finite_function_is_computable
plain-language theorem explainer
Every function between two finite sets with decidable equality admits an explicit finite lookup table that encodes its action. Workers on discrete RS dynamics cite the result to confirm that 8-tick phase transitions remain inside computable bounds. The argument constructs the table directly as the image of the universal set under the graph map of the function.
Claim. Let $A$ and $B$ be finite sets equipped with decidable equality. For any function $f:A→ B$ there exists a finite collection $T$ of pairs such that for every $a∈A$ there is $b∈B$ with $(a,b)∈T$ and $f(a)=b$.
background
The module IC-003 derives the physical Church-Turing thesis from the discrete ledger structure of Recognition Science. Upstream notions include the fundamental tick $τ_0=1$ as the time quantum and the eight phases $kπ/4$ for $k=0,…,7$. The step operation from cellular automata applies a local rule to produce successor states on a tape, while the phase map supplies the finite discrete space on which ledger transitions act.
proof idea
The proof is a direct term-mode construction. It builds the table by applying Finset.image to the universal set under the map that sends each domain element to its paired image under f. Membership follows from the image membership lemma together with universal membership and reflexivity; the equality clause is immediate from the pairing definition.
why it matters
The theorem supplies the finite-table representation required by the downstream result eight_tick_step_computable, which applies the same construction to Phase→Phase maps. It fills IC-003.10 inside the Church-Turing derivation and confirms that the eight-tick octave produces only lookup-table transitions. The result closes one link from the discrete ledger to the claim that RS processes lie in BQP.
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