phi_not_finitely_simulable
Recognition Science treats the ledger constant φ as the self-similar fixed point forced by the Recognition Composition Law. No rational number equals φ exactly. Researchers examining whether the universe could be a finite simulation cite this to conclude that exact reproduction of RS dynamics requires real arithmetic rather than any finite rational program. The proof is a direct term application of the no_exact_phi_computation lemma to the assumed existential witness.
claim$¬∃ q∈ℚ, (q:ℝ)=φ$, where φ is the Recognition Science constant satisfying the self-similar fixed-point condition of the Recognition Composition Law.
background
In the SimulationHypothesisStructure module the ledger is identified with physical reality, so the simulation hypothesis dissolves rather than being refuted. The constant φ enters as the unique positive real fixed point of the J-cost function (T5–T6), with J(x)=(x+x⁻¹)/2−1. The upstream theorem no_exact_phi_computation asserts there is no finite-precision algorithm that exactly computes φ, because any rational differs from it; its proof invokes phi_irrational after assuming equality. The sibling structure 'that' from PhiLadderLattice supplies the hypothesis interface for phi-ladder Poisson summation, treated as scaffolding until analytic infrastructure is complete.
proof idea
The declaration is a term-mode proof. It takes the assumed pair ⟨q, hq⟩ witnessing ∃q∈ℚ with (q:ℝ)=φ and feeds it directly into no_exact_phi_computation q hq, which yields the required contradiction with phi_irrational.
why it matters in Recognition Science
The result supplies IC-004.9 and is invoked by the ic004_certificate that records the derived status of the simulation hypothesis. It reinforces the framework claim that φ is forced at T6 and that its irrationality requires any substrate reproducing RS dynamics to support real numbers, consistent with the eight-tick octave and D=3. The certificate quotes the downstream consequence that simulation substrates must be real-number computers.
scope and limits
- Does not rule out approximate or finite-precision simulations of RS dynamics.
- Does not address whether infinite-precision real arithmetic could simulate the ledger.
- Does not claim φ is uncomputable in the Turing sense beyond exact finite rational representation.
- Does not engage the full Bostrom argument once the ledger-reality identification is granted.
formal statement (Lean)
179theorem phi_not_finitely_simulable : ¬ ∃ q : ℚ, (q : ℝ) = phi :=
proof body
Term-mode proof.
180 fun ⟨q, hq⟩ => no_exact_phi_computation q hq
181
182/-- **THEOREM IC-004.10**: Any universe that exactly reproduces RS dynamics
183 (including the irrational φ) must operate on real numbers, not rationals.
184 This constrains "simulation" substrates to real-number computers. -/