← BSD Conjecture · LAYER 3
The papers of this line
The released papers; each can be read on its own.
I · Precursor (five sub-lines: Phase 0, P5, Phase 1, Phase 2 and the stress test) → II · Computation and Reproduction (107 runs, RUN-001 to RUN-107) → III · Crystallization (this page, the list of papers, 11 in all).
Four nodes of the post-normalization DAG, two findings that cost a factor, and an instrument drilled to fail
The Birch–Swinnerton-Dyer conjecture is open, and nothing here closes it. What this paper does is carry one explicit point — the rank-2 curve \(389.\mathrm{a}1\) at the good ordinary prime \(p = 11\) — all the way through the determinantal (Bloch–Kato / ETNC) formalism with every finite quantity recomputed independently, every theorem-level step read in its source and its hypotheses instantiated one at a time, and the whole instrument drilled against planted defects each round.
The ten items of a determinantal comparison at $(389.\mathrm{a}1,\ p = 11)$, judged one by one, and the method that judges them
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.
What is canonical and what is a choice: a localization determinant, a local unit, a Kurihara number and a Bockstein determinant, each computed from the group law or from modular symbols built here
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. Explicit instances of Kolyvagin–Kurihara arithmetic in rank 2 are rare; this paper recomputes one completely, for the rank-2 curve \(389.\mathrm{a}1\) at the good ordinary prime \(p = 11\) with the two ramified directions \(\ell = 397\) and \(991\), and sorts every figure into what is canonical and what is a choice.
A research programme's framework documents, checked: the ladder every later claim is placed on and the five rungs every summary drops, the programme's rejection of its author's own route and why that route fails at infinite precision, a global frontier quantity whose faithfulness survives in the rationals and not in floating point, and the programme's maps scored after sixty-four rounds
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. This paper is about how evidence for it should be stated. A research programme on the conjecture opens with framework documents, and the first of them refuses to let a curve carry a bare true or false: it defines an eleven-rung certificate ladder, from the curve’s identity (C0) through numerical and rigorous analytic rank (C2, C3), algebraic bounds, weak and single-prime strong BSD, finiteness of Ш, full strong BSD, to a family theorem (C10), so that each curve and prime is placed at the rung some certificate has actually closed. Checked against the programme’s 85 curated documents, the ladder’s vocabulary is used cleanly — no document names a rung the framework does not define — but every summary of it renders six rungs, dropping C1 to C5, and with them the one boundary the framework forbids crossing: a number a computation produced against a number a proof produced.
A reproducibility note on a published computational census, read from its archived diffs and tested on its data: three isogeny rules in three commits with one output frozen at the first, a deleted predicate that nothing in the data fails, the corpus's own specifications held against what was actually produced, and a methodology's freeze conditions applied to the corpus that methodology produced
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. A published census selects base elliptic curves of conductor below 500,000 by an algorithm (Algorithm 1) and lists, for each, the quadratic twists (Algorithm 2) to which a theorem of the same paper gives strong BSD. A research programme archived three commits of the census’s code and its outputs. This note reads what changed between them, from the archived diffs, and tests each reading on the data.
A verified table of explicit invariants of the rank-two curve y² + y = x³ + x² − 2x at the prime 11 — a mod-11 p-adic L-function, formal logarithms of the generators, a local multiplier, a genus-three curve on E × E with its chain, norms and pushforwards, twisted L-values tied to a modular-symbol eigenline, and a circular unit — each recomputed independently, each attack's own statement that it is not a proof kept with it
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. For the rank-two curve \(E\): \(y^2 + y = x^3 + x^2 - 2x\) of conductor 389, with generators \(P = (0,0)\) and \(Q = (1,0)\), at the good ordinary prime 11, a research programme ran eight “proof attacks” — derivations toward the 11-part of the leading-term formula — and each attack computed something new. This paper recomputes every finite statement in them from independent arithmetic: this line’s own modular symbols, exact group law, number-field and function-field arithmetic, and point counts. Everything recomputes.
An independent recomputation of a published computational census at full scale — discriminants, conductors by Tate's algorithm, the isogeny filter decided in both directions through modular curves, every twist list rebuilt, the census's own accounting measured term by term — with every disagreement explained and the directions its own checks could not see
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. Banwait and Huang’s census selects semistable base elliptic curves of conductor below 500,000 and lists, for each, the quadratic twists to which a theorem of theirs gives strong BSD. A research programme reproduced the census as an exact artefact census, and graded its own headline count as taken from the official output and consistency-audited, with only 2 curves and their twists recomputed. This paper closes that gap for all of them, with independent code and exact integer arithmetic throughout.
A research programme's plan to prove strong BSD for infinitely many quadratic twists, checked hypothesis by hypothesis: the witness lemmas and the empty set they do not survive, membership conditions that turn out to be load-bearing, a boxed certificate refuted at p = 29 and the same argument true one quantifier over, the ordinary obstruction measured to 6000, the corpus disagreeing with itself about H3, and member 3529, where the provisional design fails at its own twisting prime
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. A research programme proposes to prove its strong form for every quadratic twist \(E^{(q)}\) of the curve 696.e1, \(y^2 = x^3 + x^2 + 8x - 16\), by a prime \(q\) in an explicit set \(\mathcal P\) of density 1/24, and routes the supersingular primes — and, in a first design, each twist’s own prime — through the theorem of Fouquet and Wan, whose three hypotheses H1, H2, H3 the programme compiles into finite tests. This paper runs those tests. The witness problem behind H3 is finite and, on this family, empty — but the route has exactly one witness, the prime 29, and an emptied witness set would make every odd prime fail, since the gcd of an empty set is 0. Two membership conditions of \(\mathcal P\) that read as bookkeeping are what keep the family off the one exact no-go and keep that lone witness alive. The programme’s boxed uniform certificate for H3 is false at exactly one prime, \(p = 29\), because it drops the clause \(\ell \ne p\); the same argument restricted to good supersingular primes is true. The ordinary obstruction to H2 fails at eight of 780 good odd primes below 6000 and has not stopped, which is the programme’s own reason to keep ordinary primes off this route. And joining four earlier measurements shows that H2 at a member’s own twisting prime reduces to a congruence on the base curve: it holds at 18 of the 19 members below 4000 and fails at \(q = 3529\), where \(a_q = 1\). The design that sends the twisting prime to Fouquet–Wan therefore fails at a member of its own family; the programme’s revised router does not use that route at \(p = q\).
Conditional algebra for calibrating a rank-two leading coefficient against a rank-zero curve — determinant lines, jets, lattice torsors, characters, holonomy and a closure into a seven-node chain — checked statement by statement on exact random instances; one theorem false in one direction, two supplements its documents leave implicit, a bookkeeping character measured rather than read, and the rank-two identity at 389.a1 closed with every factor computed here
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. A research programme on the rank-two curve 389.a1 at the good ordinary prime 11 wants the leading coefficient of an 11-adic family to be calibrated against a rank-zero curve, so that the unknown scalars common to both cancel and only arithmetic remains. Thirty symbolic rounds, written without computation, build the algebra for that: a calibration across ranks, determinant lines and their jets, lattice torsors and the limits of 11-adic descent, Euler defects of cones, a self-dual fundamental line, the characters by which every normalization choice acts, holonomy on graphs of transports, and a closing round that arranges it all into a seven-node chain and a ten-item certificate. Every statement is conditional, and the rounds say so. This paper checks each boxed identity by instantiating it in exact arithmetic — formal series with vector coefficients over the rationals, exterior algebra, integer lattices and their Smith forms, two-term complexes, ideles of the rationals, random graphs — on 40 random instances or more, with negative controls. Every identity holds except one: a theorem on ideles is false in one direction, at the sign \(-1\) its own hypothesis excludes. Two supplements are made explicit, a bookkeeping character of eight coordinates is measured from a pipeline that performs each transformation instead of being read from the text, and the round that asks for one determinant regulator is answered on 389.a1 itself: from canonical heights computed here the regulator is 0.15246017794314376, and the rank-two identity \(L''(E,1)/2 = \Omega\cdot\mathrm{Reg}\) closes with every factor computed in this line, to the floor of the quadrature.
A research programme's family for strong Birch–Swinnerton-Dyer beyond the semistable reach of a published method, recomputed: the anchor, the density and why it is not 1/48, a certificate of twenty quantities rebuilt from the curve's a-invariants, all twelve of Mazur's degrees, mod-ℓ images at 38 primes, the odd additive barrier, and the candidate theorem stated against the revised router that every prime is sent through
The Birch–Swinnerton-Dyer conjecture is open, and nothing here proves any case of it. Banwait and Huang proved strong BSD for infinite families of quadratic twists of semistable curves. A research programme proposes a family for a curve their method does not reach: the twists \(E^{(q)}\) of 696.e1, which has additive reduction at 2, by the primes \(q\) of an explicit set \(\mathcal P\), and it states, as a derived theorem candidate, strong BSD for every member, each prime of each twist routed to a published theorem. This paper recomputes everything about the family that can be computed. The anchor has conductor 696, analytic rank 0 and \(L(E,1)/\Omega = 1\) to one unit in the last place. The density of \(\mathcal P\) is \(1/24\), not the \(1/48\) its three conditions seem to multiply to, by three routes: an exact Galois computation, a Chebotarev derivation in which every field-theoretic step is computed, and a count to \(2\cdot 10^7\); one of the three conditions is implied by the other two, and that redundancy is the factor of 2. A certificate of twenty quantities is rebuilt from the a-invariants alone; all twelve of Mazur’s prime isogeny degrees are excluded; the mod-\(\ell\) image is surjective at 38 primes, and at \(\ell = 3\) provably beyond this method. The candidate is stated against the programme’s revised router, which sends every prime to exactly one theorem — checked as a partition — and not against its first design, which a companion paper shows fails at the member \(q = 3529\).
A reproduction note on a cross-curve period calibration built by a separate local verification session: every modular symbol, measure, series and cup pairing of it recomputed with independent code, and the archimedean alignment of its plus period, which it left undone, supplied
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. A research programme at the rank-two curve 389.a1 and the prime 11 wants to calibrate the leading coefficient of an 11-adic family against a curve of rank zero, so that common unknown scalars cancel. A separate local verification session built that calibration in two rounds: a rank-zero calibrator, 19.a1, with its modular symbols over the rationals and its 11-adic measures; and a relative regulator germ, the quotient \(Q(t)\) of four 11-adic series of the two curves, with a cup pairing that makes it invariant under rescaling the four eigenlines. This note reproduces both with independently written code. The calibrator’s modular-symbol space, its plus and minus eigenlines and five Hecke checks come out identical; every residue of its first layer is reproduced; and the alignment the session left undone is supplied on the archimedean side: modulo 11 the plus line is the Néron-normalised modular symbol with unit exactly 1, shown by twisted L-values that are exact integers, and the minus line agrees with the twisted L-values, with unit 2, at every discriminant tried. The four series are recomputed to \(t^{120}\) from the layer of level \(11^3\), all 1210 summand rows on each curve identical, and \(Q(t)\) has Weierstrass degree 2 and leading coefficient 7; the cup pairing gives \(J_E = 1\) and \(J_0 = 3\), and the invariance holds. What the session itself declines to claim — that \(Q\) is the regulator ratio, that its cup ratio is the adjoint period ratio — this note does not claim either.