Information Computation Limits Structure Computation Limits Structure

The structure of computation limits

Computation limits are the boundaries on how fast and how efficiently any physical system can process information. Three classical bounds frame the subject. The Landauer principle, proposed by Rolf Landauer in 1961, says that erasing one bit of information must dissipate at least k_B T ln(2) of energy, where k_B is the Boltzmann constant and T is the temperature. The Bremermann limit, named after Hans Bremermann, sets a maximum rate of computation per unit energy: no more than 2E/ℏ operations per second for energy E, with ℏ the reduced Planck constant. A third, more abstract limit is that a system with irrational constants cannot be simulated exactly by any finite rational algorithm, because rational numbers cannot represent irrational values precisely.

In Recognition Science, the framework's machine-checked library of formal theorems collects these limits under one declaration, computation_limits_structure. The declaration's core result is a theorem: the golden ratio φ, which satisfies φ² = φ + 1, is irrational. From that single fact the library derives a chain of consequences. No rational number q equals φ, so no finite rational algorithm can compute it exactly. The framework calls this the ledger constraint: the discrete record of events that Recognition Science models cannot be simulated with perfect fidelity using finite rational arithmetic, because its own fundamental scaling constant is transcendental. The library also proves that the fundamental tick τ₀, the minimum time quantum, is positive, so the maximum computation rate 1/τ₀ is positive. The Landauer energy k_B T ln(2) is proved positive for any positive temperature, and it grows linearly with temperature. The Bremermann limit 2/ℏ is proved positive, and the maximum operations per second scale with energy.

The declaration does not prove that these limits are the only limits, nor does it derive the numerical values of τ₀, k_B, or ℏ from first principles; those are definitions in the library, not theorems. It also does not claim that the Landauer or Bremermann bounds are new physics. The framework treats them as consequences of its structure, but the classical statements stand on their own experimental and theoretical grounds. What the declaration adds is a single unifying proof: the irrationality of φ, which forces the exact-simulation limit, is established in the same machine-checked framework that models the other bounds.

theorem computation_limits_structure : computation_limits_from_ledger := phi_irrational

/-- **THEOREM IC-002.7**: There is no finite-precision algorithm that exactly computes
    φ in the sense that any rational number differs from φ. -/
theorem no_exact_phi_computation (q : ℚ) : (q : ℝ) ≠ phi := by
  intro heq
  apply phi_irrational
  exact Set.mem_range.mpr ⟨q, heq⟩

/-- **THEOREM IC-002.8**: The Landauer energy k_B T ln(2) is positive for T > 0.
    This is the minimum energy cost to erase one bit of information. -/
theorem landauer_energy_pos (T : ℝ) (hT : T > 0) :
    k_B * T * Real.log 2 > 0 := by
  unfold k_B
  apply mul_pos
  apply mul_pos
  · norm_num
  · exact hT
  · exact Real.log_pos (by norm_num)

/-- **THEOREM IC-002.9**: The Landauer energy grows linearly with temperature. -/
theorem landauer_scales_with_temp (T₁ T₂ : ℝ) (hT₁ : T₁ > 0) (hT₂ : T₂ > 0) (h : T₂ > T₁) :
    k_B * T₂ * Real.log 2 > k_B * T₁ * Real.log 2 := by
  unfold k_B
  have hlog : Real.log 2 > 0 := Real.log_pos (by norm_num)
  have hkB : (1.380649e-23 : ℝ) > 0 := by norm_num
  nlinarith

/-- **THEOREM IC-002.11**: The Bremermann limit is positive and finite. -/
theorem bremermann_limit_pos : bremermann_limit > 0 := by
  unfold bremermann_limit hbar
  norm_num

/-- **THEOREM IC-002.12**: Maximum computation rate scales with energy. -/
theorem max_ops_scales_with_energy (E : ℝ) (hE : E > 0) :
    max_ops_per_sec E > 0 :=
  mul_pos bremermann_limit_pos hE

What this page does not claim

The declaration does not prove that the Landauer or Bremermann limits are the only fundamental limits on computation. The declaration does not derive the numerical values of τ₀, k_B, or ℏ from the framework's axioms; they are defined constants. The declaration does not claim that exact simulation of Recognition Science dynamics is impossible in principle, only that finite rational arithmetic cannot do it.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/ComputationLimitsStructure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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