{ "date": "2026-10-07", "scope": "Focused claim checks, not formal verification or blanket manuscript validation.", "python": "3.12.10", "numpy": "2.4.0", "scipy": "1.16.3", "sympy": "1.14.0", "script_sha256": "46f2ad2aa9b796dfb648793f9ba2ef30154f560b8e5d74a5fcee7708c9a39e54", "checks": [ { "id": "R1", "title": "Component error-budget transfer", "kind": "exact rational enumeration", "outcome": "check_satisfied", "observations": { "pairs": 6561, "bound": "|B(e)-B(f)| <= sum_i |e_i-f_i| for B(e)=1-product_i(1-e_i)" } }, { "id": "R2", "title": "Threshold mass defeats unconditional first-order decision transfer", "kind": "exact counterexample", "outcome": "counterexample_confirmed", "observations": { "construction": "All mass at scores -epsilon and +epsilon; threshold 0.", "examples": [ { "epsilon": "1/10", "score_difference": "1/5", "first_alert_probability": 0, "second_alert_probability": 1 }, { "epsilon": "1/100", "score_difference": "1/50", "first_alert_probability": 0, "second_alert_probability": 1 }, { "epsilon": "1/10000", "score_difference": "1/5000", "first_alert_probability": 0, "second_alert_probability": 1 } ] } }, { "id": "R3", "title": "A constant induced metric does not make arc and chord equal", "kind": "analytic example with numerical values", "outcome": "counterexample_confirmed", "observations": { "embedding": "f(t)=(cos t,sin t)", "induced_metric": 1, "covariant_derivative_of_metric": 0, "jacobian_norm": 1, "semicircle_arc": 3.141592653589793, "endpoint_chord": 2.0, "relative_distortion": 0.3633802276324186 } }, { "id": "R4", "title": "Population Cantelli posterior bound including class priors", "kind": "exact finite-distribution check", "outcome": "check_satisfied", "observations": { "cases": 81, "scope": "Population moments only; no plug-in confidence claim." } }, { "id": "D1", "title": "Chord identity and transverse highest-weight spin overlaps", "kind": "symbolic identities and matrix-exponential checks", "outcome": "check_satisfied", "observations": { "spin_cases": 205, "max_absolute_error": 1.1102230246251565e-15, "scope": "d=|sin(theta/2)|, K=2d; specified transverse rotation, not arbitrary dynamics." } }, { "id": "D2", "title": "Half-angle overlap does not descend to the arrow-axis quotient", "kind": "exact counterexample", "outcome": "counterexample_confirmed", "observations": { "q(0)": "[1:0]", "q(pi)": "[-1:0]=[1:0]", "S(0)": 1, "S(pi)": 0, "scope": "Keep the quotient angle and the Bloch/rotation angle distinct." } }, { "id": "D3", "title": "Finite aromatic-ring gap and the linear large-ring limit", "kind": "Hermitian eigensolver versus analytic formula", "outcome": "check_satisfied", "observations": { "cases": 84, "max_absolute_error": 5.495603971894525e-15, "formula": "sin((pi-theta)/N)/sin(pi/N), N=4k+2, 0<=theta<=pi", "max_deviations_from_linear_limit": { "6": 0.018033988749894925, "10": 0.006378671712677697, "14": 0.003238820865313796, "50": 0.0002527603285573665 } } }, { "id": "D4", "title": "Commutativity alone does not force linear gap closure", "kind": "exact counterexample", "outcome": "counterexample_confirmed", "observations": { "family": "diag(0, 1+theta^2)", "gap": "1+theta^2", "commutator": "0 for every pair of parameter values" } }, { "id": "C1", "title": "Flatness need not imply trivial global holonomy", "kind": "analytic example with numerical value", "outcome": "counterexample_confirmed", "observations": { "base": "circle", "connection": "A=(1/2) dtheta in a U(1) trivialization", "curvature": 0, "loop_holonomy_real": -1.0, "loop_holonomy_imag": 1.2246467991473532e-16, "scope": "The manuscript main theorem adds simple connectivity; its earlier unqualified lemma needs that restriction." } }, { "id": "C2", "title": "No uniform derivative L2 bound from a connection L2 norm", "kind": "analytic counterexample family", "outcome": "counterexample_confirmed", "observations": { "base": "flat torus [0,2pi]^2", "connection": "A_n=sin(nx)dy/n", "curvature": "cos(nx) dx wedge dy", "norm_A_squared": "2*pi^2/n^2", "YM_energy": "pi^2", "ratios": { "1": 2.0, "2": 0.5, "4": 0.125, "8": 0.03125, "16": 0.0078125, "32": 0.001953125 }, "scope": "Refutes the unrestricted compact-base L2 lemma; the simply-connected main claim also needs a defined, justified norm comparison." } }, { "id": "A1", "title": "A positive universal pointwise lower bound fails at interpolation nodes", "kind": "exact counterexample", "outcome": "counterexample_confirmed", "observations": { "local_function": "f(x)=x^2", "linear_interpolant_on_0_h": "h*x", "hessian": 2, "error_at_0": 0, "error_at_h": 0, "maximum_error": "h^2/4", "compact_manifold_example": "f(t)=1-cos(t) on S1 also has error 0 and Hessian 1 at an interpolated node t=0." } }, { "id": "A2", "title": "A concrete uniform-norm interpolation bound with quadratic scaling", "kind": "numerical check of exact polynomial formula", "outcome": "check_satisfied", "observations": { "examples": [ { "h": 1.0, "maximum_error": 0.25, "sup_second_derivative": 2.0, "bound": 0.25 }, { "h": 0.5, "maximum_error": 0.0625, "sup_second_derivative": 2.0, "bound": 0.0625 }, { "h": 0.25, "maximum_error": 0.015625, "sup_second_derivative": 2.0, "bound": 0.015625 }, { "h": 0.125, "maximum_error": 0.00390625, "sup_second_derivative": 2.0, "bound": 0.00390625 } ], "statement": "For C2 f on [a,b], ||f-I_h f||_infinity <= h^2 sup|f_second|/8." } }, { "id": "P1", "title": "Lift, conditioning, symmetry and independence", "kind": "exact rational enumeration", "outcome": "check_satisfied", "observations": { "joint_tables": 1230, "cell_counts": "0..5; exclude undefined marginal denominators", "scope": "Ordinary probability identities; no Riemannian curvature is inferred." } }, { "id": "Z1", "title": "Connected partitions: independent enumeration and edge-addition monotonicity", "kind": "exact exhaustive enumeration", "outcome": "check_satisfied", "observations": { "all_labeled_graphs_n_1_to_5": 1099, "single_edge_additions": 5325, "connected_graphs_with_sandwich_bound": 772, "family_counts": [ { "n": 1, "tree": 1, "cycle": null, "complete": 1 }, { "n": 2, "tree": 2, "cycle": null, "complete": 2 }, { "n": 3, "tree": 4, "cycle": 5, "complete": 5 }, { "n": 4, "tree": 8, "cycle": 12, "complete": 15 }, { "n": 5, "tree": 16, "cycle": 27, "complete": 52 }, { "n": 6, "tree": 32, "cycle": 58, "complete": 203 }, { "n": 7, "tree": 64, "cycle": 121, "complete": 877 }, { "n": 8, "tree": 128, "cycle": 248, "complete": 4140 }, { "n": 9, "tree": 256, "cycle": 503, "complete": 21147 }, { "n": 10, "tree": 512, "cycle": 1014, "complete": 115975 } ], "scope": "Finite checks supplement the injection and counting proofs; this is the established connected-set-partition object." } } ] }