proofs
Provedattentionversion-9finite-lawconstructible-state

Finite original-softmax hull control with a constructed grid certificate

For all E:Type⁡⋅\htmlData{mx=2}{E : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Type}} \cdot }}, n,d∈N\htmlData{mx=1}{n, d \in \htmlData{mx=0}{\mathbb{N}}}, A,q∈Key⁡ d\htmlData{mx=3}{A, q \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Key}}\,\htmlData{mx=1}{d}}}, w,F∈R\htmlData{mx=1}{w, F \in \htmlData{mx=0}{\mathbb{R}}}, k:Fin⁡ n→Key⁡ d\htmlData{mx=7}{k : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Key}}\,\htmlData{mx=4}{d}}}, v:Fin⁡ n→E\htmlData{mx=5}{v : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{E}} and m∈N\htmlData{mx=1}{m \in \htmlData{mx=0}{\mathbb{N}}}, if 0<n0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}, 0<w0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{w}, 0≤F0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{F} and ∀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}} (given NormedAddCommGroup⁡ E\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{NormedAddCommGroup}}\,\htmlData{mx=1}{E}} and InnerProductSpace⁡ R E\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{InnerProductSpace}}\,\htmlData{mx=1}{\mathbb{R}}\,\htmlData{mx=2}{E}}), then

(let⁡(r))(:=reference A q w k v     let a:=relative A q w k     let b:=surrogate A q w k     let ν:=surrogateProposal r b     let α:=surrogateAccept a b     let c:=controlC A q w k v     let U:=geometricU q w     let y:=finiteAccepted (scaledAccepted a b v c (1/U)) (output q k v) m     let law:=rejectionWeight ν α m     (∀ f,0≤law f) ∧ (Finset.sum Finset.univ (fun f => law f))=1 ∧ mean law y=output q k v ∧     (∀ f,y f∈convexHull R (Set.range v)) ∧     mse law y (output q k v)≤F2ˆ∗(1−1/U)2ˆ ∧     (∀ f,∥y f−c∥≤2∗F∗(1−1/U)) ∧     mean law (keyReads m)=(1−(1−scalarMean r a/scalarMean r b)mˆ)/(scalarMean r a/scalarMean r b) ∧     mean law (valueReads m)=1−(1−scalarMean r a/scalarMean r b)mˆ ∧     mean law (denseEvent m)=(1−scalarMean r a/scalarMean r b)mˆ ∧     mean law (keyReads m)≤U))\htmlData{mx=3}{(\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{let}}(\htmlData{mx=1}{r})})(\mathtt{:=reference\ A\ q\ w\ k\ v\ \ \ \ \ let\ a:=relative\ A\ q\ w\ k\ \ \ \ \ let\ b:=surrogate\ A\ q\ w\ k\ \ \ \ \ let\ {\nu}:=surrogateProposal\ r\ b\ \ \ \ \ let\ {\alpha}:=surrogateAccept\ a\ b\ \ \ \ \ let\ c:=controlC\ A\ q\ w\ k\ v\ \ \ \ \ let\ U:=geometricU\ q\ w\ \ \ \ \ let\ y:=finiteAccepted\ (scaledAccepted\ a\ b\ v\ c\ (1/U))\ (output\ q\ k\ v)\ m\ \ \ \ \ let\ law:=rejectionWeight\ {\nu}\ {\alpha}\ m\ \ \ \ \ (\forall \ f,0\leq law\ f)\ \text{∧}\ (Finset.sum\ Finset.univ\ (fun\ f\ =>\ law\ f))=1\ \text{∧}\ mean\ law\ y=output\ q\ k\ v\ \text{∧}\ \ \ \ \ (\forall \ f,y\ f\in convexHull\ \text{ℝ}\ (Set.range\ v))\ \text{∧}\ \ \ \ \ mse\ law\ y\ (output\ q\ k\ v)\leq F\^2*(1-1/U)\^2\ \text{∧}\ \ \ \ \ (\forall \ f,\| y\ f-c\| \leq 2*F*(1-1/U))\ \text{∧}\ \ \ \ \ mean\ law\ (keyReads\ m)=(1-(1-scalarMean\ r\ a/scalarMean\ r\ b)\^m)/(scalarMean\ r\ a/scalarMean\ r\ b)\ \text{∧}\ \ \ \ \ mean\ law\ (valueReads\ m)=1-(1-scalarMean\ r\ a/scalarMean\ r\ b)\^m\ \text{∧}\ \ \ \ \ mean\ law\ (denseEvent\ m)=(1-scalarMean\ r\ a/scalarMean\ r\ b)\^m\ \text{∧}\ \ \ \ \ mean\ law\ (keyReads\ m)\leq U)})}
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)
·Multiplication
Type times . (Type*)
NormedAddCommGroup
NormedAddCommGroup: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (NormedAddCommGroup)
f(x)Function application
NormedAddCommGroup applied to E. Lean writes f x; papers write f(x). (NormedAddCommGroup E)
InnerProductSpace
InnerProductSpace: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (InnerProductSpace)
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables
n, d range over natural numbers: the statement must hold for every choice. ((n d : ℕ))
Key
Key: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Key)
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)
⟹Implies
If Fin(n), then Key(d). (Between types, A → B is the type of functions from A to B.) (Fin n → Key d)
<Less than
0 is strictly smaller than n. (0<n)
≤At most
0 is less than or equal to F. (0≤F)
∀For all
The statement after the comma holds for every j. (∀ j,‖v j‖≤F)
‖v‖Norm
The length (size) of v(j). (‖v j‖)
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)
RReal numbers
The real numbers: every point on the number line. (ℝ)

Version 9 exact frozen target. Proof references the validated additive definition library. This mathematical finite-law/invariant target does not certify compiled Rust RAM runtime or the general linear-summary LP bridge.

How it was proved

Proved
7 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
26 lines of Lean
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