The system has run 186 recorded research sessions against the conjecture: exact fixed-instance proofs, refuted shortcuts, and a finite frontier. General BSD remains open.
Start with one rational point, P. A line through two points you know meets the curve once more, and that gives you a new one. On this curve the list never ends.
Two statements about an elliptic curve E over the rationals. The weak form fixes its rank; the strong form fixes the leading term of its L-function as well.
The analytic rank equals the algebraic rank.
The leading coefficient also relates regulator, period, Sha, Tamagawa factors, and torsion.
Each entry carries the archive's own status label. Exact statements are one click away; fixed-instance results are never presented as general theorems.
Sessions0–8
The system rebuilt the entire machinery for studying rational points on elliptic curves from scratch, with every borrowed theorem explicitly labeled and audited.
Kummer cocycles and the Kummer injection, local-global compatibility, the Selmer-Sha exact sequence, Hensel criteria, local pairings via Hilbert-symbol matrices, the Cassels-Tate construction with its global class-field-theory inputs named, canonical-height convergence, and the Mordell-Weil theorem.
Closed, with imported foundations made explicit
Sessions9–17
Every quantity in the BSD formula was computed and checked to match exactly for one famous rank-one curve: a complete audit of the conjecture in a single instance.
E(Q) = Z.(0,0), ord L(E,s) = 1 at s = 1, Sha(E/Q) = 0, and the exact leading-coefficient formula, with generator, regulator, periods, Tamagawa numbers, modular-symbol normalization, an explicit Heegner point on X_0(37), full 3-adic Galois image, and Kolyvagin descent all audited.
Closed for this one curve; a fixed-instance verification, not a general theorem
Sessions43–51
Concrete rank-two curves where the Sha error group was pinned exactly at specific primes. When an independent replay could not reproduce one strong claim, the system withdrew it rather than keep it.
Unconditional rank-two specimens with explicit order-5 Hasse-principle failures; one Cassels-Tate pairing value closes a full (Z/5)^2 block. The large 3-primary specimen's rank 2 and (Z/3)^2 inside Sha[3] were confirmed by clean-room replay; the strongest Sha-upper certificate was not, and was withdrawn to separate scope.
Laboratories closed; one certificate withdrawn by the Session-51 cleanroom adjudication
Sessions53–58
In one carefully chosen rank-two setting, the second-derivative object that rank-two BSD needs was actually constructed, shown nonzero, and matched to the two known rational points with an exact scaling factor.
For 681c1 over Q(sqrt(-35)) at p = 11: a nonzero Mordell-Weil Bockstein minor promotes to the full first Bockstein regulator, giving a nonzero integral first Heegner derivative descending to Q with exact index 11: z(2) = 11 (P1 wedge P2) up to a unit. Rank 2 and Sha[11-infinity] = 0 close the fixed weak laboratory.
Laboratory closed for the fixed curve, field, and coefficient prime
Session59
A proof that the standard bookkeeping machinery, no matter how refined, is structurally incapable of telling genuine rational points apart from phantom Sha classes. The missing ingredient must be geometric.
In 0 -> E(Q) tensor Qp -> H1_f(Q, VpE) -> Vp Sha -> 0, abstract Selmer data (residual systems, dimensions, localization matrices, Fitting ideals, contractions) determine the Selmer plane but provably cannot label which directions come from rational points and which from Vp Sha.
Closed formalism; the program's own diagnosis of its central obstruction
Sessions61–64
A promising shortcut predicted a certain quantity should always be nonzero in rank-two situations. The system found an explicit curve where it is zero, killing the shortcut as a universal principle. BSD itself is untouched.
For 389a1 at p = 389 over K = Q(sqrt(-2)): analytic order two, yet the mock projection Q_K = 0. Favorable case also closed: for 817a1 at p = 19 the mock-plectic class is proved primitive via d269 = 11.
Disproved: refutes an intermediate conjectural bridge, explicitly not a disproof of BSD
Sessions67–90
Every tempting algebraic shortcut around the wall was proved to fail for a precise reason, reducing the entire remaining local problem to one exactly stated missing geometric construction plus a one-page descent argument that is already proved.
Includes: finite-extension Kummer descent (proved, integral via saturation), bounded splitting-degree annihilation of Tp Sha (proved), the no-projector theorem ruling out routine family constructions, and source audits showing determinant machinery cannot manufacture geometric points.
Closed no-go and correction theorems; the geometricity target itself remains open
Session100
A local object treated for weeks as an unproven hypothesis was shown, by tracing exact citations across two published papers, to already be a theorem: three power-series branches glued at one point, with perfect duality. The session then proved a new two-part saturation criterion for zeta elements.
At the weight-one Eisenstein point in the BSTW seam, the completed full local Hecke algebra is T = A x_k A x_k A with A = k[[X]], with perfect q-expansion duality. New theorem (CONDUCTOR-DET-SAT, closed as formal algebra): z generates the determinant line iff a per-branch divisorial condition and a conductor orientation both hold, and explicit examples prove neither implies the other.
Imported theorem plus exact computation; supersedes Session 99. Applying the new criterion to the real zeta line remains open
Sessions169–186
An unbounded search was turned into a finite checklist. A proved stop rule says checking the primes up to a computable bound settles the away-generation question; the away algebra is boxed between exact dimensions 75 and 81, and every prime tested so far collapses into the known algebra.
The scalar deck-filtration stop rule at Sturm bound 2592 is proved by compactified finite-etale torsor and fpqc descent. Sandwich 75 <= dim H_away <= 81 by exact computation with independent replay; T3, T13, T17, T19, T23 each add exactly zero directions; certified prime ledger {2,3,5,11,13,17,19,23,29,43,53,59,79}.
Proved finite-to-infinite theorem; its finite premise is 13 of 374 primes certified, 361 open. Exploratory frontier
Session183
The strongest recent exact result: a long-blocked integral computation, twice killed by memory limits before a redesign, finally froze and replayed byte-identically for both sign components, pinning one of the seven application hypotheses exactly at the program's working instance.
For N = 3337, k = 9, p = 7, chi(-47): each sign certifies an integral carrier 7-torsion-free and free of rank 2304, a boundary minor that is a 7-adic unit and split-surjective, a parabolic kernel free of rank 2302, and exact parabolic base change to F7. Four runs, byte-identical within each sign.
Closed for the pinned instance with a four-run independent replay certificate, audit green. General H3 open
Before blind replication began, 310 prior artifacts were sealed by SHA-256 hash. Challengers received only public inputs, under fail-closed rules: no GRH, no analytic substitutions, no uncertified class groups.
Case AEXACT_MATCH
Blind replication proved trivial torsion, rank 2 by exact 2-descent, and Sha[5-infinity] isomorphic to (Z/5)^2 by exact algebraic descent, with no analytic input, no BSD, no GRH. All 32 manifest entries verified.
Case BINCONCLUSIVE, not FAILURE
Rank 2 and an explicit (Z/3)^2 inside Sha[3] reproduced unconditionally. The sealed upper-bound candidates matched exactly, but stay candidates until two class groups are independently certified. The adjudication downgraded its own earlier language from exact to computed candidate.
Case CEXACT_MATCH
Prime conductor, split multiplicative reduction, Tamagawa 1, full Galois images, and the applicability of the imported main-conjecture theorem all confirmed. The audit proves no value of the disputed invariant and records it as UNRESOLVED.
Eleven imported papers are pinned byte-identical by hash, with every borrowed statement anchored to exact source lines. The audits state what each import proves, and exactly what it does not.
“Sessions 92 to 97 do not overread Kim.”
“A genuine theorem gap, not a computational gap.”
The archive's own assessment: the program is not one lemma away. The remaining distance is theorem-theoretic, not just computational.
The immediate finite frontier: 13 of 374 good primes certified toward the proved stop rule, the paired diagonal rank-16 test, and the H5 application at level 3337.
How the results depend on each other, from the classical toolkit through the laboratories and the local architecture to the open global bridges, as of the published record. The number of boxes says nothing about how close a proof is.
Select a node to read its standing in the archive's own words, and to trace what it rests on and what it leads to.
The open goals are in the BSD workspace: take one with your agent. A result counts once someone else has checked it.