One operator view for the current decision, formal substrate, evidence, active work, and generated trust checks.
snapshot 307 ledger rows / ~268 distinctThe decision layer controls what work is justified; proof volume does not select an attack route.
HYP-M16-FIXED-TARGET-YIELD-001 / TASK-026 only: 3000000 primary trials across 3 frozen E_7 toy subgroups. Terminal: CLASSIFY_AS_KNOWN_LOCAL_SIMPLIFICATION; rerun authorized: false. Native Engine selection, direct secp256k1 work, solvers, and promotion remain closed. Structural selection RS-2026-07-24-001 remains unchanged.
0 selected; 0 ready. Positive toy evidence is supported, never proved; threat-model scope, route decision, and evidence remain separate.
The native queue remains empty; the exact external TASK-026 singleton is governed by the decision substrate. A future native candidate still requires both an exact mechanism and an independent raw-artifact validator.
8 mechanism/property cells; 4 decided at desk; 2 seed-eligible questions; 2 submitted; 0 quality-cleared. Typed screens, seeds, and drafts authorize nothing.
The first output is a bounded arithmetic construction or no-instance record plus a complete precomputation estimate, not an ECDLP experiment.
This open desk-cost cell has exact input presentations, an exact set-theoretic projective-tree bridge on the stated nonsingular characteristic domain, a complete named exceptional-fiber classification, deterministic control and secp256k1 GLV recovery checks, a frozen recursive projective S17 contract, kernel-checked fixed-degree projective resultant plus literal Sylvester unit-one and end-to-end common-root theorems, the actual frozenC specialization, the exact all-stage frozen projective witness chain, an exact literal finite MvPolynomial family in guarded form, an exact affine/infinity chart-polynomial cover for the frozen stage-14 predicate after injective base change, exact necessary infinity-stratum pruning, and exact conditional infinity propagation. Each fixed InfinityMask I uses 14 - I.card affine variables, no guard equations, and fifteen H equations with degree ceilings 2/4/2; the existential cover is exact and retains [1:0]. Affine external inputs reduce the exact logical cover from 16384 masks to 987 separated masks, and nonzero endpoint determinants conditionally reduce it to 377 interior masks. TASK-025 gives the nested local reductions 377 to 129 to 69 to 36 and the independent boundary-only branch 129 to 60. Under endpoint and BalancedPropagatedRegular assumptions it proves an exact single empty-mask affine chart, not a unique witness. The direct propagation applies over any field, but the source-stage bridge still requires injective base change into an algebraically closed target and separately assumed mapped-target regularity. Symbolic nonzeroness, nonemptiness, density, probability, and genericity are not proved, and source-field computation does not automatically establish mapped regularity. The 2^14 logical masks are not enumerated or materialized, and neither are the pruned families. It is not an expanded direct S17 polynomial, an unconditional base-field descent theorem, a solver run, or evidence of independent dimensions or relations. The complete cost bridge remains open. CQ-SEMAEV-S17-SYSTEM-COST remains partial; solving cost, rank, and yield remain unpriced. The scoped result has zero_retention_success and does not justify a production mask sweep, solver run, target computation, hypothesis retention, experiment authorization, cost inference, route rejection, or route promotion.
Canonical rationale, not a claim of impossibility
All routes are bound to an exact threat model, evidence gate, stop condition, and next action.
17 canonical routes
| Route | Disposition | Threat model | Priority | Next action |
|---|---|---|---|---|
| Generic-group lower boundR-GENERIC-LOWER-BOUND | Guardrail | classical-generic | P0 | Retain as a mandatory comparison and scope guard; no new experiment. |
| Baby-step giant-step and Pollard rhoR-GENERIC-BASELINE | Baseline | classical-single-target-plain, classical-generic | P0 | Use as the benchmark comparator in the evaluation harness. |
| GLV endomorphism accelerationR-GLV | Constant factor | classical-single-target-plain | P0 | Keep as target structure available to a selected route; do not treat it as a standalone attack. |
| Pohlig-Hellman subgroup reductionR-POHLIG-HELLMAN | Ruled out for target | classical-single-target-plain | P0 | Retain as a machine-checked elimination certificate. |
| MOV, Frey-Ruck, and Tate/Weil pairing transferR-PAIRING-TRANSFER | Ruled out for target | classical-single-target-plain | P0 | Keep the target-specific exclusion; defer a full pairing library unless a new route needs it. |
| Smart, Satoh-Araki, and Semaev anomalous-curve liftingR-ANOMALOUS-PADIC | Ruled out for target | classical-single-target-plain | P0 | Retain the formal exclusion; do not build an elliptic formal-group stack for this target now. |
| Weil descent and GHS-style transferR-WEIL-DESCENT | Ruled out for target | classical-single-target-plain | P1 | Document as a target-applicability exclusion; no formal stack now. |
| Prime-field index calculusR-PRIME-FIELD-INDEX-CALCULUS | Open, parked | classical-single-target-plain | P1 | Preserve evidence and prerequisites; wait for explicit route selection before any new experiment. |
| GLV-symmetrized Semaev systemsR-GLV-SEMAEV | Open, parked | classical-single-target-plain | P1 | Keep GLV-SEMAEV-ITER-001 closed without a solver run. Accept only a materially different phase-preserving mechanism with an exact recovery map, orbit tags, excluded components, a falsifiable full-cost prediction, and an independent validator through TASK-008 before any new decision. |
| Petit-style composed rational mapsR-PETIT-COMPOSED-MAPS | Open, parked | classical-single-target-plain | P2 | Keep the route parked. TASK-023 through TASK-025 completed the exact chart cover, necessary infinity-stratum filters, and conditional single-affine-chart propagation. TASK-026 then consumed its one external synthetic-toy authorization: all 3000000 trials completed, 911 exact relations were accepted, 907 were affine regular, and all six curve-arm regularity gates passed. REO-2026-07-31-001 records the result. This supports the bounded enabling claim that the named regular locus is usable in the frozen toy construction. Matched orbit/plain controls retained no H_NEW qualifying size, so the GLV-specific explanation is bounded negative and the terminal is CLASSIFY_AS_KNOWN_LOCAL_SIMPLIFICATION. The result is not source-faithful PKC relation generation, rank, solver scaling, recovery, total cost, 256-bit persistence, or an ECDLP improvement. Only formulate and review a source-faithful, non-executable HYP-M16-SOLVER-SLOPE-001 proposal with end-to-end metrics and scaling death criteria. Keep every TASK-026 rerun, production mask sweep, solver, experiment, route rejection, route promotion, cost claim, and exact-target inference closed until a separate dated authorization. Keep the auxiliary-curve cell parked until a primary source supplies a finite family or search domain with a completeness criterion. |
| Elliptic divisibility sequences and division-polynomial re-encodingsR-EDS-DIVISION-POLYNOMIAL | Open, parked | classical-single-target-plain | P2 | Keep HYP_WARD_EDS_001 parked; finish no additional point-division bridge unless a selected route needs it. |
| Isogeny, endomorphism-ring, and Frobenius transferR-ISOGENY-ENDOMORPHISM-TRANSFER | Monitor | classical-single-target-plain | P3 | Monitor literature and reuse GLV facts; no build now. |
| Multi-target and reusable-precomputation tradeoffsR-MULTI-TARGET-PRECOMPUTATION | Conditional inputs | classical-conditioned | P1 | Represent in the evaluation contract; no experiment in this phase. |
| Interval DLP, partial-key knowledge, and auxiliary-power algorithmsR-INTERVAL-AUXILIARY-INPUT | Conditional inputs | classical-conditioned | P2 | Keep as a scope category for future protocol or leakage analyses. |
| Hidden-number and lattice attacks on biased or reused ECDSA noncesR-HNP-NONCE-LEAKAGE | Separate threat model | implementation-leakage | P1 | Preserve as a separate security track; no lattice foundation build for the primary objective now. |
| Invalid-curve, twist, fault, and side-channel routesR-PROTOCOL-FAULT-SIDECHANNEL | Separate threat model | implementation-leakage | P1 | Record scope; do not expand the Lean substrate until a concrete implementation-verification goal exists. |
| Fault-tolerant quantum ECDLPR-QUANTUM-SHOR | Separate threat model | fault-tolerant-quantum | P1 | Update the registry estimate and monitor primary literature; do not build a quantum Lean stack now. |
11 of 17 critical nodes are closed. Blocked nodes retain exact resume conditions.
| Critical node | Status | Depends on | Blocker | Evidence |
|---|---|---|---|---|
| Canonical counts, registries, generated views, and active queuetruth-layer | Closed | none | none | STATUS.mdtasks/NEXT.md |
| secp256k1 field and subgroup parameterssecp-parameters | Closed | none | none | Ecdlp/Proved/Secp256k1PrimeP.leanEcdlp/Proved/Secp256k1PrimeN.lean |
| Exact rational-point cardinality and full cyclic groupsecp-rational-group | Closed | secp-parameters | none | Ecdlp/Proved/CurveCardinalityExact.leanEcdlp/Proved/CurveFullGroup.lean |
| Fixed-transcript generic-group collision coregeneric-security-core | Closed | secp-rational-group | none | Ecdlp/Proved/GenericGroupBound.leannotes/SECURITY_SCOPE.md |
| Abstract protocol algebra and concrete curve instantiationprotocol-soundness | Closed | secp-rational-group | none | Ecdlp/Proved/ProtocolInstantiation.lean |
| GLV automorphism on the full rational point groupglv-rational-scope | Closed | secp-rational-group | none | Ecdlp/Proved/GlvSubgroupEigenvalue.leannotes/SECURITY_SCOPE.md |
| Semaev S3/S4 formal foundationssemaev-foundations | Closed | secp-rational-group | none | Ecdlp/Proved/SemaevThree.leanEcdlp/Proved/SemaevFour.lean |
| Division-polynomial, EDS, degree, and coprimality substratedivision-polynomial-foundations | Closed | secp-parameters | none | Ecdlp/Proved/DivisionPolynomialCoprime.leanEcdlp/Proved/NormEDSConsecutiveZeros.lean |
| Exact closure E[n] structure for n in {2,3,4,5,7}, including the first composite case n=4small-torsion-family | Closed | division-polynomial-foundations | none | Ecdlp/Proved/TwoTorsionStructure.leanEcdlp/Proved/DoublingPointFormulaBar.lean |
| Coprime and distinct torsion locitorsion-loci | Closed | division-polynomial-foundations | none | Ecdlp/Proved/TwoTorsionCount.lean |
| Weil divisor, Miller function, and reachable evaluation layerweil-w1-w3 | Closed | small-torsion-family | none | Ecdlp/Proved/WeilMillerEval.leannotes/WEIL_LADDER.md |
| Uniform point-level multiplication coordinate formulan7-uniform | Blocked | division-polynomial-foundations, small-torsion-family | blocker-point-division-map, blocker-n7-certificates | Ecdlp/Targets/n7_uniform_carrier_induction.leantargets/n7_uniform_secp256k1_x.json |
| Uniform separability and geometric n-torsion countn10-uniform-separability | Blocked | division-polynomial-foundations | blocker-uniform-separability | notes/SEPARABILITY_ROUTES.md |
| General E[n] structure over the algebraic closuregeneral-n-torsion | Blocked | n7-uniform, n10-uniform-separability | blocker-point-division-map, blocker-uniform-separability | notes/DIVISION_POLY_TORSION_MAP.md |
| Weil reciprocity and the bilinear non-degenerate pairingweil-w4-w5 | Blocked | weil-w1-w3, general-n-torsion | blocker-weil-reciprocity | notes/WEIL_LADDER.mdBARRIERS.md |
| P-256 exact rational-point cardinalityp256-exact-cardinality | Outside release | secp-parameters | blocker-p256-cardinality | Ecdlp/Proved/P256Cardinality.lean |
| New cryptanalytic hypothesis testingexperimental-hypotheses | Parked | truth-layer | none | experiments/HYPOTHESES.yaml |
Missing foundations are recorded, but do not authorize work without a selected route.
| Blocker | What is missing | Resume condition |
|---|---|---|
| No uniform Point-to-division-polynomial multiplication mapupstream-foundation | Mathlib exposes the polynomial identities but not the theorem connecting [n]P to the phi/psi coordinate formulas for all n. | An upstream multiplication-coordinate theorem lands, or a reviewed in-repo induction closes the torsion bridge. |
| Four N7 algebra walls need large checked elimination certificatesproof-engineering | The statements are machine-elaborated and numerically/symbolically validated, but checked cofactors for the degree-heavy identities are not yet authored. | A reproducible certificate generator emits Lean-checkable cofactors for all four algebra walls. |
| Uniform separability of multiplication-by-n is not availablemissing-theory | Small n is closed, while the general prime-to-characteristic counting and differential argument remains absent. | A checked [n]*omega=n omega or equivalent general separability theorem is available. |
| Weil reciprocity and divisor-degree machinery are missingupstream-foundation | The reachable W1-W3 evaluation substrate is closed; W4 reciprocity is required before the pairing can be assembled. | Mathlib or this repository gains the needed divisor/tame-symbol reciprocity layer. |
| P-256 exact cardinality needs a general point-counting/Hasse routeout-of-scope-foundation | Only n divides the P-256 group cardinality is proved; the secp256k1 j=0 certificate does not transfer. | A reviewed Hasse/Schoof or curve-specific exact-cardinality proof becomes available. |
The public and agent-facing views resolve back to canonical machine sources and their gates.
307 ledger rows / ~268 distinct
1 structural route completed / 0 promoted / 1 exact toy run completed
8 typed cells / 2 seeds / 0 bounded experiments
307 theorem nodes in the knowledge graph
Reference deployment
What can legitimately reopen route selection
post-task026-decision-review
Research and product contracts have separate owners and KPIs; neither can count as progress in the other.
The repository needs a closed evidence -> generation -> adversarial proposal review -> selection -> bounded exploration -> validation -> retention loop without weakening the independent promotion gate. Creative model output remains untrusted; deterministic gates compile it into a candidate or retain its exact blockers.
New progress must enter through source-pinned mathematical evidence or a concrete mechanism, not by silently reviving a parked route. Generated research seeds reduce dependence on one agent noticing an intersection, while adversarial compilation prevents fluent text from becoming authorization.
The exact coordinatewise C3 action on S3 and S4 decides whether the naive invariant quotient has the symmetry it claims. This bounded structural uncertainty is now resolved; it was not route promotion or an experiment hypothesis run.
The independent audit found reproducibility, lifecycle, calibration, semantic-drift, and authorization defects. This task repairs those findings before Research Engine v0.2 is treated as stable.
A new prime-field Semaev/Petit proposal cannot be assessed honestly without claim-level source boundaries and a full-cost bridge. Yokoyama and Amadori are now bound to inspected primary artifacts and exact extracts; Kudo CANS 2018 remains full-text unread. The deterministic smooth-subgroup desk screen resolves the two arithmetic predicates but leaves solving degree, recovery, relation independence, and total work unknown, so it does not justify a solver or native experiment.
The mathematical idea space is open-ended, but the actionable frontier can be made finite relative to a versioned evidence snapshot, mechanism grammar, route model, target properties, recovery contract, and cost quantities. The existing knowledge graph already supplies that projection. Choosing the exact M16 desk-cost cell avoids both a free-form idea search and an unbounded auxiliary-curve search.
TASK-015 showed that low input degree can be obtained only by adding variables, while solving degree, fill-in, recovery, and rank remain unknown. Those quantities are not meaningful until direct S17, the recursive S3 tree, and the source-factor membership circuit are related by an exact base-field semantics with every exceptional fiber accounted for.
TASK-016 disproved the naive global affine substitution and isolated two independent exceptional mechanisms: identity prefixes and extension-only or non-liftable fibers. The next useful result is a complete set-theoretic classification of those fibers and a recovery domain, not a choice from an unbounded hypothesis space and not a solving-degree estimate.
TASK-017 proves the complete set-theoretic projective S3-tree semantics and recovery domain. TASK-018 was required to freeze the exact recursive projective S17 object before any attempt to certify its universal reverse projection or price a solver.
TASK-018 freezes the recursive projective S17 contract, records its forward algebraic argument, and replays bounded S4/S5 forward/reverse fixtures, but finite fixtures cannot replace a universal proof. TASK-019 isolates which part of that gap is generic resultant algebra and which part still depends on the actual frozen recursion.
TASK-019 proved the generic fixed-degree resultant theorem, but that theorem did not name or constrain the actual frozen recursive symbolic family. TASK-020 binds the real recursion before any universal projective witness extraction is attempted.
TASK-020 returns a non-irrelevant common projective root of the fixed homogenized predecessor and local slices at every vanishing successor. To recurse, those homogenized evaluations must be identified with the actual projective frozen-family specialization, including `[1:0]`, and the witnesses must be assembled into one exact chain.
TASK-021 proves an exact recursive chain of fourteen valid projective witnesses, but the existential `ProjectivePair` representation is not yet a finite scalar polynomial inventory. A guard `A*U + B*V - 1 = 0` expresses exactly that `(U,V)` is not `[0:0]` over a field while retaining `[1:0]`. The target representation is one literal finite `MvPolynomial` family: `∃ assignment : GuardVar → K, ∀ e : GuardedEquation, MvPolynomial.eval assignment (guardedEquation q y e) = 0`.
TASK-022 gives an exact finite polynomial system, but its four `(U,V,A,B)` scalars and one guard per projective slot retain both projective-scale and guard-witness redundancy. Every valid projective pair over a field lies in exactly the needed two-chart cover: the infinity branch uses `[1:0]`, while the affine branch uses `[X:1]`. Selecting one branch for each of fourteen intermediate slots leaves, for a fixed mask `I`, exactly `14 - |I|` scalar variables and fifteen literal `H` equations.
TASK-023 proves an exact cover over all `2^14 = 16384` affine/infinity masks but does not distinguish masks that cannot occur. The local triquadratic `H` identities at `[1:0]` expose exact necessary conditions. With affine external inputs, adjacent infinity slots would force an external projective `v` coordinate to vanish, so every solution mask is a separated subset of the fourteen-slot path. This reduces the exact logical cover to 987 masks without a solver or production enumeration.
TASK-024 conditionally leaves 377 interior separated masks and proves that every isolated infinity fixes its adjacent affine coordinates. TASK-025 substitutes those forced coordinates into the neighboring literal `H` equations, extracts explicit exceptional polynomial conditions, and proves that a precisely stated balanced nonvanishing condition makes the empty infinity mask complete. A one-chart conclusion means only one affine representation branch on that conditional locus, not one solution or an ECDLP shortcut.
TASK-025 proves that explicit endpoint and balanced nonvanishing assumptions reduce the exact projective cover to one affine chart, but it does not show that fixed-before-sampling targets admit enough such relations to make that chart usable. This task performs the cheapest precommitted test that can kill that continuation before any solver work. It does not test a secp256k1 key, a faithful PKC factor base, solver scaling, rank, recovery, or total ECDLP cost.
the source heuristic used all `D=564522` subgroup coordinates, while the exact curve factor base depends on whether `x^3+7` is a square. This task resolves that target property and records separate exact conditioned null comparators before any solver proposal can interpret a toy slope. These comparators do not establish the actual group-sum distribution and may leave additional zero-sum configurations uncounted. The task also binds the immutable proposal's own statement that its current at-most-24-bit ladder is calibration-only.
the typed cell previously cited a coarse source summary while the exact map chain, System (4), recovery boundary, and partial cost formula were distributed across prose and older certificates. This task binds those facts to exact primary-source locators and separates source statements from repository-derived completion semantics.
TASK-028 produced the current typed seed and exact source input, but it did not supply a generalized-root algorithm or complete cost bridge. TASK-029 tests whether the same scientific mechanism can now clear proposal intake without renaming an old premise or converting a design specification into evidence.
The secp256k1 repository demonstrates an owner-operated research-state loop. It does not establish that another team has the same pain, can use the contracts, or will return.
A hosted or multi-project platform is justified only after a real team exposes the minimum adapter boundary.