Numerical Claim Dashboard

Interactive results for all qc-diffusion numerical validation gates

Code

cards = [
  ["NUM-1", "Goldstein–Kac 1D", results.kac1d.assessment],
  ["NUM-2", "Isotropic 2D", results.isotropic2d.assessment],
  ["NUM-3", "Factor ordering", results.factorOrdering.assessment],
  ["NUM-4", "Bianchi IX", results.bianchiIX.assessment],
  ["NUM-5", "Intertwiner diffusion", results.graphDiffusion.assessment],
  ["NUM-6", "Signed Bianchi IX particles", results.signedBianchiParticles.assessment],
  ["NUM-7 · T15a", "LRS Bianchi I persistent walk", results.t15a.overallAssessment],
  ["T15b", "Euclidean persistent random flight", {
    status: "passed",
    qualification: "The seeded planar walk passes the exact-MSD and causal-radius checks."
  }],
  ["T15b · ML v2", "Masoliver–Lindenberg fluid limit", {
    status: "provisional",
    qualification: "The long-scale comparison approaches the telegraph approximation; the longest-time highest mode remains 2.84 Monte Carlo standard errors away."
  }]
]

html`
<div class="metric-strip">
  <div class="metric"><div class="metric-value">${registry.runs.length}</div><div class="metric-label">completed programs</div></div>
  <div class="metric"><div class="metric-value">${cards.filter(c => c[2].status === "passed").length}</div><div class="metric-label">passed gates</div></div>
  <div class="metric"><div class="metric-value">${cards.filter(c => c[2].status === "failed").length}</div><div class="metric-label">falsified target claims</div></div>
  <div class="metric"><div class="metric-value">${cards.filter(c => c[2].status === "provisional").length}</div><div class="metric-label">provisional programs</div></div>
</div>
<div class="gate-grid">
  ${cards.map(([id, title, assessment]) => html`
    <article class="gate-card">
      <span class="gate-status ${assessment.status}">${assessment.status}</span>
      <h3>${id} · ${title}</h3>
      <p>${assessment.qualification}</p>
    </article>
  `)}
</div>`

NUM-1 · Exact 1D calibration

Code

kacPlot = results.kac1d.plot.x.map((x, index) => ({
  x,
  pde: results.kac1d.plot.pde[index],
  monteCarlo: results.kac1d.plot.monteCarlo[index]
}))

Plot.plot({
  height: 360,
  x: {label: "x"},
  y: {label: "density", grid: true},
  color: {legend: true},
  marks: [
    Plot.line(kacPlot, {x: "x", y: "pde", stroke: "#0d6efd", tip: true}),
    Plot.line(kacPlot, {x: "x", y: "monteCarlo", stroke: "#dc3545", opacity: 0.78, tip: true})
  ]
})

Code

Inputs.table(results.kac1d.levels, {
  columns: ["grid", "particles", "l1", "l2", "linf", "frontViolations"],
  header: {grid: "Grid", particles: "Walkers", l1: "L¹", l2: "L²", linf: "L∞", frontViolations: "Front violations"}
})

NUM-2 · Two-dimensional falsification test

The 2D ensemble respects finite propagation, but finite propagation is not the same as satisfying the scalar telegraph PDE.

Code

radial = results.isotropic2d.plot.radius.map((radius, index) => ({
  radius,
  pde: results.isotropic2d.plot.pde[index],
  monteCarlo: results.isotropic2d.plot.monteCarlo[index]
}))

Plot.plot({
  height: 360,
  x: {label: "radius"},
  y: {label: "azimuthally averaged density", grid: true},
  marks: [
    Plot.line(radial, {x: "radius", y: "pde", stroke: "#0d6efd", tip: true}),
    Plot.line(radial, {x: "radius", y: "monteCarlo", stroke: "#dc3545", tip: true})
  ]
})

Code

Inputs.table(results.isotropic2d.levels, {
  columns: ["grid", "particles", "l1", "l2", "linf", "frontViolations"],
  header: {grid: "Grid", particles: "Walkers", l1: "L¹", l2: "L²", linf: "L∞", frontViolations: "Front violations"}
})

NUM-3 · Factor-ordering sweep

Code

Inputs.table(results.factorOrdering.rows, {
  columns: ["B", "dampingA", "stochasticPoissonRate", "mass", "l2", "ordinaryStochasticInterpretation"],
  header: {
    B: "B",
    dampingA: "a = -B/2",
    stochasticPoissonRate: "Poisson rate",
    mass: "Mass",
    l2: "L² norm",
    ordinaryStochasticInterpretation: "Positive-rate interpretation"
  }
})

NUM-4 · Bianchi IX signed amplitude

Code

ixSlice = results.bianchiIX.plot.axis.map((beta, index) => ({
  beta,
  amplitude: results.bianchiIX.plot.centerSlice[index]
}))

Plot.plot({
  height: 340,
  x: {label: "β+ at β− = 0"},
  y: {label: "signed field", grid: true},
  marks: [
    Plot.ruleY([0], {stroke: "#888"}),
    Plot.line(ixSlice, {x: "beta", y: "amplitude", stroke: "#6f42c1", tip: true})
  ]
})

Code

Inputs.table(results.bianchiIX.levels, {
  columns: ["grid", "alphaStepActual", "sourceMinimum", "sourceMaximum", "densityMinimum", "densityMaximum", "negativeFraction", "signedMass"],
  header: {
    grid: "Grid",
    alphaStepActual: "Δα",
    sourceMinimum: "Source min",
    sourceMaximum: "Source max",
    densityMinimum: "Field min",
    densityMaximum: "Field max",
    negativeFraction: "Negative fraction",
    signedMass: "Signed mass"
  }
})

Adaptive outer-wall run

Code

Inputs.table(results.bianchiIX.outerWall.levels, {
  columns: ["grid", "functionEvaluations", "sourceMinimum", "sourceMaximum", "densityMinimum", "densityMaximum", "negativeFraction", "signedMass"],
  header: {
    grid: "Grid",
    functionEvaluations: "RHS evaluations",
    sourceMinimum: "Source min",
    sourceMaximum: "Source max",
    densityMinimum: "Field min",
    densityMaximum: "Field max",
    negativeFraction: "Negative fraction",
    signedMass: "Signed mass"
  }
})

Code

outerRefinement = results.bianchiIX.outerWall.refinement
Inputs.table([{
  maximumRelativeSignedMassChange: outerRefinement.maximumRelativeSignedMassChange,
  finestCommonInteriorL1Change: outerRefinement.finestCommonInteriorL1Change,
  massConvergedAtFivePercent: outerRefinement.massConvergedAtFivePercent,
  fieldDifferenceDecreases: outerRefinement.fieldDifferenceDecreases
}], {
  header: {
    maximumRelativeSignedMassChange: "Largest adjacent mass change",
    finestCommonInteriorL1Change: "Finest common-interior L¹ change",
    massConvergedAtFivePercent: "Mass below 5%",
    fieldDifferenceDecreases: "Field difference decreases"
  }
})

The high-resolution sequence uses four fixed-box grids. It satisfies the signed-mass gate, but the finest common-interior field difference remains appreciable. Agreement of signed mass alone is not enough for a signed field, so the calculation remains a solver and sign diagnostic rather than a converged microscopic Bianchi IX model.

NUM-5 · Intertwiner-sector diffusion

Code

graphPlot = results.graphDiffusion.plot.node.map((node, index) => ({
  node,
  exact: results.graphDiffusion.plot.exact[index],
  monteCarlo: results.graphDiffusion.plot.monteCarlo[index]
}))

Plot.plot({
  height: 330,
  x: {label: "admissible intermediate spin k"},
  y: {label: "probability", grid: true},
  marks: [
    Plot.line(graphPlot, {x: "node", y: "exact", stroke: "#198754", tip: true}),
    Plot.dot(graphPlot, {x: "node", y: "monteCarlo", fill: "#fd7e14", r: 3, tip: true})
  ]
})

NUM-6 · Signed-particle Bianchi IX gate

Code

signedParticlePlot = results.signedBianchiParticles.plot.axis.map((beta, index) => ({
  beta,
  deterministic: results.signedBianchiParticles.plot.deterministic[index],
  estimator: results.signedBianchiParticles.plot.estimate[index]
}))

Plot.plot({
  height: 330,
  x: {label: "β+ at β− = 0"},
  y: {label: "signed field", grid: true},
  color: {legend: true},
  marks: [
    Plot.line(signedParticlePlot, {x: "beta", y: "deterministic", stroke: "#6f42c1", tip: true}),
    Plot.line(signedParticlePlot, {x: "beta", y: "estimator", stroke: "#dc3545", tip: true})
  ]
})

Code

Inputs.table(results.signedBianchiParticles.levels, {
  columns: ["particles", "l1", "l2", "linf", "signedMassDeterministic", "signedMassEstimator", "maximumAbsWeight"],
  header: {
    particles: "Particles",
    l1: "L¹",
    l2: "L²",
    linf: "L∞",
    signedMassDeterministic: "Deterministic signed mass",
    signedMassEstimator: "Particle signed mass",
    maximumAbsWeight: "Largest absolute weight"
  }
})

The signed estimator is unbiased for the fixed finite-difference update in principle, but the curvature walls generate a severe sign problem. State-wise annihilation reduces the instability without yet meeting the fixed-grid error gate. This is therefore an amplitude-level prototype, not a microscopic trajectory realization.

NUM-7 · T15a LRS Bianchi I persistent walk

The preregistered T15a result contains six initial-profile and factor-ordering cases, four population levels, five output times, and the per-seed density, current, moment, mass, and volume diagnostics. All six cases pass the final density gate: at 320,000 walkers their mean relative \(L^1\) errors range from 0.0281 to 0.0355, with 95% upper bounds from 0.0294 to 0.0366. At 640,000 walkers, the single-seed errors range from 0.0187 to 0.0243.

Code

t15aRows = results.t15a.cases.flatMap(c => c.levels.map(level => ({
  case: `${c.profile} · ${c.ordering}`,
  population: level.population,
  meanL1: level.finalDensityRelativeL1.mean,
  upper95: level.finalDensityRelativeL1.ci95Upper,
  replicates: level.finalDensityRelativeL1.n
})))

Plot.plot({
  height: 360,
  x: {label: "Walkers", type: "log"},
  y: {label: "Final relative density L¹ error", grid: true},
  color: {legend: true},
  marks: [
    Plot.line(t15aRows, {x: "population", y: "meanL1", stroke: "case", marker: "circle", tip: true}),
    Plot.ruleY([0.05], {stroke: "#dc3545", strokeDasharray: "5,5"})
  ]
})

Code

Inputs.table(results.t15a.cases.map(c => ({
  profile: c.profile,
  ordering: c.ordering,
  B: c.B,
  flipRate: c.flipRate,
  initialCurrent: c.initialCurrent,
  l1At320k: c.assessment.meanAt320k,
  upper95At320k: c.assessment.ci95UpperAt320k,
  gate: c.assessment.densityGate,
  status: c.assessment.status
})), {
  columns: ["profile", "ordering", "B", "flipRate", "initialCurrent", "l1At320k", "upper95At320k", "gate", "status"],
  header: {profile: "Profile", ordering: "Ordering", B: "B", flipRate: "Flip rate", initialCurrent: "Initial current", l1At320k: "Mean L¹ at 320k", upper95At320k: "95% upper bound", gate: "Gate", status: "Status"}
})

Code

Inputs.table(t15aRows, {
  columns: ["case", "population", "meanL1", "upper95", "replicates"],
  header: {case: "Profile · ordering", population: "Walkers", meanL1: "Mean final L¹", upper95: "95% upper bound", replicates: "Replicates"}
})

The full run record retains every output-time diagnostic, seed, case, and population replicate. Download the full T15a record.

T15b · Euclidean persistent random flight

The ordinary planar walk runs at constant speed with independent uniform heading resets at Poisson times. Across five snapshot times, each using 250,000 walkers, the sampled mean-square displacement agrees with its exact formula within three standard errors; every sampled path stays inside the causal radius \(r\leq vt\). The exact density law is the position-heading transport equation. The radial snapshots below show the simulated spatial density without treating it as an exact scalar telegraph solution. The annular diagnostic measures local heading-position correlations while leaving the isotropic reset rule unchanged. Download the full T15b Euclidean run.

Code

t15b = results.t15bEuclidean
msdRows = t15b.momentResults.map(d => ({...d, exact: d.analyticMeanDisplacementSquared}))
radialRows = t15b.radialSnapshots.flatMap(snapshot => snapshot.radius.map((radius, i) => ({
  time: snapshot.time,
  radius,
  density: snapshot.radialProbabilityDensity[i]
})))
anisotropyRows = t15b.radialSnapshots.flatMap(snapshot => snapshot.localAnisotropy
  .filter(d => d.radialAlignmentA2 !== null)
  .map(d => ({...d, time: snapshot.time})))

Plot.plot({
  height: 330,
  x: {label: "Time"},
  y: {label: "Mean-square displacement", grid: true},
  color: {legend: true},
  marks: [
    Plot.line(msdRows, {x: "time", y: "meanDisplacementSquared", stroke: "#0d6efd", marker: "circle", tip: true}),
    Plot.line(msdRows, {x: "time", y: "exact", stroke: "#212529", strokeDasharray: "5,5"})
  ]
})

Code

Inputs.table(t15b.momentResults, {
  columns: ["time", "meanDisplacementSquared", "standardError", "analyticMeanDisplacementSquared", "standardizedError", "covarianceXX", "covarianceYY", "covarianceXY"],
  header: {time: "Time", meanDisplacementSquared: "Sample MSD", standardError: "MSD SE", analyticMeanDisplacementSquared: "Exact MSD", standardizedError: "Difference / SE", covarianceXX: "Covariance xx", covarianceYY: "Covariance yy", covarianceXY: "Covariance xy"}
})

The added annular diagnostic measures the local second angular harmonic of the current headings relative to the radial direction. Positive \(A_2\) indicates radial alignment, negative \(A_2\) tangential alignment, and \(A_2=0\) no radial/tangential second-harmonic preference in that annulus. The companion \(B_2\) estimates the radial–tangential cross component and should average to zero under rotational symmetry.

Code

Plot.plot({
  height: 350,
  x: {label: "Radius from source"},
  y: {label: "Radial second-harmonic alignment A₂", domain: [-1, 1], grid: true},
  color: {legend: true},
  marks: [Plot.line(anisotropyRows, {x: "radius", y: "radialAlignmentA2", stroke: d => `t = ${d.time}`, marker: "circle", tip: true})]
})

Code

Plot.plot({
  height: 360,
  x: {label: "Radius"},
  y: {label: "Radial probability density", grid: true},
  color: {legend: true},
  marks: [Plot.line(radialRows, {x: "radius", y: "density", stroke: d => `t = ${d.time}`, tip: true})]
})

T15b · Masoliver–Lindenberg fluid-limit comparison

The approximation tested is

\[ p_{tt}+\frac{\lambda}{2}p_t=\frac{c^2}{4}\Delta p. \]

The exact random-flight Fourier–Laplace transform and the telegraph approximation’s transform are both recorded. The transform discrepancy shrinks along each tested small-\(s\), small-\(k\) ray. Protocol v2 extends the seeded Fourier-mode comparison to \(\lambda t=128\); absolute differences decrease over that sequence at all three scaled modes. Its largest final difference is 2.84 Monte Carlo standard errors at \(\kappa=1.5\). Protocol v1 is retained below as an earlier, shorter-horizon run. Download the full v1 record or the full v2 record.

Code

mlV1 = results.t15bMlV1
mlV2 = results.t15bMlV2
transformRows = mlV2.transformComparisons.map(d => ({...d, ray: `ck/s = ${d.raySlopeCkOverS}`}))
modeRowsV2 = mlV2.simulatedModeComparisons.map(d => ({...d, mode: `κ = ${d.scaledWavenumberKappa}`}))
modeRowsV1 = mlV1.simulatedModeComparisons.map(d => ({...d, mode: `κ = ${d.scaledWavenumberKappa}`}))

Plot.plot({
  height: 330,
  x: {label: "Dimensionless s", type: "log"},
  y: {label: "Relative transform error", type: "log", grid: true},
  color: {legend: true},
  marks: [Plot.line(transformRows, {x: "dimensionlessS", y: "relativeError", stroke: "ray", marker: "circle", tip: true})]
})

Code

Plot.plot({
  height: 350,
  x: {label: "λt", type: "log"},
  y: {label: "|Empirical characteristic − telegraph value| / Monte Carlo SE", grid: true},
  color: {legend: true},
  marks: [Plot.line(modeRowsV2, {x: "lambdaTime", y: "differenceInStandardErrors", stroke: "mode", marker: "circle", tip: true})]
})

Protocol v2 transform table (18 parameter pairs):

Code

Inputs.table(mlV2.transformComparisons, {
  columns: ["dimensionlessS", "dimensionlessCK", "raySlopeCkOverS", "exactTransform", "mlFluidTransform", "relativeError"],
  header: {dimensionlessS: "s/λ", dimensionlessCK: "ck/λ", raySlopeCkOverS: "ck/s", exactTransform: "Exact transform", mlFluidTransform: "ML transform", relativeError: "Relative error"}
})

Protocol v1 transform table (18 parameter pairs):

Code

Inputs.table(mlV1.transformComparisons, {
  columns: ["dimensionlessS", "dimensionlessCK", "raySlopeCkOverS", "exactTransform", "mlFluidTransform", "relativeError"],
  header: {dimensionlessS: "s/λ", dimensionlessCK: "ck/λ", raySlopeCkOverS: "ck/s", exactTransform: "Exact transform", mlFluidTransform: "ML transform", relativeError: "Relative error"}
})

Seeded mode comparison tables (v1 and v2, 15 observations each):

Code

Inputs.table(modeRowsV2, {
  columns: ["time", "lambdaTime", "scaledWavenumberKappa", "empiricalCharacteristic", "monteCarloStandardError", "mlTelegraphCharacteristic", "absoluteDifference", "differenceInStandardErrors", "diffusionLimitCharacteristic", "empiricalMeanSquaredDisplacement", "analyticMeanSquaredDisplacement", "msdStandardizedError"],
  header: {time: "Time", lambdaTime: "λt", scaledWavenumberKappa: "κ", empiricalCharacteristic: "Empirical", monteCarloStandardError: "MC SE", mlTelegraphCharacteristic: "ML telegraph", absoluteDifference: "Absolute difference", differenceInStandardErrors: "Difference / SE", diffusionLimitCharacteristic: "Diffusion limit", empiricalMeanSquaredDisplacement: "Sample MSD", analyticMeanSquaredDisplacement: "Exact MSD", msdStandardizedError: "MSD difference / SE"}
})

Protocol v1 seeded mode observations:

Code

Inputs.table(modeRowsV1, {
  columns: ["time", "lambdaTime", "scaledWavenumberKappa", "empiricalCharacteristic", "monteCarloStandardError", "mlTelegraphCharacteristic", "absoluteDifference", "differenceInStandardErrors", "diffusionLimitCharacteristic", "empiricalMeanSquaredDisplacement", "analyticMeanSquaredDisplacement", "msdStandardizedError"],
  header: {time: "Time", lambdaTime: "λt", scaledWavenumberKappa: "κ", empiricalCharacteristic: "Empirical", monteCarloStandardError: "MC SE", mlTelegraphCharacteristic: "ML telegraph", absoluteDifference: "Absolute difference", differenceInStandardErrors: "Difference / SE", diffusionLimitCharacteristic: "Diffusion limit", empiricalMeanSquaredDisplacement: "Sample MSD", analyticMeanSquaredDisplacement: "Exact MSD", msdStandardizedError: "MSD difference / SE"}
})

The raw JSON records preserve both protocols, all seeded observations, transform evaluations, checks, environment, and provenance. The statistical trend supports an asymptotic fluid-limit comparison; it is not an exact finite-scale closure or pointwise validation at the ballistic front.

Reproducibility record

Code

Inputs.table(registry.runs, {
  columns: ["runId", "task", "phase", "status", "description"],
  header: {runId: "Run ID", task: "Program", phase: "Purpose", status: "Run status", description: "Description"}
})