max_recognition_flux
The theorem states that for any positive area A the maximum recognition flux equals the ledger capacity limit on A divided by eight times the fundamental tick duration. Researchers deriving Bekenstein-Hawking entropy from information bounds in Recognition Science cite this result when linking surface bit capacity to the eight-tick cycle. The proof is a direct term-mode substitution that inserts the definition of the ledger capacity limit and the eight-tick period.
claimFor every real number $A>0$ there exists a real number flux such that flux equals the maximum number of recognition bits storable on a surface of area $A$, divided by eight times the fundamental tick duration $τ_0$.
background
The module derives Bekenstein-Hawking entropy from the ledger capacity limit, with the objective of showing that $S_{BH}=A/4ℓ_p^2$ arises from maximum recognition flux. LedgerCapacityLimit is defined as the maximum number of recognition bits storable on area A, given by $A/ℓ_0^2$ in RS-native units. The constants ell0 and tau0 are the fundamental length and time units, each set to 1 in RS-native units, with tau0 identified as the duration of one tick. Upstream results establish that one octave equals eight ticks, the fundamental evolution period.
proof idea
The proof is a one-line term-mode wrapper. It directly constructs the witness by substituting the definition of LedgerCapacityLimit A ell0 and dividing by the eight-tick cycle 8*tau0.
why it matters in Recognition Science
This declaration supplies the flux characterization needed to reach the saturation uniqueness and positivity results for black-hole entropy in the same module. It implements the eight-tick octave (T7) by dividing bit capacity by the period 2^3, thereby connecting the recognition composition law to the event-horizon bound. The result closes the definitional step toward recovering the Bekenstein-Hawking formula from the ledger capacity limit.
scope and limits
- Does not derive the numerical value of the entropy for any concrete black-hole mass.
- Does not incorporate quantum corrections or higher-order curvature terms.
- Does not address the stability of the flux bound under metric perturbations.
- Does not prove uniqueness of the saturation point; that is handled by sibling declarations.
formal statement (Lean)
54theorem max_recognition_flux (A : ℝ) (h_A : A > 0) :
55 ∃ (flux : ℝ), flux = LedgerCapacityLimit A ell0 / (8 * tau0) := by
proof body
Term-mode proof.
56 -- The flux is the number of bits divided by the 8-tick cycle time.
57 use LedgerCapacityLimit A ell0 / (8 * tau0)
58
59/--- **CERT(definitional)**: Bekenstein-Hawking entropy as the unique saturation point. -/