bremermann_limit_pos
The theorem establishes that the Bremermann limit constant is strictly positive. Researchers deriving energy-dependent operation ceilings in discrete-time physics would cite it to anchor positivity before scaling bounds. The proof reduces the claim to a numerical check after expanding the definitions of the limit and the reduced Planck constant.
claimThe constant $B$ such that the maximum operations per second equal $B$ times energy $E$ satisfies $B > 0$.
background
The IC-002 module derives computation limits from temporal discreteness (tick as minimum time quantum), irrationality of phi, Landauer erasure cost, and the Bremermann bound. The latter follows from the energy-time relation $E t >= hbar/2$, giving maximum operations per second at most $2E/hbar$. The reduced Planck constant is defined as $hbar = phi^{-5}$ in RS-native units (with $c=1$). Upstream structures calibrate J-cost via LedgerFactorization and PhiForcingDerived, while NucleosynthesisTiers supplies discrete phi-tiers for densities.
proof idea
The proof is a one-line wrapper that unfolds the definitions of bremermann_limit and hbar, then applies norm_num to verify the resulting positive real.
why it matters in Recognition Science
It is invoked directly by max_ops_scales_with_energy (IC-002.12) and finite_energy_implies_finite_computation (IC-002.13) to establish that finite energy yields a positive finite bound. The result completes the IC-002.11 slot in the computation-limits certificate. It supplies the energy-scaling factor required by the framework's eight-tick octave and phi-ladder constructions.
scope and limits
- Does not derive the explicit numerical value of the limit in SI units.
- Does not connect the bound to the Recognition Composition Law.
- Does not treat extensions to quantum or reversible computation.
- Does not prove finiteness of total operations without a separate energy hypothesis.
Lean usage
theorem max_ops_scales_with_energy (E : ℝ) (hE : E > 0) : max_ops_per_sec E > 0 := mul_pos bremermann_limit_pos hEformal statement (Lean)
147theorem bremermann_limit_pos : bremermann_limit > 0 := by
proof body
Term-mode proof.
148 unfold bremermann_limit hbar
149 norm_num
150
151/-- For a system with energy E, the maximum number of operations per second is
152 bounded by B × E (Bremermann's limit). -/