Information Recognition Bremermann Energy Per Resolution

The minimum energy per recognition

Bremermann's limit, proposed by Hans Bremermann in 1965, bounds how fast any physical system can compute: the maximum rate is set by the system's mass-energy. In Recognition Science, a related but stricter bound applies to recognition events. A recognition is a discrete record of an event that the framework treats as a unit of computation. The framework's library, a machine-checked collection of formal theorems, defines a quantity called energyPerResolution, which is the minimum energy required for one such recognition event.

The value is not a free parameter. It is defined as phi^5, where phi is the golden ratio, approximately 1.618. The number phi^5 is about 11.09. This definition follows from the framework's core result: the cost of recognition is forced by five plain conditions, and from that cost function the framework derives the golden ratio as the unique self-similar scaling. The energy per resolution is then the phi^5 energy quantum, which also appears in the framework's unit system where the reduced Planck constant hbar equals phi^-5.

The declaration also establishes a rate bound. The framework defines an eight-tick cycle as the minimum time for one complete resolution of a recognition debt. The bound, bremermannBound, is one resolution per eight ticks, so the maximum rate is 1/8 resolutions per tick. The theorem bound_from_phi shows that the maximum resolution rate times the energy per resolution equals phi^5 / 8, which is the power bound. The framework proves that this energy is positive and that multiple resolutions require proportionally more time.

In Recognition Science, this is a tighter bound than Bremermann's classical limit because it applies to the framework's fundamental recognition cycle rather than to arbitrary computation. The energy per resolution is a definitional choice within the framework, not a measured physical constant. It establishes what the framework models as the minimum energy cost per recognition event, and it derives the corresponding rate and power bounds from that definition.

What the declaration does not claim is that this energy has been measured in any physical experiment, or that it replaces the classical Bremermann limit for ordinary computation. The framework's claims about recognition are internal to its own model. The energy per resolution is a definition within the framework, and the theorems about it are consequences of that definition, not empirical findings.

/-- The energy per resolution is φ^5 (since ℏ = φ⁻⁵).
    This is the minimum energy quantum for one recognition event. -/
noncomputable def energyPerResolution : ℝ := phi ^ 5

/-- The bound evaluates to 1/8. -/
theorem bound_value : bremermannBound = 1 / 8 := by
  unfold bremermannBound; rw [octave_is_eight]

/-- The bound involves φ^5: the maximum resolution rate times
    the energy per resolution gives the power bound. -/
theorem bound_from_phi :
    bremermannBound * energyPerResolution = phi ^ 5 / 8 := by
  unfold bremermannBound energyPerResolution
  rw [octave_is_eight]
  ring

/-- The energy per resolution is positive. -/
theorem energy_pos : 0 < energyPerResolution := by
  exact pow_pos phi_pos 5

What this page does not claim

energyPerResolution is a measured physical constant. The framework's bound replaces Bremermann's limit for ordinary computation. The declaration claims that recognition events are the same as physical computations.

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

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