proofs
Provedattentioninnovationexternal-review-3

Affine innovation: scalar append moment quotient

For all Z,Pv,Pk,Pvk,w,k,v,s,d,u∈R\htmlData{mx=1}{Z, \mathit{Pv}, \mathit{Pk}, \mathit{Pvk}, w, k, v, s, d, u \in \htmlData{mx=0}{\mathbb{R}}}, if Z+w≠0\htmlData{mx=2}{\htmlData{mx=0}{Z} + \htmlData{mx=1}{w}} \mathrel{\htmlData{mx=3}{\neq}} 0 and Z+w+s⋅(d⋅(Pk+w⋅k)−u⋅(Z+w))≠0\htmlData{mx=18}{\htmlData{mx=2}{\htmlData{mx=0}{Z} + \htmlData{mx=1}{w}} + \htmlData{mx=17}{\htmlData{mx=3}{s} \cdot (\htmlData{mx=16}{\htmlData{mx=10}{\htmlData{mx=4}{d} \cdot (\htmlData{mx=9}{\htmlData{mx=5}{\mathit{Pk}} + \htmlData{mx=8}{\htmlData{mx=6}{w} \cdot \htmlData{mx=7}{k}}})} - \htmlData{mx=15}{\htmlData{mx=11}{u} \cdot (\htmlData{mx=14}{\htmlData{mx=12}{Z} + \htmlData{mx=13}{w}})}})}} \mathrel{\htmlData{mx=19}{\neq}} 0, then

Pv+w⋅vZ+w+s⋅(d⋅(Pvk+w⋅v⋅kZ+w)−u⋅(Pv+w⋅vZ+w))1+s⋅(d⋅(Pk+w⋅kZ+w)−u)=Pv+w⋅v+s⋅(d⋅(Pvk+w⋅v⋅k)−u⋅(Pv+w⋅v))Z+w+s⋅(d⋅(Pk+w⋅k)−u⋅(Z+w))\begin{aligned} & \htmlData{mx=54}{\frac{\htmlData{mx=37}{\htmlData{mx=9}{\frac{\htmlData{mx=5}{\htmlData{mx=1}{\mathit{Pv}} + \htmlData{mx=4}{\htmlData{mx=2}{w} \cdot \htmlData{mx=3}{v}}}}{\htmlData{mx=8}{\htmlData{mx=6}{Z} + \htmlData{mx=7}{w}}}} + \htmlData{mx=36}{\htmlData{mx=10}{s} \cdot \left(\htmlData{mx=35}{\htmlData{mx=23}{\htmlData{mx=11}{d} \cdot \left(\htmlData{mx=22}{\frac{\htmlData{mx=18}{\htmlData{mx=12}{\mathit{Pvk}} + \htmlData{mx=17}{\htmlData{mx=15}{\htmlData{mx=13}{w} \cdot \htmlData{mx=14}{v}} \cdot \htmlData{mx=16}{k}}}}{\htmlData{mx=21}{\htmlData{mx=19}{Z} + \htmlData{mx=20}{w}}}}\right)} - \htmlData{mx=34}{\htmlData{mx=24}{u} \cdot \left(\htmlData{mx=33}{\frac{\htmlData{mx=29}{\htmlData{mx=25}{\mathit{Pv}} + \htmlData{mx=28}{\htmlData{mx=26}{w} \cdot \htmlData{mx=27}{v}}}}{\htmlData{mx=32}{\htmlData{mx=30}{Z} + \htmlData{mx=31}{w}}}}\right)}}\right)}}}{\htmlData{mx=53}{1 + \htmlData{mx=52}{\htmlData{mx=38}{s} \cdot \left(\htmlData{mx=51}{\htmlData{mx=49}{\htmlData{mx=39}{d} \cdot \left(\htmlData{mx=48}{\frac{\htmlData{mx=44}{\htmlData{mx=40}{\mathit{Pk}} + \htmlData{mx=43}{\htmlData{mx=41}{w} \cdot \htmlData{mx=42}{k}}}}{\htmlData{mx=47}{\htmlData{mx=45}{Z} + \htmlData{mx=46}{w}}}}\right)} - \htmlData{mx=50}{u}}\right)}}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=99}{\frac{\htmlData{mx=79}{\htmlData{mx=59}{\htmlData{mx=55}{\mathit{Pv}} + \htmlData{mx=58}{\htmlData{mx=56}{w} \cdot \htmlData{mx=57}{v}}} + \htmlData{mx=78}{\htmlData{mx=60}{s} \cdot (\htmlData{mx=77}{\htmlData{mx=69}{\htmlData{mx=61}{d} \cdot (\htmlData{mx=68}{\htmlData{mx=62}{\mathit{Pvk}} + \htmlData{mx=67}{\htmlData{mx=65}{\htmlData{mx=63}{w} \cdot \htmlData{mx=64}{v}} \cdot \htmlData{mx=66}{k}}})} - \htmlData{mx=76}{\htmlData{mx=70}{u} \cdot (\htmlData{mx=75}{\htmlData{mx=71}{\mathit{Pv}} + \htmlData{mx=74}{\htmlData{mx=72}{w} \cdot \htmlData{mx=73}{v}}})}})}}}{\htmlData{mx=98}{\htmlData{mx=82}{\htmlData{mx=80}{Z} + \htmlData{mx=81}{w}} + \htmlData{mx=97}{\htmlData{mx=83}{s} \cdot (\htmlData{mx=96}{\htmlData{mx=90}{\htmlData{mx=84}{d} \cdot (\htmlData{mx=89}{\htmlData{mx=85}{\mathit{Pk}} + \htmlData{mx=88}{\htmlData{mx=86}{w} \cdot \htmlData{mx=87}{k}}})} - \htmlData{mx=95}{\htmlData{mx=91}{u} \cdot (\htmlData{mx=94}{\htmlData{mx=92}{Z} + \htmlData{mx=93}{w}})}})}}}} \end{aligned}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
Z, Pv, Pk, Pvk, w, k, v, s, d, u range over real numbers: the statement must hold for every choice. ((Z Pv Pk Pvk w k v s d u : ℝ))
+Addition
Z plus w. (Z + w)
≠Not equal
Z + w and 0 are different. (Z + w ≠ 0)
=Equality
((Pv + w · v)/(Z + w) + s · (d · ((Pvk + w · v · k)/(Z + w… and (Pv + w · v + s · (d · (Pvk + w · v · k) − u · (Pv + w · v… are the same. (((Pv + w * v) / (Z + w) + s * (d * ((Pv…)
a/bDivision
Pv + w · v divided by Z + w. In Lean, dividing by 0 gives 0. ((Pv + w * v) / (Z + w))
-Subtraction
d · (Pk + w · k) minus u · (Z + w). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (d * (Pk + w * k) - u * (Z + w))
·Multiplication
w times k. (w * k)

Exact scalar algebra for affine-control moments after an append, assuming both denominators nonzero. Does not verify a floating-point accumulator or vector storage implementation.

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
16 lines of Lean
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