Open, kernel-checked mathematics · Lean 4
A public programme for the full fifteen-group classification of rational elliptic-curve torsion. The Lean Pool cardinality challenge remains a release endpoint; the roadmap also makes finiteness, group structure, and every canonical construction explicit.
Programme principleConstruct the real object before scaling proof volume: progress is earned only when a reviewed API is consumed downstream.
Execution revision · 2026-08-15
Retain Mazur's degree-one formal-immersion route while recording checked but uncredited foundation slices: canonical-linear coherent H0/H1 with pointed proper-curve H1 finiteness; the associated fppf sheafification of the all-degree normalized Picard presheaf together with an honest absolute Picard degree kernel and splitting; genuine secant, tangent, and D(B₁₂) product-neighbourhood addition charts with checked overlap and diagonal comparisons; generic finite free-translation quotient geometry over affine noetherian bases; terminal proper-model specialization at five; and represented polynomial-cusp sections. Proper coherent H0 finiteness, linear connecting maps, Riemann-Roch, proper base change, a pullback-compatible relative degree and relative Pic⁰, Picard representability, gluing and infinity coverage for the global Weierstrass law, actual canonical cyclic and Eisenstein quotient instances, torsion injectivity, represented X_0, and compactification remain open.
The rational torsion subgroup is isomorphic to one of the eleven cyclic groups or four bicyclic groups in Mazur's theorem.
theorem rationalTorsion_hasMazurClassification (E : WeierstrassCurve ℚ) [E.IsElliptic] : HasMazurClassification EThis immutable challenge is a published corollary, not the project definition of Mazur's theorem. The existing finite/infinite split can also discharge it directly from allowed point orders, while the canonical release additionally proves finiteness and group structure.
theorem torsion_ncard_le (E : WeierstrassCurve ℚ) [E.IsElliptic] : (AddCommGroup.torsion (E⁄ℚ).Point : Set (E⁄ℚ).Point).ncard ≤ 16MazurTorsion.PrimeOrder.rationalPoint_primeOrder_ne_of_formalImmersionAtFiveMazurTorsion.PrimeOrder.DegreeOneFormalImmersionWitnessModularCurve.EisensteinQuotient.toDegreeOneFormalImmersionWitnessThe generic formal-immersion collision theorem is the checked route-neutral argument boundary. The witness package and its private Eisenstein constructor remain proposed. Do not add winding-quotient, modular-symbol, analytic-rank, symmetric-power, or global cyclotomic machinery to the critical path.
cohomology-picard-jacobian
A cover-independent canonical K-linear H⁰/H¹ API, including H⁰/global-sections compatibility and coefficient-morphism and connecting-map linearity, is available to the proper-curve finiteness package.
A represented Jacobian and pointed Abel–Jacobi morphism compile with base-change consumers.
represented-modular-curve
Starting from the checked over-base additions on D(x₁ - x₂) and D(B₁₂), their equality on the exact intersection, and the diagonal tangent comparison, glue their union; then add the infinity charts and coverage, prove the commutative group laws, and refactor MazurTorsion.XZeroFortyNine.orderFortyNineConcreteGroupSchemeInterface to consume the result without supplied [GrpObj (toOver E)] or CanonicalPointGroupLawCompatibility E shadow inputs.
An exact-order-49 point reaches an honest represented X₀(49) point without a supplied point-equivalence shadow.
The weighted roadmap
The programme totals 1000 points. Stage weights estimate effort across the whole theorem—not file counts, issue counts, or perceived mathematical elegance.
Stage contract
These are the concrete APIs and results that count as this stage’s scope. Node cards below show how the work is divided and connected.
MazurTorsion.RationalTorsionThe rational torsion subgroup used by every downstream cardinality and point-order theorem.
MazurTorsion.cyclicOrders and remainingKubertForbiddenOrdersThe exact allowed cyclic orders and the four composite callbacks still exposed by the baseline.
MazurTorsion.rationalTorsion_orders_mem_cyclicOrders_of_remaining_obstructionsThe compiled point-order reduction with only prime orders at least 11 and orders 18, 25, 35, and 49 left open.
MazurTorsion.torsion_ncard_le_of_explicit_arithmeticThe exact ncard-at-most-16 target once every rational torsion point has an allowed order.
WeierstrassCurve.Affine.approx_parallelogram_law / MazurTorsion.rationalLogHeightNorthcottThe checked rational height-descent inputs that prove torsion finiteness directly through Mathlib's finite-torsion descent theorem, without Mordell–Weil finite generation.
Finite-group classification and cardinality assembly; direct rational-torsion finiteness; the generic finite-abelian rank-two presentation; full 2-, 3-, 4-, 5-, and 7-torsion obstructions; exceptional products; and unconditional exclusions of orders 14, 15, 16, 20, 21, 24, and 27.
MazurTorsion.RationalTorsionThe rational torsion subgroup on which the point-order and cardinality reductions are stated.
MazurTorsion.GroupTheory.ClassificationCardinalityMazurTorsion.cyclicOrdersThe finite set of cyclic point orders allowed by the target classification.
MazurTorsion.GroupTheory.ClassificationCardinalityMazurTorsion.remainingKubertForbiddenOrdersThe exact residual composite-order callbacks exposed by the imported baseline.
MazurTorsion.Arithmetic.PointOrderMazurTorsion.rationalTorsion_orders_mem_cyclicOrders_of_remaining_obstructionsThe checked point-order reduction from the remaining prime and composite exclusions.
MazurTorsion.Arithmetic.PointOrderMazurTorsion.torsion_ncard_le_of_explicit_arithmeticThe checked cardinality bound once the explicit point-order hypotheses are supplied.
MazurTorsion.Arithmetic.CardinalityReductionWeierstrassCurve.Affine.approx_parallelogram_lawThe checked naïve-height estimate consumed by Mathlib's direct finite-torsion descent theorem.
MazurTorsion.Foundations.NaiveHeightDescentMazurTorsion.rationalLogHeightNorthcottNorthcott's property for rational logarithmic height, completing the direct torsion-finiteness input without finite generation.
MazurTorsion.NumberTheory.RatNorthcottMazurTorsion.rationalTorsion_finiteApply rational Northcott and the approximate parallelogram law to Mathlib's finite-torsion descent theorem, without full Mordell–Weil finite generation.
MazurTorsion.Arithmetic.RankTwoReductionMazurTorsion.exists_rankTwoPresentation_of_allowed_orders_and_forbiddenPut any finite abelian group satisfying the allowed-order hypothesis and exactly c2Cube, c3Square, c5Square, and c7Square into rank-two invariant-factor form; c4Square, c2c10, and c2c12 remain classification-only.
MazurTorsion.GroupTheory.FiniteClassificationStage contract
These are the concrete APIs and results that count as this stage’s scope. Node cards below show how the work is divided and connected.
MazurTorsion.XOneEleven.FiveCosetBoundThe compiled acceptance predicate for the explicit five-isogeny descent on X_1(11).
MazurTorsion.XOneEleven.fiveCosetBoundEvery rational point on the order-11 model belongs to one of the five visible classes modulo 5E(Q).
MazurTorsion.XOneThirteenDescent.no_noncuspidal_pointThe explicit X_1(13) genus-two model has no rational affine point away from its cusp abscissas.
MazurTorsion.XOneEighteenDescent.no_noncuspidal_pointThe explicit order-18 genus-two model has no noncuspidal rational affine point.
MazurTorsion.Kubert.rationalPoint_addOrderOf_ne_twentyFive / ne_thirtyFiveKernel-checked exclusions of exact rational point orders 25 and 35.
MazurTorsion.XZeroFortyNine.rationalPoint_addOrderOf_ne_fortyNineThe order-49 point-to-X_0(49) bridge and final exclusion.
finiteEndpointPointOrderOne theorem discharging order 13 and the four composite callbacks; order 11 is supplied by the prime-level formal-immersion theorem.
The reverse X_1(11) model-to-Tate bridge, its discriminant certificate, the conditional cusp classification, and the conditional five-coset proof with Q=0 are checked. The node remains open until MT-X11-JOIN supplies the unconditional order-11 exclusion and the immutable Challenge bridge compiles. The direct five-isogeny Selmer calculation remains a partial fallback.
MazurTorsion.Kubert.orderElevenModelOfRaw_inverseThe denominator-safe reverse rational functions are a checked inverse to the forward X_1(11) model map on the noncusp locus.
MazurTorsion.Kubert.OrderElevenModelInverseMazurTorsion.Kubert.exists_elliptic_tate_marked_order_eleven_of_modelA noncusp model point reconstructs an actual elliptic Tate curve with a rational point of exact order eleven; nonzero discriminant follows from the checked degree-five resultant certificate.
MazurTorsion.Kubert.OrderElevenModelInverseMazurTorsion.Kubert.model_abscissa_eq_zero_or_one_of_no_order_elevenReal consumer reducing rational X_1(11) points to the two cusp abscissae from any uniform exact-order-eleven exclusion.
MazurTorsion.Kubert.OrderElevenModelInverseMazurTorsion.XOneEleven.fiveCosetBound_of_no_order_elevenThe preferred route consumer enumerates the four affine cusp points plus infinity and proves FiveCosetBound with quotient point Q=0.
MazurTorsion.NumberTheory.XOneElevenUniformCosetMazurTorsion.XOneEleven.veluFiveMap_eq_zero_iff_five_nsmulFallback five-isogeny infrastructure: the candidate Velu point function has zero fibre exactly the points killed by five; no additivity or packaged isogeny is claimed.
MazurTorsion.NumberTheory.XOneElevenFiveIsogenyHomMazurTorsion.XOneEleven.exists_fifthPower_of_emptyFiveSelmerFallback arithmetic consumer: a rational unit with all finite valuation residues trivial modulo five is a fifth power; the local Kummer comparison and ramified factor remain open.
MazurTorsion.NumberTheory.XOneElevenFiveSelmerAdapt the formal-immersion order-11 theorem to the existing PointOrder callback; the explicit X_1(11) descent is no longer a logical prerequisite.
MazurTorsion.XOneEleven.rationalPoint_addOrderOf_ne_elevenExpose the uniform order-eleven result under the finite-endpoint namespace expected by PointOrder.
Show that the explicit genus-two order-13 model has no rational affine point with x different from 0 and -1. The checked primitive split descent clears all three cusp factors, derives the two remaining cubic coefficient identities, proves both conic parameters odd, and forces k to be -4 or 4. Its global rational-point elimination, and hence the genuine Mazur--Tate Jacobian/rank-zero conclusion, remain open.
MazurTorsion.XOneThirteenDescent.positive_split_rational_curve_pointDehomogenize every positive primitive split datum to an actual rational point on the order-thirteen sextic with its canonical positive integral ordinate.
MazurTorsion.NumberTheory.XOneThirteenPositivePellMazurTorsion.XOneThirteenDescent.homogeneous_pell_identityCheck the degree-38 homogeneous Pell identity for the explicit degree-19 and degree-16 certificates.
MazurTorsion.NumberTheory.XOneThirteenPositivePellMazurTorsion.XOneThirteenDescent.odd_prime_pell_factor_allocationProve that the two positive Pell factors have no common odd prime and allocate each odd prime divisor of b to exactly one factor.
MazurTorsion.NumberTheory.XOneThirteenPositivePellMazurTorsion.XOneThirteenDescent.PositivePellAllocatedFactorObstructionName the remaining global allocated-factor obstruction without claiming the unproved divisor-class elimination.
MazurTorsion.NumberTheory.XOneThirteenPositivePellMazurTorsion.XOneThirteenDescent.rationalPoint_addOrderOf_ne_thirteen_of_positivePellAllocatedFactorCarry the honest allocated-factor boundary through the existing descent to the exact-order-thirteen consumer.
MazurTorsion.NumberTheory.XOneThirteenPositivePellMazurTorsion.XOneThirteenDescent.positive_pell_half_factors_isCoprimeRemove the forced scalar two and prove the resulting positive Pell factors coprime, including at the prime two.
MazurTorsion.NumberTheory.XOneThirteenPellPowerSplitMazurTorsion.XOneThirteenDescent.positive_pell_factor_power_splitUse integer unique factorization to express the two halves as positive coprime thirty-eighth powers whose roots multiply to b.
MazurTorsion.NumberTheory.XOneThirteenPellPowerSplitMazurTorsion.XOneThirteenDescent.PositivePellPowerSplitObstructionName the fixed two-equation power-split cover left by the global Pell factorization.
MazurTorsion.NumberTheory.XOneThirteenPellPowerSplitMazurTorsion.XOneThirteenDescent.rationalPoint_addOrderOf_ne_thirteen_of_positivePellPowerSplitConsume the fixed-cover obstruction in the actual exact-order-thirteen exclusion path.
MazurTorsion.NumberTheory.XOneThirteenPellPowerSplitShow that the explicit genus-two order-18 model has no rational point with x different from 0 and 1. The checked primitive split descent derives the two remaining cubic coefficient identities, proves both conic parameters odd, and forces k to be -8, -4, 4, or 8. Classification of these two explicit split covers, equivalently a genus-two Jacobian rank-zero certificate or another complete descent, is still required; the Challenge remains open.
MazurTorsion.XOneEighteenDescent.splitEisensteinThreePrime_not_commonExclude simultaneous divisibility of the two split factors by the ramified prime above three using exact depth and the scalar-times-cube identity.
MazurTorsion.NumberTheory.XOneEighteenEisensteinIntegersMazurTorsion.XOneEighteenDescent.TwoPrimeSupportedEisensteinIntegerFiniteSplitCyclicCubicObstructionExpose the remaining finite Eisenstein obstruction after every common prime has been restricted to support above two.
MazurTorsion.NumberTheory.XOneEighteenEisensteinIntegersMazurTorsion.XOneEighteenDescent.rationalPoint_addOrderOf_ne_eighteen_of_twoPrimeSupportedEisensteinIntegerObstructionCarry the narrowed support-only-over-two obstruction through the checked descent to the exact-order-18 exclusion.
MazurTorsion.NumberTheory.XOneEighteenEisensteinIntegersMazurTorsion.XOneEighteenDescent.antiDiagonalExceptionalPolynomial_ne_zeroRemove the exceptional denominator of the anti-diagonal quotient by complete projective enumeration modulo five.
MazurTorsion.NumberTheory.XOneEighteenQuadraticNormBaseMazurTorsion.XOneEighteenDescent.antiDiagonalZ_sq_of_fourScalarCorrespondenceMap every surviving nondegenerate four-scalar cube correspondence to the explicit anti-diagonal genus-two curve.
MazurTorsion.NumberTheory.XOneEighteenQuadraticNormProve that no rational point on an elliptic curve over Q has exact additive order 25. The checked Tate recurrence computes (n+2)P from index n and proves every secant denominator through 13P nonzero from exact order 25. Explicit square/cube normalized coordinates through 13P reduce x(13P)=x(12P) to the cuspidal factor b-c times one fixed 234-term polynomial of total degree 40. The arbitrary-curve consumer retains b, c, and b-c nonzero together with the twelfth-power discriminant scale. Classifying rational solutions of that explicit noncuspidal factor, directly or after a checked birational reduction, remains open.
MazurTorsion.Kubert.nsmul_origin_eq_successiveCoordinatesCompute (n+2)P by a reusable Tate recurrence under exactly the preceding nonzero abscissa hypotheses.
MazurTorsion.Kubert.TateNormalFormMultiplesMazurTorsion.Kubert.tateSuccessiveX_ne_zero_of_marked_order_twentyFiveDeduce every secant denominator needed through 13P from exact order 25.
MazurTorsion.Kubert.OrderTwentyFiveMazurTorsion.Kubert.tateClearedCoordinates_specRepresent the rational Tate recurrence by a division-free numerator-denominator recurrence with proved nonzero denominators.
MazurTorsion.Kubert.TateNormalFormMultiplesMazurTorsion.Kubert.orderTwentyFiveRecurrenceEquation_eq_zero_of_marked_orderDerive the explicit rational-function collision x(13P)=x(12P) on Tate normal form.
MazurTorsion.Kubert.OrderTwentyFiveMazurTorsion.Kubert.orderTwentyFiveClearedEquation_eq_zero_of_marked_orderCross-multiply the 12P/13P collision to the fixed fraction-free X_1(25) recurrence expression.
MazurTorsion.Kubert.OrderTwentyFiveMazurTorsion.Kubert.exists_tateOrderTwentyFive_recurrence_certificateNormalize an arbitrary exact-order-25 rational point to the recurrence locus while retaining all denominators and discriminant scale.
MazurTorsion.Kubert.OrderTwentyFiveMazurTorsion.Kubert.orderTwentyFive_normalized_collision_factorizationFactor the fully normalized 12P/13P collision exactly as minus the cusp factor b-c times the explicit degree-40 noncuspidal polynomial.
MazurTorsion.Kubert.OrderTwentyFiveNormalizedModelMazurTorsion.Kubert.orderTwentyFiveNoncuspidalFactor_eq_zero_of_marked_orderUse exact marked order 25 and every checked denominator to reach the explicit noncuspidal factor with c and b-c nonzero.
MazurTorsion.Kubert.OrderTwentyFiveNormalizedModelMazurTorsion.Kubert.exists_tateOrderTwentyFive_noncuspidal_certificateSend an arbitrary exact-order-25 rational point to the fixed degree-40 model while retaining b, c, b-c, and the Tate-normalization discriminant scale.
MazurTorsion.Kubert.OrderTwentyFiveNormalizedModelUse the explicit optimal elliptic quotient X_0(35)/w_5 and Mazur's squarefree-level formal-immersion criterion at auxiliary prime 11. The source three-descent and Eisenstein ideal-support calculations prove the fixed explicit curve model has rank zero and finite rational point group. The represented modular classifying map and chart identification, the characteristic-eleven cusp local-ring generator and unit calculation, the abelian-variety identification, and Atkin--Lehner specialization inputs remain open.
MazurTorsion.OrderThirtyFive.optimalQuotientConstruct the explicit optimal quotient X_0(35)/w_5 with model y^2+y=x^3+x^2+9x+1.
MazurTorsion.OrderThirtyFive.optimalQuotient_mordellWeil_finiteProve the quotient has rank zero and rational torsion Z/3 by a checked descent.
MazurTorsion.OrderThirtyFive.formalImmersionAtInfinity_modElevenInstantiate the shared optimal-quotient formal immersion in characteristic eleven.
MazurTorsion.OrderThirtyFive.card_reductionAtEleven_le_eighteenNormalize to short form and verify the 121 coefficient pairs over F_11.
MazurTorsion.Kubert.OrderThirtyFiveFiniteFieldMazurTorsion.OrderThirtyFive.shortCurveEleven_addOrderOf_le_eighteenTurn the enumerated short-model cardinality bound into a point-order bound.
MazurTorsion.Kubert.OrderThirtyFiveFiniteFieldOrderMazurTorsion.OrderThirtyFive.shortCurveEleven_addOrderOf_ne_of_nineteen_leExclude every exact point order at least nineteen on an elliptic short model over F_11.
MazurTorsion.Kubert.OrderThirtyFiveFiniteFieldOrderMazurTorsion.OrderThirtyFive.zmod_eleven_addOrderOf_le_eighteenConsume short-Weierstrass normalization to bound point order on every elliptic curve over F_11.
MazurTorsion.Kubert.OrderThirtyFiveFiniteFieldOrderMazurTorsion.OrderThirtyFive.zmod_eleven_addOrderOf_ne_of_nineteen_leUniformly exclude every exact order at least nineteen after arbitrary-model normalization.
MazurTorsion.Kubert.OrderThirtyFiveFiniteFieldOrderMazurTorsion.Kubert.rationalPoint_addOrderOf_ne_thirtyFiveFeed the local-at-eleven collision and finite-field bound to the published endpoint.
Use the shared cyclic-subgroup moduli bridge: exact order 49 now constructs a genuine represented split finite-flat source whose recovered rational datum is the original cyclic-subgroup datum. The remaining map from that datum to a noncuspidal X_0(49)(Q) point will contradict the already checked two-cusp classification. No additivity proof for the explicit Velu point function is needed.
MazurTorsion.XZeroFortyNine.rationalPoint_addOrderOf_ne_fortyNine_of_variableChangeClassifyingMapConsume a noncuspidal classifying map from presentation-independent rational cyclic-subgroup data to the checked two-cusp X_0(49) model.
MazurTorsion.NumberTheory.XZeroFortyNineTransferMazurTorsion.XZeroFortyNine.rationalDatumOfSplitFiniteFlatSourceOfOrderFortyNineTorsionConstruct the represented split finite-flat source from exact order 49 and prove that forgetting it recovers the original raw rational Gamma_0 datum.
MazurTorsion.NumberTheory.XZeroFortyNineTransferRemove the level-13 and exact-order 18, 25, 35, and 49 callbacks; order 11 is already supplied by the uniform formal-immersion theorem.
MazurTorsion.rationalTorsion_orders_mem_cyclicOrders_of_finite_endpointsCombine order 11 from the prime route with level 13 and the four composite exclusions.
Stage contract
These are the concrete APIs and results that count as this stage’s scope. Node cards below show how the work is divided and connected.
NeronModel and torsionSpecializationKernelNeron models only through the component, rank-zero, and unramified prime-five torsion-specialization consumers used by the chosen proof.
AdmissibleFiniteFlatGroup and eisensteinRankZeroCriterionThe finite-flat filtration, fppf cohomology, and Raynaud uniqueness package actually consumed by finiteness of the Eisenstein quotient.
XZeroModuli, IntegralXZero, and cuspFormalParameterThe Gamma_0 moduli point, compactified integral curve, cusp sections, and q-parameter needed over Q, Z_(5), and Z_(11).
ModularJacobian, HeckeOperator, and OptimalQuotientJ_0(N), its Hecke-stable optimal quotients, and the cotangent/q-expansion interface proving formal immersion at infinity.
optimalQuotient_formalImmersionAtInfinity_of_residueChar_ne_twoMazur's degree-one formal-immersion theorem for every nontrivial optimal quotient away from characteristic two, consumed at 5 and 11.
EisensteinQuotientA nontrivial optimal quotient for N=11 or prime N at least 17 whose rational Mordell-Weil group is finite; no exact cusp-order API is required.
CyclotomicCharacter.no_everywhereUnramified_inverseThe inherited class-field Challenge remains a release obligation, while the formal-immersion theorem no longer depends on it.
MazurTorsion.PrimeOrder.DegreeOneFormalImmersionWitnessA proposed package for constructing the inputs of the already checked route-neutral prime-order collision from private Eisenstein data.
The canonical supplied Neron-model interface, mapping property, and section-extension equivalences compile. Separately, properness of an actual commutative group model over a valuation ring gives a terminal-point equivalence between its integral points and the points of an identified generic fibre. Over an affine noetherian base, the generic finite-translation quotient layer proves properness from a proper source and proves smoothness separately from a flat, locally-finite-type, geometrically reduced source; both results retain the supplied affine diagonal, section action, invariant affine atlas, and freeness proof. It neither identifies the arithmetic Eisenstein quotient or its generic fibre nor proves the Neron mapping property. Construct the abelian-variety Neron substrate at the required arithmetic DVRs, then realize the actual optimal Eisenstein quotient and its cusp-normalized map in that model for the rank-zero and collision consumers. The marked source point's exact-order reduction at 5 is already checked independently and does not require a source elliptic Neron model.
AlgebraicGeometry.NeronModelPackage a smooth separated model with generic-fibre recovery and the Neron mapping property.
MazurTorsion.AlgebraicGeometry.NeronModel.BasicAlgebraicGeometry.NeronModel.sectionExtensionExtend the rational sections used by the rank-zero and prime-five consumers.
MazurTorsion.AlgebraicGeometry.NeronModel.BasicAlgebraicGeometry.ProperModelBasePoint.mulEquivUse the valuative criterion for an actual proper commutative group model over a valuation ring to identify its terminal integral points with the points of an identified generic fibre; this does not assert a Neron mapping property for arbitrary smooth test schemes.
MazurTorsion.AlgebraicGeometry.NeronModel.ProperModelBasePointDefine genuine Neron identity components and component groups and prove completely toric reduction at the modular level for the Eisenstein rank-zero criterion. The canonical nonsingular-reduction subgroup, its additive map, exact formal kernel, finite cuspidal classification, unit-change transport, and the full marked tangent--secant/weighted-depth calculation already compile at five and eleven. That pointwise calculation now forces a4 in m⁴ and a6 in m⁶ and contradicts minimality, so the local prime and order-35 arithmetic endpoints no longer depend on component geometry. The open node is the represented identity-component/component-quotient and toric modular input consumed by rank zero; no Kodaira or full component-cardinality result is inferred from the checked chart algebra.
AlgebraicGeometry.NeronModel.identityComponentDefine the fibrewise identity component used by quotient specialization and the toric rank-zero argument.
AlgebraicGeometry.NeronModel.componentGroupDefine the component quotient and its specialization map.
MazurTorsion.EllipticCurve.TameAdditiveReductionDataPackage the canonical component quotient, identity-component reduction map, and exact formal-kernel comparison, with a compiled conversion to the algebraic torsion filtration.
MazurTorsion.EllipticCurve.TameAdditiveReductionDataMazurTorsion.EllipticCurve.TameAdditiveReductionDataAtFiveFix the reduction target to the actual five-adic residue field and derive component finiteness and formal-kernel torsion from checked exact-pin theorems.
MazurTorsion.EllipticCurve.TameAdditiveReductionDataMazurTorsion.EllipticCurve.TameAdditiveReductionDataAtElevenProvide the analogous canonical eleven-adic handoff consumed by the order-35 route.
MazurTorsion.EllipticCurve.TameAdditiveReductionDataWeierstrassCurve.Affine.HasNonsingularReductionDefine the canonical domain: formal-kernel points reduce to infinity, while other local points have integral coordinates reducing to the nonsingular locus.
MazurTorsion.EllipticCurve.NonsingularReductionWeierstrassCurve.Affine.nonsingularReductionConstruct actual coordinatewise reduction from the canonical domain to nonsingular points of the reduced Weierstrass cubic.
MazurTorsion.EllipticCurve.NonsingularReductionModularCurve.Jacobian.completelyToricReductionAtLevelSupply the toric special-fibre hypothesis for the Eisenstein rank-zero criterion.
Prove the specialization exact sequence, prime-to-residue torsion injection, and the unramified e < p-1 kernel lemma; exercise them on the Eisenstein quotient section at the auxiliary primes 5 and 11. For a supplied proper group model, the checked terminal specialization map extends a generic point by properness and restricts it to an arbitrary test scheme, while the checked collision reducer turns a supplied torsion-injectivity hypothesis into equality of integral model points. The reducer proves no torsion-injectivity or kernel theorem itself. A checked NeronModel consumer also transports rank zero from integral model points to the prescribed generic-fibre rational points using the actual scheme-theoretic multiplication kernel, but none of these consumers constructs the model. Exact-order reduction of the marked source point at good reduction is already integrated independently.
AlgebraicGeometry.NeronModel.torsionSpecialization_exactGive the torsion specialization sequence through the identity component and formal kernel.
AlgebraicGeometry.NeronModel.primeToResidue_torsion_injectiveInject torsion of order prime to the residue characteristic.
AlgebraicGeometry.NeronModel.torsion_eq_zero_of_specializes_zero_of_ramification_ltKill a torsion section in the formal kernel when e < p - 1, including the unramified prime-five and prime-eleven cases.
AlgebraicGeometry.NeronModel.finrank_genericBasePoint_eq_zero_of_powerKummer_kernelDataTransport the actual-kernel power-Kummer rank-zero theorem from integral model points to rational points of the supplied generic fibre through the checked Neron mapping-property equivalence.
MazurTorsion.AlgebraicGeometry.NeronModel.PowerKernelKummerRankZeroAlgebraicGeometry.ProperModelBasePoint.basePointSpecializationSpecialize a generic point by its unique extension to an actual proper commutative group model and restriction along an arbitrary test scheme over the valuation-ring base.
MazurTorsion.AlgebraicGeometry.NeronModel.SpecializationAlgebraicGeometry.ProperModelBasePoint.basePoint_eq_of_restrict_eq_of_generic_torsionTurn equality after restriction into equality of two integral model points when their generic-fibre difference is torsion and a supplied torsion-specialization injectivity predicate kills that difference; the predicate remains an input.
MazurTorsion.AlgebraicGeometry.NeronModel.SpecializationThe checked substrate packages finite-flat commutative group schemes, certified scheme-theoretic kernels, affine Hopf realizations, constant and diagonalizable examples, mu_n multiplication kernels, constant-group quotients, and an exact supplied fppf quotient presentation. Certified kernels, quotient presentations, and the named constant/multiplicative factors commute with arbitrary base change, with a compiled admissible-filtration p^2-exponent consumer.
AlgebraicGeometry.FiniteFlatCommGroupSchemeThe checked finite-flat commutative group-scheme category over an arbitrary scheme base.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.BasicAlgebraicGeometry.FiniteFlatCommGroupScheme.kernelPresentation_exists_of_finite_flatPackage inherited scheme-theoretic kernels under explicit finite-flat hypotheses.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.BasicAlgebraicGeometry.AffineFiniteFlatCommGroupScheme.point_pow_eq_one_of_constantRankThe checked Deligne-style point-exponent theorem is a real consumer of the affine finite-flat substrate.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.AffineAlgebraicGeometry.FiniteFlatCommGroupScheme.kernelConstruct the inherited scheme-theoretic kernel under the exact finite and flat hypotheses required over an arbitrary base.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.BasicAlgebraicGeometry.FiniteFlatCommGroupScheme.FppfQuotientPresentationPackage exactly the supplied finite-flat quotient, fppf projection, and certified scheme-theoretic kernel used by admissible filtrations; no unconsumed general quotient representability theorem is asserted.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.QuotientAlgebraicGeometry.FiniteFlatCommGroupScheme.KernelPresentation.baseChangeConstruct the certified scheme-theoretic kernel of a pulled-back homomorphism and prove that both its scheme and inclusion are the geometric pullbacks of the original kernel data.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.BasicAlgebraicGeometry.FiniteFlatCommGroupScheme.FppfQuotientPresentation.baseChangePresentationPull the exact quotient presentation back along an arbitrary base morphism, using geometric stability of fppf morphisms and certified kernel base change.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.QuotientAlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleSimpleFactor.baseChangeTransport the named constant Z/p and mu_p factor presentations across scalar extension.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleFiltrationAlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleFiltrationStep.baseChange_point_pow_sq_eq_oneCompile the downstream rank-zero-oriented consumer: every affine point in a base-changed exact two-factor filtration step is killed by p^2.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleFiltrationThe exact iterated constant-or-multiplicative filtration, arbitrary-base-change exponent consumer, unit Kummer quotient, and finite-p-group low-degree Euler estimate compile. Global relative fppf H1 is a genuine common-refinement quotient with its canonical commutative group law and functorial coefficient homomorphisms. Refining arbitrary multi-object fppf covers to singleton affine presentations and comparing gauges remains open.
AlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleFiniteFlatGroupPackage an honest recursive exact filtration whose graded kernels are the checked constant Z/p or multiplicative mu_p factors.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleCohomologyAlgebraicGeometry.Scheme.FppfHOneGlobalize relative cover-level nonabelian H1 over genuine fppf covers by the common-refinement quotient in Scheme.Over X.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.FppfHOneAlgebraicGeometry.CommGroupScheme.pointPresheaf_isFppfSheafApply subcanonicity to the represented point presheaf of an arbitrary ambient commutative group scheme, without a finiteness hypothesis.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.CommGroupSchemeFppfHOneAlgebraicGeometry.CommGroupScheme.FppfHOneInstantiate the checked common-refinement fppf H1 construction for an arbitrary represented commutative group-scheme coefficient.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.CommGroupSchemeFppfHOneAlgebraicGeometry.CommGroupScheme.fppfHOneMapApply an arbitrary ambient commutative group-scheme morphism to global fppf H1 by the represented point-presheaf natural transformation, with checked identity and composition laws.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.FppfHOneFunctorialityAlgebraicGeometry.AdmissibleFiniteFlatGroup.hOne_sub_hZero_leProve Mazur's filtration estimate by reduction to the elementary graded pieces.
Prove the unramified order-p uniqueness theorem needed to extend admissible Galois constituents, then assemble the bounded-Kummer-cohomology criterion that forces the Eisenstein quotient to have Mordell-Weil rank zero. The numerical endpoint now compiles: an injective Kummer map from A/pA into a finite group whose cardinality is bounded by the cardinality of A[p] forces a finitely generated abelian group A to have rank zero and hence be finite. The Eisenstein quotient's actual kernel H0/H1 certificates, multiplication flatness and surjectivity, Mordell-Weil finite generation, Neron-model construction, and unramified Raynaud input remain open.
AlgebraicGeometry.Raynaud.primeOrder_uniqueness_unramifiedExtend constant and multiplicative generic fibres uniquely over an unramified DVR.
AbelianVariety.rank_eq_zero_of_admissible_torsionDeduce rank zero from good reduction away from the level, toric level reduction, and admissible p-torsion.
AlgebraicGeometry.FiniteFlatCommGroupScheme.finrank_eq_zero_of_injective_kummer_of_card_le_torsionUse the checked finitely generated index formula to force rank zero from an injective Kummer quotient and a cohomological cardinal bound by the p-torsion.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.FppfKummerRankZeroAlgebraicGeometry.FiniteFlatCommGroupScheme.finite_of_injective_kummer_of_card_le_torsionUpgrade the numerical Kummer rank-zero conclusion to finiteness for the finitely generated abelian group.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.FppfKummerRankZeroAlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleFiltrationStep.finrank_eq_zero_of_fppfKummer_intConsume the actual represented finite-flat H1 and its checked two-factor admissible-filtration bound in the Kummer rank-zero criterion over Spec Z.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.FppfKummerRankZeroAlgebraicGeometry.FiniteFlatCommGroupScheme.AdmissibleFiltrationStep.finite_of_fppfKummer_intDerive actual finiteness of the Mordell-Weil input from the same represented finite-flat H1 consumer.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.FppfKummerRankZeroAlgebraicGeometry.FiniteFlatCommGroupScheme.finrank_additive_basePoint_eq_zero_of_powerKummer_kernelDataDerive the p-torsion cardinality internally from certified base points of the actual scheme-theoretic multiplication kernel, then consume it in the represented power-Kummer rank-zero theorem.
MazurTorsion.AlgebraicGeometry.FiniteFlatCommGroupScheme.CommGroupSchemePowerKummerTorsionRankZeroConstruct the split rational cyclic subgroup generated by a torsion point, its intrinsic divisor subgroups and degeneracy maps, and quotient raw Weierstrass data by checked admissible variable changes. For coprime divisor levels whose product is N, a checked Bezout argument proves that the two intrinsic divisor subgroups generate the original carrier; the order-five/order-seven level-35 reconstruction is its concrete consumer. The represented polynomial cusp chart and its local sections do not construct represented X_0, a generalized-elliptic compactification, or the missing Gamma_0 classifier, so this node remains blocked.
ModularCurve.XZeroModuliDefine elliptic curves with a finite locally free cyclic subgroup of level N.
ModularCurve.XZeroModuli.pointOfRationalCyclicSubgroupConstruct the X_0(N)(Q) point consumed by the prime argument.
MazurTorsion.ModularCurve.XZeroModuli.RationalCyclicSubgroupPackage a finite rational cyclic subgroup of exact level N and construct it from a rational point of exact additive order N.
MazurTorsion.ModularCurve.XZeroModuliMazurTorsion.ModularCurve.XZeroModuli.RationalDatum.variableChangeTransport a raw elliptic curve and cyclic-subgroup datum through a checked admissible Weierstrass variable change.
MazurTorsion.ModularCurve.XZeroModuliMazurTorsion.ModularCurve.XZeroModuli.RationalDatum.VariableChangeClassQuotient raw rational Gamma_0 data by the equivalence relation generated by admissible changes of Weierstrass presentation.
MazurTorsion.ModularCurve.XZeroModuliMazurTorsion.ModularCurve.XZeroModuli.RationalDatum.VariableChangeClass.liftDescend every presentation-invariant function on raw cyclic-subgroup data to the checked variable-change quotient.
MazurTorsion.ModularCurve.XZeroModuliMazurTorsion.ModularCurve.XZeroModuli.RationalCyclicSubgroup.divisorSubgroupConstruct the intrinsic order-d subgroup C[d] of a split rational cyclic subgroup of order N for d dividing N, with transport, nesting, and generator formulas.
MazurTorsion.ModularCurve.XZeroModuliCompactify X_0(N), identify the smooth cusp neighbourhood, and expose its completed local ring and q-parameter at auxiliary primes 5 and 11. On the represented one-variable polynomial chart, sectionAt constructs genuine base sections, sectionAt_closedPoint_eq_zeroSection proves their local closed-fibre collision, and valuation_j_le_one_of_polynomialCuspSectionAtFive consumes that collision in the formal-immersion valuation argument. These are completed local-chart facts only: identifying an actual integral X_0(N) infinity neighbourhood with the chart, constructing the generalized-elliptic compactification, and proving the represented quotient collision and genuine optimal-quotient q-expansion identity remain open.
ModularCurve.IntegralXZeroConstruct the compactified integral model with generic fibre X_0(N).
AlgebraicGeometry.IsFormalImmersionAtDefine formal immersion by surjectivity on the functorial completed-stalk map; on locally Noetherian schemes, the checked residue-field and cotangent criterion implies this predicate.
MazurTorsion.AlgebraicGeometry.FormalCompletionIsLocalRing.QuotientCotangentCertificatePackage compatible source and target quotient ideals, target quotient maximal-ideal finiteness, the containment needed to lift quotient equality, and surjectivity of the induced quotient cotangent map.
MazurTorsion.AlgebraicGeometry.FormalImmersionSpecialFiberIsLocalRing.cotangentMap_surjective_of_quotientCotangentCertificateLift quotient cotangent surjectivity and a surjective residue-field map to surjectivity of the total local cotangent map.
MazurTorsion.AlgebraicGeometry.FormalImmersionSpecialFiberAlgebraicGeometry.Scheme.Hom.isFormalImmersionAt_of_quotientCotangentCertificateConsume a quotient cotangent certificate on the actual stalk map and a residue-field isomorphism to prove completed-stalk formal immersion.
MazurTorsion.AlgebraicGeometry.FormalImmersionSpecialFiberAlgebraicGeometry.Scheme.Hom.isFormalImmersionAt_of_mappedIdealCotangentSurjectiveSpecialize the lift to quotienting the target stalk by one ideal and the source stalk by its extension, giving the characteristic-five special-fibre consumer.
MazurTorsion.AlgebraicGeometry.FormalImmersionSpecialFiberMazurTorsion.ModularCurve.AffineCuspPolynomialChart.sectionAtConstruct the genuine affine structural section obtained by evaluating the represented polynomial cusp coordinate at a chosen base-ring element; this is a local chart section, not a represented X_0 point.
MazurTorsion.ModularCurve.AffineCuspPolynomialChartMazurTorsion.ModularCurve.AffineCuspPolynomialChart.sectionAt_closedPoint_eq_zeroSectionProve that a polynomial-chart section whose coordinate lies in the maximal ideal collides with the zero section at the local base's closed point.
MazurTorsion.ModularCurve.AffineCuspPolynomialChartMazurTorsion.ModularCurve.AffineCuspPolynomialChart.valuation_j_le_one_of_polynomialCuspSectionAtFiveConsume the constructed chart sections and their closed-fibre collision in the formal-immersion argument at five, while retaining the specialization and equal-quotient-image hypotheses.
MazurTorsion.ModularCurve.AffineCuspPolynomialChartModularCurve.IntegralXZero.completedLocalRingAtInfinity_of_auxiliaryPrimeIdentify the odd prime-to-level cusp completion with the q-power-series ring.
Construct the rational cusp sections, transport formal immersion between them by Atkin-Lehner, and prove that potentially multiplicative reduction at a prime-to-level auxiliary prime sends the classifying point to a cusp.
ModularCurve.XZero.infinityCuspConstruct the rational cusp used to normalize Abel-Jacobi.
ModularCurve.XZero.atkinLehnerTransport either prime-level cusp to infinity.
ModularCurve.XZero.specializesToCusp_iff_potentiallyMultiplicativeRelate cusp specialization at an auxiliary prime to potentially multiplicative elliptic reduction.
Construct J_0(N) and the Abel-Jacobi morphism x |-> [x]-[infinity], with the base-change and Neron-model interfaces used by the quotient and local proof.
ModularCurve.ModularJacobianSpecialize the generic Jacobian API to X_0(N).
ModularCurve.XZero.abelJacobiAtInfinityMap x to the divisor class [x]-[infinity].
Construct the Hecke action on J_0(N) and prove its q-expansion recursion. The abstract degree-one argument now proves that every nonzero simultaneous eigen-expansion has nonzero first coefficient and consumes that fact in an actual completed-stalk formal-immersion theorem. A represented rational section is a second compiled consumer: its section law automatically discharges the source non-genericity premise. The modular/Jacobian Hecke operators and their genuine q-expansion identity at characteristics 5 and 11 remain open.
ModularCurve.HeckeOperatorConstruct prime-to-level Hecke correspondences and the level operator on J_0(N).
ModularCurve.HeckeOperator.qExpansion_firstCoefficient_ne_zeroUse the Hecke recursions and q-expansion principle to detect a nonzero cotangent vector at infinity.
MazurTorsion.ModularCurve.DegreeOneCotangentCertificate.isFormalImmersionAt_of_smoothRelativeCurve_rationalPoint_of_normalizedQExpansionConclude actual completed-stalk formal immersion from a target local parameter whose pullback has normalized expansion c q plus q squared times a series with c nonzero.
MazurTorsion.ModularCurve.QExpansionFirstCoefficientMazurTorsion.ModularCurve.DegreeOneCotangentCertificate.specMap_fromStalk_eq_of_normalizedQExpansionUse a normalized first q-coefficient to cancel canonical local-spectrum points through the resulting formal immersion.
MazurTorsion.ModularCurve.QExpansionFirstCoefficientMazurTorsion.ModularCurve.DegreeOneCotangentCertificate.specMap_fromStalk_eq_of_completeDVR_normalizedQExpansionConstruct the complete-DVR coordinate, prove the normalized-q formal immersion, and cancel its actual canonical local-spectrum points in one checked consumer.
MazurTorsion.ModularCurve.CompleteDVRCoordinateMazurTorsion.ModularCurve.HeckeFirstCoefficient.coeff_one_ne_zero_of_simultaneousEigenvectorUse the first-coefficient Hecke recursion to prove that a nonzero simultaneous eigen-expansion cannot have zero q coefficient in degree one.
MazurTorsion.ModularCurve.HeckeFirstCoefficientMazurTorsion.ModularCurve.DegreeOneCotangentCertificate.isFormalImmersionAt_of_heckeEigen_qExpansionFeed the detected first q coefficient to the real completed-stalk formal-immersion predicate as a named downstream consumer.
MazurTorsion.ModularCurve.HeckeFirstCoefficientMazurTorsion.ModularCurve.DegreeOneCotangentCertificate.isFormalImmersionAt_of_rationalSection_heckeEigen_qExpansionApply the Hecke first-coefficient criterion to an actual rational section, using its derived non-genericity instead of a caller hypothesis.
MazurTorsion.ModularCurve.HeckeFirstCoefficientDefine Hecke-stable optimal quotients of the new modular Jacobian and prove Mazur's Proposition 3.1 away from characteristic two, with consumers at 5 and 11.
ModularCurve.OptimalNewQuotientPackage a connected-kernel quotient of the new part of J_0(N) with its induced Hecke action.
ModularCurve.OptimalNewQuotient.formalImmersionAtInfinity_of_residueChar_ne_twoProve the cusp Abel-Jacobi projection is a formal immersion in every residue characteristic other than two whenever the quotient is nontrivial.
Construct an optimal Eisenstein quotient for N=11 or prime N at least 17, prove that it is nontrivial and that its rational Mordell-Weil group is finite, and feed it directly to the characteristic-five formal-immersion theorem. Exact cusp-order and equality of the whole rational group with the cusp subgroup are not consumers.
MazurTorsion.PrimeOrder.DegreeOneFormalImmersionWitnessPackage the represented modular and cusp sections, normalized quotient map, formal immersion, generic distinctness, and the two bad-branch collision implications consumed directly by the checked theorem. Torsion remains a private constructor input used to derive the whole-section collision.
ModularCurve.EisensteinQuotient.toDegreeOneFormalImmersionWitnessConstruct the route-neutral witness privately from the nontrivial optimal Eisenstein quotient and its specialized finite-Mordell–Weil theorem.
ModularCurve.EisensteinQuotientConstruct the nontrivial optimal quotient in exactly the levels used by the torsion theorem.
ModularCurve.EisensteinQuotient.nontrivial_of_level_eleven_or_ge_seventeenProve that the quotient is nonzero for N=11 and prime N at least 17.
ModularCurve.EisensteinQuotient.mordellWeil_finiteApply the admissible finite-flat rank-zero criterion and Mordell-Weil finite generation.
ModularCurve.EisensteinQuotient.formalImmersionAtInfinity_modFiveInstantiate the optimal-quotient formal-immersion theorem at the quotient used downstream.
Close the inherited locally-primary pseudo-unit reciprocity Challenge and preserve the checked cyclotomic infrastructure as an independent release obligation. The checked Hilbert-94 layer exposes the actual equivariant capitulation map, a nontrivial Galois-stable exponent-p kernel, and a division-field consumer. The chosen proof of Mazur's theorem does not consume this result.
NumberTheory.CyclotomicCharacter.inverseExtensionPackage the inverse-cyclotomic character extension over the p-th cyclotomic field.
NumberTheory.CyclotomicCharacter.unramifiedAtFinitePlacesGive the local criterion showing that the relevant extension is unramified at every finite place.
NumberTheory.CyclotomicCharacter.noEverywhereUnramifiedExclude an everywhere-unramified inverse-cyclotomic extension using the required class-field input.
NumberTheory.CyclotomicCharacter.locallyPrimaryPseudoUnitKummerReciprocityPrincipleProve integral one-sided Kummer reciprocity for locally-primary pseudo-units; checked comparison and normalization reductions then supply the inverse-character class-group quotient.
NumberTheory.CyclotomicCharacter.InverseExtension.capitulationHomExtend ideal classes from the prime cyclotomic field to a supplied inverse extension through the genuine class-group extended-ideal homomorphism.
MazurTorsion.NumberTheory.CyclotomicCapitulationNumberTheory.CyclotomicCharacter.InverseExtension.capitulationHom_equivariantProve that capitulation commutes with the cyclotomic action on the base class group and the chosen lifted action on the extension class group.
MazurTorsion.NumberTheory.CyclotomicCapitulationNumberTheory.CyclotomicCharacter.InverseExtension.exists_nontrivial_p_torsion_capitulating_orbitUse Hilbert 94 to produce a nontrivial exponent-p ideal class whose full cyclotomic Galois orbit capitulates in every finite-place-unramified inverse extension.
MazurTorsion.NumberTheory.CyclotomicCapitulationMazurTorsion.PrimeOrder.divisionField_exists_nontrivial_p_torsion_capitulating_orbitConsume the actual division-field unramifiedness datum in the equivariant exponent-p capitulation theorem without asserting the missing inverse-character quotient.
MazurTorsion.PrimeOrder.CyclotomicObstructionStage contract
These are the concrete APIs and results that count as this stage’s scope. Node cards below show how the work is divided and connected.
eisensteinImage_isTorsionThe rational prime-torsion point defines an X_0(N) point whose image in the finite Eisenstein quotient is torsion.
primeTorsion_potentiallyGoodReductionAtFiveCusp reduction, torsion specialization, and formal immersion exclude potentially multiplicative reduction at 5.
primeTorsion_goodReductionAtFivePrime-to-five specialization and the tame additive component bound upgrade potentially good reduction to good reduction.
card_reductionAtFive_le_tenNormalize to one of 25 short models over F_5 and check that the special fibre has at most ten points.
rationalPoint_addOrderOf_ne_eleven / ne_prime_ge_seventeenInject the marked point into the at-most-ten-element fibre and exclude order 11 and every prime order at least 17.
A rational point of prime order N gives a rational cyclic subgroup, an X_0(N) point, and therefore a finite-order point in the rank-zero Eisenstein quotient.
MazurTorsion.PrimeOrder.xZeroPoint_of_rationalPrimeTorsionConstruct the classifying X_0(N)(Q) point from the generated cyclic subgroup.
MazurTorsion.PrimeOrder.eisensteinImage_isTorsionUse finiteness of the quotient's rational points to make the modular image torsion.
If the elliptic curve is potentially multiplicative at 5, its X_0(N) point specializes to a cusp; Atkin-Lehner transport reduces the collision to infinity.
MazurTorsion.PrimeOrder.xZeroPoint_specializesToInfinity_of_potentiallyMultiplicativeAtFiveMove the specialized cusp to infinity without changing the quotient argument.
A finite-order quotient point specializing to zero is zero by the unramified specialization lemma. The modular section and cusp would then cross with equal images, contradicting formal immersion. The completed local collision and quotient-cotangent consumers compile over the actual five-adic integer ring. A checked proper-model consumer now derives whole-section quotient equality from an actual proper commutative quotient model, generic-fibre torsion, closed-fibre collision, and a supplied torsion-specialization injectivity hypothesis before calling the route-neutral formal-immersion endpoint. It does not construct the model, quotient map, represented sections, formal immersion, collision, torsion, or injectivity input. Constructing those objects and the actual quotient certificate from the modular q-expansion remains open.
MazurTorsion.PrimeOrder.primeTorsion_potentiallyGoodReductionAtFiveImplement Mazur 1978 Corollary 4.3 at the single unramified prime five.
MazurTorsion.PrimeOrder.valuation_j_le_one_of_formalImmersionAtFiveTurn the explicit modular/cusp closed-point collision and quotient-section equality into the five-adic j-valuation bound using the checked completed-stalk formal immersion.
MazurTorsion.PrimeOrder.FormalImmersionAtFiveMazurTorsion.PrimeOrder.valuation_j_le_one_of_quotientCotangentCertificateAtFiveDerive formal immersion from the quotient-cotangent certificate on the actual five-adic cusp stalk and feed it to the rational j collision.
MazurTorsion.PrimeOrder.FormalImmersionSpecialFiberAtFiveMazurTorsion.PrimeOrder.rationalPoint_addOrderOf_ne_of_quotientCotangentAtFive_of_nonsingularReductionConsume the cusp-stalk quotient calculation and canonical nonsingular-reduction inputs in the checked prime-order F_5 contradiction.
MazurTorsion.PrimeOrder.FormalImmersionSpecialFiberAtFiveMazurTorsion.PrimeOrder.valuation_j_le_one_of_mappedIdealCotangentAtFiveDerive the five-adic j bound directly from the extended special-fibre ideal and its quotient cotangent calculation.
MazurTorsion.PrimeOrder.FormalImmersionSpecialFiberAtFiveMazurTorsion.PrimeOrder.rationalPoint_addOrderOf_ne_of_mappedCotangentAtFive_of_nonsingularReductionFeed the mapped characteristic-five cotangent calculation directly to the canonical prime-order F_5 contradiction.
MazurTorsion.PrimeOrder.FormalImmersionSpecialFiberAtFiveMazurTorsion.PrimeOrder.valuation_j_le_one_of_mappedIdealDegreeOneCotangentAtFiveDerive the five-adic j bound from the concrete degree-one quotient-fibre outputs detected by q-expansion.
MazurTorsion.PrimeOrder.FormalImmersionSpecialFiberAtFiveMazurTorsion.PrimeOrder.rationalPoint_addOrderOf_ne_of_mappedDegreeOneCotangentAtFive_of_nonsingularReductionConsume degree-one special-fibre data and checked canonical reduction additivity in the prime-order endpoint.
MazurTorsion.PrimeOrder.FormalImmersionSpecialFiberAtFiveMazurTorsion.PrimeOrder.rationalPoint_primeOrder_ne_of_properModelSpecializationAtFiveFeed an actual proper commutative quotient model, its identified generic fibre, section collision, generic-fibre torsion, and a supplied torsion-specialization injectivity predicate to the checked formal-immersion prime-order contradiction; no full Neron mapping property or specialization theorem is constructed.
MazurTorsion.ModularCurve.NeronSectionSpecializationThe checked good-reduction specialization homomorphism preserves the exact finite additive order of the marked point at 5. Its concrete ZMod 5 form feeds the finite-field cardinality contradiction directly; no Neron model or division field is used.
MazurTorsion.PrimeOrder.specializedPoint_addOrderOf_eq_atFive_of_goodReductionPreserve the exact additive order of every finite-order point under good reduction at 5.
MazurTorsion.PrimeOrder.TorsionSpecializationMazurTorsion.PrimeOrder.specializedPointZMod_addOrderOf_eq_atFive_of_goodReductionTransport exact order to the concrete ZMod 5 reduction used by the finite-field endpoint.
MazurTorsion.PrimeOrder.TorsionSpecializationMazurTorsion.PrimeOrder.rationalPoint_addOrderOf_ne_of_eleven_le_of_goodReductionAtFiveConsume exact-order specialization and the ten-point enumeration to exclude every marked order at least 11 under good reduction.
MazurTorsion.PrimeOrder.GoodReductionAtFiveThe checked marked Tate-depth argument excludes additive reduction at 5 by forcing a forbidden weighted rescaling of the selected minimal equation. It uses no cyclotomic, division-field, Herbrand–Kummer, or Néron-component premise.
MazurTorsion.PrimeOrder.not_additiveReductionAtFiveExclude additive reduction by the checked marked weighted-depth and minimality contradiction; the good-reduction theorem is its compiled consumer.
MazurTorsion.PrimeOrder.GoodReductionAtFiveMazurTorsion.PrimeOrder.addOrderOf_ne_prime_ge_eleven_of_tameAdditiveFiltrationAtFiveProve the prime-order contradiction from the component, additive-residue, and torsion-free formal-kernel filtration.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.addOrderOf_ne_prime_ge_eleven_of_componentExponentTwelveAtFiveExclude a marked prime-order point using only its component exponent, the five-element identity-component reduction target, and the torsion-free formal kernel.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.addOrderOf_ne_prime_ge_eleven_of_nonsingularReduction_of_componentExponentTwelveAtFiveInstantiate the component-exponent contradiction on canonical coordinatewise nonsingular reduction and discharge the formal-kernel theorem at five.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.EllipticCurve.TameAdditiveReductionDataAtFive.toTameAdditiveFiltrationDataConvert the canonical five-adic handoff to the checked torsion filtration while deriving finite component and torsion-free formal-kernel inputs.
MazurTorsion.EllipticCurve.TameAdditiveReductionDataMazurTorsion.PrimeOrder.minimalCompletionAtFive_reduction_invariants_of_hasAdditiveReductionSpecialize vanishing of the discriminant and c₄ to Mathlib's selected minimal five-adic integral model, starting the cuspidal nonsingular-locus classification.
MazurTorsion.PrimeOrder.FormalImmersionNeronAtFiveMazurTorsion.PrimeOrder.tateAlgorithm_valuationInput_of_hasAdditiveReductionPackage the additive-reduction and integral-j valuation inequalities consumed by the tame Tate-algorithm route.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.tateAlgorithm_coefficientObstruction_of_hasAdditiveReductionApply selected-equation minimality to rule out another pure weighted scaling while retaining all translation, blowup, and component steps as open work.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.tateAlgorithm_hasAdditiveReduction_variableChange_of_valuation_u_eq_oneKeep the minimal additive branch stable under every integral valuation-unit coordinate change used by the residue-translation stage.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.tateAlgorithm_translatedCoefficientObstruction_of_hasAdditiveReductionRule out simultaneous integrality of all five exactly transformed coefficients after arbitrary translations and a scale of valuation below one.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveIntegral j excludes multiplicative reduction, while the checked marked Tate-depth theorem excludes additive reduction; the exhaustive reduction trichotomy therefore gives good reduction on the selected five-adic minimal equation. Exact-order transport from the rational point and checked F_5 reduction yield the unconditional local contradiction, consumed by both the formal-immersion and affine Hecke/q-expansion endpoints. Represented modular/cusp data remain in the upstream formal-immersion node, not as a premise of this completed local result.
MazurTorsion.PrimeOrder.goodReductionAtFiveCombine the checked multiplicative and weighted-depth additive exclusions; the rational finite-field and formal-immersion endpoints consume this theorem.
MazurTorsion.PrimeOrder.GoodReductionAtFiveMazurTorsion.PrimeOrder.valuation_j_gt_one_of_hasMultiplicativeReductionCalculate that multiplicative reduction forces valuation of the j-invariant strictly above one.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.not_hasMultiplicativeReduction_of_valuation_j_le_oneDeduce that an integral j-invariant excludes the multiplicative branch.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.hasGoodReduction_of_valuation_j_le_one_of_additiveOrderObstructionShare the reduction trichotomy between residue characteristics five and eleven by isolating the additive branch as an exact-order obstruction.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.hasGoodReduction_of_valuation_j_le_one_of_tameAdditiveFiltrationAtFiveConsume integral j, an actual tame additive filtration, and a marked large-prime-order point in the reduction-type trichotomy to prove good reduction.
MazurTorsion.PrimeOrder.TameAdditiveAtFiveMazurTorsion.PrimeOrder.completionPoint_addOrderOf_ne_of_eleven_le_of_hasGoodReductionAtFiveReduce a torsion point on a minimal completed equation with good reduction, transport it to ZMod 5, and consume the checked finite-field order bound.
MazurTorsion.PrimeOrder.GoodReductionAtFiveThe checked exhaustive enumeration proves every elliptic curve over F₅ has at most ten rational points. It is an independent finite computation, not a Shafarevich input.
MazurTorsion.PrimeOrder.card_reductionAtFive_le_tenNormalize in characteristics other than two and three, then verify all 25 short models over F_5.
MazurTorsion.PrimeOrder.FiniteFieldFiveMazurTorsion.PrimeOrder.zmod_five_addOrderOf_ne_of_eleven_leConsume the finite enumeration to rule out every point whose exact additive order is at least eleven.
MazurTorsion.PrimeOrder.FiniteFieldFiveOrderJoin potential-good reduction from formal immersion, the checked additive exclusion, prime-to-five exact-order specialization, and the ten-point enumeration to exclude order 11 and every prime order at least 17.
MazurTorsion.PrimeOrder.rationalPoint_addOrderOf_ne_elevenExclude exact rational order eleven by the formal-immersion-at-five argument.
MazurTorsion.PrimeOrder.rationalPoint_addOrderOf_ne_prime_ge_seventeenExclude every rational prime order at least seventeen by the same argument.
Stage contract
These are the concrete APIs and results that count as this stage’s scope. Node cards below show how the work is divided and connected.
Lean 4.33.0-rc1 / mathlib 79d0395 lockThe root and Tau Ceti contract workspaces share the exact Lean toolchain and complete resolved dependency graph, and the downstream contracts build under the permanent quality, axiom, linter, and style gates.
rationalTorsion_orders_mem_cyclicOrdersThe unconditional point-order theorem combining the finite endpoints and the prime-order argument.
Challenge.Mazur.torsion_ncard_leThe final Lean Pool statement: the rational torsion set of an elliptic curve over Q has ncard at most 16.
MazurTorsion.rationalTorsion_hasMazurClassificationThe canonical release theorem: rational torsion is isomorphic to one of Mazur's fifteen groups.
kernelAndProvenanceAuditReproducible build, permitted-axiom report, source pins, exact blueprint topology, and downstream API audit.
The root and Tau Ceti contract workspaces use one exact Lean toolchain and complete resolved package-revision graph. The Tau Ceti aggregator builds as a downstream consumer and passes the contract-axiom, Mathlib-linter, style, and repository quality gates.
MazurTheorem.Release.sharedDependencyGraphEmbed the exact root and Tau Ceti toolchains and manifests whose complete resolved package-revision graphs are checked for equality.
MazurTorsion.Release.PinMigrationMazurTheorem.Release.tauCetiConsumerBuildRecord the exact Tau Ceti downstream build and audit commands exercised by the permanent quality and CI gates.
MazurTorsion.Release.PinMigrationExpose one unconditional theorem that every rational torsion point has an allowed order, using the formal-immersion theorem for 11 and primes at least 17.
MazurTorsion.rationalTorsion_orders_mem_cyclicOrdersCombine the exceptional finite endpoints and the formal-immersion prime theorem.
The checked rationalTorsion_finite theorem uses rational Northcott, the approximate parallelogram law, and Mathlib finite-torsion descent without full Mordell–Weil finite generation. The checked generic rank-two theorem uses the finite-abelian decomposition, allowed-order hypothesis, and the four elementary rank obstructions c2Cube, c3Square, c5Square, and c7Square. Its compiled rational adapter remains conditional on the point-order, h55, and h77 inputs; the later classification also consumes c4Square, c2c10, and c2c12 before deriving the immutable ncard-at-most-16 corollary.
MazurTorsion.rationalTorsion_hasRankTwoPresentationThe compiled cross-module adapter applies the generic theorem to finite rational torsion, conditional on the point-order, h55, and h77 inputs still owned by API integration.
MazurTorsion.Arithmetic.RankTwoReductionMazurTorsion.rationalTorsion_hasMazurClassificationClassify the rational torsion subgroup up to group isomorphism as one of Mazur's fifteen groups.
Challenge.Mazur.torsion_ncard_leDerive the immutable Lean Pool ncard-at-most-16 statement from the full classification.
Verify every blueprint edge, close every retained Challenge including the now-noncritical X_1(11) and cyclotomic contracts, reproduce the clean build, and publish the final axiom and source audit.
MazurTheorem.Release.kernelAndProvenanceAuditRecord the final axiom report, exact source pins, declaration provenance, and reproducible build evidence.
MazurTheorem.Release.allChallengesClosedVerify that every registered Challenge is a checked bridge with no open contract.
MazurTheorem.Release.versoBlueprintPublish a Verso blueprint exactly matching the formal-immersion dependency graph.
Interactive dependency graph
Read left to right across the six programme stages. Select any node to inspect its definitions, theorem contracts, dependencies, and downstream consumers.
Prefer a theorem-first view? The canonical Verso blueprint includes proof-status summaries and its own interactive dependency graph. Open the formal blueprint
Current work-package frontier
Nested packages make large nodes navigable, but they do not create extra progress credit. The parent node still closes only when its reviewed API reaches the stated downstream boundary.
WP-MT-TC-B1-COHERENT-COHOMOLOGY-COHERENT-CORE
A cover-independent canonical K-linear H⁰/H¹ API, including H⁰/global-sections compatibility and coefficient-morphism and connecting-map linearity, is available to the proper-curve finiteness package.
WP-MT-EC-ISOGENY-WEIL-WEIERSTRASS-GROUP-SCHEME
Starting from the checked over-base additions on D(x₁ - x₂) and D(B₁₂), their equality on the exact intersection, and the diagonal tangent comparison, glue their union; then add the infinity charts and coverage, prove the commutative group laws, and refactor MazurTorsion.XZeroFortyNine.orderFortyNineConcreteGroupSchemeInterface to consume the result without supplied [GrpObj (toOver E)] or CanonicalPointGroupLawCompatibility E shadow inputs.
Open contracts
Every card is an acceptance contract. There are 1 ordinary claims worth 10 points and 1 nonexclusive research intentions worth 18 points. Paused contracts are preserved in a separate section and cannot be claimed.
MT-O49-TOWER · 10 points
An elliptic curve over Q has no rational point of exact order 49.
MazurTheorem.Challenge.no_rational_point_of_order_fortyNinetheorem no_rational_point_of_order_fortyNine
(E : WeierstrassCurve ℚ) [E.IsElliptic]
(P : (E⁄ℚ).Point) :
addOrderOf P ≠ 49 := sorryMazurTorsion.NumberTheory.XZeroFortyNineTransferMazurTorsion.rationalTorsion_orders_mem_cyclicOrders_of_remaining_obstructionsMazurTorsion.NumberTheory.XZeroFortyNineTransferMazurTorsion.XZeroFortyNine.rationalPoint_addOrderOf_ne_fortyNineMT-TC-A3-DIVISOR-LINE-BUNDLE · 18 points
For the geometric order system of a smooth proper integral curve, prove the full invertible-sheaf/Picard comparison and identify divisor classes with the scheme Picard group. Checked code proves these two outputs are equivalent to the complete dictionary. Affinely, it constructs the actual tensor-additive tilde line bundles, a canonical Picard embedding with exact principal kernel, an unconditional range equivalence, and a full equivalence under exactly the reverse tensor-unit/local-rank-one comparison.
Parallel approaches are welcome for this research-open boundary.
MazurTheorem.Challenge.divisorLineBundleDictionarytheorem divisorLineBundleDictionary
(K : Type u) [Field K]
(X : Scheme.{u}) [IsIntegral X] [IsNoetherian X]
(π : X ⟶ Spec (.of K)) [IsProper π] [SmoothOfRelativeDimension 1 π]
(S : TauCeti.AlgebraicGeometry.WeilDivisor.OrderSystem
(TauCeti.AlgebraicGeometry.CodimensionOnePoint X)
(Additive X.functionFieldˣ))
(hord : S.ord = TauCeti.AlgebraicGeometry.SchemeWeilDivisor.orderAt) :
Nonempty (MazurTorsion.AlgebraicGeometry.DivisorPicard.Dictionary S X) := sorryMazurTorsion.Upstream.DivisorLineBundleMathlib.AlgebraicGeometry.Morphisms.ProperMathlib.AlgebraicGeometry.Morphisms.SmoothTauCeti.AlgebraicGeometry.WeilDivisor.Scheme.OrderMazurTheorem.Challenge.divisorClassEquivPicardMazurTheorem.Challenge.divisorPicardCoreDataMazurTorsion.Upstream.DivisorLineBundleMazurTorsion.AlgebraicGeometry.divisorLineBundleDictionaryDeliberately paused
Contracts remain immutable and compiled, but they are not claimable and receive no maintainer proof volume while the canonical foundation lanes are unfinished.
MT-X11-COSET
The reverse X_1(11) model-to-Tate bridge, its discriminant certificate, the conditional cusp classification, and the conditional five-coset proof with Q=0 are checked. The node remains open until MT-X11-JOIN supplies the unconditional order-11 exclusion and the immutable Challenge bridge compiles. The direct five-isogeny Selmer calculation remains a partial fallback.
MT-X13-NONCUSP
Show that the explicit genus-two order-13 model has no rational affine point with x different from 0 and -1. The checked primitive split descent clears all three cusp factors, derives the two remaining cubic coefficient identities, proves both conic parameters odd, and forces k to be -4 or 4. Its global rational-point elimination, and hence the genuine Mazur--Tate Jacobian/rank-zero conclusion, remain open.
MT-X18-NONCUSP
Show that the explicit genus-two order-18 model has no rational point with x different from 0 and 1. The checked primitive split descent derives the two remaining cubic coefficient identities, proves both conic parameters odd, and forces k to be -8, -4, 4, or 8. Classification of these two explicit split covers, equivalently a genus-two Jacobian rank-zero certificate or another complete descent, is still required; the Challenge remains open.
MT-O25-EXCLUDE
Prove that no rational point on an elliptic curve over Q has exact additive order 25. The checked Tate recurrence computes (n+2)P from index n and proves every secant denominator through 13P nonzero from exact order 25. Explicit square/cube normalized coordinates through 13P reduce x(13P)=x(12P) to the cuspidal factor b-c times one fixed 234-term polynomial of total degree 40. The arbitrary-curve consumer retains b, c, and b-c nonzero together with the twelfth-power discriminant scale. Classifying rational solutions of that explicit noncuspidal factor, directly or after a checked birational reduction, remains open.
MT-O35-EXCLUDE
Use the explicit optimal elliptic quotient X_0(35)/w_5 and Mazur's squarefree-level formal-immersion criterion at auxiliary prime 11. The source three-descent and Eisenstein ideal-support calculations prove the fixed explicit curve model has rank zero and finite rational point group. The represented modular classifying map and chart identification, the characteristic-eleven cusp local-ring generator and unit calculation, the abelian-variety identification, and Atkin--Lehner specialization inputs remain open.
MT-CYCLOTOMIC-UNRAMIFIED
Close the inherited locally-primary pseudo-unit reciprocity Challenge and preserve the checked cyclotomic infrastructure as an independent release obligation. The checked Hilbert-94 layer exposes the actual equivariant capitulation map, a nontrivial Galois-stable exponent-p kernel, and a division-field consumer. The chosen proof of Mazur's theorem does not consume this result.
How progress is scored
A node earns no theorem-progress credit for a statement alone. A proof node earns 70% for an isolated kernel proof, 85% after API review, and 100% only after a downstream consumer compiles. Infrastructure earns 40% when its definitions compile, 70% after mathematical sanity checks, and 100% after a real downstream consumer compiles. The headline is the weighted sum.
The formal declaration faithfully matches the mathematics.
Lean accepts it without project-specific axioms or placeholders.
Definitions pass mathematical sanity checks and review.
A named downstream consumer compiles against the result.
It is an effort-weighted planning estimate, not a theorem of project management. The denominator is uncertain because Jacobians, integral modular curves, Néron models, and the Eisenstein quotient do not yet share a tested Lean spine.
The ≈18% ecosystem-ready figure is deliberately shown separately. It counts work that may be reusable from Lean Pool, Tau Ceti, FLT, and other audited repositories, but it earns integrated credit only after porting and consumer tests. Percentages will move as interfaces reveal the real work.
Work-package allocations only divide a stable parent node into reviewable steps. They never add to the denominator and never earn credit independently. Paused contracts keep their weight and statement while remaining unavailable for claims.
Join the programme
Contributors get an exact declaration, pinned dependencies, a named consumer, and review criteria. Maintainers coordinate the shared architecture so independent work composes.