horizonBandwidth_pos
The horizon bandwidth for a Schwarzschild black hole of mass M is positive for every M > 0. Workers on holographic bounds and black-hole thermodynamics in the Recognition Science program cite this result to confirm that recognition capacity stays strictly positive at the horizon. The argument reduces to the positivity of Bekenstein-Hawking entropy divided by the product of k_R and the eight-tick cadence.
claimFor every real number $M > 0$, the horizon bandwidth satisfies $R_{max}(M) > 0$, where $R_{max}(M) = 4πM² / (k_R · 8τ₀)$ and $k_R = ln(φ)$.
background
Black holes represent the limiting case of maximal recognition saturation in which the recognition operator is fully occupied at every scale. Horizon bandwidth is defined as Bekenstein-Hawking entropy divided by the product of the recognition constant k_R and the eight-tick cadence 8τ₀, with entropy given by 4πM² from the horizon area law. The constant k_R equals ln(φ) and is positive because φ > 1, while the eight-tick cadence follows from the octave period 2³ in the forcing chain T7.
proof idea
The proof is a one-line term-mode wrapper. It unfolds the definition of horizonBandwidth and applies div_pos to the positivity of bhEntropy (from hM) together with the product of k_R_pos and eightTickCadence_pos.
why it matters in Recognition Science
This result feeds the parent theorem saturationRatio_pos, which establishes that the saturation ratio is positive for any M > 0 and thereby confirms every black hole is saturated at its horizon. It supplies the key positivity step for the module claims that black holes exhibit no hair because bandwidth is fully consumed and that Hawking temperature contains the factor 8π from the eight-tick cadence. The construction ties directly to the eight-tick octave landmark T7 and the Recognition Composition Law.
scope and limits
- Does not compute the numerical value of horizonBandwidth.
- Does not extend to rotating or charged black holes.
- Does not incorporate quantum gravity corrections to entropy.
- Does not prove the area-entropy relation itself.
Lean usage
exact horizonBandwidth_pos hMformal statement (Lean)
74theorem horizonBandwidth_pos {M : ℝ} (hM : 0 < M) : 0 < horizonBandwidth M := by
proof body
Term-mode proof.
75 unfold horizonBandwidth
76 exact div_pos (bhEntropy_pos hM) (mul_pos k_R_pos eightTickCadence_pos)
77
78/-! ## §3. Gravitational Demand at Horizon -/
79
80/-- The gravitational recognition demand at the horizon.
81
82 At the Schwarzschild radius, the dynamical time is T_dyn = 4πM (light-crossing).
83 Each Planck-mass unit requires one update per T_dyn.
84 Demand = M / T_dyn = 1/(4π). -/