feat(Analysis/SpecialFunctions): add the Chase-Lovett diagonal reduction for binEntropy

[Harmenszoon]

This PR adds the two-variable entropy inequality of Chase and Lovett (arXiv:2211.11689, Lemma 2.2): for x, y ∈ [0, 1],

x * binEntropy y + y * binEntropy x ≤ 2 * sqrt (x * y) * binEntropy (sqrt (x * y))

in a new leaf file Mathlib/Analysis/SpecialFunctions/BinaryEntropy/Diagonal.lean, together with its negMulLog (1 - ·) half (Real.mul_negMulLog_one_sub_add_le) and a private concavity layer for the comparison function η u = exp u * negMulLog (1 - exp (-u)). Two entries are added to docs/references.bib (Chase–Lovett arXiv:2211.11689, Boppana arXiv:2301.09664).

Motivation. This inequality is the reusable reduction step behind the recent entropic lower bounds for Frankl's union-closed sets conjecture (Gilmer; Alweiss–Huang–Sellke; Chase–Lovett; Sawin; Boppana): it reduces two-variable inequalities of Boppana type K * (x * h y + y * h x) ≤ h (x * y) to one-variable inequalities 2 * K * t * h t ≤ h (t ^ 2). Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean currently contains calculus basics for binEntropy only, and no inequality of this family; I checked current master for collisions before opening this. A kernel-checked downstream application — a complete Lean formalization of the union-closed lower bound at the constant (3 - √5)/2 that uses exactly this reduction — is public at https://github.com/demonstrandum-research/artifacts (the UCFrankl development), which I maintain.

Proof shape (one page of calculus): substituting x = exp (-u), the inequality becomes midpoint concavity of η on [0, ∞); the negMulLog (x * y) parts of the two sides cancel exactly, the negMulLog (1 - ·) parts transfer to η via the identity x * η (-log x) = negMulLog (1 - x), and η'' ≤ 0 reduces to log t ≤ t - 1. Stating η through negMulLog absorbs the 0 * log 0 boundary behaviour, so no separate edge-case argument is needed at x = 1.

Design notes / open questions for reviewers.

• Hypotheses are stated as 0 ≤ x/x ≤ 1 pairs rather than x ∈ Set.Icc 0 1; happy to convert if the membership form is preferred.
• The η-machinery is private; it could be exposed if judged independently useful.
• File placement: a new BinaryEntropy/Diagonal.lean leaf keeps the import footprint of BinaryEntropy.lean unchanged (the proof needs Analysis.Convex.Deriv); merging into the main file is also possible if preferred.
• Naming: mul_binEntropy_add_mul_binEntropy_le / mul_negMulLog_one_sub_add_le follow the conclusion-based convention but I am glad to rename.

AI disclosure (per the mathlib AI policy): the Lean code in this PR was generated by LLM agents (Anthropic's Claude, running in an agent pipeline that I operate under the name Demonstrandum), and was then compile-verified against mathlib and reviewed by me. This PR description was also prepared with AI assistance and reviewed by me. The proof is elementary calculus as summarized above; I understand it and take full responsibility for the contribution. I am adding the LLM-generated label as required. The downstream UCFrankl development linked above (kernel-checked, axioms [propext, Classical.choice, Quot.sound] only) is the evidence that the lemma carries real weight in applications.

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• depends on: nothing

[github-actions]

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[Harmenszoon]

LLM-generated

[github-actions]

PR summary 98831c4466

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.SpecialFunctions.BinaryEntropy.Diagonal (new file)	2021

---

Declarations diff (regex)

+ concaveOn_eta
+ continuous_eta
+ eta
+ eta''_nonpos
+ eta_add_le
+ hasDerivAt_eta
+ hasDerivAt_eta'
+ hasDerivAt_inner
+ mul_binEntropy_add_mul_binEntropy_le
+ mul_eta_neg_log
+ mul_negMulLog_one_sub_add_le
+ one_sub_exp_neg_pos

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 98831c4).

• +2 new declarations
• −0 removed declarations

+Real.mul_binEntropy_add_mul_binEntropy_le
+Real.mul_negMulLog_one_sub_add_le

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 98831c4466
Reference commit 80ffd59621

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[Timeroot]

I did some golfs of your proofs. :) (posting it this way because it was little ones all up and down, would've cluttered the git I feel)

[Harmenszoon]

Thanks @Timeroot, much appreciated! Applied all seven in 98831c4. I kept the original docstrings but took everything else as-is.