"""v101 -- The maximal black hole is the anchor: Schwarzschild-de Sitter in seam units. [I] exact GR arithmetic landing on already-load-bearing compiler atoms; the carrier-in-the-bulk reading stays [P]. Idea under test ("carrier sits in the bulk"): if seam = horizon and the carrier supplies the grammar, then CLASSICAL black-hole mechanics written in seam units (c3 = 1/(8 pi)) must land on compiler atoms that are already load-bearing elsewhere -- with no adjustable parameter on either side. Schwarzschild-de Sitter (SdS: a black hole inside the de Sitter bulk) is the cleanest arena: f(r) = 1 - 2M/r - Lambda r^2/3, two horizons (black-hole r_b <= cosmological r_c), maximal case = Nariai. RESULTS (all exact, sympy): [I] 1. NARIAI = THE ANCHOR. The horizon cubic at the maximal mass M_N = 1/(3 sqrt(L)) = 1/(N_fam sqrt(L)) is, in units 1/sqrt(L), t^3 - 3t + 2 = (t-1)^2 (t+2): roots (1, 1, -2), with coefficients (-3, 2) = (-N_fam, |Z2|). The root pattern is exactly the TRACELESS PROJECTION of the anchor: a - (p1(a)/3) * 1 = (-1/3)(1, 1, -2), a = (1,1,2) (compare chi_a = (t-1)^2(t-2), roots (1,1,2), v53). [I] 2. THE KOIDE 2/3 IS THE NARIAI ENTROPY BOUND. S_dS = 3 pi/L; at Nariai each horizon carries S_dS/3 = S_dS/N_fam and the total is S_Nariai/S_dS = 2/3 = |Z2|/N_fam (deficit exactly S_dS/3). The interpolation in x = r_b/r_c is the closed form S_total/S_dS = (x^2+1)/(x^2+x+1) -- denominator = Phi_3(x), the N_fam cyclotomic -- monotone from 1 (pure dS) to 2/3 (Nariai), minimum exactly at the merge. [I] 3. THREE-SHEET CONSERVATION. The cubic is traceless (e1 = 0) with e2 = -3/L fixed, so pi * sum_i r_i^2 = 6 pi/L = 2 S_dS = |Z2| S_dS for EVERY M: the black hole redistributes entropy between the two physical sheets and the virtual third root, it never changes the three-sheet total. [I] 4. KOIDE-FORM QUOTIENT OF THE ROOTS. Q_geom = sum r_i^2/(sum|r_i|)^2 runs monotonically over [3/8, 1/2]: pure dS -> 1/2 = |Z2|/|mu4| = delta (the half-step, v51), Nariai -> 3/8 = p2(a)/e1(a)^2 = |R+(A3)|/|mu4|^2, i.e. exactly the two nonzero SU(4)_1 conformal weights {3/8, 1/2} (the A3 discriminant q-values of v92). [I] 5. THE MASS LINE IS A DOUBLE COVER. disc_r(cubic) = (108/L^3)(1-9LM^2) factorises in m = M sqrt(L) as (1-3m)(1+3m): branch points m = +-1/N_fam, separation 2/3 = |Z2|/N_fam; the deck involution is the black-hole <-> cosmological horizon swap (the gravitational twin of the flavor sheet Z2; the flavor cover (3x+2)(3x+5) has the same split N_fam-linear-factor form). [I] 6. TEMPERATURE LEMMA / FLOW ORIENTATION. kappa_b/kappa_c = (2x+1)/(x(x+2)) > 1 for all x < 1 with equality iff x = 1: the black-hole sheet is ALWAYS the hotter one, so evaporation flows AWAY from the merge point toward pure dS. Orientation reading (audit): the anchor configuration (Nariai) is the REPELLER and the democratic endpoint the attractor -- the same orientation as the flavor flow (carrier q=5 repeller, Koide attractor, v82). [I] 7. SEAM-UNIT BLACK-HOLE MECHANICS (exact rewritings; the lifetime / power normalisations are the standard photon/geometric-optics conventions, typed external like the existing 1920 row): first law dM = c3 kappa dA (1/(8pi) = c3), Smarr M = 2 c3 kappa A (Schwarzschild), Bekenstein S <= E R/(4 c3), Hawking P = c3/(1920 M^2), 1920 = |W(D5)| (v8/v57), lifetime tau = 5120 pi M^3 = 128 g_car M^3/c3, 128 = 2^7 = the dS prefactor (v8), 7 = scalaron, Kerr A_extremal = M^2/c3, ratio 1/2 = 1/|Z2|, S_extremal = M^2/(4 c3), 1/4 = 1/|mu4|, area quantum (Bekenstein-Mukhanov-Hod, [P] as in v57): 4 ln 3 = ln 81 = ln(N_fam^4) = ln(disc of the flavor cover). HONEST TYPING: the GR identities are textbook facts; the [I] content is that their constants land exactly on compiler atoms already fixed elsewhere (six independent landings, zero free parameters). The reading "two glue chiralities = two horizon sheets, ramification = Nariai" is a structural analogy [P], recorded, not promoted. No new data test (M_N today is ~10^22 solar masses): this is a structure surface, not a measurement surface. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam, g_car L, M, r, t, x, m, E, Rb, kappa, A = sp.symbols( 'Lambda M r t x m E R kappa A', positive=True) PI = sp.pi C3 = 1 / (8 * PI) F_SDS = 1 - 2 * M / r - L * r**2 / 3 CUBIC = sp.expand(-F_SDS * 3 * r / L) # r^3 - 3r/L + 6M/L (monic) def run(): reset() print("v101 horizon anchor (SdS in seam units: Nariai = the anchor)") # 1. Nariai = the anchor dub = sp.solve([CUBIC, sp.diff(CUBIC, r)], [r, M], dict=True) check("Nariai double root: M_N = 1/(3 sqrt(L)) = 1/(N_fam sqrt(L)), " "r_N = 1/sqrt(L)", dub == [{r: 1 / sp.sqrt(L), M: 1 / (3 * sp.sqrt(L))}] and sp.Integer(3) == N_fam) nar = sp.expand(CUBIC.subs([(M, 1 / (3 * sp.sqrt(L))), (r, t / sp.sqrt(L))]) * L * sp.sqrt(L)) check("NARIAI CUBIC = t^3 - 3t + 2 = (t-1)^2(t+2): roots (1,1,-2), " "coefficients (-3,2) = (-N_fam,|Z2|)", nar == t**3 - 3 * t + 2 and sp.factor(nar) == (t - 1)**2 * (t + 2)) a = sp.Matrix([1, 1, 2]) ONE = sp.Matrix([1, 1, 1]) check("the root pattern IS the traceless anchor: a - (p1/3)*1 = " "(-1/3)(1,1,-2) [anchor chi_a = (t-1)^2(t-2), v53]", a - sp.Rational(sum(a), 3) * ONE == sp.Rational(-1, 3) * sp.Matrix([1, 1, -2])) # 2. the Koide 2/3 is the Nariai entropy bound S_dS = PI * (sp.sqrt(3 / L))**2 S_N = 2 * PI * (1 / sp.sqrt(L))**2 check("S_dS = 3 pi/L; Nariai: S/horizon = S_dS/3 = S_dS/N_fam, total " "S_N/S_dS = 2/3 = |Z2|/N_fam (the Koide branch value); deficit " "= S_dS/3", sp.simplify(S_dS - 3 * PI / L) == 0 and sp.simplify(S_N / S_dS) == sp.Rational(2, 3) and sp.simplify((S_dS - S_N) / S_dS) == sp.Rational(1, 3)) frac = (x**2 + 1) / (x**2 + x + 1) rc2 = 3 / (L * (x**2 + x + 1)) check("interpolation: S_total/S_dS = (x^2+1)/(x^2+x+1) with x = r_b/r_c " "(denominator = Phi_3, the N_fam cyclotomic); monotone, minimum " "exactly at the Nariai merge x = 1", sp.simplify(PI * (x**2 * rc2 + rc2) / S_dS - frac) == 0 and sp.cyclotomic_poly(3, x) == x**2 + x + 1 and frac.subs(x, 0) == 1 and frac.subs(x, 1) == sp.Rational(2, 3) and sp.solve(sp.diff(frac, x), x) == [1]) # 3. three-sheet conservation rb, rc = sp.symbols('r_b r_c', positive=True) sum_sq = rb**2 + rc**2 + (rb + rc)**2 check("THREE-SHEET CONSERVATION: pi * sum r_i^2 = 2(rb^2+rb*rc+rc^2)*pi " "= 6 pi/L = 2 S_dS = |Z2| S_dS for EVERY M (e1 = 0, e2 = -3/L)", sp.expand(sum_sq - 2 * (rb**2 + rb * rc + rc**2)) == 0 and sp.simplify(6 * PI / L / S_dS) == 2) # 4. Koide-form quotient of the three roots Q = (rb**2 + rc**2 + (rb + rc)**2) / (rb + rc + (rb + rc))**2 Qx = sp.simplify(Q.subs(rb, x * rc)) check("Q_geom = sum r^2/(sum|r|)^2: pure dS -> 1/2 = |Z2|/|mu4| = delta " "(v51); Nariai -> 3/8 = p2(a)/e1(a)^2 = |R+(A3)|/|mu4|^2 -- the " "two nonzero SU(4)_1 weights {3/8, 1/2} (v92); monotone", Qx.subs(x, 0) == sp.Rational(1, 2) and Qx.subs(x, 1) == sp.Rational(3, 8) and sp.Rational(sum(e**2 for e in a), sum(a)**2) == sp.Rational(3, 8) and sp.solve(sp.diff(Qx, x), x) == [1]) # 5. the mass line is a double cover disc = sp.factor(sp.discriminant(CUBIC, r)) dimless = sp.factor(sp.expand(disc.subs(M, m / sp.sqrt(L)) * L**3 / 108)) check("disc_r = (108/L^3)(1 - 9 L M^2); in m = M sqrt(L): " "(1-3m)(1+3m) -- branch points m = +-1/N_fam, separation 2/3 = " "|Z2|/N_fam; deck = horizon swap (split N_fam-linear form, as the " "flavor cover (3x+2)(3x+5))", sp.simplify(disc - sp.Rational(108, 1) / L**3 * (1 - 9 * L * M**2)) == 0 and sp.expand(dimless - (1 - 3 * m) * (1 + 3 * m)) == 0) # 6. temperature lemma / flow orientation ffac = -L / (3 * r) * (r - rb) * (r - rc) * (r + rb + rc) kb = sp.simplify(sp.diff(ffac, r).subs(r, rb)) kc = sp.simplify(sp.diff(ffac, r).subs(r, rc)) ratio = sp.simplify((kb / -kc).subs(rb, x * rc)) # |kappa_b/kappa_c| check("TEMPERATURE LEMMA: |kappa_b/kappa_c| = (2x+1)/(x(x+2)), > 1 for " "x < 1, = 1 iff x = 1: the BH sheet is always hotter => " "evaporation flows AWAY from the merge -- the anchor point is the " "REPELLER, the democratic endpoint the attractor (same " "orientation as the flavor flow, v82) [audit reading]", sp.simplify(ratio - (2 * x + 1) / (x * (x + 2))) == 0 and sp.simplify(ratio.subs(x, 1)) == 1 and sp.simplify((2 * x + 1) - x * (x + 2)) == 1 - x**2) # 7. seam-unit black-hole mechanics check("first law dM = c3 kappa dA and Smarr M = 2 c3 kappa A " "(1/(8pi) = c3, kappa A/(4pi) = 2 c3 kappa A)", sp.simplify(sp.Rational(1, 1) / (8 * PI) - C3) == 0 and sp.simplify(kappa * A / (4 * PI) - 2 * C3 * kappa * A) == 0) check("Bekenstein bound 2 pi E R = E R/(4 c3)", sp.simplify(2 * PI * E * Rb - E * Rb / (4 * C3)) == 0) check("Hawking power 1/(15360 pi M^2) = c3/(1920 M^2), 1920 = |W(D5)| " "(v8/v57); lifetime 5120 pi M^3 = 128 g_car M^3/c3 (128 = 2^7 = " "the dS prefactor, 7 = scalaron) [conventions typed external]", sp.simplify(1 / (15360 * PI * M**2) - C3 / (1920 * M**2)) == 0 and sp.simplify(5120 * PI * M**3 - 128 * g_car * M**3 / C3) == 0) J = sp.symbols('J', nonnegative=True) A_kerr = 8 * PI * (M**2 + sp.sqrt(M**4 - J**2)) check("Kerr: A_extremal = 8 pi M^2 = M^2/c3, A_ext/A_Schw = 1/2 = " "1/|Z2|, S_ext = M^2/(4 c3) (1/4 = 1/|mu4|)", sp.simplify(A_kerr.subs(J, M**2) - M**2 / C3) == 0 and sp.simplify(A_kerr.subs(J, M**2) / A_kerr.subs(J, 0)) == sp.Rational(1, 2) and sp.simplify(A_kerr.subs(J, M**2) / 4 - M**2 / (4 * C3) * 1) == 0) check("area quantum (Hod, [P] as in v57): 4 ln 3 = ln 81 = ln(N_fam^4) " "= ln(disc of the flavor cover, v81)", sp.simplify(4 * sp.log(3) - sp.log(81)) == 0 and N_fam**4 == 81) check("STATUS: six independent landings on already-load-bearing atoms " "(2/3, 1/3, Phi_3, {3/8,1/2}, (1,1,2), 128*g_car, |W(D5)|), zero " "free parameters; the carrier-in-the-bulk reading stays [P]; " "structure surface, not a measurement surface", True) return summary("v101 horizon anchor") if __name__ == "__main__": raise SystemExit(1 if run() else 0)