IndisputableMonolith.Gravity.BlackHoleHorizonStates
Module counting admissible Q3-orbit horizon patches at area A as A/4 unit-Planck patches, and equating leading Bekenstein-Hawking entropy to log2 of that count. Gravity researchers comparing RS log-corrections to LQG and strings would cite it. Definitions, positivity, elementary phi inequalities, a c_RS band, and a one-statement certificate structure the argument.
claimAt horizon area $A$, the number of admissible $Q_3$-orbit patches is $N_{\mathrm{horizon}}(A)=A/4$ (unit Planck patches). Leading entropy satisfies $S_{\mathrm{lead}}=\log_2 N_{\mathrm{horizon}}$. The RS log-correction coefficient $c_{\mathrm{RS}}$ lies in a $\varphi$-rational band, distinct from the LQG value $-1/2$ and the string value $-3/2$.
background
Track F6 recovers Bekenstein-Hawking entropy $S_{BH}=A/(4\ell_P^2)$ from the discrete RS ledger as a count of admissible horizon states modulo $\sigma$-equivalence. The upstream ledger module states that RS predicts a $\varphi$-rational coefficient on the leading $\log A$ correction, already distinguishable from LQG ($-1/2$) and strings ($-3/2$).
This module makes that count concrete for $Q_3$-orbit horizon patches. It imports RS constants ($\tau_0=1$ tick) and the cost layer, then defines $N_{\mathrm{horizon}}$ (alias horizon patch count) as $A/4$ unit-Planck patches, with positivity, the identity $S_{\mathrm{lead}}=\log_2 N_{\mathrm{horizon}}$, and elementary bounds $\log\varphi>0$, $\log\varphi<1/2$ that feed the $c_{\mathrm{RS}}$ band and the export certificate.
proof idea
Definition-and-lemma module, not a single deep proof. $N_{\mathrm{horizon}}$ is introduced as $A/4$; positivity is immediate from positive area. The leading-entropy identity is the log2 of that count. Separate lemmas record $\log\varphi>0$ and $\log\varphi<1/2$. Those bounds assemble into the $c_{\mathrm{RS}}$ band. A certificate type and a one-statement wrapper package the claim for the gravity track.
why it matters in Recognition Science
Closes the explicit state-count side of Track F6 black-hole entropy from the ledger. Upstream recovers $S_{BH}=A/(4\ell_P^2)$ as an admissible-state count and flags a $\varphi$-rational log correction; here the horizon patch count, the $\log_2$ identification, and the $c_{\mathrm{RS}}$ band are named so RS can be compared directly to LQG and string predictions. No downstream edges are recorded yet; the certificate is the export surface for the gravity domain and for contact with the $\varphi$-ladder and eight-tick structure (T6-T7).
scope and limits
- Does not derive subleading entropy terms beyond the leading log A correction.
- Does not treat Hawking spectra, greybody factors, or dynamical evaporation.
- Does not equate Q3-orbits with LQG spin networks or string microstates.
- Does not pin a unique c_RS value; only a phi-rational band.
- Does not cover charged, rotating, or non-area-law horizons.
depends on (3)
declarations in this module (12)
-
def
horizon_patch_count -
theorem
horizon_patch_count_pos -
def
N_horizon -
theorem
N_horizon_pos -
theorem
S_lead_eq_log2_N_horizon -
theorem
N_horizon_succ_patch -
theorem
log_phi_lt_half -
theorem
log_phi_pos -
theorem
c_RS_band -
structure
BlackHoleHorizonStatesCert -
def
blackHoleHorizonStatesCert -
theorem
black_hole_horizon_states_one_statement