Numerical Check: Falsify a Self-Authored Math Claim by Sweep

Empirically stress-test a mathematical claim you wrote but have not proven. The goal is falsification: throw many random instances at the claim and try to break it. A single genuine counterexample kills the claim; a large clean sweep is evidence, never proof.

When to Use

• You wrote a Proposition / Theorem / Conjecture (monotonicity, threshold, comparative-static, inequality, closed-form, limit) and want to know if it's actually true before claiming it.
• numerical-check, "stress-test my conjecture", "find a counterexample to X", "is Q(ρ) really monotone", "does the threshold hold for all …".
• The write-time empirical arm of the mark-unverified rule (self-authored math must be checked before assertion).

When NOT to Use

Position in the verification spectrum

R1 — numerical falsification. Can FALSIFY definitively (a confirmed counterexample refutes the claim) but can never VERIFY (no counterexample ≠ proof). The strongest positive result is INCONCLUSIVE (supported): no counterexample in N draws. Pair with lean-check (R3) to prove the claim once it survives.

Procedure

1. Formalize the claim as a predicate over a domain

Restate the claim as P(x) that must hold for all x in a domain D. Make the failure condition explicit and quantitative.

• "Q(ρ) is monotone decreasing in ρ" → P(instance) := max_i (Q(ρ_{i+1}) − Q(ρ_i)) ≤ tol over a ρ-grid.
• "threshold ρ* separates help/hurt" → P := (Q<p_max) iff (ρ>ρ*).
• Write down the domain D precisely (which parameters, which ranges, which side-conditions — e.g. "mean competence > ½, dispersed").

2. Sample the domain to approximate the TRUE object — not finite-n atoms

This is the step that fools people. If the claim is about a continuous or large-n limit object, a tiny discrete instance is NOT that object — it carries finite-n artifacts (ties, atoms, degenerate medians, staircase discontinuities) that manufacture fake violations.

• If the claim is a large-n / continuous-distribution statement, represent each random instance with dense sampling (hundreds of points), so the computed quantity approximates the limit.
• Generate instances from varied shapes (uniform, skewed, bimodal, heavy-tailed) so the sweep is adversarial, not cherry-picked.

3. Evaluate on an INTERIOR grid, smoothly, with a noise-aware tolerance

• Grid the parameter on the interior (e.g. ρ ∈ [0.02, 0.98]); endpoints breed boundary/degeneracy artifacts.
• Prefer a smooth evaluation (e.g. root-find the crossing) over indicator-quadrature, which quantization-jitters and creates false steps.
• Set the violation tolerance an order of magnitude above the numerical noise floor (measure the floor on a case you believe holds). Too tight → false positives; too loose → misses real breaks.

4. Sweep, count, capture the worst — seeded

• Run over thousands of random instances (uv run --no-project --with numpy --with scipy python; never bare python3). Seed the RNG.
• Count genuine violations; capture the worst counterexample (the instance + violation magnitude) for reporting and for the figure.

5. Diagnose a surprise BEFORE trusting it

If you find violations (or a suspiciously high/low rate), do not report the raw number yet. Check it is not an artifact:

• Re-plot the worst case on a fine grid — is the "violation" a real interior feature, or a jump at an endpoint / a finite-n tie?
• Re-run with denser sampling and odd vs even n — does the rate persist as you approach the continuous limit? An artifact shrinks; a real effect stays.
• Only once it survives these is it real. (Document the surprise + the diagnosis — design-before-results.)

6. Characterize the violators — mechanism, not just a rate

A bare "X% violate" is weak; find when it breaks. Break the sweep down by instance feature (shape, skew, competence-gap, bimodality) and report the driver: "non-monotonicity is a bimodal phenomenon — 18% of bimodal vs ~0% unimodal." This turns a number into a result.

7. Emit the verification report + wire numbers if feeding a paper

Write the report (shape below). If the result feeds a LaTeX paper, emit every number via a generated macro file (results-numbers.tex, no-hardcoded-results) and keep the seeded script in experiments/.

Script skeleton (adapt; keep it seeded + uv-run)

# uv run --no-project --with numpy --with scipy --with matplotlib python <script>.py
import numpy as np; from scipy.stats import norm; from scipy.optimize import brentq
RNG = np.random.default_rng(0)
def quantity(instance, t):        # smooth eval of the claimed object at parameter t
    ...                           # prefer root-find over indicator-quadrature
def sweep(n_trials, nsamp):       # dense instances, interior grid, noise-aware tol
    grid = np.linspace(0.02, 0.98, 90); viol = 0; worst = (-1, None); by = {}
    for _ in range(n_trials):
        inst = draw_instance(RNG, nsamp)          # varied shapes, dense
        if not in_domain(inst): continue
        Q = np.array([quantity(inst, t) for t in grid])
        up = np.diff(Q).max()                     # violation statistic
        if up > worst[0]: worst = (up, inst)
        if up > 1e-4: viol += 1; bump(by, feature(inst))
    return viol, worst, by                        # characterize by feature

Anti-Patterns

• Don't test a continuous/large-n claim with tiny discrete instances — finite-n ties/atoms fabricate violations. (2026-07-04: a small-n sweep read 29%; dense sampling read the real 6%.)
• Don't evaluate at exact parameter endpoints — degeneracies live there.
• Don't set the tolerance near the numerical-noise floor (false positives) or absurdly loose (misses real breaks).
• Don't trust the first surprising number — diagnose artifact-vs-real first.
• Don't report a bare violation rate — characterize the driver.
• Don't claim VERIFIED — numerical can only FALSIFY or SUPPORT. Say "no counterexample in N draws".
• Don't use bare python3 — it's blocked by the uv-only allowlist; use uv run.
• Don't confirm "clean" from a compile log or a grep alone — check the actual object (e.g. grep the rendered PDF for ??, not the build log).

Output — Verification Report (shared *-check shape)

Write to reviews/<scope>/verify-numerical/<YYYY-MM-DD-HHMM>.md:

claim:    <the exact statement tested, with its domain>
method:   R1 numerical falsification (N=<trials>, nsamp=<density>, grid=<interior>, tol=<t>)
verdict:  FALSIFIED | INCONCLUSIVE (supported: no counterexample in N) | INCONCLUSIVE | ERROR
evidence: <worst counterexample instance + magnitude>  OR  <"no counterexample; sweep params">
mechanism:<what feature drives violations, if any>
reproduce: uv run --no-project --with ... python experiments/<script>.py   (seed=<s>)

Verification (did this skill work?)

• The script runs seeded via uv run and prints a violation count + worst case.
• The verdict is one of the four values; VERIFIED is never emitted.
• If a paper consumes the result, numbers come from a generated macro file (nothing hand-typed).

Worked example — 2026-07-04 (median-collapse paper)

Claim: "Q₀(∞;ρ) is monotone decreasing in ρ for all competence distributions (mean>½, dispersed)" → single-threshold conjecture.

• Naive small-n sweep: 29% violations → suspicious; worst case was a balanced even-n council.
• Diagnosed: finite-even-n ties + endpoint staircase — an artifact.
• Corrected (dense sampling, interior grid, smooth root-find eval, tol=10⁻⁴): 6% real violations, worst step +0.013.
• Characterized: non-monotonicity is a bimodal phenomenon (18% of bimodal vs 3/3363 unimodal).
• Verdict: FALSIFIED — the conjecture was removed and replaced with the characterized finding.