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Document 5 of the TFPT 5.0 setAppendix H — reframe
TFPT 5.02026-06-08377 KBSHA-256 eb7e9aeb…3a91
Paper 5Appendix H — reframe

Appendix H — The Horizon Unit System

One seam constant c₃ = 1/(8π) as the universal horizon thermal code

A change of bookkeeping, not new gravitational physics: if gravity is the geometry-channel readout of the seam, then all horizons read the same boundary constant c₃ = 1/(8π). This note collects the readouts — Hawking, de Sitter and Unruh temperature, black-hole thermodynamics, the Page time, scrambling, the Nariai bound, v_GW = c and cosmic birefringence — in seam units, with two genuine compiler fingerprints (1920 = |W(D₅)|, |μ₄| = 4).

Inputs
  • ›The seam constant c₃ = 1/(8π) from P1, read as the horizon normaliser.
Contribution
  • ›All horizon temperatures share one factor, 1/(2π) = 4c₃; black-hole, de Sitter and Unruh share one thermal grammar.
  • ›Two genuine compiler fingerprints: 1920 = |W(D₅)| in the Hawking power, and |μ₄| = 4 in the scrambling time.
  • ›The boundary transport sub-leading eigenvalue λ₂ = (2/3)⁶ governs both the SM flavor gap and the horizon Page recovery.
Not claimed here
  • ›Nothing here is new gravitational physics — it is a reframe that exposes shared structure. The search ansätze are explicitly [A], not results.
Falsification surface
  • ›As a reframe it cannot be falsified by new gravity; the compiler fingerprints (1920, |μ₄|) and the shared λ₂ fail only if the underlying lattice numbers are wrong.
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Highlights
Factor1/(2π) = 4c₃Universal horizon temperature factor
Hawking1920 = |W(D₅)|Compiler fingerprint in the power
S_dS≈ 3.32×10¹²²De Sitter entropy from the Λ closure
β_rad0.2424°Cosmic birefringence (ACT DR6: 0.4σ)

Key formulas

  • Universal factor
    12π=4c3,TH=c3/M\tfrac{1}{2\pi} = 4c_3, \qquad T_H = c_3/M
    One seam constant behind every horizon temperature. [I]
  • Hawking fingerprint
    PH=c31920 M2,1920=∣W(D5)∣P_H = \frac{c_3}{1920\,M^2}, \quad 1920 = |W(D_5)|
    Compiler Weyl-group order in the Hawking power. [I]
  • Shared transport
    λ2=(2/3)6\lambda_2 = (2/3)^6
    Same eigenvalue fixes flavor gap and Page recovery. [I]

The universal horizon temperature factor

The factor that appears in every horizon temperature is the seam constant itself. Black holes, de Sitter and Unruh therefore share one thermal grammar.

12π=4c3,18π=c3\frac{1}{2\pi} = 4c_3, \qquad \frac{1}{8\pi} = c_3
Thor=4c3 ℏκc kBT_{\mathrm{hor}} = 4c_3\,\frac{\hbar\kappa}{c\,k_B}

Schwarzschild thermodynamics in four c₃-lines

Temperature, entropy, power and lifetime all read off c₃, with the Hawking power denominator carrying the compiler fingerprint 1920 = |W(D₅)| (the Weyl group order of D₅).

TH=c3M,SBH=M22c3,PH=c31920 M2,τevap=640c3M3T_H = \frac{c_3}{M}, \quad S_{BH} = \frac{M^2}{2c_3}, \quad P_H = \frac{c_3}{1920\,M^2}, \quad \tau_{\mathrm{evap}} = \frac{640}{c_3}M^3
1920=∣W(D5)∣1920 = |W(D_5)|

Page time and scrambling

The Page time is a fixed fraction of the evaporation time, and the scrambling time carries the second fingerprint |μ₄| = 4. The Page-recovery kernel decays at the same λ₂ = (2/3)⁶ that sets the SM flavor gap.

tscr∼∣μ4∣ Mlog⁡S,∣μ4∣=4t_{\mathrm{scr}} \sim |\mu_4|\,M\log S, \qquad |\mu_4| = 4
In∼λ2 n=(2/3)6n,Δgap=−log⁡(2/3)6=6log⁡32I_n \sim \lambda_2^{\,n} = (2/3)^{6n}, \qquad \Delta_{\mathrm{gap}} = -\log(2/3)^6 = 6\log\tfrac{3}{2}

De Sitter, Nariai and cosmic birefringence

The de Sitter entropy and the cosmic-birefringence angle are the same seam readouts; v_GW = c follows with no measurable dispersion.

SdS=e2α−1128 c34=32π4e2α−1≈3.32×10122S_{dS} = \frac{e^{2\alpha^{-1}}}{128\,c_3^4} = 32\pi^4 e^{2\alpha^{-1}} \approx 3.32\times 10^{122}
βrad=φ04π≈0.2424∘,vGW=c\beta_{\mathrm{rad}} = \frac{\varphi_0}{4\pi} \approx 0.2424^\circ, \qquad v_{\mathrm{GW}} = c

Key formulas at a glance

  • Universal factor
    12π=4c3,TH=c3/M\tfrac{1}{2\pi} = 4c_3, \qquad T_H = c_3/M

    One seam constant behind every horizon temperature. [I]

  • Hawking fingerprint
    PH=c31920 M2,1920=∣W(D5)∣P_H = \frac{c_3}{1920\,M^2}, \quad 1920 = |W(D_5)|

    Compiler Weyl-group order in the Hawking power. [I]

  • Shared transport
    λ2=(2/3)6\lambda_2 = (2/3)^6

    Same eigenvalue fixes flavor gap and Page recovery. [I]