{"checks":[{"claim":"The submitted source is exactly the reviewed v12 file, not a previous manuscript","evidence":{"actual_sha256":"ea27de068cd78871211d0a1ac102e76b65c7bd1f033267fb24a2c5f8dc423f9e","reviewed_sha256":"ea27de068cd78871211d0a1ac102e76b65c7bd1f033267fb24a2c5f8dc423f9e"},"id":"S01","module":"source_binding","status":"PASS"},{"claim":"Every inventoried labeled display matches the separately frozen source contract","evidence":{"display_blocks":80,"labels":92},"id":"S02","module":"source_binding","status":"PASS"},{"claim":"This v12 release does not silently change the supplied manuscript","evidence":"Both source copies are preserved byte-for-byte.","id":"S03","module":"source_binding","status":"PASS"},{"claim":"Quaternion multiplication is associative","evidence":"Four polynomial identities over QQ in 12 independent real variables.","id":"T01","module":"topology","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"Quaternion norm is multiplicative and conjugation reverses products","evidence":"Hamilton multiplication was expanded from components; no sampled values.","id":"T02","module":"topology","nonzero_residuals":[],"residual_count":5,"status":"PASS"},{"claim":"The quotient coordinate u^{-1} x u is invariant under (x,u)->(q x q^{-1},qu)","evidence":"Four generic denominator-cleared polynomial identities; nonzero norms required for inverses.","id":"T03","module":"topology","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"Bundle transition and right principal action commute","evidence":"Associativity applied to g_n(x), u, h; smoothness of g_n is a separate analytic premise.","id":"T04","module":"topology","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"The marked transition is conjugation by g_n(y), after imposing the equivariance premise","evidence":"Four exact denominator-cleared identities in arbitrary quaternion components; u and g_n(x) are unit in the geometric application.","id":"T09","module":"topology","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"Opposite conjugations are inverse when the conjugating quaternion is unit","evidence":"The polynomial identity is C_b(C_bbar(x))=|b|^4 x. The manifold inverse formula additionally uses b(sigma(x))=b(x), proved using equivariance in the manuscript.","id":"T10","module":"topology","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"The induction step for iterated conjugation has the required multiplication order","evidence":"A generic two-step conjugation identity, not sampling of the nonlinear Blakers--Massey map. The all-integer induction is human reasoning from this group law and invariance of b along sigma-orbits.","id":"T11","module":"topology","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"K^* K = 4((|p|^2+|Im w|^2)I-p p^T-Im(w) Im(w)^T)","evidence":"Exact 3-by-3 matrix identity.","id":"T05","module":"topology","nonzero_residuals":[],"residual_count":9,"status":"PASS"},{"claim":"On |x|=1, ||K_x v||^2 <= 4||v||^2, including all orbit ranks","evidence":"The deficit is the displayed sum of squares, valid for every real vector v.","id":"T06","module":"topology","nonzero_residuals":[],"residual_count":1,"status":"PASS"},{"claim":"Blakers--Massey scalar factor has endpoint limits pi and pi/2","evidence":"Exact symbolic limits. Limits alone are not an all-orders smoothness proof.","id":"T07","module":"topology","nonzero_residuals":[],"residual_count":2,"status":"PASS"},{"claim":"Residues 0,...,27 give 28 labels and orientation reversal gives 15 orbits","evidence":"Finite modular bookkeeping only; the geometric interpretation uses DPR Theorem 1.","id":"T08","module":"topology","nonzero_residuals":[],"residual_count":2,"status":"PASS"},{"claim":"Audited connection formulas are bound to the submitted v12 source","evidence":{"audited_sha256":"ea27de068cd78871211d0a1ac102e76b65c7bd1f033267fb24a2c5f8dc423f9e","label_lines":{"eq:HHHH":[1031],"eq:HHHV":[989],"eq:HHVV":[979],"eq:HVHV":[974],"eq:HVVV":[983],"eq:Koszul1":[955],"eq:Koszul2":[959],"eq:Lambda":[890],"eq:ONeill":[1376],"eq:Plucker":[1072],"eq:VVVV":[1041],"eq:boundarydata":[1282],"eq:bracketcontraction":[1100],"eq:brackets":[944],"eq:master":[1170],"eq:omega":[1318],"eq:potentials":[1307],"eq:reserve":[1227],"eq:sharp-master":[1160],"eq:smallA":[1194],"eq:smallB":[1197],"eq:smallC":[1201],"eq:southlowerbound":[1233],"eq:southsourceform":[1296],"eq:verticalderivative":[947],"prop:concave":[880],"thm:southfilling":[1260]},"scope":"Version binding and unique labels; this does not parse or prove TeX semantics.","sha256":"ea27de068cd78871211d0a1ac102e76b65c7bd1f033267fb24a2c5f8dc423f9e"},"id":"connection.v12_source_binding","module":"connection","status":"PASS"},{"claim":"Quaternion vector fields independently confirm bracket and equivariance signs","evidence":{"bracket":"[F_a,F_b]=F_[e_a,e_b]","equivariance_derivative":"d(Ad(exp(-t e_a))W)/dt=-[e_a,W]","exact_component_checks":45,"fundamental_field":"F_a(q)=q e_a"},"id":"connection.right_action_sign","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with Koszul formula","evidence":{"base_dimension":7,"exact_component_comparisons":594,"failures":[]},"id":"connection.koszul","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with HVHV formula","evidence":{"base_dimension":7,"exact_component_comparisons":441,"failures":[]},"id":"connection.hvhv","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with HHVV formula","evidence":{"base_dimension":7,"exact_component_comparisons":441,"failures":[]},"id":"connection.hhvv","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with HVVV formula","evidence":{"base_dimension":7,"exact_component_comparisons":189,"failures":[]},"id":"connection.hvvv","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with HHHV formula","evidence":{"base_dimension":7,"exact_component_comparisons":1029,"failures":[]},"id":"connection.hhhv","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with HHHH formula","evidence":{"base_dimension":7,"exact_component_comparisons":2401,"failures":[]},"id":"connection.hhhh","module":"connection","status":"PASS"},{"claim":"Koszul reconstruction agrees with VVVV formula","evidence":{"base_dimension":7,"exact_component_comparisons":81,"failures":[]},"id":"connection.vvvv","module":"connection","status":"PASS"},{"claim":"A deliberately reversed HVHV bracket sign is rejected","evidence":{"deliberate_error":"Replace -<[e_0,e_1],Omega_01>/4 by its positive opposite.","rejection_residual":"-O0_1_2"},"id":"connection.curvature_sign_negative_control","module":"connection","status":"PASS"},{"claim":"The printed Pluecker identity has the correct sign","evidence":{"symbolic_identities":441},"id":"connection.plucker","module":"connection","status":"PASS"},{"claim":"Linear curvature terms sum to -3 J/2","evidence":{"alpha_gamma":"-J","beta_beta":"-J/2","orientation_witness":"Omega_12=e_3, X=e_1,Y=e_2,U=e_1,V=e_2 gives J=2 and total=-3.","total":"-3*J/2"},"id":"connection.bracket_contraction","module":"connection","status":"PASS"},{"claim":"Derived horizontal correction contracts to -3 r^2 |Omega(X,Y)|^2/4","evidence":{"exact_polynomial_residual":"0"},"id":"connection.horizontal_section","module":"connection","status":"PASS"},{"claim":"h^2+m^2+k^2=|(X+U) wedge (Y+V)|^2","evidence":{"exact_polynomial_residual":"0"},"id":"connection.bivector_norm","module":"connection","status":"PASS"},{"claim":"The SU(2) bracket bound is an exact norm identity","evidence":{"fibre_normalization":"[e_a,e_b]=2 epsilon_abc e_c","identity":"|[U,V]|^2=4|U wedge V|^2"},"id":"connection.quaternion_bracket_norm","module":"connection","status":"PASS"},{"claim":"Cauchy--Schwarz and the fibre dimension-three array factor are exact","evidence":{"HVVV_array":"sum_abc(delta_ab w_c-delta_ac w_b)^2=4|w|^2","cauchy_sos":"|a|^2|b|^2-<a,b>^2=sum_(i<j)(a_i b_j-a_j b_i)^2","ordered_skew_pair_norm":"Full squared norm = 2 * restricted squared norm, per skew pair"},"id":"connection.norm_kernels","module":"connection","status":"PASS"},{"claim":"C=4 dominates every sharp-master loss","evidence":{"nonnegative_gap":"13/4*r^2*M0^2*h^2 + 7/2*r^2*M0^2*m^2 + 3*r^2*M0^2*h*k + 2*r*M0*T*m*k + 2*r*M1*h*m + M0*h*k"},"id":"connection.common_constant","module":"connection","status":"PASS"},{"claim":"Every absorption uses a certified nonnegative square","evidence":{"identity":"p x^2 + a^2 y^2/(4p)-a x y = (2p x-a y)^2/(4p), p>0"},"id":"connection.young_identity","module":"connection","status":"PASS"},{"claim":"The written losses leave at least the claimed reserve","evidence":{"claimed_mixed":"1/8","horizontal_over_kappa":"1/2","mixed_over_eps_Lambda":"1/8","vertical_times_eps":"1/8"},"id":"connection.reserve_arithmetic","module":"connection","status":"PASS"},{"claim":"All three printed Young coefficients are sufficient","evidence":{"assumptions":["r^2 <= 2 eps","T <= eps D0/2","eps D0 M0 <= 1","2 eps M0 <= 1"],"hk_required_coefficient":"8 C^2 M0^2/kappa; printed 8","hm_required_coefficient":"4 C^2 eps/kappa (M1+1)^2; printed 8","km_required_coefficient":"C^2 eps^4 D0^2 M0^2; printed 2"},"id":"connection.young_coefficients","module":"connection","status":"PASS"},{"claim":"Each smallness hypothesis supplies its assigned loss budget","evidence":{"cleared_denominator_identities":7,"failures":[],"premises":{"derivative_mixed_loss":"kap Lam-64 C^2 (M1+1)^2 >= 0; denominator 8 kap > 0","hessian_loss":"Lam-4 eps D0^2 >= 0; denominator 16 > 0","hk_to_vertical":"64 C^2 eps M0^2 <= kap; denominator 8 eps kap > 0","horizontal_initial":"kap-8 C eps M0^2 >= 0; 2 eps-r2 >= 0","km_to_mixed":"Lam-32 C^2 eps^3 D0^2 M0^2 >= 0; denominator 16 > 0","omega_mixed_loss":"Lam-16 C M0^2 >= 0; 2 eps-r2 >= 0; denominator 8 > 0","vertical_initial":"2 eps^3 D0^2 <= 1; denominator 8 eps > 0"}},"id":"connection.smallness_implications","module":"connection","status":"PASS"},{"claim":"Differentiating r=sqrt(eps) exp(eps phi/2) gives the claimed Hessian tensor","evidence":{"formula":"-Hess(r)/r = -eps Hess(phi)/2 - eps^2 dphi tensor dphi/4","verification":"Symbolic arbitrary phi(z0,z1), including mixed derivatives, in normal coordinates"},"id":"connection.radius_chain_rule","module":"connection","status":"PASS"},{"claim":"Southern r, logarithmic slope and Hessian have the printed signs","evidence":{"Hess_phi_angular":"A0*cos(t)","Hess_phi_radial":"A0*cos(t)","N_radial":"-1/4*A0^2*ee^2*sin(t)^2 - 1/2*A0*ee*cos(t)","concavity":"On a<=t<=pi with a>pi/2: -A0 cos(t)>=A0|cos(a)|.","r_prime_over_r":"1/2*A0*ee*sin(t)"},"id":"south.radius_derivatives","module":"connection","status":"PASS"},{"claim":"Pole parity, mixed pole coefficient, boundary form and slope scaling agree","evidence":{"limit_eps_to_zero":"A0 sin(a)/2","outward_form":"-(cos(a)/sin(a)) |X|^2 - (eps A0 sin(a)/2) |U|^2","polar_radius":"sqrt(ee)*e^(1/2*A0*ee*cos(rho))","pole_N":"eps A0/2","q_s_over_r_a_squared":"1/2*A0*e^(A0*ee*cos(t))*sin(t)","smoothness_scope":"Even radial function and vanishing potential near pole; global bundle smoothness is a geometric premise."},"id":"south.pole_and_boundary","module":"connection","status":"PASS"},{"claim":"Maurer--Cartan gives the stated cutoff curvature","evidence":{"curvature":"chi^2*theta_square - chi*theta_square + dchi_theta","gauge_transform":"Ad(g^-1)((chi-1) dg g^-1)+g^-1 dg=chi g^-1 dg"},"id":"south.connection_curvature","module":"connection","status":"PASS"},{"claim":"Quaternion multiplication verifies the southern gauge transformation","evidence":{"denominator_cleared":"|g|^4, positive for g != 0","exact_component_checks":4,"identity":"g^-1 ((chi-1) dg g^-1) g + g^-1 dg = chi g^-1 dg","limitation":"The actual smooth global g_n and its equivariance are geometric inputs."},"id":"south.noncommutative_gauge","module":"connection","status":"PASS"},{"claim":"Global marked-disk identification, action freeness, and O'Neill descent are external to this symbolic module","evidence":"This module certifies the displayed local algebra; it does not replace bundle topology or a proof of the Riemannian-submersion theorem.","id":"south.external_geometry","module":"connection","status":"UNVERIFIED"},{"claim":"The infinitesimal conjugation map is tangent and ||K_x|| <= 2.","evidence":"Exact polynomial Gram identity and sum-of-squares certificate on |x|=1.","id":"N01","module":"north","status":"PASS"},{"claim":"The star-horizontal graph and boundary cometric have the stated factors and signs.","evidence":"Generic 6 by 3 K; all matrix identities are exact over a rational function field.","id":"N02","module":"north","status":"PASS"},{"claim":"Both area estimates are valid, including the mixed-area factor 8.","evidence":"Exact tensor, Gram determinant, radial linearity and determinant-one shear identities; apply operator norm after orthogonalization.","id":"N03","module":"north","status":"PASS"},{"claim":"The complete doubly warped curvature operator has exactly the five claimed blocks.","evidence":"Independently reconstructed from the full 10-dimensional metric 2-jet: 45 diagonal entries match and 1980 off-diagonal entries vanish exactly.","id":"N04","module":"north","status":"PASS"},{"claim":"The profile derivatives, curvature coefficients and center Taylor jet are correct.","evidence":"Exact symbolic differentiation and vanishing ODE Taylor defect.","id":"N05","module":"north","status":"PASS"},{"claim":"The global elementary lower bound is valid for every x>0.","evidence":"Exact minimizer factorization and rational fourth-power comparison; positive convergent sinh series supplies the preceding inequality.","id":"N06","module":"north","status":"PASS"},{"claim":"The radial criterion and the claimed uniform P bound follow with strict reserve.","evidence":"Rational positive-coefficient certificate; elementary exponential-series bounds avoid numerical approximations.","id":"N07","module":"north","status":"PASS"},{"claim":"The angular curvature criterion holds uniformly and leaves at least half the angular coefficient.","evidence":"Exact coefficient arithmetic combines delta^3 A0 Fa <= 1/(128(1+C_eta)), the density bound and N06.","id":"N08","module":"north","status":"PASS"},{"claim":"The small-parameter northern inequalities and matching boundary data coexist.","evidence":"An explicit positive eps_N is supplied after delta and finite ell_N are fixed; no lower bound on eps conflicts with the southern upper bounds.","id":"N09","module":"north","status":"PASS"},{"claim":"There is an explicit exact witness for the northern parameter inequalities.","evidence":"All final comparisons are rational; e<4 gives the finite length and boundary derivative bounds.","id":"N10","module":"north","status":"PASS"},{"claim":"The outward shape terms cancel exactly in the fibre directions; the stated boundary margin is positive and sharp.","evidence":"Exact rational eigenvalue identity and nonnegative gap factorization for 0<=b<=4; sharp eigenvalue 4 verified.","id":"N11","module":"north","status":"PASS"},{"claim":"The announced positive quotient lower bound follows from the verified coefficients.","evidence":"N07/N08 plus elementary exponential monotonicity and the second exterior-power operator norm inequality.","id":"N12","module":"north","status":"PASS"},{"claim":"Global analytic/topological hypotheses needed to turn the exact north certificates into a complete filling and gluing theorem.","evidence":"Not formalized: smooth eta and inverse profile, global quotient marking, earlier southern theorem and the subsequent gluing theorem. These are distinguished from the exact checks above.","id":"N13","module":"north","status":"UNVERIFIED"},{"claim":"Cubic interpolant matches values and first normal derivatives at both interfaces.","evidence":"Four identities in QQ(z,t,a,b,p,q), with t != 0; each residual is exactly zero.","id":"gluing.hermite.endpoint_jets","module":"gluing","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"Displayed first and second derivatives are exact.","evidence":"Formal differentiation of the interpolant over the exact rational-function field.","id":"gluing.hermite.derivatives","module":"gluing","nonzero_residuals":[],"residual_count":2,"status":"PASS"},{"claim":"Finite Taylor jets have the claimed scaled leading orders H=h+O(t), H'=((1-y)P+(1+y)Q)/2+O(t), H''=-(P-Q)/(2t)+O(1).","evidence":"Arbitrary independent one-sided normal Taylor jets through degree four; z=t*y. All three normalized remainders coerce to a polynomial ring, so have no negative t powers. Uniform smooth-family remainder estimates remain analytic obligations.","id":"gluing.hermite.formal_orders","module":"gluing","nonzero_residuals":[],"residual_count":3,"status":"PASS"},{"claim":"The 2-by-2 determinant interpolation identity has the sign stated in the manuscript.","evidence":"Universal polynomial identity for two arbitrary symmetric 2-by-2 matrices.","id":"gluing.determinant","module":"gluing","nonzero_residuals":[],"residual_count":1,"status":"PASS"},{"claim":"Subtracting the interpolated determinant gives the nonnegative tangential correction +l(1-l) det(P-Q)/4 when 0<=l<=1 and P-Q is positive definite.","evidence":"Exact identity; positive definiteness and the interval restriction are explicit mathematical premises.","id":"gluing.tangential_gain","module":"gluing","nonzero_residuals":[],"residual_count":1,"status":"PASS"},{"claim":"The displayed all-lowered coordinate curvature formula has the correct convention and separates linear second derivatives from nonlinear first derivatives.","evidence":"2401 exact polynomial component identities from independent differentiation of connection coefficients, for arbitrary collar 2-jets in the manuscript's dimension seven at slice normal coordinates.","id":"gluing.curvature.coordinate_formula","module":"gluing","nonzero_residuals":[],"residual_count":2401,"status":"PASS"},{"claim":"Radial block equals -H''/2+H' H^{-1} H'/4.","evidence":"36 exact entries; H=I at the evaluation point, so H^{-1}=I. The quadratic term is a sum of squares on diagonal entries.","id":"gluing.curvature.radial","module":"gluing","nonzero_residuals":[],"residual_count":36,"status":"PASS"},{"claim":"Tangential Gauss block has the sign that yields K_G=K_H-det(H'|Pi)/(4 det(H|Pi)).","evidence":"1296 arbitrary formal tangential components in dimension seven, independently calculated from the Levi-Civita connection.","id":"gluing.curvature.gauss","module":"gluing","nonzero_residuals":[],"residual_count":1296,"status":"PASS"},{"claim":"Mixed curvature block equals ((D_j H')_ik-(D_i H')_jk)/2 and contains no H''.","evidence":"216 arbitrary formal mixed components; tangential slice Christoffels vanish at the point, so covariant derivatives there equal partial derivatives.","id":"gluing.curvature.codazzi","module":"gluing","nonzero_residuals":[],"residual_count":216,"status":"PASS"},{"claim":"The all-plane lower bound follows by an exact nonnegative square, including angles depending on tau.","evidence":"Identity for x=|cos(theta)|, y=|sin(theta)| with a0>0. Gap=(a0*x-2*M*y)^2/(2*a0). A sufficient explicit choice is 0<tau<=beta/(C0+2*M^2/a0+a0/2), also subject to all earlier smallness bounds.","id":"gluing.mixed.schur","module":"gluing","nonzero_residuals":[],"residual_count":2,"status":"PASS"},{"claim":"Jump coefficients of the metric and first derivative vanish at both interfaces.","evidence":"The four exact zero jumps eliminate delta-prime and delta contributions in a one-dimensional distributional second derivative. Distribution theory itself is an analytic input.","id":"gluing.interface.distributional_jump_coefficients","module":"gluing","nonzero_residuals":[],"residual_count":4,"status":"PASS"},{"claim":"Uniform C_X^j estimates, positivity for sufficiently small tau, and compact minima on all plane bundles.","evidence":"UNVERIFIED by finite algebra. Analytic audit finds the supplied argument valid for fixed smooth families on compact X with Delta>=2*b_* h and positive input sectional curvature; these hypotheses are not established by this module.","id":"gluing.analytic.uniformity","module":"gluing","status":"UNVERIFIED"},{"claim":"Common tensor convolution and localized smoothing preserve positive sectional curvature.","evidence":"UNVERIFIED by finite algebra. Analytic audit finds no gap: C1 matching removes singular measures; R=L(D2G)+Q(G,DG); C1 convergence controls Q; nonnegative convolution preserves the decomposable-bivector cone; C2 convergence controls the fixed smooth cutoff annuli. Bounds depend on tau after tau is fixed.","id":"gluing.analytic.smoothing","module":"gluing","status":"UNVERIFIED"},{"claim":"The particular disk metrics satisfy the global positive sectional-curvature and strictly positive boundary-jump hypotheses.","evidence":"Outside this module: no finite Appendix B identity verifies the earlier geometric construction or its premises.","id":"gluing.application.input_metrics","module":"gluing","status":"UNVERIFIED"},{"claim":"Section 6 obtains K>min(c_S,c_N)/2 from Reiser-Wraith Corollary B.","evidence":"Primary source arXiv:2308.06996v2, Theorem A(i) and Corollary B, was read on 2026-09-25: k=1 is sectional curvature and the lower threshold may be any real kappa. This is a checked external theorem interface, not a machine proof. Appendix B alone does not assert this exact half-min constant.","id":"gluing.external.quantitative_bound","module":"gluing","status":"UNVERIFIED"},{"claim":"A/(B(1+x)) gives slack A/(1+x) in A-B*epsilon*x at the threshold","evidence":"Universal rational identity for A,B>0 and x>=0; expressions below instantiate it.","id":"P01","module":"parameter_logic","nonzero_residuals":[],"status":"PASS"},{"claim":"For 0<=epsilon<=1, epsilon^3<=epsilon","evidence":"Exact factorization epsilon-epsilon^3=epsilon(1-epsilon)(1+epsilon).","id":"P02","module":"parameter_logic","nonzero_residuals":[],"status":"PASS"},{"claim":"Every displayed scalar upper bound satisfies its corresponding linear constraint","evidence":"11 exact rational identities. Cubic constraints also use P02. Finiteness and signs of geometric constants are explicit premises.","id":"P03","module":"parameter_logic","nonzero_residuals":[],"status":"PASS"},{"claim":"Northern d<1 follows with a fixed strict reserve from the chosen thresholds","evidence":"d^2=epsilon*(epsilon A0 Fa)^2 exp(epsilon A0 c)/4 < epsilon/8 <=1/8.","id":"P04","module":"parameter_logic","nonzero_residuals":[],"status":"PASS"},{"claim":"The allocated mixed-loss budget leaves exactly epsilon Lambda/8","evidence":"The mixed coefficient budget is exactly epsilon Lambda/8. Other reserves are kappa/2 and 1/(8 epsilon).","id":"P05","module":"parameter_logic","nonzero_residuals":[],"status":"PASS"},{"claim":"The concrete n-dependent curvature suprema and smooth cutoff derivative bounds are finite, and all interval/exponential monotonicity hypotheses apply","evidence":"This is established by the smooth compact geometric construction and elementary analysis; no finite CAS assertion formalizes these global statements.","id":"P06","module":"parameter_logic","status":"UNVERIFIED"},{"claim":"The rational cells cover [1/2,4] without a gap.","evidence":"224 closed rational intervals of width 1/64, exact adjacency checks.","id":"I01","module":"interval_bounds","status":"PASS"},{"claim":"The elementary ratio is greater than 1/2 on every point of [1/2,4].","evidence":"All 224 outward-rounded interval lower endpoints are greater than the exact rational 1/2.","id":"I02","module":"interval_bounds","status":"PASS"},{"claim":"Exact endpoint comparisons close the small and large tails, conditional on the positive sinh-series bound.","evidence":"Small tail R>=2 and large tail R>=8/3; both exceed 1/2. Tail reasoning is analytic, not finite sampling.","id":"I03","module":"interval_bounds","status":"PASS"},{"claim":"The stated strict exponential and fourth-root reserves have rigorous interval separation.","evidence":"Three outward-rounded enclosures separated from their exact rational thresholds.","id":"I04","module":"interval_bounds","status":"PASS"},{"claim":"The concrete northern-only parameter witness obeys both epsilon conditions.","evidence":"Rigorous enclosures show q_s ell_N<1/2 and 0<d<1; delta comparisons are exact rational.","id":"I05","module":"interval_bounds","status":"PASS"},{"claim":"The witness gives strictly positive certified scalar c_N and c_B.","evidence":"Exact rational lower endpoints of both outward-rounded intervals are positive; no underflow or floating-point sign test.","id":"I06","module":"interval_bounds","status":"PASS"},{"claim":"Revised LaTeX labels are unique","evidence":{"duplicates":[],"labels":115},"id":"D01","module":"document","status":"PASS"},{"claim":"All ref/eqref/hyperref targets exist","evidence":{"missing":[]},"id":"D02","module":"document","status":"PASS"},{"claim":"All citation keys have bibliography entries","evidence":{"missing":[]},"id":"D03","module":"document","status":"PASS"},{"claim":"All nine theorem, lemma and proposition statements agree byte-for-byte with the supplied v12","evidence":{"entire_source_unchanged":true,"statement_blocks":9},"id":"D04","module":"document","status":"PASS"},{"claim":"The standard-library-only checker accepts all exported supported certificates","evidence":{"checked":48,"failures":[],"scope":"encoded rational polynomial claims only"},"id":"independent_exact_replay","module":"certificate_replay","status":"PASS"},{"claim":"Adding 1 to the right side of each unexpanded identity is detected","evidence":{"accepted_mutations":[],"all_rejected":true,"mutations_tested":33},"id":"incorrect_certificates_rejected","module":"certificate_replay","status":"PASS"}],"computation_status":"PASS","counts":{"PASS":80,"UNVERIFIED":7},"elapsed_seconds":16.4113,"failed_checks":[],"finished_utc":"2026-09-25T13:34:58.716926+00:00","limitations":{"connection":["Koszul computation assumes the printed principal-frame brackets and curvature equivariance; their geometric derivation is not formalized in a proof assistant.","Normal base-frame curvature must be restored to HHHH; its sectional lower bound kappa is an assumption.","Array estimates use exact Cauchy sum-of-squares kernels plus explicit index contractions and triangle inequalities, not quantifier elimination over all tensors.","Existence and uniformity on the compact disk use smoothness and compactness arguments outside symbolic algebra.","No explicit global M0 or M1 is computed for g_n; they depend on the chosen smooth cutoff and clutching map.","The source proposition is conditional on a uniformly concave phi; on a closed boundaryless base that hypothesis cannot hold. This is vacuity, not a contradiction.","The global theorem also requires the separate north construction, boundary matching, and gluing arguments."],"document":["Document consistency checks do not assess the truth of mathematical statements or the correctness of external references."],"gluing":["PASS denotes exact listed algebraic identities, not global existence or an automatic proof of the positive gluing theorem.","The formal Taylor certificate uses finite jets; smooth-family Taylor remainder bounds, tangential regularity, compactness, and convergence are analytic reasoning.","Curvature components are checked for generic jets in the actual ambient dimension seven at normal slice coordinates; the universal all-dimension index derivation remains the mathematical argument.","Input metrics and strict boundary-jump positivity are hypotheses, not outputs of these checks.","No external theorem or reference content is certified here, and no numeric samples are used."],"interval_bounds":["This module certifies specified scalar inequalities, not every plane of an actual metric by interval subdivision.","Global x>0 control uses the stated analytic positive-series tails in addition to the finite interval domain.","The independent standard-library checker cannot recompute exp, sinh or sqrt; interval output is replayed only by rerunning Sage/MPFI/MPFR.","Smooth cutoff existence, ODE inverse existence, geometric reduction to scalar bounds, and the southern and gluing theorems remain analytic dependencies.","The witness is not a closed-manifold metric and does not select a valid southern connection or A0 for any smooth sphere type."],"north":["Sage checks algebraic identities, exact inequalities and jets, not a formal proof of smooth ODE inversion, smooth cutoff existence, compactness, or smooth polar extension. These steps have been reviewed analytically and no defect was found.","The north proof uses standard submersion and warped-product geometry. The quotient's global disk marking and the identity-to-sigma^n transition are inherited from the earlier topology section, not independently certified by this module.","Boundary equality is proved conditional on the southern connection vanishing in the stated common north gauge and the southern boundary formulas. This module does not certify the southern positive-curvature theorem.","The explicit rational witness is for north alone. General coexistence with a selected southern filling follows conditionally from that filling supplying a positive eps_S and fixed A0.","No claim of complete manuscript correctness or a formal proof assistant verification follows from these finite exact checks."],"parameter_logic":["The bounds are symbolic functions of fixed finite constants. This is not a numerical evaluation of M0 or M1 for 28 individual bundles.","log(2)>1/2 follows by integrating 1/t>1/2 over [1,2); inverse-function and compactness arguments are supplied in the human proof."],"source_binding":["Hashes establish source identity, not formula truth or a correct translation. Human review establishes correspondence to the independently derived code.","The inventory covers labeled equation/align/gather environments, not every inline or unlabeled formula. The full-source hash binds all remaining text."],"topology":["DPR Theorem 1 and its Section 4 identify the attaching map with class n mod 28; this is an externally cited theorem, not a Sage classification.","Equivariance of the exponential, smooth removable extensions, smooth free quotients, and the integer iterate argument are mathematical proofs documented in docs/PROOF_SCOPE.md.","This module does not infer all-28 curvature existence from enumeration of residues."]},"main_theorem_machine_status":"NOT_FULLY_FORMALIZED_OR_MACHINE_CERTIFIED","meaning_of_pass":"All implemented checks, rigorous scalar interval enclosures and exact certificate replays passed; this is not a certificate for the complete all-28 smooth-existence theorem.","revised_sha256":"ea27de068cd78871211d0a1ac102e76b65c7bd1f033267fb24a2c5f8dc423f9e","runs":[{"counts":{"PASS":3},"elapsed_seconds":0.025,"module":"source_binding","returncode":0},{"counts":{"PASS":11},"elapsed_seconds":4.202,"module":"topology","returncode":0},{"counts":{"PASS":26,"UNVERIFIED":1},"elapsed_seconds":4.1813,"module":"connection","returncode":0},{"counts":{"PASS":12,"UNVERIFIED":1},"elapsed_seconds":3.8021,"module":"north","returncode":0},{"counts":{"PASS":11,"UNVERIFIED":4},"elapsed_seconds":1.4611,"module":"gluing","returncode":0},{"counts":{"PASS":5,"UNVERIFIED":1},"elapsed_seconds":1.3225,"module":"parameter_logic","returncode":0},{"counts":{"PASS":6},"elapsed_seconds":1.3156,"module":"interval_bounds","returncode":0},{"counts":{"PASS":4},"elapsed_seconds":0.0266,"module":"document","returncode":0}],"schema_version":1,"source_sha256":"ea27de068cd78871211d0a1ac102e76b65c7bd1f033267fb24a2c5f8dc423f9e","started_utc":"2026-09-25T13:34:42.305207+00:00"}