekovář's needs no Hypothesis 4.1 and allows any k ≥ 1 nondegeneracy UNMEASURED — at k = 1 it is Schneider's conjecture; det(h_p) ≠ 0 not available det(h_p) D_MW,11^{⊗2} → (Q^k)^{⊗2}; the target is NOT presumed to be Z₁₁ local line (Track C) = PASS as a computation, with a finding v₁₁(log_ω(g)) = 1 log_ω(H¹_f(Q₁₁,T)) = 11·Z₁₁ — index 11, NOT integrally trivial C6 η_pdR = 1 is a different normalisation; the inference is refused, and refusing it is worth exactly a factor 11 here boundary p-adic ≠ real: MCS §4 identifies two p-adic heights and reaches no Reg_NT; C_p PASS ⇏ C_NT PASS, and C_p PARTIAL does not hold DAG-C back handed to D/E the p-adic realization, with its gap named, and the local index 11 ``` ## 1. Track A — the pairing, its convention, and the regulator Canonical heights as `4^{-k}h(x(2^k·))` at `k = 10` (the limit's own residual at this depth, `6.5/4¹⁰ ≈ 6e-6`, is reported rather than hidden): | quantity | value | | --- | --- | | `ĥ(P)`, `P = (0,0)` | `0.327000763` | | `ĥ(Q)`, `Q = (1,0)` | `0.476711547` | | `⟨P,Q⟩` = `½(ĥ(P+Q) − ĥ(P) − ĥ(Q))` | `0.058522646` | | the parallelogram route `¼(ĥ(P+Q) − ĥ(P−Q))` | agrees to `2.4e-9` | | `Reg = det(h♭) = ĥ(P)ĥ(Q) − ⟨P,Q⟩²` | `0.152460139368` (residual `3.9e-8` against the external value) | `h♭ : M_E → M_E^∨`, `v ↦ ⟨v,·⟩`; its determinant is a quadratic form on `D_MW`, i.e. an element of `(D_MW^∨)^{⊗2}` — which is why the rank-2 regulator is quadratic in the determinant generator and the square in RUN-105's line is exactly what it trivialises. **The convention is discriminated, not declared.** `Ω·Reg·∏c_v/#tors²` against `L''(E,1)/2` gives `0.99999975`; the same computation with a doubled pairing gives `0.9326` and with a halved one `1.0168`. That is the check a factor-2 slip fails. It is a *consistency check on the convention* and is never used as the bridge itself — taking it for the bridge is planted and caught (`dag-c-nt`). ## 2. Track A — the global index, proved twice **(i) Hermite + one cited bound + an exhaustive box.** In a rank-2 lattice of Gram determinant `D` the shortest nonzero vector satisfies `λ₁ ≤ γ₂√D` with `γ₂ = 2/√3`. If `[Λ_E : ZP+ZQ] = m` then `det Gram(Λ_E) = Reg/m²`, so `m ≥ 2` forces a point of canonical height `≤ (2/√3)√Reg/2 = 0.225433`. A Silverman-type bound `ĥ(R) ≥ h(x(R)) − [¼h(j) + ⅙h(Δ) + 1.946]` — **the single cited ingredient of this round** — turns that into `h(x) ≤ 6.704`, i.e. `x = a/d²` with `|a|, d² ≤ 816`. The box is searched exhaustively: **27 points up to sign, the smallest canonical height `0.326965`**, which exceeds `0.225433`. Hence `m = 1`. The cited constant is itself measured rather than trusted: over the 120 points `aP+bQ`, `|a|,|b| ≤ 5`, **no point falls short at all** — `ĥ ≥ h(x)` at every one — so none of the `6.479` the box allows is used, and the box is generous by at least a factor `e^{6.48}` in coordinate size, so a misread normalisation would have to be worse than that before the search could miss anything. **(ii) An ℓ-sieve with no citation at all.** If `(aP+bQ)/ℓ ∈ E(Q)` then for every prime `q` of good reduction `(aP+bQ) mod q ∈ ℓ·E(F_q)`. For each `ℓ ∈ {2,3,5,7,11,13,17,19}` and each of the `ℓ+1` projective classes `(a:b)` mod `ℓ`, the gate exhibits a prime `q` with `ℓ | #E(F_q)`, cyclic `ℓ`-torsion (so that "`S ∈ ℓE(F_q)`" is exactly "`(N/ℓ)S = O`"), and `(N/ℓ)·S ≠ O`. Every class dies for every `ℓ`: **`ℓ ∤ m` for all `ℓ ≤ 19`**, by a finite argument that cites nothing. LMFDB/Cremona also record `P, Q` as generators; that is written down as **agreement**, not used as a step. Taking 11-adic saturation for global primitivity, or independence for primitivity, is planted and caught (`dag-c-lattice`). ## 3. Track B — MCS §4 as a literal type | datum | what the paper says | | --- | --- | | domain | `S_p^{(k)} × S_p^{(k)}`, `S_p(E/Q) = lim_n Sel_{p^n}(E/Q)`, with Bertolini–Darmon's filtration `S_p = S^{(1)} ⊃ … ⊃ S^{(p)}` | | Selmer filtration | (4.2.1) `Sel^{(k)} = Sel_{p^n}(E/Q) ∩ 𝓘^{k−1}Sel_{p^n}(E/Q_n)`, through Lemma 4.4's `Sel_{p^n}(E/K) ≅ Sel_{p^n}(E/L)^G` | | codomain | `Q^k = I^k/I^{k+1}` — **a line, not `Z₁₁`**; it becomes `Z₁₁` only after a topological generator of `Γ` is chosen, which is a gauge choice | | order `k` | `1 ≤ k ≤ p − 1 = 10`; `k = 1` is the usual `p`-adic height | | symmetry / construction | symmetric; Nekovář's is `⟨x,y⟩_k = β^{(k)}(x) ∪_k y`, with `β^{(k)}` the `k`-th derived Bockstein of the `I`-adic filtration on the Iwasawa–Selmer complex | | the theorem | Theorem 4.2: under Hypothesis 4.1, `E_k^{0,1} = S_p^{(k)}` and `⟨,⟩^BD_k = −⟨,⟩^Nek_k` | | hypotheses | Hypothesis 4.1 (i) `p ∤ Tam·∏#Ẽ_v(F_v)`, (ii) the image of `ρ` contains a Cartan subgroup; Remark 4.3: Nekovář's pairing needs neither | Instantiated at `(389.a1, 11)`: (i) `11 ∤ 1·16` — computed; (ii) `ρ̄₁₁` surjective (RUN-069) and `GL₂(F₁₁)` contains its own Cartan subgroups — derived. So the identification holds, **by cited theorem**. **What it is not.** An equality of two pairings is not a statement that either is nondegenerate: it fixes no rank and no radical. For `k = 1` nondegeneracy of the `p`-adic height on a rank-2 curve is Schneider's conjecture — open. Therefore `det(h_p) : D_MW,11^{⊗2} → (Q^k)^{⊗2}` is defined but `≠ 0` is **UNMEASURED**, and `C_p` is PARTIAL. Reading the equality as nondegeneracy, rewriting the codomain as `Z₁₁`, or taking a determinant at an order outside `1 ≤ k ≤ 10` is planted and caught (`dag-c-padic`). ## 4. Track C — the local Bloch–Kato line, realised Since `11 ∤ #Ẽ(F₁₁) = 16`, `E(Q₁₁)⊗̂Z₁₁ = E₁(Q₁₁)` and `H¹_f(Q₁₁,T) = E(Q₁₁)⊗̂Z₁₁` (Bloch–Kato, cited). The formal group of the minimal model, computed here by the standard recursion: ```text ω = (1 + 0·t + t² + 2t³ − 3t⁴ + …) dt log(t) = t + 0·t² + t³/3 + t⁴/2 − 3t⁵/5 + … ``` Evaluated at `t(16P)` (16 kills `E(F₁₁)`, so `16P ∈ E₁`): `v₁₁(t) = 1`, `v₁₁(log_ω) = 1`, `log/11 ≡ 328 (mod 11³)`; the same for `Q`. The valuation is exact for a structural reason the gate states: `log(t) = t + Σ_{n≥2}(a_n/n)t^n` with integral `a_n` on a minimal model, and `v(t^n/n) ≥ n − log₁₁(n) > 1` for `n ≥ 2`. ```text log_ω( H¹_f(Q₁₁,T) ) = 11 · Z₁₁ index 11 — the local line is NOT integrally trivial ``` **The C6 audit.** C6 settled `η_pdR = 1` for the *period/de Rham comparison pairing* `⟨γ, ω⟩` of the normalisation route (Eichler–Shimura; `Ω^±` is its value on the chosen Betti basis). This round uses the *local* identification `D_dR(V)/Fil⁰ ≅ Q₁₁` via `ω_E`, i.e. the formal-group logarithm. They share the differential and nothing else that any theorem in this tree establishes. The inference "`η_pdR = 1`, therefore the local log normalisation is 1" is refused — and refusing it is not pedantry here: it is worth exactly the factor 11 just computed. Taking it, or reading the rational isomorphism as an integral unit, is planted and caught (`dag-c-local`). **The ledger.** Three indices, three different maps, each entering once: the global index `[Λ_E : ZP+ZQ] = 1` in the free factor; this local index `11` in the trivialisation of `H¹_f(Q₁₁,T)^{-1}`; RUN-104's `11⁰` (the index of `loc₁₁(E(Q)⊗Z₁₁)`) in the torsion of `H²`. Entering the local index twice is planted and caught (`dag-c-ledger`). ## 5. The boundary, and what DAG-C's verdict rests on MCS §4 proves that two **p-adic** constructions agree. It does not prove nondegeneracy, and it reaches no real-valued regulator; no comparison theorem between a `p`-adic height and the Néron–Tate height appears in anything read on this line. So `C_p PASS ⇏ C_NT PASS` — and, by the instruction's own rule, `C_p PARTIAL` does not hold DAG-C back, because DAG-C is the classical bridge. Equating the two heights is planted and caught (`dag-c-boundary`). ## 6. The eleven mutations and two controls | mutation | check | | --- | --- | | the derived `p`-adic height equated with the Néron–Tate height | `dag-c-boundary` | | 11-adic saturation taken for global primitivity | `dag-c-lattice` | | independence taken for a primitive basis | `dag-c-lattice` | | the numerical regulator match taken as the theorem bridge | `dag-c-nt` | | the Néron–Tate pairing off by a factor 2 | `dag-c-nt` | | Theorem 4.2's equality read as nondegeneracy | `dag-c-padic` | | a determinant taken at an order `k` outside `1 ≤ k ≤ p−1` | `dag-c-padic` | | the rational Bloch–Kato log isomorphism read as an integral unit | `dag-c-local` | | `η_pdR = 1` taken for the local log normalisation | `dag-c-local` | | the local index `11` entered twice | `dag-c-ledger` | | `ε` demanded this round | `dag-c-verdict` | Controls: **A** — the same Hermite-plus-search method, pointed at `ZP + Z(2Q)` (a genuine index-2 sublattice), must **fail** to prove primitivity, and it does: the cap rises to `0.4509` and `P` itself sits under it. A method that proved primitivity of everything would prove nothing. **B** — the quadratic-form identity `ĥ(aP+bQ) = a²ĥ(P) + b²ĥ(Q) + 2ab⟨P,Q⟩` must hold at a second point (`2P − Q`), with its requirement actually enforced; switching the requirement off is itself planted, because a control that cannot fail measures nothing. Also planted: the declared index, the declared local valuation, the declared codomain and order range, control A aimed at the full lattice, a multiplier that does not reach the formal group, and a generator off the curve. Five drill controls (other multipliers, another control point, another non-saturated sublattice, a shorter sieve) must stay green. ## 7. What this round does not claim * `Reg_NT` reached from any `p`-adic height — no comparison theorem exists in the sources read, and none is assumed; * nondegeneracy of the `p`-adic height pairing, hence `det(h_p) ≠ 0` — Schneider's conjecture, open; * the unit `ε` of RUN-105 — this round was told not to touch it, and does not; * the height-difference bound re-proved here (CITED; measured against 120 points, not re-derived); Hermite's inequality (classical); * `Ш(E/Q)[11] = 0` — inherited, and **not used by C_NT at all**: the index argument and the regulator are unconditional; * anything about D, E, F, G; BSD; * that the repaired `line_diff` was wrong about the map: its numbers, when it answers, are and were `1899 / 53404`, exactly what Round 13 states — the fault was in how a non-answer was reported, not in the measurement. ## 8. The drill, and what its first run found in the instrument **The first run went red — and the finding is about this tree's own apparatus, not about BSD.** `algorithm2-mirror-and-diff` turned red roughly eleven defects into *every one of the eight shards*, at the same position rather than on the same defect, and stayed red: which made the post-drill restore check and about twenty later controls read as disturbed. Neither the subject nor any planted defect had anything to do with it. The cause: that check calls `src68.line_diff()`, which shells out to `git diff --no-index --numstat` on the 370,000-line condition map **on every single call** — several hundred times per shard, eight shards at once. On the loaded machine the subprocess began returning no output, the function's `except` turned that into `(-1, -1)`, and `(-1, -1) ≠ (1899, 53404)` was reported as a **disagreement**. A check about a map went red because the machine was busy. Two separate faults, both repaired in `src68_algorithm2_mirror_and_diff.py`: * **A measurement spawned once per call although its inputs cannot change during a run.** `line_diff()` is now memoised on the two files' identities and modification times — one subprocess per process instead of several hundred (50 calls: 6 ms, previously 50 subprocesses) — and retried before it gives up. * **A subprocess that did not answer, reported as a subject that disagreed.** The record now carries `measured`, which separates "git said 1899/53404" from "git did not answer". `agrees` stays False when nothing was measured, because this line's standing rule is that a check which cannot run has no verdict and must not be green. Both are now drilled: a planted defect makes `line_diff` report an unanswered subprocess as an agreement (caught by `algorithm2-mirror-and-diff`), and a control clears the memo and requires that nothing moves — the memo must not be load-bearing for any verdict. The run that counts, after the repair: **661 planted defects, 661 caught by the check named for them, 0 uncaught, 0 caught by the wrong check, 174 controls undisturbed, state restored, over 209 checks** (as 8 shards). Twenty-one of the defects and six of the controls are this round's. The report and package were written before the drill was launched; the only thing touched while it ran was this section, after the first run's own evidence was read. ## 9. Status ```text normalization certificate C1–C10 CLOSED (architecture), frozen DAG A PASS by cited theorem (RUN-104) B PASS (RUN-105) — det⁻¹RΓ(Z_S,T) ≅ D_MW^{⊗2} ⊗ H¹_f(Q₁₁,T)⁻¹ C PASS — C_NT PASS (m = 1 proved; Reg_NT placed), C_p PARTIAL (identification cited, nondegeneracy open), local line index 11 D–G untouched BSD OPEN ```