---
title: "Closing a normalization certificate against actual objects"
subtitle: "The ten items of a determinantal comparison at $(389.\\mathrm{a}1,\\ p = 11)$, judged one by one, and the method that judges them"
author:
  - "Aletheia — the certificate, the DAG and every round's instruction; model: ChatGPT, GPT series"
  - "Neo.K — AMRAL Research Lab"
  - "墜衡 (Zhuì Héng) — AMRAL Research Lab, verification; model: Claude, Opus series"
date: 2026-10-18
version: "0.1"
standard: "1"
headline_problem: "Birch–Swinnerton-Dyer conjecture"
evidence:
  branch: agent/bsd-verification-zhuiheng
  commit: "6f9c8cc10139ee05f235bf9a22085479c3d2966a"
  dir: bsd-verification-zhuiheng
  sources: "data/gate-logs/src82-*.json, data/gate-logs/src9[2-9]-*.json, data/gate-logs/src10[0-4]-*.json, data/gate-logs/src83-*.json, data/results.v1.json"
abstract: |
  The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. This
  paper concerns the step a determinantal approach to it must take before any statement about
  $L$-values: fixing the normalization of a comparison between an auxiliary Eisenstein family and
  an arithmetic line, for the rank-2 curve $389.\mathrm{a}1$ at $p = 11$ with the rank-0 curve
  $19.\mathrm{a}1$ as calibrator. A symbolic programme reduced that step to a certificate of ten
  items, C1–C10. We judge each item against objects that actually exist — constructed, recomputed,
  or supplied by a theorem whose hypotheses we instantiate on these curves — and close all ten.

  The closure has three ingredients. The late character of the comparison is a function of the
  auxiliary family alone, by Loeffler–Rivero's theorems, every inherited hypothesis instantiated;
  the auxiliary leading line exists and is the Kummer line of $\mathbf{Q}(\sqrt2)$, whose local
  conditions are vacuous at every place; and the comparison route is realized as an actual chain
  of line maps — injective but not surjective for the target, because its $p$-adic $L$-function has
  a double zero — along which the pairing scalar, the determinant scalar and the residual holonomy
  are all exactly 1, so that the reduced comparison is itself a canonical section. The method
  matters as much as the verdicts: one rule judges every item and never turns an unmeasured item
  into a pass, and the rule was itself corrected twice when it misjudged the kind of claim an item
  makes. At the pinned commit the line's whole-tree drill stands at 692 planted defects, 687 caught
  by their named check, 5 detected only by a crash of it, 179 controls undisturbed, over 218 checks.
keywords: [Birch and Swinnerton-Dyer, Beilinson–Flach, Eisenstein degeneration, Euler systems, verification]
---

> **Status of this document.** Preprint, version 0.1: complete enough to stand in public, and not
> final; a later version may revise it. Not prepared for journal submission. Section 13 records who
> did which part, including the parts this line did not do.

# 0. What is claimed, and what is not

Six kinds of statement are kept apart and never merged; the distinction is not presentational, it
is the content.

| label | meaning |
| --- | --- |
| `COMPUTED_HERE` | recomputed in this line with independently written code, never by running the producers' scripts |
| `DERIVED_FROM_VERIFIED_INPUTS` | follows from quantities this line already computed |
| `CITED_THEOREM` | a published theorem, with its hypotheses instantiated here one at a time |
| `INHERITED` | taken from another document and never verified here; every use is marked |
| `UNMEASURED` | well posed, and not established here |
| `FAILED` | attempted and refused |

In particular `PASS` and `PASS by cited theorem` are **different verdicts** and are never written
as one: an item discharged by Loeffler–Rivero's theorem is not an item this line proved, and the
table in §2 says which is which.

**Not claimed anywhere in this paper:** the Birch–Swinnerton-Dyer conjecture; any statement about
$L(E,s)$; any node of the post-normalization DAG; the values of $C_\pi$, of the period ratio
$\Pi_E/\Pi_0$, of the comparison scalar $c_{\mathrm{ES}/\mathrm{Kum}}$, of the lattice index $q$ or of
the jet order $n$; and any Beilinson–Flach class in coordinates. "Closed" means closed **as
normalization architecture**: the comparison's normalization is fixed canonically, which is the
precondition for the DAG and nothing more.

# 1. The setting

**The comparison.** Let $E = 389.\mathrm{a}1$ (the target, rank 2) and $E_0 = 19.\mathrm{a}1$, with
a-invariants $[0, 1, 1, -769, -8470]$ (the calibrator, rank 0; Cremona's 19a2, not Cremona's 19a1, which
is $X_0(19)$), both ordinary at $p = 11$, and let $g = E_2(1, \chi_8)$ be the weight-2
Eisenstein series with $\chi_8 = (2/\cdot)$, with critical-slope refinement at 11. A cross-curve
calibration — built in a separate session and reproduced coefficient for coefficient by this line —
writes, for each partner $i \in \{E, 0\}$, the auxiliary family's projected leading class as
$$\hat B_i = C_\pi\, A_i\, j\, z_i,$$
with a late character $C_\pi$, a partner factor $A_i$, the cyclotomic coordinate $j$ and the Kato
class $z_i$; the calibration's purpose is the partner ratio, in which $C_\pi$ cancels **if** it is
common to both. Whether it is, and whether nothing else hides in the comparison, is a question of
normalization.

**The certificate.** A symbolic programme reduced that question to ten conditions, stated in its
thirtieth round as a conjunction — "only after C1–C10 pass should the normalization ledger be
frozen":

| | condition (verbatim) | the claim it makes |
| --- | --- | --- |
| C1 | Target and calibrator use the same exact late character. | equality |
| C2 | No partner arithmetic is hidden in the common line. | absence |
| C3 | No analytic coordinate weight is hidden in exact normalization. | absence |
| C4 | The auxiliary leading tensor or equivalent q-blind source exists. | existence |
| C5 | Every defect factor is tagged E, L, G, or P. | enumeration |
| C6 | Either canonical cancellation or one explicit residual pairing scalar. | existence |
| C7 | Either canonical cohomological equality or explicit $\kappa_{\det}$. | existence |
| C8 | No exact unit ambiguity is attributed to a length / index. | absence |
| C9 | No residual holonomy. | absence |
| C10 | Canonical section or one exact anchor. | existence |

The tags in C5 sort every factor of the comparison's defect $\delta = a_D\, q_D\, \eta_D$: G a frame
or convention, L a lattice or finite Euler magnitude, P a partner factor, E an exact comparison.

# 2. The method, and how it was corrected

**One rule, never `UNMEASURED` → `PASS`.** The first audit judged all ten items with one rule:
`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. Applied to
the objects that existed it gave C3 `PASS`, the other nine `UNMEASURED`, no `FAIL` — and the
rule's two negations (promoting `UNMEASURED` to `PASS`; declaring closure when no item fails) are
planted in the drill.

**First correction: the kind of claim.** That rule assumes every item asks for an exhibited
object. The ten sentences do not. The last column of the table in §1 is computed by reading each
sentence out of the corpus file at gate time and classifying it by a stated lexicon ("every" ⇒
enumeration; "exists", "either", "or" ⇒ existence; "the same" ⇒ equality; "no" ⇒ absence; first
match wins). An equality, an absence or an existence can be discharged by a theorem; an
enumeration cannot — one cannot tag every factor of a decomposition nobody has. So the corrected
rule admits a `CITED_THEOREM` or `DERIVED_FROM_VERIFIED_INPUTS` discharge only for a criterion
that does not demand an exhibited object, refuses an absence whose subject does not exist ("no
holonomy" on a graph nobody has built is vacuous, not true), and carries a positive control on
itself: handed a theorem discharge for C5 it must refuse, handed an admissible discharge it must
accept, because a guard that refuses everything is not a guard.

**Second correction: what blocks what.** The phrase "the first blocker" gave three different
answers under three stated orders, which is the signal that it had no single-item answer: the
question it stood for — what can be worked on now — is a set. And the dependency graph the first
audit recorded had one node for two things: C5's enumeration cannot be certified until the
comparison route (C7's *object*) exists, while C7's *verdict* comes after the enumeration. Read
with the objects separated, the executable frontier was the comparison route itself (§6).

**Outside the certificate.** Obligations that are real but that no sentence asks about are kept in
a separate ledger and can change no verdict either way: the Beilinson–Flach classes in
coordinates, the value of $\Pi_E/\Pi_0$, the comparison scalar $c_{\mathrm{ES}/\mathrm{Kum}}$, the
lattice index $q$'s 11-adic valuation, and the jet order $n$.

**The verdicts, and what discharges each:**

| item | verdict | discharged by | label |
| --- | --- | --- | --- |
| C1 | PASS | Loeffler–Rivero C1.9, C1.10, C1.11, C1.13, §A8.3, hypotheses instantiated (§4) | `CITED_THEOREM` |
| C2 | PASS | C1.13's factorization and the smoothing factor computed common (§4) | `DERIVED_FROM_VERIFIED_INPUTS` |
| C3 | PASS | the coordinate audit on the actual series (§3) | `COMPUTED_HERE` |
| C4 | PASS | the line computed; non-splitness by Loeffler–Rivero A6.1 (§5) | `CITED_THEOREM` |
| C5 | PASS | the 23-record ledger, re-run against the actual chain (§7) | `COMPUTED_HERE` |
| C6 | PASS, $\eta_{\mathrm{pdR}} = 1$ | one comparison pairing; frame laws derived (§8) | `DERIVED_FROM_VERIFIED_INPUTS` |
| C7 | PASS, $\kappa_{\det} = 1$ | one route, one category, functoriality (§9) | `DERIVED_FROM_VERIFIED_INPUTS` |
| C8 | PASS | the target's lines computed in the Manin lattice (§7) | `COMPUTED_HERE` |
| C9 | PASS, monodromy trivial | the complete transport graph (§10) | `COMPUTED_HERE` |
| C10 | PASS, canonical, $\rho = 1$ | the reduced comparison is choice-free and natural (§11) | `DERIVED_FROM_VERIFIED_INPUTS` |

# 3. C3: no coordinate weight hidden

Both partners' four 11-adic series — $L_E, \lambda_E, L_0, \lambda_0$ to $t^{120}$ — are built from
one exponent map $e(u) = \log_{12}\langle u\rangle$, one dictionary over the 1210 units mod $11^3$
(`COMPUTED_HERE`). Rebuilt with the generator $\gamma' = 23$ and mapped back by
$S_{23} = S_{12} \circ ((1+t)^{k} - 1)$, where $k = \log_{23} 12$ is the inverse of
$\log_{12} 23 \equiv 112 \pmod{121}$, they agree in all 121 coefficients; the leading coefficient of
the order-2 series $L_E$ moves $2 \mapsto 6 = 2 \cdot 2^{-2}$, by the law $u^{-r}$ with
$u = \log_{12} 23 \equiv 2 \pmod{11}$, and nothing else in the leading data moves. Twelve random
formal coordinate changes give $u^{-r}$ on the leading coefficient and the predicted correction on
the next. The calibration's identity is covariant of weight $-1$ on both sides, so no coordinate
weight is hidden. A defect that rebuilds only the calibrator with the other generator, and calls it
the same coordinate, is caught.

# 4. C1 and C2: the late character is auxiliary

**C1.** The calibration's $C_\pi$ is a quotient $c_{\mathrm{lead}}/q_f$. Its numerator is the
leading coefficient of $c_f$, which Loeffler–Rivero's Theorem C1.9 states is "a meromorphic function
on $V_1$ alone" — $V_1$ the auxiliary family's disc. Its denominator is $b_f^+(0)$, a trivialisation
defined in §A8.3 from the auxiliary family alone. Theorem C1.10 gives a unique $\omega_f$, and
Remark C1.11 that the cuspidal partner used in constructing it "is uniquely determined by $f$";
Theorem C1.13 — the calibration's formula, matched slot by slot — opens its proof by taking
$C = c^*_f(r)$, nonzero. So $C_\pi$ is a functional of the auxiliary family alone, and the
"common frame" the calibration assumed is a free normalization, not an arithmetic claim.

The theorems carry hypotheses, including some inherited from the paper's Part B that the setup
sentence does not restate. All are instantiated on both curves (`COMPUTED_HERE`): prime-to-11 level
($11 \nmid 389$, $11 \nmid 19$); distinct roots of the Hecke polynomial at 11 (discriminants $-28$
and $-35$, nonzero mod 11), hence noble; ordinary ($a_{11} = -4$ and $3$, unit roots 7 and 3);
residually irreducible at 11, by semistability computed from the Weierstrass invariants
($389 \nmid c_4 = 112$, $19 \nmid c_4 = 36928$) and Mazur's theorem on rational isogenies of prime
degree; and the smoothing integer $d = 5$ coprime to $6S$. Full mod-11 surjectivity is **not**
among them, and adding it is planted as an over-strong hypothesis.

**C2.** The smoothing factor is partner-dependent as a formula — Loeffler–Rivero's equation (8),
$C_d(f,g,j) = d^2 - d^{\,j-k_1-k_2}\,\varepsilon_f^{-1}(d)\,\varepsilon_g^{-1}(d)$ — and common as a
value: the auxiliary nebentypus is $\chi_8$ with $\chi_8(5) = -1$, both partners have trivial
nebentypus and weight parameter 0, so $C_d = 25 + 5^j$ for both, at every $j$
(`DERIVED_FROM_VERIFIED_INPUTS`). The calibration's independently computed value $26$ pins $j = 0$.
Giving the calibrator a nontrivial nebentypus, or weight 4, splits $C_d$ between the partners and
turns the check red: the commonality is a measurement, not a label.

**The partner ratio, measured.** With $D_i = \Pi_i E_i$, the adjoint Euler halves are
$E_E \equiv 1 - 7^{-2} \equiv 3$ and $E_0 \equiv 7 \pmod{11}$, so
$A_0(0)/A_E(0) \equiv 6 \cdot (\Pi_E/\Pi_0) \pmod{11}$ — an earlier round had carried only the
$\lambda$ half. And $\Pi_E/\Pi_0$ is an 11-adic unit (`COMPUTED_HERE` with a `CITED_THEOREM`): by
Hida the adjoint value generates the congruence ideal; at level 389 the generalised mod-11
eigenspace of the target's Hecke system in the 65-dimensional dual Manin space is 2-dimensional,
equal to the ordinary eigenspace, so no form is congruent to it mod 11; at level 19 there is one
newform. So the partner ratio is a nonzero unit. Its residue is not claimed, and a planted residue
is caught.

# 5. C4: the auxiliary leading line exists

**The target is a line.** $\chi_8$ is the Kronecker symbol $(2/\cdot)$, the character of
$K = \mathbf{Q}(\sqrt2)$ (checked at twelve arguments). $K$ is real quadratic with unit rank 1; in
$\mathbf{Z}[\sqrt2]$, $\varepsilon = 1 + \sqrt2$ has $\sigma(\varepsilon) = 1 - \sqrt2 =
-\varepsilon^{-1}$, so the nontrivial automorphism acts by $-1$ and the whole unit rank lies in the
$\chi_8$-part: $\dim(O_K^\times \otimes \mathbf{Q})^{\chi_8} = 1$, derived from the rank and the
sign (`COMPUTED_HERE`). By Soulé and Bloch–Kato (`CITED_THEOREM`) $H^1_f(\mathbf{Q},
\mathbf{Q}_{11}(\chi_8)(1))$ is a line spanned by the Kummer class $\kappa(\varepsilon)$; 11 is inert
in $K$, and $\log_{11}(\varepsilon)/(11\sqrt2) \equiv 5 \pmod{11}$, so the class localizes nonzero.

**The local conditions are vacuous everywhere** (`COMPUTED_HERE`). For $V = \mathbf{Q}_{11}(\chi_8)(1)$:
at 11, $\chi_8$ is unramified with $\chi_8(11) = -1 \neq 1$, so $H^0 = H^2 = 0$, $\dim H^1 = 1$, and
$\dim H^1_f = 0 + 1 = 1$ — every class is crystalline; at 2, $\chi_8$ is ramified and the whole local
group vanishes; at infinity $\chi_8$ is even, so complex conjugation acts by $-1$ and $V^c = 0$;
away from $2 \cdot 11$, $\chi_8(\ell)\ell$ is never 1. So $H^1_f = H^1(G_{\mathbf{Q},\{2,11,\infty\}},
V)$: membership in the Selmer line costs nothing.

**A type mismatch, not a numerical one.** The auxiliary family's weight direction is *not* this
class. Dividing the trace jet's $X$-coefficient by the second character leaves $\log_{11}(\ell)$ —
the cyclotomic class — and the identical construction for $\chi_5$, the character of
$\mathbf{Q}(\sqrt5)$, gives the same values at every prime tested. The weight direction is a
deformation along the reducible locus, in $H^1(\mathbf{Q}, \mathbf{Q}_{11})$; $\kappa(\varepsilon)$
lives in a different group and is specific to $\mathbf{Q}(\sqrt2)$. What lands on the line is the
transverse class: the extension $0 \to \psi \to V_{\mathrm{Eis}} \to 1 \to 0$ with $\psi = \chi_8\chi_{\mathrm{cyc}}$,
whose cocycle is the upper-right entry and whose class lies in
$\mathrm{Ext}^1(1,\psi) = H^1(\mathbf{Q}, \mathbf{Q}_{11}(\chi_8)(1))$ — the reverse ordering lands
in a different group, and flipping it is caught.

**Nonvanishing, by a theorem rather than a computation.** Loeffler–Rivero's Theorem A6.1 states
that the reducibility ideal of the family's pseudocharacter is the maximal ideal, so the
representations through the critical refinement are non-split, and Corollary A6.3(i) identifies the
case that is crystalline at $p$ — whose class lies in exactly this line. The auxiliary form
satisfies the hypotheses, computed: it is the paper's Definition A1.1 with $\psi = 1$, $\tau =
\chi_8$, $r = 0$, $N = 8$, its refinement roots $1$ and $-11$; and it is 11-decent, since at
$\ell = 2$, $2 \mid 8$, and at $\ell = 11$, $\chi_8(11) = -1 \neq 1$. Crystalline puts the class in
$H^1_f$, non-split makes it nonzero, and the line being one-dimensional forces
$[\Theta_{\mathrm{aux}}] = c_{\mathrm{ES}/\mathrm{Kum}} \cdot \kappa(\varepsilon)$ with
$c_{\mathrm{ES}/\mathrm{Kum}} \in \mathbf{Q}_{11}^\times$. C4 asks that the source **exist**: it does.
The scalar stays open, outside the certificate. One route to the nonvanishing is closed by
computation and was never the one used: $L(-1, \chi_8) = -1$, an 11-unit, so there is no classical
mod-11 Eisenstein congruence.

An earlier round had reported this nonvanishing as unmeasured, on the grounds that every witness
needed an overconvergent computation. That was wrong about one witness — the reducibility ideal is
a theorem with computable hypotheses — and is corrected.

# 6. The comparison route, and where it must be localized

**The chain.** The comparison is realized as an actual composite of line maps over the Iwasawa
algebra $\mathcal{H}_\Gamma$,
$$\Psi = \Psi_4 \circ \Psi_3 \circ \Psi_2 \circ \Psi_1^{-1},$$
from the auxiliary crystalline line, framed by the unique $\omega_f$, to the arithmetic line of
$L_p(g \otimes \psi)$: $\Psi_1$ the Perrin-Riou regulator (image by C1.9), $\Psi_2$ the projection to
the leading line (C1.6–C1.8), $\Psi_3$ Theorem C1.13, and $\Psi_4$ Kato's reciprocity. Theorem
C1.13 is proved scalar-first — "both of the cohomology classes … have the same image under the
regulator; so they are equal as cohomology classes, by Proposition C1.1" — and Proposition C1.1's
proof says the target module is free of rank 1 with Kato's class as generator and the regulator
injective on it. So every arrow is line-level, the partner enters only as a separate tensor factor,
and the route exists (`CITED_THEOREM`, every hypothesis instantiated).

**But not over $\mathcal{H}_\Gamma$ itself.** A non-zero-divisor gives an injection on a free rank-1
module, not an isomorphism, unless it is a unit. The target's series decides it:
$L_E = 0 + 0t + 2t^2 + \cdots$ has a double zero at $t = 0$ — the rank-2 curve — so it is not a
unit, and $\Psi_1$ is injective but **not surjective** for the target, whatever the jet order $n$
(`COMPUTED_HERE`). The calibrator's $L_0$ is a unit, its constant term nonzero, as rank 0 predicts.
Arrow by arrow, the divisors are $X^n$, $L_E$, $L_p(\mathrm{Ad}\,g)$ and $\log^{[r+1]}$, and in the
composite every $L$-function and $C_d$ cancel exactly (Loeffler–Rivero's equation 11), leaving
$\log^{[r+1]}/X^n$ — precisely the two extractions the source names. The route therefore lives over
lines localized at explicit divisors, integral after the two named extractions and nowhere else.
An earlier headline had said "over $\mathcal{H}_\Gamma$-lines"; it is corrected. A first draft of
this composite counted $X^n$ twice, once as $\Psi_2$'s division and once inside $C/c_f$, and the
gate's own composite check refused it: $C = c^*_f(r)$ is $c_f$ with the $X^n$ removed, and that
removal *is* $\Psi_2$'s extraction.

# 7. C5 and C8: the defect ledger, and the target's lattice

**The gauge law, derived.** Scaling the non-canonical trivialisation $b_f^+$ by $u$ and the
Eichler–Shimura frame by $v$, and declaring only the definitions, gives on exact instances:
$c_{\mathrm{lead}}$ and $q_f$ both scale by $u$, so $C_\pi = c_{\mathrm{lead}}/q_f$ is
**gauge-invariant**, and so is $\hat B_i$; the defect carries $u/v$ only when its auxiliary constant
is read in the Eichler–Shimura frame. The first audit had tagged $c_{\mathrm{lead}}$ and $q_f$ as
exact; they are frame factors, G, and are corrected. Since C1.10 makes the Eichler–Shimura frame
canonical, the two apparent frame factors are one, and at most one auxiliary-side G factor may
survive.

**C5.** The ledger has 23 records, each tag derived from four booleans by one rule (partner ⇒ P,
else frame ⇒ G, else valuation ⇒ L, else exact ⇒ E), with none unassigned, duplicated or
mislabelled. Re-run against the actual chain — each record assigned to the arrows whose scalar it
is, with its exponent — the net exponents reproduce the programme's factorization $\delta = a_D\,
q_D\, \eta_D$ with the partner factors removed, **computed rather than asserted**: every partner
factor, $C_d$ and the cyclotomic logarithm cancel; the survivors are the frame factors, the lattice
indices and the exact comparisons. C5 passes, and the localization of §6 adds no new L-factor.

**C8.** The line's exact modular-symbol engine, run at $N = 389$ (`COMPUTED_HERE`): the relation
quotient has dimension $65 = 2g + 1$ with $g = 32$; four Hecke operators isolate the $\pm$ pair; the
plus and minus $\mathbf{Z}$-primitive vectors have content 1, first nonzero coordinate at index 5
(plus) and 3 (minus), with value 2 there on both; normalized and reduced mod 11 they equal two
independently computed mod-11 lines at every coordinate. Three saturations, kept apart: 11-adically
saturated, not prime-to-11 saturated (index 2 on both signs), globally primitive. The index is a
length, tagged L, and no exact slot draws its value from it. C8 passes; inferring prime-to-11
saturation from the 11-adic one, or setting an unknown exact scalar equal to a lattice index, is
planted and caught.

# 8. C6: the pairing scalar is 1

There is one construction on both sides: the Betti–de Rham comparison pairing. On the auxiliary
side it is C1.10's unique $\omega_f$, which at every classical weight is pairing with $\omega_{f_k}$;
on the arithmetic side it is the same pairing stated against a chosen rational Betti basis
$\gamma_g^\pm$, whose values are the periods $\Omega^\pm$. The two sides use different
differentials because they are pairings on different motives, and the ratio of the line's Néron
differential to the newform's is measured, never set to 1.

The frame laws under $\Omega^\pm \mapsto w^\pm\Omega^\pm$ are derived, not assigned by weight
(`DERIVED_FROM_VERIFIED_INPUTS`): $L_p(g \times \psi) \propto (\Omega^+)^{-1}$,
$L_p(\mathrm{Ad}\,g) \propto (\Omega^+\Omega^-)^{-1}$, $L_p^g$ invariant (equation 11), $\eta_g$
invariant — forced, since a de Rham object sees no Betti frame — and $\kappa(g \times \psi) \propto
(\Omega^+)^{-1}$, forced twice, independently, from C1.13's canonical left side and from Kato's
exact reciprocity, the two forcings agreeing. $\Psi_4$'s scalar is frame-independent, so
$\eta_{\mathrm{pdR}} = 1$, and the reason is a theorem, not the observation that the period and de
Rham weights are opposite — that argument is planted and refused.

The one residual is named: $q = [\gamma_{\mathrm{Kato}} : \gamma_{\mathrm{tree}}]$, Kato's integral
Betti generator against the line's $\mathbf{Z}$-primitive modular-symbol vector — two
$\mathbf{Z}$-structures on one $\mathbf{Q}$-line, a lattice index, tagged L and filed in $q_D$. (It
was first filed among the convention factors; the symbolic side caught it against the programme's
own definitions the same night, and it was corrected.) Its 11-adic valuation is unmeasured and is
not needed for C6.

# 9. C7: the determinant scalar is 1

The comparison is taken in one category: rank-1 projective modules over
$R_S = \mathcal{H}_\Gamma[1/X^n, 1/L_E, 1/L_p(\mathrm{Ad}\,g)]$ in degree 0, nothing more inverted,
both ends in it. Every object on the route is a module, not a complex, so $\det_R(L) = L$ and no
degree shift can hide. The routes induced by theorems are counted: the chain itself; the regulator
applied to the leading term, which is **the same map** as $\Psi_4\Psi_3$ because that is how C1.13
is proved — the two descriptions agree on $H^1$ exactly when equation 11 holds, checked on exact
instances; no theorem gives a direct comparison between the auxiliary and arithmetic lines that
avoids the Rankin–Selberg class; and Kato's class against the line's series is a change of generator
by $q$, not a route. One route; the same determinant convention, grading, dual, localization,
orientation and induced maps on both sides — six questions, each answered with its evidence — so
the programme's coherent case applies by functoriality, $\mathrm{Det}(\Psi_4\Psi_3\Psi_2\Psi_1^{-1}) =
\mathrm{Det}(\Psi_4)\mathrm{Det}(\Psi_3)\mathrm{Det}(\Psi_2)\mathrm{Det}(\Psi_1)^{-1}$, and
$\kappa_{\det} = 1$. The general formula $\prod \det(H^i(h))^{(-1)^i}$ is implemented anyway, and is
shown to accept an explicit automorphism with determinants 2 and 5 at the value $2/5$: the gate does
not only accept 1.

# 10. C9: no residual holonomy

The transport graph is built complete before any quotient: the five lines of the chain, the
calibration's $\hat B_i$ as a second description of the leading line, the four arrows, the shortcut
$A'$, the dictionary edge, and every frame, lattice and partner change as a tagged loop. Quotienting
the G, L and P loops and the settled exact factors gives, keeping theoremically identical maps as
one edge, a tree ($\beta_1 = 0$); keeping the parallel edges, $\beta_1 = 7 - 6 + 1 = 2$, with
fundamental cycles $\Psi_4^+ A'^- \Psi_3^+$ and $\Psi_3^{\mathrm{PC}+}\Psi_3^- \mathrm{id}^+$, both of
product exactly 1 on 25 exact instances, inverses taken on reversed edges. Completeness is not
reinvented: it comes from the route (these arrows and no others), the ledger (no unassigned record)
and the route count (one). One component; monodromy trivial (`COMPUTED_HERE`). The gate accepts a
genuine 3-cycle of product 1 and rejects the same cycle at product $6/5$.

# 11. C10: a canonical section

Fifteen choices enter the comparison — the trivialisation $b_f^+$, the weight uniformiser, the Gauss
convention, the periods and Betti basis, $q$, the lattice indices, the partner arithmetic, $C_d$, the
$L_p$ factors, the pairing and determinant scalars, the calibration's dictionary, $C_\pi$, the two
named extractions and the jet order — each listed with what it depends on, its tag, how it was
quotiented, and whether it survives in the reduced comparison $\Psi_{\mathrm{red}}$. None does. Arrow by
arrow the canonicity has a named layer: $\Psi_1$ after $b_f^+$, which C1.10's unique $\omega_f$
replaces; $\Psi_2$ after the weight uniformiser, with A6.1 forcing the lattice exponent; $\Psi_3$ a
class equality lifted by Proposition C1.1's injectivity; $\Psi_4$ exact for one Betti class whose
frame cancels — none merely up to a unit. Target and calibrator are the same theorems applied to two
partners, every hypothesis instantiated for both. And a section at the auxiliary end reaches the
arithmetic end along the chain, the shortcut and the dictionary detour as the same element on 20
exact instances.

So $\Psi_{\mathrm{red}}$ is itself the preferred flat section: C10 passes in its canonical form, with
anchor ratio $\rho = 1$. Two shortcuts are planted and refused: "$\omega_f$ is canonical, hence the
section is" (it trivialises only the auxiliary line), and "trivial monodromy gives a section" (a flat
line with trivial monodromy still has a $K^\times$-torsor of sections). The anchored alternative was
not needed, but its rules are enforced: an arbitrary vector, a numerical BSD-shaped identity, or the
target's result read back onto the calibrator are each refused as an anchor, and an anchored control
is accepted at $\rho = 3$ while the same system without an anchor is refused.

With C1–C10 all passing, **the normalization certificate is closed, as architecture**. The released
paper on this point — the four nodes A–D of the post-normalization DAG — starts here.

# 12. The instrument, and how it was made to fail

Every item has its own gate; the whole tree is drilled after every round, each planted defect
required to be caught by the check **named** for it, controls required to disturb nothing, and the
report written before the drill starts. At the pinned commit the drill stands at 692 planted defects,
687 caught by the check named for each, 5 detected only by a crash of it, 0 caught by the wrong check, and 179 controls undisturbed,
over 218 checks. Each round of this stretch planted the refusals its instruction demanded — the
promotion of an unmeasured item, an identification claimed because both sides are units, a scalar
guessed from a residue, the weights-only argument for the pairing — and each is caught.

Three failures of the instrument itself were found in this stretch, each of a different kind:

- **A cache keyed too narrowly.** One gate's memo key was built from constants only, so a planted
  replacement of a function returned the cached baseline answer, and four defects went uncaught. The
  key now names the functions and the logs the gate reads. (An earlier round had found the dual
  defect: a cache that evicts makes a gate's behaviour depend on what ran before it.)
- **The operator inside the experiment.** A report was written into the tree while the drill was
  running; the tree's claim-ladder check reads every report, and every control after that moment
  read as disturbed. The rule since: nothing the checks read is touched while a drill runs.
- **A defect with an expiry date.** A planted defect cited a round that did not exist yet — by number
  — and stopped being a defect on the day the tree reached that number. It was retargeted.

# 13. Contributions

The three contributions behind this paper are different in kind, and the paper is worth less if
they are blurred. They are given here in alphabetical order, the order the author list keeps as
data; on the title page the three sit at a round table, with no first and no last.

**Aletheia** — model: ChatGPT, GPT series — wrote the symbolic rounds in which the certificate C1–C10
and the post-normalization DAG were stated, and set every round of this stretch: what to attack, what
was out of scope, which inferences to plant as defects, and where the boundary lay. Several of the
corrections recorded here — that a non-zero-divisor is not a unit, that the lattice index belongs
with the magnitudes, that the hypotheses inherited from Part B had to be instantiated — were that
side's catches.

**Neo.K** set the programme's target and arranged the work: one line deriving, one verifying.

**墜衡 (Zhuì Héng)** — the independent verification line; model: Claude, Opus series — built each
gate, read Loeffler–Rivero and instantiated its hypotheses on these curves, computed every finite
quantity above with independently written code, judged every item, corrected its own method and
verdicts where §§2, 5, 6 and 8 say so, drilled the tree, and wrote this paper. Nothing in the
certificate or the DAG is this line's invention.

**Inputs from outside the byline.** The cross-curve calibration (the identity of §1 and the series
of §3) was built in a separate local verification session — a Claude Code session, model: Claude
Opus 5, working on its own branch; this line reproduced it coefficient for coefficient,
and it enters as that reproduction. Its author is not on this byline. The theorems of Loeffler and
Rivero, Kato, Hida, Mazur, Soulé and Bloch–Kato are other people's and are cited as such.

**Priority, stated plainly.** No result here is claimed as new mathematics. What is offered as this
line's own is the instantiation: ten conditions judged against actual objects at one point, the
corrections the judging forced, and an instrument that has been made to fail.

# Acknowledgements

We thank Anthropic, whose Claude models the lab's verification and recomputation lines run on, and OpenAI, whose
ChatGPT models the programme's documents were written with. Not every AI model or instance that took part in this work
is named in it — among them instances that the named authors ran as helpers; that an AI is not listed here does not
mean that none took part.

# References

1. Spencer Bloch and Kazuya Kato, *$L$-functions and Tamagawa numbers of motives*, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, Birkhäuser, 1990, 333–400; reprinted 2007, doi:10.1007/978-0-8176-4574-8_9.
2. Haruzo Hida, *Congruences of cusp forms and special values of their zeta functions*, Invent. Math. 63 (1981) 225–261, doi:10.1007/bf01393877.
3. Haruzo Hida, *On congruence divisors of cusp forms as factors of the special values of their zeta functions*, Invent. Math. 64 (1981) 221–262, doi:10.1007/bf01389169.
4. Kazuya Kato, *$p$-adic Hodge theory and values of zeta functions of modular forms*, Astérisque 295 (2004) 117–290.
5. David Loeffler and Óscar Rivero, *Eisenstein degeneration of Euler systems*, J. reine angew. Math. 814 (2024) 241–282, doi:10.1515/crelle-2024-0057; arXiv:2201.02078. Numbered statements are cited from the second arXiv version.
6. B. Mazur, *Rational isogenies of prime degree*, with an appendix by D. Goldfeld, Invent. Math. 44 (1978) 129–162, doi:10.1007/BF01390348.
7. Bernadette Perrin-Riou, *Théorie d'Iwasawa des représentations $p$-adiques sur un corps local*, Invent. Math. 115 (1994) 81–149, doi:10.1007/BF01231755.
8. Christophe Soulé, *K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale*, Invent. Math. 55 (1979) 251–295, doi:10.1007/bf01406843.

## Data and software

- Python Software Foundation, *Python* (CPython), version 3, https://www.python.org/ — the language of every gate; the line's tree does not record the minor version.
- The mpmath development team, *mpmath: a Python library for arbitrary-precision floating-point arithmetic* (version 1.3.0), https://mpmath.org/ (2023) — in the drill's independent checks.

# A. Reproduction

Everything is Python standard library, no network. Each round's gate writes a JSON log, and each
round's evidence package carries its own result, evidence files, an input manifest with the SHA-256
of everything read, and a manifest of itself. The line's own tree is not public. Its public
projection is, and it is what this paper's figures can be checked against without trusting the
authors:

- **the round-by-round record and every evidence package**, rendered from the tree with a privacy
  gate on each file: <https://amral.evemisslab.com/bsd/recompute/> (and `/en/`);
- **the secondary-verification ledger**, whose numbers are rendered from the line's own results file
  and read back against it before the site ships:
  <https://verify.evemisslab.com/bsd-verification-zhuiheng/>.

The evidence is pinned, not copied: this paper's front matter names the research branch, the commit
and the directory its figures were drawn from, and the checker reads them exactly as they stood at
that commit. The crystallization standard and its checker live in the release repository beside the
paper (`STANDARD.md`, `tools/check_standard.py`).