proofs
Provedattentionfinite-historytilted-momentsprecision

Tilted midpoint state: finite original-softmax accuracy and work

For all n∈N\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}, h∈N\htmlData{mx=1}{h \in \htmlData{mx=0}{\mathbb{N}}}, q,R,F,epsilon∈R\htmlData{mx=1}{q, R, F, \mathit{epsilon} \in \htmlData{mx=0}{\mathbb{R}}} and k,v:Fin⁡ n+1→R\htmlData{mx=6}{k, v : \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=2}{\htmlData{mx=1}{n} + 1}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\mathbb{R}}}, if 0≤q0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{q}, q≤R\htmlData{mx=0}{q} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{R}, 0≤F0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{F}, 0<epsilon0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\mathit{epsilon}}, ∀j, 0≤k(j)∧k(j)≤1\htmlData{mx=9}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{k}(\htmlData{mx=1}{j})} \mathbin{\htmlData{mx=8}{\land}} \htmlData{mx=6}{\htmlData{mx=4}{k}(\htmlData{mx=5}{j})} \mathrel{\htmlData{mx=7}{\leq}} 1}, ∀j, ∣v(j)∣≤F\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{v}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{F}} and 0<relativePolynomial⁡(evenOrder⁡(h),−R4)0 \mathrel{\htmlData{mx=8}{<}} \htmlData{mx=7}{\htmlData{mx=0}{\operatorname{relativePolynomial}}\left(\htmlData{mx=3}{\htmlData{mx=1}{\operatorname{evenOrder}}(\htmlData{mx=2}{h})}, \htmlData{mx=6}{-\htmlData{mx=5}{\frac{\htmlData{mx=4}{R}}{4}}}\right)}, then

(let⁡(m))(:= evenOrder h     let D := polynomialDeadline n R     let H := tiltedCopies h R F epsilon     let r := uniformRow n     let a := polynomialAccept q R k     let c := storedPolynomialControl m (q−R/2)       (tiltedMomentSums m R (List.ofFn (fun j : Fin (n+1) => (k j,v j))))     let o := softmaxOutput q k v     let y := finiteAccepted (tiltedAccepted m q R k v c) o D     let law := rejectionWeight r a D     (∀ z,0 ≤ copiesWeight law H z) ∧ Finset.sum Finset.univ (copiesWeight law H)=1 ∧     mean (copiesWeight law H) (average y H)=o ∧     mse (copiesWeight law H) (average y H) o ≤ epsilon2ˆ ∧     mean (copiesWeight law H) (tiltedQueryWork m H D R) ≤       15∗(m : R)+18+(H : R)∗(6∗Real.exp R+10∗(m : R)+28)))\htmlData{mx=3}{(\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{let}}(\htmlData{mx=1}{m})})(\mathtt{:=\ evenOrder\ h\ \ \ \ \ let\ D\ :=\ polynomialDeadline\ n\ R\ \ \ \ \ let\ H\ :=\ tiltedCopies\ h\ R\ F\ epsilon\ \ \ \ \ let\ r\ :=\ uniformRow\ n\ \ \ \ \ let\ a\ :=\ polynomialAccept\ q\ R\ k\ \ \ \ \ let\ c\ :=\ storedPolynomialControl\ m\ (q-R/2)\ \ \ \ \ \ \ (tiltedMomentSums\ m\ R\ (List.ofFn\ (fun\ j\ :\ Fin\ (n+1)\ =>\ (k\ j,v\ j))))\ \ \ \ \ let\ o\ :=\ softmaxOutput\ q\ k\ v\ \ \ \ \ let\ y\ :=\ finiteAccepted\ (tiltedAccepted\ m\ q\ R\ k\ v\ c)\ o\ D\ \ \ \ \ let\ law\ :=\ rejectionWeight\ r\ a\ D\ \ \ \ \ (\forall \ z,0\ \leq \ copiesWeight\ law\ H\ z)\ \text{∧}\ Finset.sum\ Finset.univ\ (copiesWeight\ law\ H)=1\ \text{∧}\ \ \ \ \ mean\ (copiesWeight\ law\ H)\ (average\ y\ H)=o\ \text{∧}\ \ \ \ \ mse\ (copiesWeight\ law\ H)\ (average\ y\ H)\ o\ \leq \ epsilon\^2\ \text{∧}\ \ \ \ \ mean\ (copiesWeight\ law\ H)\ (tiltedQueryWork\ m\ H\ D\ R)\ \leq \ \ \ \ \ \ \ 15*(m\ :\ \text{ℝ})+18+(H\ :\ \text{ℝ})*(6*Real.exp\ R+10*(m\ :\ \text{ℝ})+28))})}
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)
x ∈ TVariables
q, R, F, epsilon range over real numbers: the statement must hold for every choice. ((q R F epsilon : ℝ))
≤At most
0 is less than or equal to q. (0 ≤ q)
<Less than
0 is strictly smaller than epsilon. (0 < epsilon)
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 n + 1. Lean writes f x; papers write f(x). (Fin (n+1))
⟹Implies
If Fin(n + 1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin (n+1) → ℝ)
x : TVariables
k, v range over Fin (n+1) → ℝ: the statement must hold for every choice. ((k v : Fin (n+1) → ℝ))
∀For all
The statement after the comma holds for every j. (∀ j,0 ≤ k j ∧ k j ≤ 1)
let
let: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (let)
∧And
Both hold: 0 ≤ k(j), and k(j) ≤ 1. (0 ≤ k j ∧ k j ≤ 1)
|x|Absolute value
The size of v(j) without its sign. (|v j|)
relativePolynomial
relativePolynomial: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (relativePolynomial)
+Addition
n plus 1. (n+1)
evenOrder
evenOrder: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (evenOrder)
−xNegation
The negative of R/4. (-(R/4))
a/bDivision
R divided by 4. In Lean, dividing by 0 gives 0. (R/4)

Derived from actual key/value arrays and tilted moments, with whole-support symmetric certificate, finite original normalized expectation, epsilon MSE, fixed averaging and explicit mathematical operation bill. Assumes only scalar endpoint positivity as an evaluated range eligibility test. Full RAM/lifecycle bridge remains written; no sharp finite frontier or universal actual-variance dominance.

How it was proved

Proved
9 October 2026
By
Ajax
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
28 lines of Lean
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