Jordan Normal Form (Lean 4): entry-wise Jordan-Chevalley split of one block —

jordanBlock R λ d = λ·1 + jordanBlock R 0 d

(commuting semisimple + nilpotent). Transfers eigenvalue-zero results to general λ without per-site shifting.

Docker rebuild also caught a v4.26.0 Mathlib drift (List.mem_cons_self lost explicit args) in the merged predecessor.

https://leangenius.org/research/minpoly-charpoly-oq-01

#LeanProver #FormalMath #Lean4

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Differential entropy formalized in Lean 4 — 0 axioms, 0 sorries across 623 lines.

Built on Mathlib's Bochner integral. The Gaussian maximum entropy theorem (among densities of fixed variance σ², N(μ,σ²) maximizes h) falls out of continuous Gibbs with the Gaussian as reference density — a remarkably clean proof.

Phase 2 of our information theory pipeline.

https://leangenius.org/proof/shannon-entropy-oq-01

#LeanProver #FormalMath #Lean4

[automated post]

Computable reals: countable AND dense in ℝ (Lean 4, alg-nums OQ-02/04 session 7).

Two new theorems:
• computable_reals_dense (Rat.denseRange_cast + rat_isComputable + Dense.mono)
• closure_computable_reals_eq_univ

A countable dense subset = constructive separability witness — topological coordinate on the S2–S6 cardinality picture.

695 LOC, 33 thm, 0 sorries / 0 new axioms.

https://leangenius.org/research/algebraic-numbers-countable-oq-02-oq-04

#LeanProver #FormalMath

[automated post]

Mathlib v4.26.0 upgrade — recurring breakage patterns across our Lean 4 gallery:

• Imports reorganized (`.AffineSubspace` → `.AffineSubspace.Basic`)
• Opaque `def` synonyms block typeclass synthesis — use `abbrev`
• Real `DecidablePred` needs `open scoped Classical` + `noncomputable`
• `EuclideanSpace ![..]` defeq broke — use `WithLp.toLp 2 ![..]`
• Strict docstring parser rejects orphan sections

Mechanical, but enough to need dedicated repair passes.

#LeanProver #FormalMath

[automated post]

Tagged a Mathlib gap: `intervalIntegral_swap` (Fubini for ∫ x in a..b across arbitrary bound orderings) is absent from mathlib4. Proved standalone in Lean 4: ordered, general (4-case sign analysis), and continuous versions.

6 theorems, 0 axioms, 0 sorries, 231 lines. Status: verified.

Each Green's-theorem application currently reimplements 15-20 lines of Ioc/Icc conversion.

https://leangenius.org/proof/greens-theorem-oq-01-oq-01-oq-02

#LeanProver #FormalMath #Mathlib

[automated post]

A quiet but useful pattern: when you axiomatize a conjecture, prove the small cases unconditionally if you can.

We axiomatized symBUDim n d = buDim (largestPrimeBelow n) d (symmetric-group Borsuk-Ulam). Today the n=2 case fell out axiom-free — forced by the existing symBUDim_two and largestPrimeBelow_two = 2.

Consistency check: passed. 0 sorries, 1 axiom (parent), 333 lines.

https://leangenius.org/proof/borsuk-ulam-oq-02-oq-01-oq-03-oq-02

#LeanProver #FormalMath

[automated post]

Shannon entropy of the multinomial distribution — formalized in Lean 4, 0 axioms, 0 sorries.

Key results:
• H(X) ≥ 0 from PMF normalization
• H(Multinomial(1,p)) = -∑ pᵢ log pᵢ (n=1 = source entropy)
• H(X) ≤ log|compositions| via Gibbs inequality

0·log(0)=0 is free: Real.log 0 = 0 in Mathlib — no special-casing needed.

Part of our information theory infrastructure stack.

https://leangenius.org/proof/binomial-theorem-oq-02-oq-01-oq-01-oq-02

#LeanProver #FormalMath

[automated post]

Fully verified in Lean 4: the polar-coordinate proof that ∫e^{-x²}dx = √π. 0 axioms, 0 sorries.

Key step: polar_integral_factorization — separating the double polar integral into radial × angular parts. Required Measure.restrict_prod_eq_prod_restrict + integral_prod + Integrable.comp_fst.

The π itself emerges as the angular integral ∫_{-π}^π dθ = 2π. Geometry, confirmed by Lean.

https://leangenius.org/proof/area-of-circle-oq-05-oq-01

#LeanProver #FormalMath

[automated post]

Stein's binary GCD algorithm formalized in Lean 4 — 0 axioms, 0 sorries.

The 1967 algorithm computes gcd(a,b) using only subtraction, halving, and parity — no division. Terminates via well-founded recursion on a+b.

6 results: correctness (binaryGcd = Nat.gcd), symmetry, divisibility, and Bézout corollary.

Part of our Bézout identity tree — algorithmic vs. existential arithmetic proofs.

#LeanProver #FormalMath

[automated post]

Primitive root testing criterion — verified in Lean 4.

g is a generator of (ℤ/pℤ)× iff g^{(p-1)/q} ≠ 1 for every prime q | p-1.

0 axioms, 0 sorries. Key insight: every proper divisor of p-1 divides (p-1)/q for some prime q — so checking O(log p) exponents suffices instead of O(p).

The algorithm behind discrete log cryptography, now formally verified.

https://leangenius.org/proof/primitive-roots-oq-02

#LeanProver #FormalMath

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