proofs
Provedproposal-principleattentionunifiedcrux

Prefix masses on the key tree: root mass 1, leaf mass π_b, and the two children of a node (bit at ℓ set true/false) partition the parent mass

For all L∈N\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}, π:(Fin⁡ L→Bool⁡)→R\htmlData{mx=7}{\pi : (\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\operatorname{Bool}}) \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\mathbb{R}}} and b:Fin⁡ L→Bool⁡\htmlData{mx=5}{b : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\operatorname{Bool}}}, if ∑bπ(b)=1\htmlData{mx=3}{\sum_{b} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{b})}} \mathrel{\htmlData{mx=4}{=}} 1, then

(∑b′:Fin⁡ L→Bool⁡({π(b′)if (∀i:Fin⁡ L, i<0  ⟹  b′(i)=b(i))0otherwise))=1∧(∑b′:Fin⁡ L→Bool⁡({π(b′)if (∀i:Fin⁡ L, i<L  ⟹  b′(i)=b(i))0otherwise))=π(b)∧(∀ℓ, h:ℓ<L,(∑b′:Fin⁡ L→Bool⁡({π(b′)if (∀i:Fin⁡ L, i<ℓ+1  ⟹  b′(i)=update⁡(b,⟨ℓ,h⟩,true⁡,i))0otherwise))+(∑b′:Fin⁡ L→Bool⁡({π(b′)if (∀i:Fin⁡ L, i<ℓ+1  ⟹  b′(i)=update⁡(b,⟨ℓ,h⟩,false⁡,i))0otherwise))=∑b′:Fin⁡ L→Bool⁡({π(b′)if (∀i:Fin⁡ L, i<ℓ  ⟹  b′(i)=b(i))0otherwise))\begin{aligned} & \left(\htmlData{mx=25}{\sum_{b' : \htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{L}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\operatorname{Bool}}} \left(\htmlData{mx=24}{\begin{cases} \htmlData{mx=8}{\htmlData{mx=6}{\pi}(\htmlData{mx=7}{b'})} & \text{if } (\htmlData{mx=23}{\forall i : \htmlData{mx=22}{\htmlData{mx=20}{\operatorname{Fin}}\,\htmlData{mx=21}{L}},\ \htmlData{mx=10}{\htmlData{mx=9}{i}} \mathrel{\htmlData{mx=11}{<}} 0 \mathrel{\htmlData{mx=19}{\implies}} \htmlData{mx=14}{\htmlData{mx=12}{b'}(\htmlData{mx=13}{i})} \mathrel{\htmlData{mx=18}{=}} \htmlData{mx=17}{\htmlData{mx=15}{b}(\htmlData{mx=16}{i})}}) \\ 0 & \text{otherwise} \end{cases}}\right)}\right) \mathrel{\htmlData{mx=26}{=}} 1 \\ & \htmlData{mx=0}{\land} \left(\htmlData{mx=52}{\sum_{b' : \htmlData{mx=29}{\htmlData{mx=27}{\operatorname{Fin}}\,\htmlData{mx=28}{L}} \mathrel{\htmlData{mx=31}{\to}} \htmlData{mx=30}{\operatorname{Bool}}} \left(\htmlData{mx=51}{\begin{cases} \htmlData{mx=34}{\htmlData{mx=32}{\pi}(\htmlData{mx=33}{b'})} & \text{if } (\htmlData{mx=50}{\forall i : \htmlData{mx=49}{\htmlData{mx=47}{\operatorname{Fin}}\,\htmlData{mx=48}{L}},\ \htmlData{mx=36}{\htmlData{mx=35}{i}} \mathrel{\htmlData{mx=38}{<}} \htmlData{mx=37}{L} \mathrel{\htmlData{mx=46}{\implies}} \htmlData{mx=41}{\htmlData{mx=39}{b'}(\htmlData{mx=40}{i})} \mathrel{\htmlData{mx=45}{=}} \htmlData{mx=44}{\htmlData{mx=42}{b}(\htmlData{mx=43}{i})}}) \\ 0 & \text{otherwise} \end{cases}}\right)}\right) \mathrel{\htmlData{mx=56}{=}} \htmlData{mx=55}{\htmlData{mx=53}{\pi}(\htmlData{mx=54}{b})} \\ & \htmlData{mx=0}{\land} \left(\right. \htmlData{mx=57}{\forall} \ell,\ h : \htmlData{mx=58}{\ell} \mathrel{\htmlData{mx=60}{<}} \htmlData{mx=59}{L}, \\ & \quad \htmlData{mx=126}{\left(\htmlData{mx=93}{\sum_{b' : \htmlData{mx=64}{\htmlData{mx=62}{\operatorname{Fin}}\,\htmlData{mx=63}{L}} \mathrel{\htmlData{mx=66}{\to}} \htmlData{mx=65}{\operatorname{Bool}}} \left(\htmlData{mx=92}{\begin{cases} \htmlData{mx=69}{\htmlData{mx=67}{\pi}(\htmlData{mx=68}{b'})} & \text{if } (\htmlData{mx=91}{\forall i : \htmlData{mx=90}{\htmlData{mx=88}{\operatorname{Fin}}\,\htmlData{mx=89}{L}},\ \htmlData{mx=71}{\htmlData{mx=70}{i}} \mathrel{\htmlData{mx=74}{<}} \htmlData{mx=73}{\htmlData{mx=72}{\ell} + 1} \mathrel{\htmlData{mx=87}{\implies}} \htmlData{mx=77}{\htmlData{mx=75}{b'}(\htmlData{mx=76}{i})} \mathrel{\htmlData{mx=86}{=}} \htmlData{mx=85}{\htmlData{mx=78}{\operatorname{update}}(\htmlData{mx=79}{b}, \htmlData{mx=82}{\left\langle \htmlData{mx=80}{\ell}, \htmlData{mx=81}{h} \right\rangle}, \htmlData{mx=83}{\operatorname{true}}, \htmlData{mx=84}{i})}}) \\ 0 & \text{otherwise} \end{cases}}\right)}\right) + \left(\htmlData{mx=125}{\sum_{b' : \htmlData{mx=96}{\htmlData{mx=94}{\operatorname{Fin}}\,\htmlData{mx=95}{L}} \mathrel{\htmlData{mx=98}{\to}} \htmlData{mx=97}{\operatorname{Bool}}} \left(\htmlData{mx=124}{\begin{cases} \htmlData{mx=101}{\htmlData{mx=99}{\pi}(\htmlData{mx=100}{b'})} & \text{if } (\htmlData{mx=123}{\forall i : \htmlData{mx=122}{\htmlData{mx=120}{\operatorname{Fin}}\,\htmlData{mx=121}{L}},\ \htmlData{mx=103}{\htmlData{mx=102}{i}} \mathrel{\htmlData{mx=106}{<}} \htmlData{mx=105}{\htmlData{mx=104}{\ell} + 1} \mathrel{\htmlData{mx=119}{\implies}} \htmlData{mx=109}{\htmlData{mx=107}{b'}(\htmlData{mx=108}{i})} \mathrel{\htmlData{mx=118}{=}} \htmlData{mx=117}{\htmlData{mx=110}{\operatorname{update}}(\htmlData{mx=111}{b}, \htmlData{mx=114}{\left\langle \htmlData{mx=112}{\ell}, \htmlData{mx=113}{h} \right\rangle}, \htmlData{mx=115}{\operatorname{false}}, \htmlData{mx=116}{i})}}) \\ 0 & \text{otherwise} \end{cases}}\right)}\right)} \\ & \quad \quad \htmlData{mx=61}{=} \htmlData{mx=152}{\sum_{b' : \htmlData{mx=129}{\htmlData{mx=127}{\operatorname{Fin}}\,\htmlData{mx=128}{L}} \mathrel{\htmlData{mx=131}{\to}} \htmlData{mx=130}{\operatorname{Bool}}} \left(\htmlData{mx=151}{\begin{cases} \htmlData{mx=134}{\htmlData{mx=132}{\pi}(\htmlData{mx=133}{b'})} & \text{if } (\htmlData{mx=150}{\forall i : \htmlData{mx=149}{\htmlData{mx=147}{\operatorname{Fin}}\,\htmlData{mx=148}{L}},\ \htmlData{mx=136}{\htmlData{mx=135}{i}} \mathrel{\htmlData{mx=138}{<}} \htmlData{mx=137}{\ell} \mathrel{\htmlData{mx=146}{\implies}} \htmlData{mx=141}{\htmlData{mx=139}{b'}(\htmlData{mx=140}{i})} \mathrel{\htmlData{mx=145}{=}} \htmlData{mx=144}{\htmlData{mx=142}{b}(\htmlData{mx=143}{i})}}) \\ 0 & \text{otherwise} \end{cases}}\right)} \left.\right) \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)
∑Summation
Add up π(b) for every b. (∑ b, π b)
=Equality
∑ b, π(b) and 1 are the same. (∑ b, π b = 1)
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 Bool. (Between types, A → B is the type of functions from A to B.) (Fin L → Bool)
πPi
The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
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)
RReal numbers
The real numbers: every point on the number line. (ℝ)
{ a if c; b otherwiseCases (if … then … else)
π(b') when ∀ i, i < 0 → b'(i) = b(i), and 0 otherwise. (if (∀ i : Fin L, (i : ℕ) < 0 → b' i = b…)
∀For all
The statement after the comma holds for every i. (∀ i : Fin L, (i : ℕ) < 0 → b' i = b i)
(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 0. ((i : ℕ) < 0)
∧And
Both hold: (∑ b', π(b') if ∀ i, i < L → b'(i) = b(i), else 0) = π(b), and ∀ ℓ, h, (∑ b', π(b') if ∀ i, i < ℓ + 1 → b'(i) = update(b,…. ((∑ b' : Fin L → Bool, (if (∀ i : Fin L,…)
+Addition
ℓ plus 1. (ℓ + 1)
update
Function.update: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Function.update)
⟨a, b⟩Anonymous constructor
Builds a structure (a pair, a subtype element, …) from ℓ, h. (⟨ℓ, h⟩)
true
true: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (true)
false
false: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (false)

Defines m ℓ b = Σ_{b' agreeing with b below ℓ} π b' and proves the three structural facts the descent needs. With cv-node-mass-inheritance, cv-end-to-end-abstract and cv-tree-proposal-sums-to-one, the binary-tree instantiation of the Conversion Theorem is formal link by link; only their assembly into one statement remains. 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
43 lines of Lean
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