Provedalphabaguettebaguette-zreversible

An addition that absorbs any increment admits no exact inverse

For all $\htmlData{mx=5}{\mathit{add} : \htmlData{mx=0}{\mathbb{R}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\mathbb{R}} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=1}{a, b, c \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=0}{a} \mathrel{\htmlData{mx=2}{\neq}} \htmlData{mx=1}{b}$ and $\htmlData{mx=3}{\htmlData{mx=0}{\mathit{add}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{c})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\htmlData{mx=4}{\mathit{add}}(\htmlData{mx=5}{b}, \htmlData{mx=6}{c})}$ , then

\htmlData{mx=16}{\lnot \htmlData{mx=15}{\exists \mathit{sub} : \htmlData{mx=10}{\mathbb{R}} \mathrel{\htmlData{mx=14}{\to}} \htmlData{mx=11}{\mathbb{R}} \mathrel{\htmlData{mx=13}{\to}} \htmlData{mx=12}{\mathbb{R}},\ \htmlData{mx=9}{\forall x, y,\ \htmlData{mx=6}{\htmlData{mx=0}{\mathit{sub}}(\htmlData{mx=4}{\htmlData{mx=1}{\mathit{add}}(\htmlData{mx=2}{x}, \htmlData{mx=3}{y})}, \htmlData{mx=5}{y})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{x}}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: a, b, c range over real numbers: the statement must hold for every choice. ((a b c : ℝ))
≠Not equal: a and b are different. (a ≠ b)
f(x)Function application: add applied to a, c. Lean writes f x; papers write f(x). (add a c)
=Equality: add(a, c) and add(b, c) are the same. (add a c = add b c)
¬Not: It is not the case that ∃ sub, ∀ x, y, sub(add(x, y), y) = x. (¬ ∃ sub : ℝ → ℝ → ℝ, ∀ x y, sub (add x …)
∃There exists: At least one sub makes the statement after the comma true. (∃ sub : ℝ → ℝ → ℝ, ∀ x y, sub (add x y)…)
⟹Implies: If ℝ, then ℝ. (Between types, A → B is the type of functions from A to B.) (ℝ → ℝ)
∀For all: The statement after the comma holds for every x, y. (∀ x y, sub (add x y) y = x)

theorem bz-absorbing-add-not-invertible (add : ℝ → ℝ → ℝ) (a b c : ℝ) (hab : a ≠ b) (habs : add a c = add b c) :
    ¬ ∃ sub : ℝ → ℝ → ℝ, ∀ x y, sub (add x y) y = x

If a rounded addition satisfies add a c = add b c for some a ≠ b (as BF16/FP32 addition does when c dwarfs a − b), no subtraction can recover the first argument. This is why reversible blocks need integer (group) residual streams, not floating point. Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

How it was proved

Proved: 3 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: 14 lines of Lean

An addition that absorbs any increment admits no exact inverse

How it was proved

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