proofs
Provedattentioninnovationexternal-review-3

Innovation control: one-trial coefficient error identity

For all J∈Type⁡\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}, r,a,v:J→R\htmlData{mx=3}{r, a, v : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}} and m,o,p,Vr,Vπ,β,star∈R\htmlData{mx=1}{m, o, p, \mathit{Vr}, \mathit{V{\pi}}, \beta, \mathit{star} \in \htmlData{mx=0}{\mathbb{R}}}, if ∑jr(j)⋅(v(j)−m)2=Vr\htmlData{mx=10}{\sum_{j} \htmlData{mx=9}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=8}{(\htmlData{mx=7}{\htmlData{mx=5}{\htmlData{mx=3}{v}(\htmlData{mx=4}{j})} - \htmlData{mx=6}{m}})^{2}}}} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{\mathit{Vr}}, ∑jr(j)⋅a(j)⋅(v(j)−o)2=p⋅Vπ\htmlData{mx=14}{\sum_{j} \htmlData{mx=13}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{a}(\htmlData{mx=4}{j})}} \cdot \htmlData{mx=12}{(\htmlData{mx=11}{\htmlData{mx=9}{\htmlData{mx=7}{v}(\htmlData{mx=8}{j})} - \htmlData{mx=10}{o}})^{2}}}} \mathrel{\htmlData{mx=18}{=}} \htmlData{mx=17}{\htmlData{mx=15}{p} \cdot \htmlData{mx=16}{\mathit{V{\pi}}}}, ∑jr(j)⋅a(j)⋅(v(j)−o)=0\htmlData{mx=13}{\sum_{j} \htmlData{mx=12}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{a}(\htmlData{mx=4}{j})}} \cdot (\htmlData{mx=11}{\htmlData{mx=9}{\htmlData{mx=7}{v}(\htmlData{mx=8}{j})} - \htmlData{mx=10}{o}})}} \mathrel{\htmlData{mx=14}{=}} 0 and Vr⋅star=p⋅Vπ\htmlData{mx=2}{\htmlData{mx=0}{\mathit{Vr}} \cdot \htmlData{mx=1}{\mathit{star}}} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\htmlData{mx=3}{p} \cdot \htmlData{mx=4}{\mathit{V{\pi}}}} (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}), then

(∑jr(j)⋅(((a(j)−β)⋅(v(j)−m))2−2a(j)⋅(o−m)⋅((a(j)−β)⋅(v(j)−m))+a(j)⋅(o−m)2))−(∑jr(j)⋅(((a(j)−star)⋅(v(j)−m))2−2a(j)⋅(o−m)⋅((a(j)−star)⋅(v(j)−m))+a(j)⋅(o−m)2))=Vr⋅(β−star)2\begin{aligned} & \htmlData{mx=95}{\left(\htmlData{mx=47}{\sum_{j} \htmlData{mx=46}{\htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})} \cdot (\htmlData{mx=45}{\htmlData{mx=36}{\htmlData{mx=15}{(\htmlData{mx=14}{(\htmlData{mx=8}{\htmlData{mx=6}{\htmlData{mx=4}{a}(\htmlData{mx=5}{j})} - \htmlData{mx=7}{\beta}}) \cdot (\htmlData{mx=13}{\htmlData{mx=11}{\htmlData{mx=9}{v}(\htmlData{mx=10}{j})} - \htmlData{mx=12}{m}})})^{2}} - \htmlData{mx=35}{\htmlData{mx=23}{\htmlData{mx=19}{2\htmlData{mx=18}{\htmlData{mx=16}{a}(\htmlData{mx=17}{j})}} \cdot (\htmlData{mx=22}{\htmlData{mx=20}{o} - \htmlData{mx=21}{m}})} \cdot (\htmlData{mx=34}{(\htmlData{mx=28}{\htmlData{mx=26}{\htmlData{mx=24}{a}(\htmlData{mx=25}{j})} - \htmlData{mx=27}{\beta}}) \cdot (\htmlData{mx=33}{\htmlData{mx=31}{\htmlData{mx=29}{v}(\htmlData{mx=30}{j})} - \htmlData{mx=32}{m}})})}} + \htmlData{mx=44}{\htmlData{mx=39}{\htmlData{mx=37}{a}(\htmlData{mx=38}{j})} \cdot \htmlData{mx=43}{(\htmlData{mx=42}{\htmlData{mx=40}{o} - \htmlData{mx=41}{m}})^{2}}}})}}\right) - \left(\htmlData{mx=94}{\sum_{j} \htmlData{mx=93}{\htmlData{mx=50}{\htmlData{mx=48}{r}(\htmlData{mx=49}{j})} \cdot (\htmlData{mx=92}{\htmlData{mx=83}{\htmlData{mx=62}{(\htmlData{mx=61}{(\htmlData{mx=55}{\htmlData{mx=53}{\htmlData{mx=51}{a}(\htmlData{mx=52}{j})} - \htmlData{mx=54}{\mathit{star}}}) \cdot (\htmlData{mx=60}{\htmlData{mx=58}{\htmlData{mx=56}{v}(\htmlData{mx=57}{j})} - \htmlData{mx=59}{m}})})^{2}} - \htmlData{mx=82}{\htmlData{mx=70}{\htmlData{mx=66}{2\htmlData{mx=65}{\htmlData{mx=63}{a}(\htmlData{mx=64}{j})}} \cdot (\htmlData{mx=69}{\htmlData{mx=67}{o} - \htmlData{mx=68}{m}})} \cdot (\htmlData{mx=81}{(\htmlData{mx=75}{\htmlData{mx=73}{\htmlData{mx=71}{a}(\htmlData{mx=72}{j})} - \htmlData{mx=74}{\mathit{star}}}) \cdot (\htmlData{mx=80}{\htmlData{mx=78}{\htmlData{mx=76}{v}(\htmlData{mx=77}{j})} - \htmlData{mx=79}{m}})})}} + \htmlData{mx=91}{\htmlData{mx=86}{\htmlData{mx=84}{a}(\htmlData{mx=85}{j})} \cdot \htmlData{mx=90}{(\htmlData{mx=89}{\htmlData{mx=87}{o} - \htmlData{mx=88}{m}})^{2}}}})}}\right)} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=101}{\htmlData{mx=96}{\mathit{Vr}} \cdot \htmlData{mx=100}{(\htmlData{mx=99}{\htmlData{mx=97}{\beta} - \htmlData{mx=98}{\mathit{star}}})^{2}}} \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)
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 J. Lean writes f x; papers write f(x). (Fintype J)
⟹Implies
If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
x : TVariables
r, a, v range over J → ℝ: the statement must hold for every choice. ((r a v : J → ℝ))
x ∈ TVariables
m, o, p, Vr, Vπ, β, star range over real numbers: the statement must hold for every choice. ((m o p Vr Vπ β star : ℝ))
∑Summation
Add up r(j) · (v(j) − m)^2 for every j. (∑ j, r j * (v j - m) ^ 2)
=Equality
∑ j, r(j) · (v(j) − m)^2 and Vr are the same. (∑ j, r j * (v j - m) ^ 2 = Vr)
RReal numbers
The real numbers: every point on the number line. (ℝ)
·Multiplication
r(j) times (v(j) − m)^2. (r j * (v j - m) ^ 2)
aᵇPower
v(j) − m raised to the power 2. ((v j - m) ^ 2)
-Subtraction
v(j) minus m. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (v j - m)
+Addition
((a(j) − β) · (v(j) − m))^2 − 2 · a(j) · (o − m) · ((a(j) … plus a(j) · (o − m)^2. (((a j - β) * (v j - m)) ^ 2 - 2 * a j *…)

Finite scalar algebra for the conditional second-moment penalty under the optimal coefficient relation. Full predictable stopped-run martingale formalization remains a paper proof.

How it was proved

Proved
6 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
13 lines of Lean
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