{
  "generated_at_utc": "2026-04-24T14:00:01.997809+00:00",
  "python": "3.13.5 | packaged by Anaconda, Inc. | (main, Jun 12 2025, 16:37:03) [MSC v.1929 64 bit (AMD64)]",
  "platform": "Windows-11-10.0.22631-SP0",
  "results": [
    {
      "slug": "poincare-sharpness",
      "title": "Poincar\u00e9 / Wirtinger sharpness",
      "category": "exact symbolic",
      "method": "sympy",
      "passed": true,
      "summary": "The first cosine eigenfunction attains equality in the time-domain bound.",
      "details": [
        "Used r(t) = cos(pi t / T), the first zero-mean eigenfunction on [0, T].",
        "Verified the temporal mean is exactly zero.",
        "Verified Var_U(r(U)) and (T / pi^2) int_0^T (r'(t))^2 dt simplify to the same closed form."
      ],
      "metrics": {
        "mean": "0",
        "variance": "1/2",
        "rhs": "1/2",
        "gap": "0"
      },
      "math_blocks": [
        "\\bar r = \\frac{1}{T}\\int_0^T \\cos\\!\\left(\\frac{\\pi t}{T}\\right) dt = 0",
        "\\mathrm{Var}_U(r(U)) = \\frac{1}{T}\\int_0^T \\cos^2\\!\\left(\\frac{\\pi t}{T}\\right) dt = \\frac{1}{2}",
        "\\frac{T}{\\pi^2}\\int_0^T (r'(t))^2 dt = \\frac{1}{2}"
      ]
    },
    {
      "slug": "jv-equality",
      "title": "Jacobian-velocity theorem on a deterministic path",
      "category": "exact symbolic",
      "method": "sympy",
      "passed": true,
      "summary": "For X_t = t, g = f, and f(x) = cos(pi x / T), the theorem is exact with beta = 1.",
      "details": [
        "The deployment path is deterministic, so the expectation operators collapse exactly.",
        "Because g = f, Assumption A3 holds with beta = 1 and |grad g dot Xdot| = |J_f Xdot|.",
        "The Jacobian-velocity right-hand side equals the exact volatility from the lemma-sharp example."
      ],
      "metrics": {
        "variance": "1/2",
        "jv_rhs": "1/2",
        "gap": "0"
      },
      "math_blocks": [
        "X_t = t,\\qquad \\dot X_t = 1,\\qquad f(x) = g(x) = \\cos\\!\\left(\\frac{\\pi x}{T}\\right)",
        "J_f(X_t)\\dot X_t = -\\frac{\\pi}{T}\\sin\\!\\left(\\frac{\\pi t}{T}\\right)",
        "\\mathrm{Var}_U(r(U)) = \\frac{T}{\\pi^2}\\int_0^T \\left\\|J_f(X_t)\\dot X_t\\right\\|^2 dt = \\frac{1}{2}"
      ]
    },
    {
      "slug": "composition-chain-rule",
      "title": "Composition case in Remark A3",
      "category": "exact symbolic",
      "method": "sympy",
      "passed": true,
      "summary": "Directional derivatives of g = h o f factor exactly into h'(f) and the score Jacobian term.",
      "details": [
        "Used a nontrivial smooth score field f(x1, x2) = x1^2 + x1 x2 + sin(x2).",
        "Used h(z) = atan(z), so g(x1, x2) = atan(f(x1, x2)).",
        "Simplified grad g dot v - h'(f) grad f dot v to zero exactly."
      ],
      "metrics": {
        "difference": "0"
      },
      "math_blocks": [
        "g(x) = h(f(x)),\\qquad h(z) = \\arctan(z)",
        "\\nabla g(x)\\cdot v = h'(f(x))\\, \\nabla f(x)\\cdot v"
      ]
    },
    {
      "slug": "hazard-rank1-bookkeeping",
      "title": "Rank-1 hazard-score bookkeeping",
      "category": "exact symbolic",
      "method": "sympy",
      "passed": true,
      "summary": "The pointwise algebra behind Proposition 1 is exact; the expectation form follows by averaging these identities.",
      "details": [
        "Worked in an orthonormal basis with v = e1, u = e2, Delta mu / Delta = (a, r), and a generic 2 x 2 Jacobian matrix.",
        "Verified s_t^2 = |a|^2 + |r|^2 from the orthogonal block-drift decomposition.",
        "Verified the directional energy splits into aligned, orthogonal, and overlap terms with coefficients c^2, s^2, and 2cs."
      ],
      "metrics": {
        "s_sq_gap": "0",
        "g_gap": "0",
        "h_gap": "0"
      },
      "math_blocks": [
        "\\frac{\\Delta \\mu_t}{\\Delta} = a v + r u,\\qquad v_t = c v + s u,\\qquad v^\\top u = 0",
        "s_t^2 = \\left\\|\\frac{\\Delta \\mu_t}{\\Delta}\\right\\|^2 = |a|^2 + |r|^2",
        "\\|J_f v_t\\|^2 = c^2\\|J_f v\\|^2 + s^2\\|J_f u\\|^2 + 2cs\\langle J_f v, J_f u\\rangle"
      ]
    },
    {
      "slug": "cross-entropy-derivative",
      "title": "Bernoulli cross-entropy derivative bound",
      "category": "exact symbolic",
      "method": "sympy + dense numeric sweep",
      "passed": true,
      "summary": "The soft-target Bernoulli cross-entropy derivative is sigma(z) - q, so its magnitude never exceeds 1.",
      "details": [
        "Differentiated the scalar loss h(z) = -q log sigma(z) - (1-q) log(1-sigma(z)) exactly.",
        "Verified the closed form h'(z) = sigma(z) - q symbolically.",
        "Checked the beta = 1 bound over a dense grid in z and q."
      ],
      "metrics": {
        "identity_gap": "0",
        "max_abs_derivative_on_grid": 0.9999938558253978
      },
      "math_blocks": [
        "h(z) = -q \\log \\sigma(z) - (1-q)\\log(1-\\sigma(z))",
        "h'(z) = \\sigma(z) - q,\\qquad |h'(z)| \\le 1 \\text{ for } q \\in [0,1]"
      ]
    },
    {
      "slug": "corollary-randomized",
      "title": "Low-rank corollary inequalities",
      "category": "numerical",
      "method": "numpy randomized stress test",
      "passed": true,
      "summary": "The triangle and Frobenius inequalities used in the low-rank corollary hold across randomized orthonormal decompositions.",
      "details": [
        "Generated random orthonormal drift bases V with QR factorization.",
        "Projected residual drift rho into the orthogonal complement of span(V).",
        "Checked the split inequality, the Frobenius inequality, and the combined corollary bound trial by trial."
      ],
      "metrics": {
        "trials": 5000,
        "max_split_violation": -0.5533667361070016,
        "max_frobenius_violation": -0.04947205657132647,
        "max_combined_violation": -0.9995369035779569,
        "min_combined_slack": 0.9995369035779569,
        "max_orthogonality_error": 3.02715132833159e-15
      },
      "math_blocks": [
        "\\|u+w\\|^2 \\le 2\\|u\\|^2 + 2\\|w\\|^2",
        "\\|MVa\\|^2 \\le \\|MV\\|_F^2 \\|a\\|^2",
        "\\begin{aligned} \\|M(Va+\\rho)\\|^2 &\\le 2\\|MV\\|_F^2\\|a\\|^2 \\\\ &\\quad + 2\\|M\\rho\\|^2 \\end{aligned}"
      ]
    },
    {
      "slug": "theorem-numeric-expectation",
      "title": "Jacobian-velocity theorem with a nontrivial expectation",
      "category": "numerical",
      "method": "dense quadrature on a smooth finite-mixture path",
      "passed": true,
      "summary": "A smooth mixture path satisfies the full inequality chain Var <= derivative-energy bound <= Jacobian-velocity bound.",
      "details": [
        "Used X_t = t + Z with a three-point discrete random offset Z and deterministic velocity Xdot = 1.",
        "Used f(x) = cos(pi x) and g(x) = atan(f(x)), so A3 holds with beta <= 1.",
        "Integrated the continuous-time quantities on a dense grid to verify the theorem in a genuine expectation setting."
      ],
      "metrics": {
        "beta_upper": 1.0,
        "volatility": 0.2965336847373633,
        "chain_bound": 0.2988423861194997,
        "jv_bound": 0.5,
        "chain_minus_volatility": 0.002308701382136402,
        "jv_minus_chain": 0.2011576138805003
      },
      "math_blocks": [
        "X_t = t + Z,\\qquad Z \\in \\{-0.2, 0, 0.15\\}",
        "f(x) = \\cos(\\pi x),\\qquad g(x) = \\arctan(f(x)),\\qquad \\beta \\le 1",
        "\\begin{aligned} \\mathrm{Var}_U(r(U)) &\\le \\frac{T}{\\pi^2}\\int_0^T (r'(t))^2 dt \\\\ &\\le \\frac{T}{\\pi^2}\\int_0^T \\mathbb{E}\\!\\left[\\|J_f(X_t)\\dot X_t\\|^2\\right] dt \\end{aligned}"
      ]
    },
    {
      "slug": "synthetic_theorem_summary",
      "title": "Synthetic theorem summary bounds",
      "category": "artifact consistency",
      "method": "pandas cached-summary validation",
      "passed": true,
      "summary": "Every cached synthetic theorem run satisfies the finite-difference, chain-rule, and Jacobian-velocity bounds.",
      "details": [
        "Loaded 80 rows from figures/synthetic_theorem_summary.csv.",
        "Checked volatility <= bound_fd, volatility <= bound_chain, and volatility <= bound_jv wherever those columns exist.",
        "Allowed an absolute tolerance of 1e-4 for cached numerical summaries, since those bounds come from finite grids and Monte Carlo estimates rather than exact algebra."
      ],
      "metrics": {
        "rows": 80,
        "absolute_tolerance": 0.0001,
        "min_slack_bound_fd": 0.0010709632679066,
        "min_slack_bound_chain": -6.602278903039644e-05,
        "min_slack_bound_jv": 0.042822847963994104
      },
      "math_blocks": [
        "\\begin{aligned} \\mathrm{volatility} &\\le \\mathrm{bound\\_fd} \\\\ \\mathrm{volatility} &\\le \\mathrm{bound\\_chain} \\\\ \\mathrm{volatility} &\\le \\mathrm{bound\\_jv} \\end{aligned}"
      ]
    },
    {
      "slug": "synthetic_directional_comparison_summary",
      "title": "Directional comparison bounds",
      "category": "artifact consistency",
      "method": "pandas cached-summary validation",
      "passed": true,
      "summary": "Every cached directional-comparison run preserves the volatility upper bounds used in the paper.",
      "details": [
        "Loaded 140 rows from figures/synthetic_directional_comparison_summary.csv.",
        "Checked volatility <= bound_fd, volatility <= bound_chain, and volatility <= bound_jv wherever those columns exist.",
        "Allowed an absolute tolerance of 1e-4 for cached numerical summaries, since those bounds come from finite grids and Monte Carlo estimates rather than exact algebra."
      ],
      "metrics": {
        "rows": 140,
        "absolute_tolerance": 0.0001,
        "min_slack_bound_fd": 0.0007220234292298001,
        "min_slack_bound_chain": -6.602278903039644e-05,
        "min_slack_bound_jv": 0.042822847963994104
      },
      "math_blocks": [
        "\\begin{aligned} \\mathrm{volatility} &\\le \\mathrm{bound\\_fd} \\\\ \\mathrm{volatility} &\\le \\mathrm{bound\\_chain} \\\\ \\mathrm{volatility} &\\le \\mathrm{bound\\_jv} \\end{aligned}"
      ]
    },
    {
      "slug": "synthetic_directional_misspecification_summary",
      "title": "Misspecification bounds",
      "category": "artifact consistency",
      "method": "pandas cached-summary validation",
      "passed": true,
      "summary": "Every cached misspecification run preserves the volatility upper bounds under subspace rotation.",
      "details": [
        "Loaded 60 rows from figures/synthetic_directional_misspecification_summary.csv.",
        "Checked volatility <= bound_fd, volatility <= bound_chain, and volatility <= bound_jv wherever those columns exist.",
        "Allowed an absolute tolerance of 1e-4 for cached numerical summaries, since those bounds come from finite grids and Monte Carlo estimates rather than exact algebra."
      ],
      "metrics": {
        "rows": 60,
        "absolute_tolerance": 0.0001,
        "min_slack_bound_fd": 0.0009769828884500001,
        "min_slack_bound_chain": 7.011414604999764e-07,
        "min_slack_bound_jv": 0.1150300553307467
      },
      "math_blocks": [
        "\\begin{aligned} \\mathrm{volatility} &\\le \\mathrm{bound\\_fd} \\\\ \\mathrm{volatility} &\\le \\mathrm{bound\\_chain} \\\\ \\mathrm{volatility} &\\le \\mathrm{bound\\_jv} \\end{aligned}"
      ]
    }
  ]
}