A registered result in its source context
Original ledger classification: Identity [E] (exact variation-free entropy bound)
Recorded statement: THE ONE-LINE NARIAI ENTROPY BOUND (external-review proposal 2026-06-14): the SdS entropy ratio S_tot/S_dS = (x^2+1)/Phi3(x) (x=r_b/r_c; Phi3=x^2+x+1 the N_fam=3 cyclotomic; HOR.NARIAI.01) satisfies S_tot >= (2/3) S_dS with equality exactly…
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HOR.NARIAI.02
Original ledger classification: Identity [E] (exact variation-free entropy bound)
Original claim text
THE ONE-LINE NARIAI ENTROPY BOUND (external-review proposal 2026-06-14): the SdS entropy ratio S_tot/S_dS = (x^2+1)/Phi3(x) (x=r_b/r_c; Phi3=x^2+x+1 the N_fam=3 cyclotomic; HOR.NARIAI.01) satisfies S_tot >= (2/3) S_dS with equality exactly at the Nariai merge x=1, via the perfect square 3(x^2+1)-2*Phi3(x)=(x-1)^2 >= 0 -- a variation-free proof of the entropy floor (the global minimum on x>0 is 2/3=|Z2|/N_fam). Consistent with d/dx=(x-1)(x+1)/Phi3^2 vanishing at x=1. Exact; Wolfram-mirrored
Original classification detail
Identity [E]: S_tot/S_dS >= 2/3 via (x-1)^2 >= 0, equality at the Nariai merge
Dependencies as recorded
HOR.NARIAI.01; v101_horizon_anchor.py; v8_horizon.py
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- 2026-06-14 (geometry round: Riemann-Roch carrier, Nariai bound, branch line, energy clock — v189–v192)Research changelog; TFPT — Introduction and reading guide
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- The `` [formula in source] '' is triply forced: geometry [formula in source] lattice [formula in source] gravityinput 4 origin theory
- The local sibling: an EHT achromatic polarization interceptHorizon readouts
- The `` [formula in source] '' is triply forced: geometry [formula in source] lattice [formula in source] gravityinput 4 origin theory
- The celestial and twistor continuum routeinput 5 tfpt 3 e8 audit bootstrap
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- The `` [formula in source] '' is triply forced: geometry [formula in source] lattice [formula in source] gravityorigin theory
- The celestial and twistor continuum routeinput 5 tfpt 3 e8 audit bootstrap
- Front matter / supportE8 audit and bootstrap
Registered verification modules
- v190_nariai_entropy_bound.pyRegistered verification module · registry
Read the description recorded in the module registry
the one-line Nariai entropy bound: the SdS entropy ratio S_tot/S_dS=(x²+1)/Φ₃(x) (x=r_b/r_c; Φ₃=x²+x+1 the N_fam=3 cyclotomic; already in tfpt_horizon_readouts) satisfies S_tot ≥ (2/3)S_dS with equality exactly at the Nariai merge x=1, via the perfect square 3(x²+1)−2Φ₃(x)=(x−1)²≥0 — a variation-free proof of the entropy floor (global minimum on x>0 is 2/3=|Z₂|/N_fam).
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