proofs
Provedattentionversion-9finite-lawconstructible-state

Dimension-aware certificate for a positive affine mixture on Euclidean balls

For all d∈N\htmlData{mx=1}{d \in \htmlData{mx=0}{\mathbb{N}}}, q∈Key⁡ d\htmlData{mx=3}{q \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Key}}\,\htmlData{mx=1}{d}}} and Q,δ,Δ∈R\htmlData{mx=1}{Q, \delta, \Delta \in \htmlData{mx=0}{\mathbb{R}}}, if 0<Q0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{Q}, 0≤δ0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\delta}, 0≤Δ0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\Delta}, Δ≤δ\htmlData{mx=0}{\Delta} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{\delta}, sum⁡(univ⁡,i↦(q(i))2)≤Q2\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{sum}}(\htmlData{mx=1}{\operatorname{univ}}, \htmlData{mx=6}{i \mapsto \htmlData{mx=5}{(\htmlData{mx=4}{\htmlData{mx=2}{q}(\htmlData{mx=3}{i})})^{2}}})} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{\htmlData{mx=8}{Q}^{2}} and δ⋅(1+d)≤2\htmlData{mx=4}{\htmlData{mx=0}{\delta} \cdot \left(\htmlData{mx=3}{1 + \htmlData{mx=2}{\sqrt{\htmlData{mx=1}{d}}}}\right)} \mathrel{\htmlData{mx=5}{\leq}} 2, then

(secantSlope⁡(Δ)⋅(δ2Q)⋅l1⁡(q)≤1−secantSlope⁡(Δ)⋅(Δ2))\begin{aligned} & \left(\right. \htmlData{mx=12}{\htmlData{mx=8}{\htmlData{mx=3}{\htmlData{mx=1}{\operatorname{secantSlope}}(\htmlData{mx=2}{\Delta})} \cdot \left(\htmlData{mx=7}{\frac{\htmlData{mx=4}{\delta}}{\htmlData{mx=6}{2\htmlData{mx=5}{Q}}}}\right)} \cdot \htmlData{mx=11}{\htmlData{mx=9}{\operatorname{l1}}(\htmlData{mx=10}{q})}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=19}{1 - \htmlData{mx=18}{\htmlData{mx=15}{\htmlData{mx=13}{\operatorname{secantSlope}}(\htmlData{mx=14}{\Delta})} \cdot \left(\htmlData{mx=17}{\frac{\htmlData{mx=16}{\Delta}}{2}}\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)
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)
f(x)Function application
Key applied to d. Lean writes f x; papers write f(x). (Key d)
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
Q, δ, Δ range over real numbers: the statement must hold for every choice. ((Q δ Δ : ℝ))
<Less than
0 is strictly smaller than Q. (0<Q)
≤At most
0 is less than or equal to δ. (0≤δ)
sum
Finset.sum: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.sum)
·Multiplication
δ times 1 + √(d). (δ*(1+Real.sqrt d))
+Addition
1 plus √(d). (1+Real.sqrt d)
√Square root
The non-negative number whose square is d. (Real.sqrt d)
secantSlope
secantSlope: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (secantSlope)
univ
Finset.univ: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ)
a/bDivision
δ divided by 2 · Q. In Lean, dividing by 0 gives 0. (δ/(2*Q))
x ↦ f(x)Function (λ-abstraction)
The function sending i to q(i)^2. Lean writes fun x => …; papers write x ↦ …. (fun i => (q i)^2)
l1
l1: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (l1)
aᵇPower
q(i) raised to the power 2. ((q i)^2)
-Subtraction
1 minus secantSlope(Δ) · (Δ/2). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1-secantSlope Δ*(Δ/2))

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
10 lines of Lean
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