proofs
Provedvariational-cost-lawadaptivefirst-principles

Sequential form of the variational cost law: for an adaptive scheme whose proposal after history h is r_h, the expected next-draw cost is ≥ Σ_h P(h)·(Σ_j √(Σ_i λ_{h,i} a_ij))²

For all H,I,J∈Type⁡\htmlData{mx=1}{H, I, J \in \htmlData{mx=0}{\operatorname{Type}}}, P:H→R\htmlData{mx=3}{P : \htmlData{mx=0}{H} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}, lam:H→I→R\htmlData{mx=5}{\mathit{lam} : \htmlData{mx=0}{H} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{I} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}, a:I→J→R\htmlData{mx=5}{a : \htmlData{mx=0}{I} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}} and r:H→J→R\htmlData{mx=5}{r : \htmlData{mx=0}{H} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}, if ∀h, 0≤P(h)\htmlData{mx=4}{\forall h,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{P}(\htmlData{mx=1}{h})}}, ∀h,i, 0≤lam(h,i)\htmlData{mx=5}{\forall h, i,\ 0 \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=0}{\mathit{lam}}(\htmlData{mx=1}{h}, \htmlData{mx=2}{i})}}, ∀i,j, 0≤a(i,j)\htmlData{mx=5}{\forall i, j,\ 0 \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=0}{a}(\htmlData{mx=1}{i}, \htmlData{mx=2}{j})}}, ∀h,j, 0<r(h,j)\htmlData{mx=5}{\forall h, j,\ 0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{\htmlData{mx=0}{r}(\htmlData{mx=1}{h}, \htmlData{mx=2}{j})}} and ∀h, ∑jr(h,j)=1\htmlData{mx=6}{\forall h,\ \htmlData{mx=4}{\sum_{j} \htmlData{mx=3}{\htmlData{mx=0}{r}(\htmlData{mx=1}{h}, \htmlData{mx=2}{j})}} \mathrel{\htmlData{mx=5}{=}} 1} (given Fintype⁡ H\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{H}}, Fintype⁡ I\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{I}}, Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}} and Nonempty⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}), then

∑hP(h)⋅(∑j∑ilam(h,i)⋅a(i,j))2≤∑hP(h)⋅∑ilam(h,i)⋅∑ja(i,j)r(h,j)\begin{aligned} & \htmlData{mx=18}{\sum_{h} \htmlData{mx=17}{\htmlData{mx=3}{\htmlData{mx=1}{P}(\htmlData{mx=2}{h})} \cdot \htmlData{mx=16}{\left(\htmlData{mx=15}{\sum_{j} \htmlData{mx=14}{\sqrt{\htmlData{mx=13}{\sum_{i} \htmlData{mx=12}{\htmlData{mx=7}{\htmlData{mx=4}{\mathit{lam}}(\htmlData{mx=5}{h}, \htmlData{mx=6}{i})} \cdot \htmlData{mx=11}{\htmlData{mx=8}{a}(\htmlData{mx=9}{i}, \htmlData{mx=10}{j})}}}}}}\right)^{2}}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=39}{\sum_{h} \htmlData{mx=38}{\htmlData{mx=21}{\htmlData{mx=19}{P}(\htmlData{mx=20}{h})} \cdot \htmlData{mx=37}{\sum_{i} \htmlData{mx=36}{\htmlData{mx=25}{\htmlData{mx=22}{\mathit{lam}}(\htmlData{mx=23}{h}, \htmlData{mx=24}{i})} \cdot \htmlData{mx=35}{\sum_{j} \htmlData{mx=34}{\frac{\htmlData{mx=29}{\htmlData{mx=26}{a}(\htmlData{mx=27}{i}, \htmlData{mx=28}{j})}}{\htmlData{mx=33}{\htmlData{mx=30}{r}(\htmlData{mx=31}{h}, \htmlData{mx=32}{j})}}}}}}}} \end{aligned}
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)
x ∈ TVariables
H, I, J range over Type: the statement must hold for every choice. ((H I J : Type))
Fintype
Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application
Fintype applied to H. Lean writes f x; papers write f(x). (Fintype H)
Nonempty
Nonempty: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Nonempty)
⟹Implies
If H, then ℝ. (Between types, A → B is the type of functions from A to B.) (H → ℝ)
∀For all
The statement after the comma holds for every h. (∀ h, 0 ≤ P h)
RReal numbers
The real numbers: every point on the number line. (ℝ)
≤At most
0 is less than or equal to P(h). (0 ≤ P h)
∑Summation
Add up r(h, j) for every j. (∑ j, r h j)
=Equality
∑ j, r(h, j) and 1 are the same. (∑ j, r h j = 1)
<Less than
0 is strictly smaller than r(h, j). (0 < r h j)
aᵇPower
∑ j, √(∑ i, lam(h, i) · a(i, j)) raised to the power 2. ((∑ j, Real.sqrt (∑ i, lam h i * a i j))…)
√Square root
The non-negative number whose square is ∑ i, lam(h, i) · a(i, j). (Real.sqrt (∑ i, lam h i * a i j))
·Multiplication
lam(h, i) times a(i, j). (lam h i * a i j)
a/bDivision
a(i, j) divided by r(h, j). In Lean, dividing by 0 gives 0. (a i j / r h j)

Repairs disproval loop 1 of the variational cost law (an adaptive sampler can learn the row from early samples). With H the set of histories, P(h) their probabilities, λ_h the posterior over rows given h, and r_h the proposal used after h, the expected cost of the next draw is at least the law's optimum at each history, averaged. So adaptivity buys exactly the information the samples gather, and the law applies step by step. Proof: the Cauchy–Schwarz bound at each history. First-principles reformulation, 2026-10-05.

How it was proved

Proved
5 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
37 lines of Lean
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