horizonArea_pos
Physicists modeling black-hole entropy via recognition bandwidth cite the result that horizon area stays positive whenever mass is positive. The claim guarantees that derived quantities such as entropy and saturation ratios remain well-defined and positive. Its proof is a direct term application of multiplication and square positivity after unfolding the area expression.
claimFor any real number $M > 0$, the horizon area $A = 16 pi M^2$ satisfies $A > 0$.
background
The module treats a Schwarzschild black hole as the limiting case of full recognition saturation: every holographic bit on the horizon is consumed by gravitational dynamics, leaving no excess bandwidth for additional structure. Horizon area enters the Bekenstein-Hawking entropy $S_{BH} = A/4 = 4 pi M^2$ (in units with Planck length 1), which is identified with total recognition bandwidth at the horizon. Upstream, entropy of a configuration is defined as proportional to its total defect, with the initial state having minimum entropy; the active edge count per fundamental tick is fixed at 1.
proof idea
The proof is a term-mode one-liner. It unfolds the horizonArea definition to a product containing a positive constant (via linarith), pi > 0, and M squared. Two applications of mul_pos together with sq_pos_of_pos on the hypothesis M > 0 close the argument.
why it matters in Recognition Science
The result supplies the basic positivity needed for the module's downstream statements on maximal saturation, Hawking temperature, and entropy-bandwidth equality. It sits inside the chain that links the eight-tick octave to the area-entropy identification and supports the claim that black holes realize the recognition composition law at its saturation limit. The declaration touches the open question of how finite recognition bandwidth can saturate exactly at the horizon without violating information conservation.
scope and limits
- Does not derive the area formula from the recognition operator.
- Does not incorporate quantum corrections or higher-curvature terms.
- Does not treat rotating or charged black holes.
- Does not supply numerical values for specific masses.
formal statement (Lean)
53theorem horizonArea_pos {M : ℝ} (hM : 0 < M) : 0 < horizonArea M := by
proof body
Term-mode proof.
54 unfold horizonArea
55 exact mul_pos (mul_pos (by linarith) Real.pi_pos) (sq_pos_of_pos hM)
56
57/-- Bekenstein-Hawking entropy: S_BH = A/(4ℓ_P²) = 4πM² (with ℓ_P = 1). -/