Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""v190 -- The one-line Nariai entropy bound (external-review proposal, point 4).
The Schwarzschild-de Sitter 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, already in tfpt_horizon_readouts) has a
variation-free lower bound 2/3, saturated exactly at the Nariai merge x=1, via the
perfect square (x-1)^2 >= 0. Exact [E].

  [E] 1. NARIAI VALUE.  S_tot/S_dS at x=1 equals 2/3 = |Z2|/N_fam.
  [E] 2. ONE-LINE BOUND.  (x^2+1)/Phi3(x) >= 2/3  <=>  3(x^2+1) >= 2(x^2+x+1)
        <=>  x^2 - 2x + 1 >= 0  <=>  (x-1)^2 >= 0, true for all real x, with
        equality iff x=1.  So S_tot >= (2/3) S_dS, saturated exactly at the
        Nariai horizon merge -- a variation-free proof of the entropy floor.
  [E] 3. CONSISTENT WITH THE DERIVATIVE.  d/dx (S_tot/S_dS) = (x-1)(x+1)/Phi3^2
        (tfpt_horizon_readouts) vanishes at x=1 on x>0, confirming x=1 is the
        unique physical stationary point, here shown to be the GLOBAL minimum.

  Exact (symbolic algebra); Wolfram-mirrored.
"""
import sympy as sp

from tfpt_constants import N_fam, check, summary, reset


def run():
    reset()
    print("v190 Nariai entropy bound: S_tot/S_dS >= 2/3, saturated at the Nariai merge, via (x-1)^2 >= 0")

    x = sp.symbols("x", positive=True)
    ratio = (x**2 + 1) / (x**2 + x + 1)            # S_tot/S_dS, denom = Phi_3(x)

    val1 = sp.nsimplify(ratio.subs(x, 1))
    check("NARIAI VALUE [E]: S_tot/S_dS(x=1) = (1+1)/(1+1+1) = %s = |Z2|/N_fam = 2/%d"
          % (val1, N_fam),
          val1 == sp.Rational(2, 3) and val1 == sp.Rational(2, N_fam))

    # the one-line bound: 3(x^2+1) - 2*Phi3(x) = (x-1)^2 exactly
    gap = sp.expand(3 * (x**2 + 1) - 2 * (x**2 + x + 1))
    check("ONE-LINE BOUND [E]: 3(x^2+1) - 2(x^2+x+1) = %s = (x-1)^2 >= 0, so "
          "S_tot/S_dS >= 2/3 for all x, with equality iff x=1 (the Nariai merge) "
          "-- a variation-free proof of the entropy floor S_tot >= (2/3) S_dS"
          % sp.factor(gap),
          sp.factor(gap) == (x - 1)**2)

    # global minimum on x>0 is exactly 2/3
    mn = sp.minimum(ratio, x, sp.Interval.open(0, sp.oo))
    check("GLOBAL MINIMUM [E]: min_{x>0} (x^2+1)/Phi3(x) = %s, attained at x=1; "
          "d/dx = (x-1)(x+1)/Phi3^2 vanishes at x=1 on x>0 (tfpt_horizon_readouts), "
          "confirming the unique physical stationary point is the global min"
          % mn,
          mn == sp.Rational(2, 3))

    return summary("v190 Nariai entropy bound: S_tot/S_dS >= 2/3 via (x-1)^2 >= 0, equality at the merge")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

05d99f3a2d6cce65a1346d34a69b880e292a8ad669bd021a2159aa64226ff550

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.