{ "researchRevision": "5c304fd9957eafc781969703b4bcc75abe0c0201", "statements": [ { "id": "identities", "number": "3.1", "htmlAnchor": "lem:identities", "title": "Lemma 3.1 · Identities", "status": "partial", "declarations": [ { "module": "Identities", "name": "JacobiData.collapse_beta" }, { "module": "Identities", "name": "JacobiData.collapse_alpha" }, { "module": "Identities", "name": "JacobiData.collapse_diff" }, { "module": "Identities", "name": "JacobiData.hasDerivAt_f1_div_c" }, { "module": "Identities", "name": "JacobiData.hasDerivAt_β1_div_c" }, { "module": "Identities", "name": "JacobiData.hasDerivAt_α1_div_d" }, { "module": "Identities", "name": "JacobiData.hasDerivAt_Γ" }, { "module": "Identities", "name": "JacobiData.β1_deriv_eq_Γ" }, { "module": "Monotone", "name": "JacobiInit.Γ_zero" }, { "module": "Monotone", "name": "JacobiInit.Γ_pos" }, { "module": "ZerosF1", "name": "JacobiPeriodic.exists_unique_f1_zero" }, { "module": "ZerosF1", "name": "JacobiPeriodic.f1_sign_before" }, { "module": "ZerosF1", "name": "JacobiPeriodic.f1_sign_after" }, { "module": "Zeros", "name": "JacobiPeriodic.exists_unique_β1_zero" }, { "module": "CellCount", "name": "JacobiFull.signed_α1_div_d_strictAntiOn" }, { "module": "CellCount", "name": "JacobiFull.α1_zero_unique" }, { "module": "CellCount", "name": "JacobiPeriodic.α1_even" } ], "covered": [ "Coefficient collapse, the three quotient derivatives and the Γ identity.", "Existence, uniqueness and signs of f₁/β₁ zeros; monotonicity and uniqueness for α₁ and its even-period value." ], "missing": [ "Simplicity of the α₁ zero and identification of K = 2E with a unique numerical modulus k₀ are not proved in this snapshot." ], "assumptions": [ "Jacobi differential rules, initial/period data and positivity hypotheses are supplied through the Jacobi structures." ] }, { "id": "riccati", "number": "3.2", "htmlAnchor": "lem:riccati", "title": "Lemma 3.2 · Riccati slope bound", "status": "partial", "declarations": [ { "module": "Riccati", "name": "JacobiData.a_deriv_form" }, { "module": "Riccati", "name": "JacobiData.b_deriv_form" }, { "module": "Riccati", "name": "JacobiData.hasDerivAt_z" }, { "module": "Riccati", "name": "JacobiData.T1_mul_T2" }, { "module": "Riccati", "name": "JacobiData.T1_T2_same_sign" }, { "module": "Riccati", "name": "JacobiData.slope_bound_abs" }, { "module": "Riccati", "name": "JacobiData.z_eq_neg_beta_div_ell" }, { "module": "SignedBracket", "name": "SBracket.one_le_ε_deriv_u" } ], "covered": [ "The Riccati pair, z derivative, T₁T₂ product, common sign and AM–GM slope inequality.", "The oriented phase-lift derivative bound on signed brackets." ], "missing": [ "The displayed exact square-gap formula for |u′| − 1 and its equality-case characterization are not exported as matching theorems in this snapshot." ], "assumptions": [ "Nonzero sn/dn and q/ℰ denominators, common sign and bracket conditions remain explicit in the theorem parameters." ] }, { "id": "crossing", "number": "3.3", "htmlAnchor": "lem:crossing", "title": "Lemma 3.3 · Crossing principle", "status": "partial", "declarations": [], "covered": [ "The counting clause is linked in the phase, counting, bracket and arc passages below." ], "missing": [ "Analytic zero orders and finiteness of tangency phases are not proved by Crossing.crossing." ], "assumptions": [ "CrossingData assumes continuity, endpoint values and strict branch monotonicity; the article derives these analytically." ] }, { "id": "first", "number": "3.4", "htmlAnchor": "thm:first", "title": "Theorem 3.4 · Exact first conjugate time", "status": "partial", "declarations": [ { "module": "BeforeFirst", "name": "JacobiFull.exact_first_time" }, { "module": "BeforeFirst", "name": "JacobiFull.first_time_complete" }, { "module": "BeforeFirst", "name": "JacobiFull.J2_pos_before_p0" } ], "covered": [ "Exactly one reduced-Jacobian zero in [p₀,p₁] and in (0,p₁]; positivity excludes zeros before p₀." ], "missing": [ "Continuity and phase extrema of p* are explicitly outside the published Lean scope.", "The geometric identification of conjugate/cut time and the scale t = 2kp are not certified by these scalar root-counting declarations." ], "assumptions": [ "exact_first_time and first_time_complete require hfin: finite tangencies on the first bracket.", "JacobiFull hypotheses; the reduced base is constructed by JacobiCore.toJacobiFull." ] }, { "id": "cells", "number": "4.1", "htmlAnchor": "thm:cells", "title": "Theorem 4.1 · Conjugate times in every cell", "status": "partial", "declarations": [ { "module": "CellCount", "name": "JacobiFull.cell_roots" }, { "module": "CellCount", "name": "JacobiFull.two_per_cell_of_ne" }, { "module": "CellCount", "name": "JacobiFull.eL_eq_eR_iff" }, { "module": "CellCount", "name": "JacobiFull.coincide_imp" }, { "module": "LeftBracket", "name": "JacobiFull.left_exact_time" }, { "module": "RightBracket", "name": "JacobiFull.right_exact_time" }, { "module": "BeforeFirst", "name": "JacobiFull.J2_even_eq_zero_iff" }, { "module": "BeforeFirst", "name": "JacobiFull.roots_ncard" }, { "module": "SignedTime", "name": "SBracket.J2_eq_zero_iff" }, { "module": "SignedTime", "name": "SBracket.exact_time" } ], "covered": [ "One zero on each closed bracket, exhaustion of the half-open cell, two distinct roots for K ≠ 2E, bracket contact iff K = 2E and the necessary location of coincident roots.", "The phase of an even-period zero and the cumulative 2N root count." ], "missing": [ "Simplicity, interior tangency zero orders and the quartic order of the coincident root are not formalized.", "The unique critical modulus k₀ is not identified; Lean uses the algebraic condition K = 2E." ], "assumptions": [ "The counting declarations require finite tangencies on the left and right brackets (hL/hR or hfin).", "Lean cell index m corresponds to article n = m + 1. JacobiFull is conditional data, not a construction of SE(2) geodesics." ] }, { "id": "coincident", "number": "4.2", "htmlAnchor": "rem:coincident", "title": "Remark 4.2 · Coincident zero order", "status": "not-formalized", "declarations": [], "covered": [], "missing": [ "The quartic expansion and exact order four are not formalized in this snapshot. Cell-counting theorems do not establish this zero order." ], "assumptions": [] }, { "id": "floquet", "number": "5.1", "htmlAnchor": "thm:floquet", "title": "Theorem 5.1 · Floquet form and large-index law", "status": "not-formalized", "declarations": [], "covered": [], "missing": [ "Both the Floquet determinant identity and the large-index asymptotic law are outside the published Lean scope (Appendix A excludes Section 5)." ], "assumptions": [] }, { "id": "late-folds", "number": "5.2", "htmlAnchor": "remark-late-folds", "title": "Remark 5.2 · Late conjugate points are folds", "status": "not-formalized", "declarations": [], "covered": [], "missing": [ "This is a claim from the companion manuscript, not a theorem of these fifteen Lean modules. No formal proof of the stated late-fold conclusion is indexed." ], "assumptions": [] }, { "id": "separatrix", "number": "5.3", "htmlAnchor": "rem:separatrix", "title": "Remark 5.3 · Separatrix limits", "status": "not-formalized", "declarations": [], "covered": [], "missing": [ "The k → 1 limits and their rate are outside the published Lean scope." ], "assumptions": [] } ], "supportDeclarations": [ { "module": "Reduction", "name": "JacobiCore.toJacobiFull" } ] }