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Add Erdős Problem 524 (Salem–Zygmund sup norm)#3758

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kieranmcshane wants to merge 58 commits intogoogle-deepmind:mainfrom
kieranmcshane:add-erdos-524
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Add Erdős Problem 524 (Salem–Zygmund sup norm)#3758
kieranmcshane wants to merge 58 commits intogoogle-deepmind:mainfrom
kieranmcshane:add-erdos-524

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@kieranmcshane kieranmcshane commented Apr 16, 2026 •
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Summary

Formalizes Erdős Problem #524: the almost-sure order of magnitude of the sup norm of random polynomials with i.i.d. Rademacher coefficients on [-1, 1], per Chojecki's January 2026 resolution.

What is fully proven (no sorry)

Two-walk sandwich (Corollary 3) — FULLY PROVEN

  • Lower bound: max(|S_n|, |T_n|) ≤ M_n via endpoint evaluation
  • Upper bound: M_n ≤ max(max_k |S_k|, max_k |T_k|) via Abel summation identity + weight telescoping

Subgaussian tails (Proposition 4) — FULLY PROVEN

  • ℙ(M_n ≥ u√n) ≤ 4 exp(-u²/2) for n ≥ 1
  • Set containment via contrapositive of Abel bound + Finset.sup'_lt_iff
  • Union bound assembly with measure_union_le
  • Note: added 0 < n hypothesis (the original statement was false for n=0)

Running-max tail bound — FULLY PROVEN

  • ℙ(max_k |S_k| ≥ u√n) ≤ 2 exp(-u²/2) via union bound on one-sided bounds

Rademacher walk symmetry — FULLY PROVEN

  • ℙ(S_m ≥ 0) ≥ 1/2 via IdentDistrib.pi + measure_mem_eq
  • Rademacher symmetry: each a_k and -a_k identically distributed (four-case analysis on {±1} ae support)

Submartingale construction (partial)

  • Adapted: walk adapted to natural filtration + exp composition
  • Integrable: iIndepFun.integrable_exp_mul_sum with integrable_exp_mul_of_mem_Icc
  • One-step condition: pullout (condExp_mul_of_aestronglyMeasurable_left) + independence (condExp_indep_eq)
  • Remaining sorry: E[exp(λ·X)] ≥ 1 for Rademacher X (cosh(λ) ≥ 1)

Supporting infrastructure

  • randomPoly_at_one/neg_one, abs_randomPoly_le_supNorm, walk/alternatingWalk_le_supNorm
  • supNorm_le, abel_identity, weight_eq, abel_bound_nonneg
  • randomPoly_neg, walk_neg_eq_alternatingWalk, isRademacherSequence_neg_mul
  • rademacher_ae_mem_pm_one

What remains sorry and why

Theorem # sorrys Blocker
erdos_524 (meta) 1 Wraps sharp_upper_envelope + sparse_lower_envelope
sharp_upper_envelope (Thm 6) 1 Kolmogorov LIL not in Mathlib
levy_maximal_ineq 1 First-crossing-time decomposition + independence (~80 lines)
one_sided_running_max 2 cosh(λ) ≥ 1 computation + Doob assembly
sparse_lower_envelope (Thm 18) 1 2D KMT + Gao-Li-Wellner (multi-year)
exact_lower_constant (Rmk 19) 1 Open in mathematics

Verification

lake env lean FormalConjectures/ErdosProblems/524.lean — zero errors, 6 sorry warnings.

🤖 Generated with Claude Code

@github-actions github-actions bot added the erdos-problems Erdős Problems label Apr 16, 2026
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kieranmcshane commented Apr 16, 2026

@googlebot I signed it!

@kieranmcshane kieranmcshane mentioned this pull request Apr 16, 2026
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@kieranmcshane kieranmcshane force-pushed the add-erdos-524 branch 2 times, most recently from b9309e1 to 4158425 Compare April 16, 2026 09:25
@kieranmcshane
feat: Add Erdős Problem 524 with machine-checked proofs
a4f8960 
Formalize and partially certify Chojecki (Jan 2026) resolution of
Erdős Problem 524 on sup norm of random ±1 polynomials on [-1,1].

Machine-verified proofs (sorry-free, Lean kernel checked):
- Two-walk sandwich (Corollary 3): Abel summation + weight telescoping
- Subgaussian tails assembly: measure_mono + union bound + 2+2=4
- Running-max tail bound: set decomposition + neg Rademacher + arithmetic
- Submartingale adapted + integrable cases: Filtration.natural + exp composition
- isRademacherSequence_neg_mul: iIndepFun.comp + parity case split
- rademacher_ae_mem_pm_one, rademacher_walk_nonneg_prob, identDistrib helpers

Remaining sorry's (infrastructure gaps, not paper errors):
- one_sided_running_max: Doob maximal_ineq wiring (~50 lines)
- sharp_upper_envelope: Kolmogorov LIL (not in Mathlib)
- sparse_lower_envelope: 2D KMT coupling (multi-year project)
- exact_lower_constant: open problem in mathematics

Paper: https://www.ulam.ai/research/erdos524.pdf
kieranmcshane and others added 7 commits April 16, 2026 11:43
@kieranmcshane @claude
feat: restore proof version and add Doob assembly skeleton
ee95f22 
Restore the 1073-line proof version from reflog (831ff19).
Break the one_sided_running_max sorry into two modular pieces:
- hdobo: Doob's maximal inequality application (sorry)
- hmgf: sub-Gaussian MGF bound via Hoeffding's lemma (sorry)
- Combination step: fully proven via exp_sub + field_simp algebra

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove sub-Gaussian MGF bound for one_sided_running_max
8112438 
Prove E[exp(λ·S_n)] ≤ exp(λ²n/2) via:
- iIndepFun.mgf_sum: factorize MGF of walk as product of individual MGFs
- hasSubgaussianMGF_of_mem_Icc_of_integral_eq_zero (Hoeffding's lemma):
  each Rademacher a_j ∈ [-1,1] with E[a_j]=0 gives mgf ≤ exp(lam²/2)
- Product of exp = exp of sum: ∏ exp(lam²/2) = exp(n·lam²/2)

Also prove the Doob+MGF combination step:
  P(walk≥t) ≤ E[f_n]/exp(λt) ≤ exp(λ²n/2)/exp(λt) = exp(-t²/(2n))

Only sorry remaining in one_sided_running_max: the Doob maximal
inequality application (connecting maximal_ineq to the probability bound).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove Doob maximal inequality assembly — one_sided_running_max …
84703a1 
…is sorry-free

Complete the final sorry in one_sided_running_max by applying Doob's
maximal inequality (maximal_ineq) to the exponential submartingale:
- Set containment: {walk ≥ t} ⊆ {sup' f ≥ exp(λt)} via exp monotonicity
- Doob bound: ε·ℙ(sup'≥ε) ≤ E[f_n·1_{sup'≥ε}] ≤ E[f_n]
- ENNReal→ℝ conversion: ℙ(A) ≤ E[f_n]/exp(λt)

Combined with the previously proven sub-Gaussian MGF bound and
combination algebra, one_sided_running_max is now fully proven.

Sorry count: 6 → 5. Remaining sorrys:
- erdos_524 (meta-wrapper)
- sharp_upper_envelope (Kolmogorov LIL not in Mathlib)
- levy_maximal_ineq (first-crossing decomposition)
- sparse_lower_envelope (2D KMT, multi-year)
- exact_lower_constant (open in mathematics)

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: add LIL upper bound skeleton with walk_tail_bound
a6fab72 
Add section LIL with:
- lilNorm: normalizing function √(2n log log n)
- walk_tail_bound: ℙ(S_n ≥ t) ≤ exp(-t²/(2n)), proven via
  one_sided_running_max (corollary of Doob's maximal inequality)
- kolmogorov_lil_upper_bound: limsup S_n/√(2n log log n) ≤ 1 a.s.
  (sorry'd — proof sketch via sparse subsequence + first BC +
  interpolation documented in docstring)

The walk_tail_bound is the key input for the Borel-Cantelli argument.
The full LIL proof requires ~300 more lines of real-analysis estimates.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: add LIL upper bound proof skeleton with modular structure
194c427 
Add lil_upper_for_eps (core: sparse subsequence + first BC + interpolation)
and kolmogorov_lil_upper_bound (assembly: take ε → 0).
Both sorry'd pending ~300 lines of real-analysis estimates.

walk_tail_bound is proven (ℙ(S_n ≥ t) ≤ exp(-t²/(2n)) via one_sided_running_max).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove lil_tail_at_scale — exponential Chebyshev at LIL scale
b936dae 
Prove ℙ(S_n ≥ (1+ε)·√(2n log log n)) ≤ exp(-(1+ε)²·log log n)
via walk_tail_bound + algebraic simplification of the exponent.

This is the key input for the first Borel-Cantelli argument on
the sparse subsequence in the LIL upper bound proof.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: modular LIL upper bound structure with 4 sorry'd components
6b292c7 
Break kolmogorov_lil_upper_bound into 4 modular components:

1. lil_tail_at_scale (PROVEN): ℙ(S_n ≥ (1+ε)·φ(n)) ≤ exp(-(1+ε)²·log log n)
2. lil_sparse_bc (sorry): a.s. eventually S_{n_k} < (1+ε)·φ(n_k)
   on sparse subsequence, via summability + first Borel-Cantelli
3. lil_interpolation (sorry): |S_n - S_{n_k}| ≤ ε·φ(n) a.s. eventually,
   via running-max bound on increments + Borel-Cantelli
4. lil_upper_for_eps (sorry): assembly combining 2+3
5. kolmogorov_lil_upper_bound (sorry): take ε → 0

Each sorry has a clear proof sketch. The main blockers are:
- Summability estimates (∑ k^{-p} for p > 1)
- Real-analysis bounds on ⌊c^k⌋ growth and log log asymptotics
- Independence of walk increments for the interpolation step

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@Smetalo
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Smetalo commented Apr 16, 2026

Is it planned to merge this PR or #3587 (that contains problem 524, among other things)?

@kieranmcshane @claude
feat: prove Borel-Cantelli structure for LIL sparse subsequence
c515bf7 
In lil_sparse_bc, prove the first Borel-Cantelli application:
- Define events E_k = {S_{n_k} ≥ (1+ε)·φ(n_k)}
- Apply measure_setOf_frequently_eq_zero (first BC)
- Convert "not frequently E_k" to "eventually S_{n_k} < bound"

Remaining sorry: hsum (∑ ℙ(E_k) < ∞), which requires:
- lil_tail_at_scale to bound each ℙ(E_k)
- lil_tail_summable (sorry) for the series comparison

Also added lil_tail_summable as a separate sorry for the
real-analysis estimate that ∑ exp(-(1+ε)²·log log ⌊c^k⌋₊) < ∞.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane
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kieranmcshane commented Apr 16, 2026 •
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@Smetalo My PR updates 524 to reflect Chojecki's 2026 resolution (solved, not open) and includes proven lemmas. Happy to coordinate with @ryantuck

kieranmcshane and others added 9 commits April 16, 2026 13:01
@kieranmcshane @claude
feat: prove kolmogorov_lil_upper_bound assembly (countable intersection)
47155cf 
Prove the ε → 0 step: if for each m ≥ 1 a.s. eventually f(n) ≤ 1+1/m,
then a.s. limsup f ≤ 1. Uses ae_all_iff for the countable intersection.

The final limsup_le step is sorry'd pending IsCobounded/IsBounded
conditions for the walk ratio (straightforward from walk growth bounds).

Also proves the Borel-Cantelli logic in lil_sparse_bc: the conversion
from "not frequently" to "eventually not" is machine-verified.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: refine LIL sorry structure — isolate real-analysis estimates
e8643c1 
Refine the sorry decomposition in the LIL upper bound:
- lil_sparse_bc: BC logic proven, hsum sorry isolated to
  pointwise ℙ(E_k) ≤ ofReal(exp(...)) comparison
- kolmogorov_lil_upper_bound: countable intersection proven,
  final limsup_le sorry isolated to IsCobounded/IsBounded conditions
- lil_tail_summable: sorry isolated to p-series convergence estimate

The logical backbone (Doob → tail → Chebyshev → BC → countable ∩)
is fully machine-verified. Remaining sorrys are pure real-analysis
estimates (floor asymptotics, p-series, log growth, toReal↔ofReal).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: document hsum sorry route (toReal↔ofReal + split at large k)
6b9ac89 
The hsum comparison requires splitting the tsum at k=N where n_k
becomes large enough for lil_tail_at_scale. For k < N, use ℙ ≤ 1
(finite sum). For k ≥ N, use the exponential tail bound + comparison
with lil_tail_summable.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove lil_upper_for_eps assembly structure
4c38bdb 
Show how sparse BC + interpolation combine to give the full result:
- Choose δ = ε/3 and c = 1 + δ
- Apply lil_sparse_bc (a.s. eventually S_{n_k} < (1+δ)·φ(n_k))
- Apply lil_interpolation (a.s. eventually |increment| ≤ δ·φ(n))
- Combine: S_n ≤ S_{n_k} + |increment| < (1+2δ)·φ(n) ≤ (1+ε)·φ(n)

Remaining sorry: the arithmetic combining step
(S_n decomposition + φ monotonicity + 2δ ≤ ε).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: document interpolation proof route and refine limsup assembly
c473265 
- lil_interpolation: detailed 5-step proof route documented
  (shifted walk independence, running-max tail, summability, asymptotics, BC)
- kolmogorov_lil_upper_bound: by_contra + Archimedean structure
  (find m with 1+1/m < limsup, contradict eventually ≤ 1+1/m)
- lil_upper_for_eps: assembly structure proven (δ=ε/3, c=1+δ, combine)

All sorry routes are clearly documented. The logical backbone is verified.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove p-series convergence structure in lil_tail_summable
e8f639b 
Apply Summable.of_norm_bounded_eventually_nat with comparison function
k^{-(1+ε)²} and Real.summable_nat_rpow for the convergence (since
(1+ε)² > 1 implies -(1+ε)² < -1).

Remaining sorry: the pointwise eventually bound
  ‖exp(-(1+ε)²·log log ⌊c^k⌋₊)‖ ≤ k^{-(1+ε)²}
which is a floor/log asymptotic estimate (log ⌊c^k⌋₊ ≥ k for large k).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: clean up lil_tail_summable and fix build errors
3b4d719 
Simplify lil_tail_summable sorry to a clean single sorry with
documented proof route (p-series comparison via summable_nat_rpow
+ floor/log asymptotics).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: clean hsum sorry
0fd9d5b 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: remove unused levy_maximal_ineq — sorry count 10 → 9
d0dbe69 
Remove levy_maximal_ineq (unused standalone sorry) since
one_sided_running_max provides the stronger exp(-t²/(2n)) bound
via Doob's inequality. This eliminates one sorry warning.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
kieranmcshane and others added 26 commits April 16, 2026 15:41
@kieranmcshane @claude
feat: prove floor_c_pow_lower and log_floor_c_pow_lower
817c924 
Two new sorry-free lemmas for the LIL summability chain:
- floor_c_pow_lower: eventually ⌊c^k⌋₊ ≥ c^k/2 (via Nat.sub_one_lt_floor)
- log_floor_c_pow_lower: eventually log ⌊c^k⌋₊ ≥ k·log(c)/2
  (via floor bound + Real.log_div + Real.log_pow + field_simp arithmetic)

Also refactored lil_tail_summable to use these lemmas.
Added log_log_floor_ge_log (sorry) as the remaining bridge.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove exp_neg_p_log_log_floor_le — rpow comparison sorry-free!
85665e1 
Prove that exp(-p·log log ⌊c^k⌋₊) ≤ (log c/2)^{-p} · k^{-p} eventually,
using:
- log_floor_c_pow_lower for the floor/log bound
- Real.rpow_def_of_pos for exp(log x · y) = x^y
- mul_rpow for (a·b)^p = a^p · b^p

This replaces the old log_log_floor_ge_log which required p ≥ 2.
The new comparison works for ALL p > 0, avoiding the p < 2 issue.

Sorry count: 10 → 9. Three new sorry-free lemmas in this session:
floor_c_pow_lower, log_floor_c_pow_lower, exp_neg_p_log_log_floor_le.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: lil_tail_summable is sorry-free! Full summability chain verified.
a77e3e8 
Close the last sorry in lil_tail_summable by chaining:
- exp_neg_p_log_log_floor_le: exp(-p·log log n_k) ≤ C·k^{-p} eventually
- Real.summable_nat_rpow: ∑ k^{-p} converges for p > 1
- Summable.of_norm_bounded_eventually_nat: comparison test

The full chain floor_c_pow_lower → log_floor_c_pow_lower →
exp_neg_p_log_log_floor_le → lil_tail_summable is machine-verified.

Sorry count: 9 → 8.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: lil_sparse_bc is sorry-free! BC summability chain complete.
d75be29 
Close the hsum sorry in lil_sparse_bc by pointwise comparison:
- ℙ(E k) ≤ ofReal(exp(-(1+ε)²·log log n_k)) for ALL k
- When log log n_k ≤ 0: exp(nonneg) ≥ 1 ≥ ℙ(E k)
- When log log n_k > 0 and n_k ≥ 1: lil_tail_at_scale
- When n_k = 0: log 0 = 0, so exp(0) = 1 ≥ ℙ(E k)
- Sum comparison via lil_tail_summable (now sorry-free)

Uses le_ofReal_iff_toReal_le for the ENNReal↔ℝ conversion.

Sorry count: 8 → 7.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: document lil_interpolation blockers + IsCobounded analysis
e9ae181 
The 3 remaining LIL sorrys form a chain:
  lil_interpolation → lil_upper_for_eps → kolmogorov_lil_upper_bound

lil_interpolation needs:
  (1) shifted walk Rademacher property (iIndepFun.comp with injective shift)
  (2) running_max_tail on the increment
  (3) second BC application with summable tails
  (4) asymptotic comparison C·√(Δn_k·log k) vs ε·φ(n)

kolmogorov_lil_upper_bound needs IsCoboundedUnder, which requires
applying lil_upper_for_eps to -walk (Rademacher symmetry).

7 sorrys, 0 errors. ~30 sorry-free theorems.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: clean lil_interpolation sorry
f9d11e8 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove isRademacherSequence_shift via iIndepFun.precomp
91a39b8 
The shifted Rademacher sequence (fun j => a (m+j)) is i.i.d. Rademacher
because iIndepFun.precomp with the injective shift (m+·) preserves
independence, and the marginals are unchanged.

This is the key ingredient for lil_interpolation's shifted walk argument.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: document lil_interpolation proof route with all ingredients
3675ec0 
All ingredients for lil_interpolation are now proven and available:
- isRademacherSequence_shift (sorry-free): shifted walk is Rademacher
- running_max_tail (sorry-free): bounds running max of Rademacher walk
- measure_setOf_frequently_eq_zero (Mathlib): first Borel-Cantelli
- floor_exp_tendsto (sorry-free): ⌊c^k⌋₊ → ∞

The sorry is pure assembly: reindex the walk sum, apply running_max_tail,
BC summability, and asymptotic comparison (~50 lines).

7 sorrys, 0 errors, ~35 sorry-free theorems.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: consolidate lil_interpolation documentation
bf1eeee 
All ingredients proven. Sorry is pure assembly (~50 lines):
- Finset sum reindexing for walk difference identity
- running_max_tail on shifted Rademacher walk
- BC summability with 2/(k+2)²
- Asymptotic: Δn_k·log(k+2)/(n·log log n) → 0

1465 lines, 7 sorrys, 0 errors, ~35 sorry-free theorems.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: simplify lil_interpolation sorry
d92bc87 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove lil_interpolation BC structure
83eb6ec 
The outer Borel-Cantelli structure of lil_interpolation is verified:
- Define events F_k (increment running max exceeds ε·lilNorm)
- hFsum: ∑ ℙ(F_k) < ∞ (sorry — needs running_max_tail + comparison)
- measure_setOf_frequently_eq_zero: a.s. eventually ¬F_k
- calc chain: convert ¬(eventually bounded) to frequently F_k (sorry)

The BC logic itself is machine-verified. Two inner sorrys remain:
(1) hFsum summability (running_max_tail application)
(2) set inclusion (walk difference = shifted walk + lilNorm monotonicity)

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: document hFsum proof route
8959719 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: consolidate all LIL sorrys with proof routes
8be8a66 
1483 lines, 8 sorrys, 0 errors, ~35 sorry-free theorems.

LIL sorry chain (4 sorrys, all pure assembly):
- hFsum (1199): running_max_tail on shifted walk + summability comparison
- set inclusion (1212): walk difference identity + lilNorm monotonicity
- lil_upper_for_eps (1239): covering + decomposition + δ-arithmetic
- kolmogorov_lil_upper_bound (1272): IsCoboundedUnder + Archimedean

Non-LIL sorrys (4):
- erdos_524 (385): meta-wrapper
- sharp_upper_envelope (414): needs full LIL + Chung
- sparse_lower_envelope (1448): blocked (2D KMT)
- exact_lower_constant (1481): open mathematics

All mathematical ingredients are sorry-free. Remaining work is
Lean API plumbing for connecting proven components.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: refine kolmogorov_lil_upper_bound route
98dbead 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: redefine F_k with 2√(Δn_k·log(k+2)) threshold
8f56fc4 
Redefine the bad events in lil_interpolation to use threshold
2√(Δn_k·log(k+2)) instead of ε·lilNorm(n_k). This gives:
- ℙ(F_k) ≤ 2/(k+2)² via running_max_tail with u=2√(log(k+2))
- Summable tails (∑ 2/(k+2)² converges) for ALL ε > 0 and c > 1
- Eventually 2√(Δn_k·log(k+2)) ≤ ε·lilNorm(n) (asymptotic comparison)

The old threshold ε·lilNorm(n_k) required ε²/c > 1 which fails for small ε.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: document hFsum proof
8181178 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: simplify hFsum sorry doc
3283940 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: refine set inclusion sorry
f1652d6 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove hFsum summability framework
0d9f2eb 
The tsum comparison structure for hFsum in lil_interpolation is verified:
- ne_top_of_le_ne_top with summable bound
- ENNReal.tsum_le_tsum for pointwise comparison
- Summable bound: 2 * (summable_one_div_nat_pow shifted by +2)
  via comp_injective with Nat.add_right_cancel

Remaining sorry: pointwise ℙ(F k) ≤ ofReal(2*exp(-2*log(k+2)))
(running_max_tail on shifted walk + exp↔rpow conversion).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove hFsum Δ=0 case (F k = ∅ when no steps)
a3bcfe7 
When Δn_k = ⌊c^(k+1)⌋₊ - ⌊c^k⌋₊ = 0, the Finset.Icc 1 0 is empty,
so F k = ∅ and ℙ(F k) = 0. Proved via Finset.mem_Icc + omega.

The Δ > 0 case (running_max_tail application) remains sorry'd.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove running_max_tail application compiles in hFsum
450ac73 
The running_max_tail on the shifted Rademacher walk (isRademacherSequence_shift)
compiles with u = 2√(log(k+2)) and Δ steps.

Remaining sorry: set equality (F k = running_max event) + ENNReal conversion
+ exp(-2log(k+2)) = 1/(k+2)² arithmetic.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove exp arithmetic in hFsum (exp(-2log(k+2)) = 1/(k+2)²)
6877a04 
Close the exp↔rpow conversion step in the hFsum calc:
- u² = 4·log(k+2) via sq_sqrt
- exp(-2·log(k+2)) = (k+2)^{-2} via rpow_def_of_pos
- (k+2)^{-2} = 1/(k+2)² via rpow_neg + rpow_natCast + push_cast

Remaining sorry in hFsum: connecting F_k set to running_max_tail
(set containment + measure_mono + toReal conversion).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: prove exp arithmetic + framework in hFsum
cc411a6 
The hFsum calc proof is nearly complete:
- Δ=0 case: proven (empty Finset.Icc)
- Δ>0 case: running_max_tail compiles, exp(-2log(k+2))=1/(k+2)² proven
- Remaining: set containment F_k = running_max event (~5 lines, Nat.cast issue)

The exp↔rpow conversion chain is fully verified:
u²=4log(k+2) → exp(-2log(k+2)) → rpow_neg → rpow_natCast → 1/(k+2)²

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: isolate hFsum sorry to Nat.cast_sub + sqrt identity
bf5a5d2 
The hFsum proof structure is nearly complete:
- Δ=0 case: proven
- Δ>0 case: running_max_tail compiles, exp arithmetic proven
- Remaining: threshold equality (Nat.cast_sub + √(a·b)=√a·√b)
  and measure_mono from set containment.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
feat: hFsum is sorry-free! Interpolation BC summability complete.
b2fe58d 
Close the hFsum sorry in lil_interpolation:
- Set equality: F_k = running_max event via hthresh (Nat.cast_sub +
  sqrt_mul + mul_assoc) and hthresh ▸ for both Iff directions
- Running_max_tail on shifted walk: isRademacherSequence_shift
- Exp arithmetic: exp(-2log(k+2)) = 1/(k+2)² via rpow chain
- Summability: ∑ 2/(k+2)² via summable_one_div_nat_pow + comp_injective
- First BC: measure_setOf_frequently_eq_zero

Sorry count: 8 → 7.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane @claude
chore: refine set inclusion sorry
9d44f88 
Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane kieranmcshane mentioned this pull request Apr 17, 2026
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@kieranmcshane @claude
feat: complete Kolmogorov LIL upper bound proof chain
c6a9d39 
All closeable proof obligations discharged. The full LIL upper bound
chain is machine-verified: Rademacher infrastructure, submartingale
construction, Doob maximal inequality, sub-Gaussian MGF, running max
tail bound, LIL summability, sparse Borel-Cantelli, interpolation,
assembly, and limsup bound. 4 blocked sorrys remain (Chung's exact
LIL / LIL lower bound — not in Mathlib).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
@kieranmcshane
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kieranmcshane commented Apr 17, 2026

Closing in favor of a new PR from the statements-only branch (per repo guidelines). Full proofs remain at https://github.com/kieranmcshane/formal-conjectures/tree/add-erdos-524

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@kieranmcshane @Smetalo