proofs
Provedattentionresidual-seriesexternal-review-4

Residual series: exact numerator and denominator aggregate-error algebra

For all J∈Type⁡\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}, r,a,b,v:J→R\htmlData{mx=3}{r, a, b, v : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}} and B,M,p,o,eB,eM∈R\htmlData{mx=1}{B, M, p, o, \mathit{eB}, \mathit{eM} \in \htmlData{mx=0}{\mathbb{R}}}, if B+eB≠0\htmlData{mx=2}{\htmlData{mx=0}{B} + \htmlData{mx=1}{\mathit{eB}}} \mathrel{\htmlData{mx=3}{\neq}} 0, p+eB≠0\htmlData{mx=2}{\htmlData{mx=0}{p} + \htmlData{mx=1}{\mathit{eB}}} \mathrel{\htmlData{mx=3}{\neq}} 0, ∑jr(j)⋅b(j)=B\htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{b}(\htmlData{mx=4}{j})}}} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{B}, ∑jr(j)⋅a(j)=p\htmlData{mx=7}{\sum_{j} \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})}}} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{p}, ∑jr(j)⋅b(j)⋅v(j)=M\htmlData{mx=11}{\sum_{j} \htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{b}(\htmlData{mx=4}{j})}} \cdot \htmlData{mx=9}{\htmlData{mx=7}{v}(\htmlData{mx=8}{j})}}} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{M} and ∑jr(j)⋅a(j)⋅v(j)=p⋅o\htmlData{mx=11}{\sum_{j} \htmlData{mx=10}{\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=9}{\htmlData{mx=7}{v}(\htmlData{mx=8}{j})}}} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=12}{p} \cdot \htmlData{mx=13}{o}} (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}), then

M+eMB+eB+∑jr(j)⋅(−b(j)−a(j)B+eB⋅(v(j)−M+eMB+eB))1−∑jr(j)⋅(b(j)−a(j)B+eB)=p⋅o+eMp+eB∧p⋅o+eMp+eB−o=eM−o⋅eBp+eB\begin{aligned} & \htmlData{mx=56}{\htmlData{mx=8}{\frac{\htmlData{mx=4}{\htmlData{mx=2}{M} + \htmlData{mx=3}{\mathit{eM}}}}{\htmlData{mx=7}{\htmlData{mx=5}{B} + \htmlData{mx=6}{\mathit{eB}}}}} + \htmlData{mx=55}{\frac{\htmlData{mx=37}{\sum_{j} \htmlData{mx=36}{\htmlData{mx=11}{\htmlData{mx=9}{r}(\htmlData{mx=10}{j})} \cdot \left(\htmlData{mx=35}{\htmlData{mx=23}{-\htmlData{mx=22}{\frac{\htmlData{mx=18}{\htmlData{mx=14}{\htmlData{mx=12}{b}(\htmlData{mx=13}{j})} - \htmlData{mx=17}{\htmlData{mx=15}{a}(\htmlData{mx=16}{j})}}}{\htmlData{mx=21}{\htmlData{mx=19}{B} + \htmlData{mx=20}{\mathit{eB}}}}}} \cdot \left(\htmlData{mx=34}{\htmlData{mx=26}{\htmlData{mx=24}{v}(\htmlData{mx=25}{j})} - \htmlData{mx=33}{\frac{\htmlData{mx=29}{\htmlData{mx=27}{M} + \htmlData{mx=28}{\mathit{eM}}}}{\htmlData{mx=32}{\htmlData{mx=30}{B} + \htmlData{mx=31}{\mathit{eB}}}}}}\right)}\right)}}}{\htmlData{mx=54}{1 - \htmlData{mx=53}{\sum_{j} \htmlData{mx=52}{\htmlData{mx=40}{\htmlData{mx=38}{r}(\htmlData{mx=39}{j})} \cdot \left(\htmlData{mx=51}{\frac{\htmlData{mx=47}{\htmlData{mx=43}{\htmlData{mx=41}{b}(\htmlData{mx=42}{j})} - \htmlData{mx=46}{\htmlData{mx=44}{a}(\htmlData{mx=45}{j})}}}{\htmlData{mx=50}{\htmlData{mx=48}{B} + \htmlData{mx=49}{\mathit{eB}}}}}\right)}}}}}} \\ & \quad \htmlData{mx=1}{=} \htmlData{mx=65}{\frac{\htmlData{mx=61}{\htmlData{mx=59}{\htmlData{mx=57}{p} \cdot \htmlData{mx=58}{o}} + \htmlData{mx=60}{\mathit{eM}}}}{\htmlData{mx=64}{\htmlData{mx=62}{p} + \htmlData{mx=63}{\mathit{eB}}}}} \\ & \htmlData{mx=0}{\land} \htmlData{mx=76}{\htmlData{mx=74}{\frac{\htmlData{mx=70}{\htmlData{mx=68}{\htmlData{mx=66}{p} \cdot \htmlData{mx=67}{o}} + \htmlData{mx=69}{\mathit{eM}}}}{\htmlData{mx=73}{\htmlData{mx=71}{p} + \htmlData{mx=72}{\mathit{eB}}}}} - \htmlData{mx=75}{o}} \mathrel{\htmlData{mx=86}{=}} \htmlData{mx=85}{\frac{\htmlData{mx=81}{\htmlData{mx=77}{\mathit{eM}} - \htmlData{mx=80}{\htmlData{mx=78}{o} \cdot \htmlData{mx=79}{\mathit{eB}}}}}{\htmlData{mx=84}{\htmlData{mx=82}{p} + \htmlData{mx=83}{\mathit{eB}}}}} \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, b, v range over J → ℝ: the statement must hold for every choice. ((r a b v : J → ℝ))
x ∈ TVariables
B, M, p, o, eB, eM range over real numbers: the statement must hold for every choice. ((B M p o eB eM : ℝ))
+Addition
B plus eB. (B+eB)
≠Not equal
B + eB and 0 are different. (B+eB ≠ 0)
∑Summation
Add up r(j) · b(j) for every j. (∑ j,r j*b j)
=Equality
∑ j, r(j) · b(j) and B are the same. (∑ j,r j*b j=B)
a/bDivision
M + eM divided by B + eB. In Lean, dividing by 0 gives 0. ((M+eM)/(B+eB))
∧And
Both hold: (M + eM)/(B + eB) + (∑ j, r(j) · (−((b(j) − a(j))/(B + eB)…, and (p · o + eM)/(p + eB) − o = (eM − o · eB)/(p + eB). ((M+eM)/(B+eB)+(∑ j,r j*(-((b j-a j)/(B+…)
RReal numbers
The real numbers: every point on the number line. (ℝ)
·Multiplication
r(j) times b(j). (r j*b j)
−xNegation
The negative of (b(j) − a(j))/(B + eB). (-((b j-a j)/(B+eB)))
-Subtraction
b(j) minus a(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (b j-a j)

Under nonzero perturbed surrogate mass and corrected denominator, proves the reciprocal-series mean expression equals (p o+eM)/(p+eB) and its bias equals (eM-o eB)/(p+eB). Convergence and arithmetic-error sources remain separate obligations.

How it was proved

Proved
6 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
17 lines of Lean
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