trivial_free_choice
plain-language theorem explainer
The theorem establishes that the list [1, 1] satisfies the free-choice predicate by exhibiting two positive paths each with zero J-cost. Researchers working on compatibilist foundations in Recognition Science would cite it to exhibit degeneracy in the cost landscape. The proof is a direct term construction that applies the constructor to HasFreeChoice, discharges the length check by decide, and verifies the universal quantifier via simplification to r = 1 together with the unit J-cost lemma and a lower bound on phi.
Claim. The predicate asserting free choice holds for the list consisting of the real number 1 repeated twice: the list has length at least 2, every entry $r$ is positive, and $J(r) = 0$ for each such $r$.
background
In the module on free will from sigma conservation the J-cost is the function $J(r) = (r-1)^2/(2r)$, which vanishes exactly at $r=1$. The predicate HasFreeChoice(paths) is the conjunction of paths.length ≥ 2 with the statement that every entry $r$ in paths is positive and satisfies Jcost r = 0. This formalizes the RS mechanism in which free choice arises from genuine degeneracy when multiple paths attain the minimum cost of zero, while a determined decision occurs when exactly one path achieves that minimum.
proof idea
The proof applies the constructor tactic to the definition of HasFreeChoice. The length conjunct is discharged immediately by decide. For the universal quantifier the variable r is introduced, the membership hypothesis is simplified to r = 1, and the required pair is constructed by linarith using phi_gt_onePointFive for positivity together with rewriting by hr and the lemma Jcost_unit0 for the cost equality.
why it matters
This theorem supplies the free_choice_exists field inside the FreeWillCert structure that certifies the five compatibilist positions. It directly illustrates the module's claim that free will is compatible with sigma-conservation precisely because of J-cost degeneracy at zero. The result closes the trivial case in the chain that begins with J-uniqueness and the phi-ladder and leads to optionality within the eight-tick octave.
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