proofs
Provedproposal-principleattentionunifiedcrux

Binary-tree proposals are distributions: prefix-dependent, complementary branch probabilities give Σ_b ∏_ℓ Q ℓ b = 1 over bit-string leaves

∀L∈N, Q:N→(Fin⁡ L→Bool⁡)→R,(∀ℓ, h:ℓ<L, b,b′:Fin⁡ L→Bool⁡,(∀i:Fin⁡ L, i<ℓ  ⟹  b(i)=b′(i))  ⟹  (b(⟨ℓ,h⟩)=b′(⟨ℓ,h⟩)  ⟹  Q(ℓ,b)=Q(ℓ,b′))∧(b(⟨ℓ,h⟩)≠b′(⟨ℓ,h⟩)  ⟹  Q(ℓ,b)+Q(ℓ,b′)=1))  ⟹  ∑b:Fin⁡ L→Bool⁡∏ℓ=0L−1Q(ℓ,b)=1\begin{aligned} & \htmlData{mx=0}{\forall} L \in \htmlData{mx=1}{\mathbb{N}},\ Q : \htmlData{mx=2}{\mathbb{N}} \mathrel{\htmlData{mx=10}{\to}} (\htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{L}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\operatorname{Bool}}) \mathrel{\htmlData{mx=9}{\to}} \htmlData{mx=8}{\mathbb{R}}, \\ & \quad \left(\right. \htmlData{mx=12}{\forall} \ell,\ h : \htmlData{mx=13}{\ell} \mathrel{\htmlData{mx=15}{<}} \htmlData{mx=14}{L},\ b, b' : \htmlData{mx=18}{\htmlData{mx=16}{\operatorname{Fin}}\,\htmlData{mx=17}{L}} \mathrel{\htmlData{mx=20}{\to}} \htmlData{mx=19}{\operatorname{Bool}}, \\ & \quad \quad (\htmlData{mx=37}{\forall i : \htmlData{mx=36}{\htmlData{mx=34}{\operatorname{Fin}}\,\htmlData{mx=35}{L}},\ \htmlData{mx=23}{\htmlData{mx=22}{i}} \mathrel{\htmlData{mx=25}{<}} \htmlData{mx=24}{\ell} \mathrel{\htmlData{mx=33}{\implies}} \htmlData{mx=28}{\htmlData{mx=26}{b}(\htmlData{mx=27}{i})} \mathrel{\htmlData{mx=32}{=}} \htmlData{mx=31}{\htmlData{mx=29}{b'}(\htmlData{mx=30}{i})}}) \\ & \quad \quad \htmlData{mx=21}{\implies} (\htmlData{mx=43}{\htmlData{mx=39}{b}(\htmlData{mx=42}{\left\langle \htmlData{mx=40}{\ell}, \htmlData{mx=41}{h} \right\rangle})} \mathrel{\htmlData{mx=49}{=}} \htmlData{mx=48}{\htmlData{mx=44}{b'}(\htmlData{mx=47}{\left\langle \htmlData{mx=45}{\ell}, \htmlData{mx=46}{h} \right\rangle})} \mathrel{\htmlData{mx=59}{\implies}} \htmlData{mx=53}{\htmlData{mx=50}{Q}(\htmlData{mx=51}{\ell}, \htmlData{mx=52}{b})} \mathrel{\htmlData{mx=58}{=}} \htmlData{mx=57}{\htmlData{mx=54}{Q}(\htmlData{mx=55}{\ell}, \htmlData{mx=56}{b'})}) \\ & \quad \quad \htmlData{mx=38}{\land} (\htmlData{mx=64}{\htmlData{mx=60}{b}(\htmlData{mx=63}{\left\langle \htmlData{mx=61}{\ell}, \htmlData{mx=62}{h} \right\rangle})} \mathrel{\htmlData{mx=70}{\neq}} \htmlData{mx=69}{\htmlData{mx=65}{b'}(\htmlData{mx=68}{\left\langle \htmlData{mx=66}{\ell}, \htmlData{mx=67}{h} \right\rangle})} \mathrel{\htmlData{mx=81}{\implies}} \htmlData{mx=79}{\htmlData{mx=74}{\htmlData{mx=71}{Q}(\htmlData{mx=72}{\ell}, \htmlData{mx=73}{b})} + \htmlData{mx=78}{\htmlData{mx=75}{Q}(\htmlData{mx=76}{\ell}, \htmlData{mx=77}{b'})}} \mathrel{\htmlData{mx=80}{=}} 1) \left.\right) \\ & \quad \htmlData{mx=11}{\implies} \htmlData{mx=93}{\sum_{b : \htmlData{mx=84}{\htmlData{mx=82}{\operatorname{Fin}}\,\htmlData{mx=83}{L}} \mathrel{\htmlData{mx=86}{\to}} \htmlData{mx=85}{\operatorname{Bool}}} \htmlData{mx=92}{\prod_{\ell=0}^{\htmlData{mx=87}{L} - 1} \htmlData{mx=91}{\htmlData{mx=88}{Q}(\htmlData{mx=89}{\ell}, \htmlData{mx=90}{b})}}} \mathrel{\htmlData{mx=94}{=}} 1 \end{aligned}
What the notation means · or point at any symbol
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
RReal numbers
The real numbers: every point on the number line. (ℝ)
Bool
Bool: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Bool)
∀For all
The statement after the comma holds for every i. (∀ i : Fin L, (i : ℕ) < ℓ → b i = b' i)
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)
(x : T)As natural numbers
i read as an element of natural numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((i : ℕ))
<Less than
i is strictly smaller than ℓ. ((i : ℕ) < ℓ)
⟹Implies
If i < ℓ, then b(i) = b'(i). (Between types, A → B is the type of functions from A to B.) ((i : ℕ) < ℓ → b i = b' i)
f(x)Function application
b applied to i. Lean writes f x; papers write f(x). (b i)
=Equality
b(i) and b'(i) are the same. (b i = b' i)
∧And
Both hold: b(⟨ℓ, h⟩) = b'(⟨ℓ, h⟩) → Q(ℓ, b) = Q(ℓ, b'), and b(⟨ℓ, h⟩) ≠ b'(⟨ℓ, h⟩) → Q(ℓ, b) + Q(ℓ, b') = 1. ((b ⟨ℓ, h⟩ = b' ⟨ℓ, h⟩ → Q ℓ b = Q ℓ b')…)
⟨a, b⟩Anonymous constructor
Builds a structure (a pair, a subtype element, …) from ℓ, h. (⟨ℓ, h⟩)
≠Not equal
b(⟨ℓ, h⟩) and b'(⟨ℓ, h⟩) are different. (b ⟨ℓ, h⟩ ≠ b' ⟨ℓ, h⟩)
+Addition
Q(ℓ, b) plus Q(ℓ, b'). (Q ℓ b + Q ℓ b')
∑Summation
Add up ∏ ℓ, Q(ℓ, b) for every b in Fin(L) → Bool. (∑ b : Fin L → Bool, ∏ ℓ ∈ Finset.range …)
∏Product
Multiply together Q(ℓ, b) for ℓ = 0, 1, …, L − 1. (∏ ℓ ∈ Finset.range L, Q ℓ b)

The tree instantiation of the Conversion Theorem's proposal r_j = ∏_ℓ q_ℓ(j): if the branch probability at level ℓ depends only on the length-ℓ prefix and the bit at ℓ, with the two bits at a node complementary, then r is a probability distribution on the 2^L leaves. Proof by induction on depth via Fin.consEquiv (peel the first bit; the two subtrees satisfy the same structural hypothesis). With cv-end-to-end-abstract this makes the induced proposal a genuine sampleable distribution with 1 + χ² ≤ (1+2ε)^L. 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
56 lines of Lean
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