Provedproposal-principleattentionunifiedcrux

Path product: per-level under-proposal ≤ c gives leaf ratio π(j)/r(j) ≤ c^L

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{p, q : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{q}(\htmlData{mx=1}{l})}}$ , $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{l})}}$ and $\htmlData{mx=9}{\forall l,\ \htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{l})} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\htmlData{mx=3}{c} \cdot \htmlData{mx=6}{\htmlData{mx=4}{q}(\htmlData{mx=5}{l})}}}$ , then

\htmlData{mx=3}{\prod_{l} \htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{l})}} \mathrel{\htmlData{mx=12}{\leq}} \htmlData{mx=11}{\htmlData{mx=6}{\htmlData{mx=4}{c}^{\htmlData{mx=5}{L}}} \cdot \htmlData{mx=10}{\prod_{l} \htmlData{mx=9}{\htmlData{mx=7}{q}(\htmlData{mx=8}{l})}}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to L. Lean writes f x; papers write f(x). (Fin L)
⟹Implies: If Fin(L), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin L → ℝ)
x : TVariables: p, q range over Fin L → ℝ: the statement must hold for every choice. ((p q : Fin L → ℝ))
∀For all: The statement after the comma holds for every l. (∀ l, 0 < q l)
∏Product: Multiply together p(l) for every l. (∏ l, p l)
<Less than: 0 is strictly smaller than q(l). (0 < q l)
≤At most: 0 is less than or equal to p(l). (0 ≤ p l)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: c times q(l). (c * q l)
aᵇPower: c raised to the power L. (c ^ L)

theorem cv-path-ratio {L : ℕ} (p q : Fin L → ℝ) (c : ℝ) (hq : ∀ l, 0 < q l) (hp : ∀ l, 0 ≤ p l) (hratio : ∀ l, p l ≤ c * q l) :
    ∏ l, p l ≤ c ^ L * ∏ l, q l

Along a depth-L = ⌈log₂ n⌉ descent the induced proposal satisfies π ≤ (1 + 2ε)^L r. Part of 'The Proposal Principle: attention complexity is proposal complexity'.

How it was proved

Proved: 4 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 13 lines of Lean

Path product: per-level under-proposal ≤ c gives leaf ratio π(j)/r(j) ≤ c^L

How it was proved

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