OneD/QuarticBoundedTest: Lipschitz unnormalised estimate (Tide B1)#15
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Step 2 of the grammar paper precursor sequence: introduces a Lipschitz
test function φ against the bounded-prior quartic Gibbs measure.
Headline `quartic_lipschitz_unnormalised_bounded_prior`:
| ∫_{[-a,a]} φ(w)·exp(-tw⁴/24) − φ(0) · ∫_{[-a,a]} exp(-tw⁴/24) |
≤ K · (1/2) · √(24π/t)
with rate t^{-1/2} in the unnormalised form. Per the Step 2 deliberation
(GPT-5.5 Pro vote: β, "prove unnormalised first"), the proof reduces
the Lipschitz bound to the closed-form full-line first absolute moment
∫_ℝ |w|·exp(-tw⁴/24) dw = (1/2)·√(24π/t), then bounds the bounded-prior
analogue by the full-line value via setIntegral_le_integral.
Sublemmas:
- quartic_integral_w_exp_Ioi: half-line first moment, closed form
- quartic_integral_abs_w_exp_full: full-line first absolute moment
- quartic_integrable_abs_w: integrability
- quartic_integral_abs_w_bounded_prior_le: bounded-prior ≤ full-line
Tide log: sri/projects/primer/tide-log/2026-05-06-tide-bounded-prior-continuous-test.md.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Six-page LaTeX retrospective on the B1 tide step. Covers Step 2's
pivot from normalised to unnormalised (GPT's load-bearing architectural
contribution, with a small correction to its justification noted), the
three roadblocks (Pi.sub vs lambda twice, rpow-to-sqrt cast chain), and
follow-ups (normalised corollary, IsLittleO packaging, generalisation
to L = w^{2k}/(2k)!).
Tide log: sri/projects/primer/tide-log/2026-05-06-tide-bounded-prior-continuous-test.md.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Adds the normalised companion to the unnormalised Lipschitz estimate:
| (∫_{[-a,a]} φ(w)·exp(-tw⁴/24) dw) / Z_a(t) − φ(0) |
≤ K · (1/2) · √(24π/t) / Z_a(t)
where Z_a(t) = ∫_{[-a,a]} exp(-tw⁴/24) dw. The closed-form rate t^{-1/4}
falls out asymptotically once Z_a(t) is comparable to its full-line value
(O(t^{-1/4})); the explicit-constant version with C = 24^{1/4}·√π/Γ(1/4)
≈ 1.082 is now a one-line corollary using quartic_partition_bounded_prior.
Also adds a positivity lemma for the bounded-prior partition function
via the constant lower bound exp(-(t·a⁴/24)) on Icc(-a,a).
Two new theorems:
- quartic_partition_bounded_prior_pos
- quartic_lipschitz_normalised_bounded_prior
Tide log: sri/projects/primer/tide-log/2026-05-06-tide-bounded-prior-continuous-test.md.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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Summary
Step 2 of the grammar paper precursor sequence. Adds a Lipschitz test function
φagainst the bounded-prior quartic Gibbs measure, with closed-form rate constant.Headline:
for
K-Lipschitzφ(i.e.|φ(w) − φ(0)| ≤ K·|w|on[-a,a]). Ratet^{-1/2}in the unnormalised form; the normalisedO(t^{-1/4})follows as a corollary using the existingquartic_partition_bounded_prior.New file:
Laplace/OneD/QuarticBoundedTest.lean(211 lines, 0 sorry / 0 axiom / 0 native_decide).Sublemmas:
quartic_integral_w_exp_Ioi: half-line first moment, closed formquartic_integral_abs_w_exp_full: full-line first absolute momentquartic_integrable_abs_w: integrability witnessquartic_integral_abs_w_bounded_prior_le: bounded-prior ≤ full-line by set monotonicityStep 2 deliberation (Claude ↔ GPT-5.5 Pro) at
sri/projects/primer/tide-log/2026-05-06-tide-bounded-prior-continuous-test.md. GPT vote = β with the tactical refinement "prove unnormalised first" (which dodged the conditional-expectation gymnastics that would have inflated the proof to ~400 lines).Per-tide retrospective at
retrospectives/2026-05-06-tide-bounded-prior-continuous-test.tex(6 pages, 0 overfull hboxes).Test plan
lake buildclean (full Laplace target builds in 129s).scripts/sorriesclean on the new file./tmp/b1_sanity.py: rate constantC ≈ 1.082matches predicted value to machine precision.🤖 Generated with Claude Code