finite_energy_implies_finite_computation
Finite positive energy E guarantees a positive finite upper bound on the maximum operations per second. Researchers deriving computational speed limits in discrete spacetime or Recognition Science models would cite this when applying the Bremermann bound. The proof is a one-line term that constructs the witness directly as max_ops_per_sec E, verifies positivity via scaled multiplication of positive constants, and closes the inequality by reflexivity.
claimFor any real number $E > 0$, there exists a real number $B > 0$ such that the maximum operations per second permitted by energy $E$ satisfies that rate at $E$ is at most $B$.
background
The module IC-002 derives computation limits from temporal discreteness (tick as minimum time quantum), the irrationality of phi, Landauer erasure cost, and the Bremermann limit (maximum operations bounded by $2E/ℏ$ per second in RS units). The function max_ops_per_sec E encodes this energy-proportional rate. The local setting treats these as emergent from the Recognition framework rather than imposed externally.
proof idea
The proof is a one-line term that supplies the witness max_ops_per_sec E, applies mul_pos to bremermann_limit_pos and the hypothesis E > 0 to obtain positivity of the bound, and invokes le_refl to establish the required inequality.
why it matters in Recognition Science
This result (IC-002.13) supplies the finite-energy implication inside the computation-limits section, directly supporting the Bremermann bound stated in the module documentation. It anchors the information-domain claims that link energy to bounded operations and prepares for later results on tick discreteness and phi irrationality. No downstream uses are recorded yet.
scope and limits
- Does not compute an explicit numerical value for the bound.
- Does not incorporate Landauer erasure or phi irrationality.
- Does not treat multi-particle or entanglement corrections.
- Does not extend the claim to zero or infinite energy.
formal statement (Lean)
161theorem finite_energy_implies_finite_computation (E M : ℝ) (hE : E > 0) :
162 ∃ bound : ℝ, bound > 0 ∧ max_ops_per_sec E ≤ bound := by
proof body
Term-mode proof.
163 exact ⟨max_ops_per_sec E, mul_pos bremermann_limit_pos hE, le_refl _⟩
164
165/-! ## V. The RS Computation Bound from φ -/
166
167/-- **THEOREM IC-002.14**: φ > 1 (φ is greater than 1). -/