Provedproposal-principleattentionunifiedcrux

Conversion Theorem, end-to-end (abstract level form): per-level under-proposal ≤ c along every root-to-leaf path ⇒ π_j ≤ c^L r_j and Σ π_j²/r_j ≤ c^L for the induced proposal r_j = ∏_ℓ q_ℓj

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{m, q : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ , $\htmlData{mx=3}{\pi : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{c}$ , $\htmlData{mx=4}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{m}(0, \htmlData{mx=1}{j})} \mathrel{\htmlData{mx=3}{=}} 1}$ , $\htmlData{mx=8}{\forall j,\ \htmlData{mx=3}{\htmlData{mx=0}{m}(\htmlData{mx=1}{L}, \htmlData{mx=2}{j})} \mathrel{\htmlData{mx=7}{=}} \htmlData{mx=6}{\htmlData{mx=4}{\pi}(\htmlData{mx=5}{j})}}$ , $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $\htmlData{mx=5}{\forall \ell, j,\ 0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{\htmlData{mx=0}{q}(\htmlData{mx=1}{\ell}, \htmlData{mx=2}{j})}}$ and $\htmlData{mx=21}{\forall \ell, j,\ \htmlData{mx=0}{\ell} \mathrel{\htmlData{mx=2}{<}} \htmlData{mx=1}{L} \mathrel{\htmlData{mx=20}{\implies}} \htmlData{mx=7}{\htmlData{mx=3}{m}(\htmlData{mx=5}{\htmlData{mx=4}{\ell} + 1}, \htmlData{mx=6}{j})} \mathrel{\htmlData{mx=19}{\leq}} \htmlData{mx=18}{\htmlData{mx=13}{\htmlData{mx=8}{c} \cdot \htmlData{mx=12}{\htmlData{mx=9}{q}(\htmlData{mx=10}{\ell}, \htmlData{mx=11}{j})}} \cdot \htmlData{mx=17}{\htmlData{mx=14}{m}(\htmlData{mx=15}{\ell}, \htmlData{mx=16}{j})}}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\begin{aligned} & \left(\htmlData{mx=15}{\forall j,\ \htmlData{mx=3}{\htmlData{mx=1}{\pi}(\htmlData{mx=2}{j})} \mathrel{\htmlData{mx=14}{\leq}} \htmlData{mx=13}{\htmlData{mx=6}{\htmlData{mx=4}{c}^{\htmlData{mx=5}{L}}} \cdot \htmlData{mx=12}{\prod_{\ell=0}^{\htmlData{mx=7}{L} - 1} \htmlData{mx=11}{\htmlData{mx=8}{q}(\htmlData{mx=9}{\ell}, \htmlData{mx=10}{j})}}}}\right) \\ & \htmlData{mx=0}{\land} \htmlData{mx=27}{\sum_{j} \htmlData{mx=26}{\frac{\htmlData{mx=19}{\htmlData{mx=18}{\htmlData{mx=16}{\pi}(\htmlData{mx=17}{j})}^{2}}}{\htmlData{mx=25}{\prod_{\ell=0}^{\htmlData{mx=20}{L} - 1} \htmlData{mx=24}{\htmlData{mx=21}{q}(\htmlData{mx=22}{\ell}, \htmlData{mx=23}{j})}}}}} \mathrel{\htmlData{mx=31}{\leq}} \htmlData{mx=30}{\htmlData{mx=28}{c}^{\htmlData{mx=29}{L}}} \end{aligned}

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to J. Lean writes f x; papers write f(x). (Fintype J)
NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x : TVariables: m, q range over ℕ → J → ℝ: the statement must hold for every choice. ((m q : ℕ → J → ℝ))
≤At most: 0 is less than or equal to c. (0 ≤ c)
∀For all: The statement after the comma holds for every j. (∀ j, m 0 j = 1)
∑Summation: Add up π(j) for every j. (∑ j, π j)
∧And: Both hold: ∀ j, π(j) ≤ c^L · (∏ ℓ, q(ℓ, j)), and (∑ j, π(j)^2/(∏ ℓ, q(ℓ, j))) ≤ c^L. ((∀ j, π j ≤ c ^ L * ∏ ℓ ∈ Finset.range …)
⟹Implies: If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
=Equality: m(0, j) and 1 are the same. (m 0 j = 1)
<Less than: 0 is strictly smaller than q(ℓ, j). (0 < q ℓ j)
RReal numbers: The real numbers: every point on the number line. (ℝ)
aᵇPower: c raised to the power L. (c ^ L)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
+Addition: ℓ plus 1. (ℓ + 1)
∏Product: Multiply together q(ℓ, j) for ℓ = 0, 1, …, L − 1. (∏ ℓ ∈ Finset.range L, q ℓ j)
·Multiplication: c times q(ℓ, j). (c * q ℓ j)
a/bDivision: π(j)^2 divided by ∏ ℓ, q(ℓ, j). In Lean, dividing by 0 gives 0. (π j ^ 2 / ∏ ℓ ∈ Finset.range L, q ℓ j)

theorem cv-end-to-end-abstract {J : Type} [Fintype J] {L : ℕ} (m q : ℕ → J → ℝ) (π : J → ℝ) (c : ℝ) (hc : 0 ≤ c) (hroot : ∀ j, m 0 j = 1) (hleaf : ∀ j, m L j = π j) (hπ : ∀ j, 0 ≤ π j) (hπs : ∑ j, π j = 1) (hq : ∀ ℓ j, 0 < q ℓ j) (hstep : ∀ ℓ j, ℓ < L → m (ℓ + 1) j ≤ c * q ℓ j * m ℓ j) :
    (∀ j, π j ≤ c ^ L * ∏ ℓ ∈ Finset.range L, q ℓ j) ∧ ∑ j, π j ^ 2 / ∏ ℓ ∈ Finset.range L, q ℓ j ≤ c ^ L

Single-statement composition of cv-path-ratio and cv-ratio-to-chi-square by induction along the descent: masses m ℓ j (root 1, leaf π_j) and branch probabilities q ℓ j with m(ℓ+1) ≤ c·q·m(ℓ). With c = 1+2ε from smoothed additive-ε branch estimates (cv-smoothed-branch-ratio), L = ⌈log₂ n⌉ and ε = 1/L the χ² factor is ≤ e² (cv-log-depth-constant). The binary-tree instantiation (prefix masses telescoping; Σ_j r_j = 1 from complementary branch probabilities) remains prose; the technique is Jerrum–Valiant–Vazirani's counting ⇒ sampling descent. 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: 34 lines of Lean

Conversion Theorem, end-to-end (abstract level form): per-level under-proposal ≤ c along every root-to-leaf path ⇒ π_j ≤ c^L r_j and Σ π_j²/r_j ≤ c^L for the induced proposal r_j = ∏_ℓ q_ℓj

How it was proved

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