"""v103 -- The trisection normal form: the canonical coordinate of the SdS moduli line exists, and in it the entropy is ONE cosine made of glue atoms. [I] exact; answers the v102 curvature-matching question at the geometric level; the remaining [P] is one clock. v102 left the sharpening question: is there a canonical coordinate in which the flavor invariant (the gap Delta) and the gravity invariant (the anchor curvature 2/9) become commensurable? Answer: yes -- the SdS horizon cubic is uniformized by ANGLE TRISECTION, and in the trisection angle everything collapses to atoms: [I] 1. TRISECTION UNIFORMIZATION. With r = 2 cos(theta) (units 1/sqrt(L)) the horizon cubic r^3 - 3r + 6m = 0 becomes exactly cos(3 theta) = -3m : the mass line is uniformized by trisecting one angle; the three horizon roots are the three trisection branches (deck = the cyclic root rotation = a Z_3 -- the same triality deck group as coker Q = Z/N_fam, v72). In the centered angle psi := phi - pi (cos phi = -3m): m = cos(psi)/N_fam, Nariai psi = 0, pure dS psi = pi/2; the v101 deck involution x -> 1/x is psi -> -psi (m and sigma both even). [I] 2. THE NORMAL FORM. The entropy functional becomes ONE cosine, sigma(psi) = S_tot/S_dS = 4/3 - (2/3) cos(2 psi/3), i.e. mean = |mu4|/N_fam, amplitude = |Z2|/N_fam, frequency = |Z2|/N_fam -- every constant is a glue atom over the family count. Checks: sigma(0) = 2/3 (Nariai), sigma(pi/2) = 1 (dS). [I] 3. CANONICAL CURVATURE = THE KOIDE CONSTANT TO THE FAMILY POWER. sigma''(psi = 0) = 8/27 = (2/3)^3 = (|Z2|/N_fam)^{N_fam}. [I] 4. INVARIANT BASE SLOPE. Both sigma and m are quadratic in psi at the anchor, so the base slope is finite and coordinate-invariant: d sigma/d m |_Nariai = sigma''/m'' = (8/27)/(-1/3) = -8/9 = -rank(E8)/N_fam^2, generally d sigma/dm = -(4/3) sin(2 psi/3)/sin(psi); in dimensionful form dS_tot/dM|_N = -(8 pi/3) r_N = -r_N/(N_fam c3). (Coordinate footnote: in the v101 sheet coordinate x the mass map has m''(1) = -1/4 = -1/|mu4| -- a different atom, as expected for a non-invariant second derivative; the SLOPE -8/9 is the same in both coordinates, as it must be.) [I] 5. THE BRIDGE (what the matching question becomes). The flavor invariant is a RATE: multiplier (2/3)^6 = (2/3)^{2 N_fam} per transport step (eigenvalue -Delta). The gravity invariant is a CURVATURE: (2/3)^{N_fam} in the canonical angle. Same base 2/3 = |Z2|/N_fam; exponent ratio 2 = |Z2|. What is still missing for a full identification is the gravity-side CLOCK (the near-Nariai evaporation generator that converts curvature into a rate) -- the same class of missing object as P2's continuous transfer generator: ONE clock, two known geometries. That identification stays [P]. HONEST TYPING: 1-4 are exact GR/trigonometric arithmetic landing on atoms fixed elsewhere; the triality cross-reference and the base-2/3 bridge are structural readings typed [P]/audit; no open gate is closed here. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam PSI, TH, M_ = sp.symbols('psi theta m', real=True) L, X = sp.symbols('Lambda x', positive=True) def run(): reset() print("v103 trisection normal form (canonical coordinate of the SdS line)") # 1. trisection uniformization r = 2 * sp.cos(TH) check("TRISECTION: with r = 2 cos(theta), r^3 - 3r = 2 cos(3 theta) " "exactly => horizon cubic <=> cos(3 theta) = -3m", sp.simplify(r**3 - 3 * r - 2 * sp.cos(3 * TH)) == 0) phi = PSI + sp.pi rc = 2 * sp.cos(phi / 3) rb = 2 * sp.cos((phi - 2 * sp.pi) / 3) r3 = 2 * sp.cos((phi + 2 * sp.pi) / 3) check("centered angle psi = phi - pi: m = cos(psi)/N_fam; " "Nariai psi = 0 (m = 1/3), pure dS psi = pi/2 (m = 0)", sp.simplify(-sp.cos(phi) / 3 - sp.cos(PSI) / N_fam) == 0 and sp.simplify(sp.cos(0) / 3 - sp.Rational(1, 3)) == 0) check("the three roots are the three trisection branches: sum = 0, " "e2 = -3 (Vieta), Nariai roots (1,1,-2) at psi = 0", sp.simplify(rc + rb + r3) == 0 and sp.simplify(sp.expand_trig(rc * rb + rc * r3 + rb * r3) + 3) == 0 and [sp.simplify(v.subs(PSI, 0)) for v in (rc, rb, r3)] == [1, 1, -2]) # 2. the normal form sigma = sp.simplify(sp.expand_trig((rb**2 + rc**2) / 3)) NF = sp.Rational(4, 3) - sp.Rational(2, 3) * sp.cos(2 * PSI / 3) check("NORMAL FORM: sigma(psi) = 4/3 - (2/3) cos(2 psi/3) -- mean = " "|mu4|/N_fam, amplitude = |Z2|/N_fam, frequency = |Z2|/N_fam " "(every constant a glue atom over the family count)", sp.simplify(sigma - NF) == 0) check("endpoints: sigma(0) = 2/3 (Nariai bound), sigma(pi/2) = 1 " "(pure dS)", NF.subs(PSI, 0) == sp.Rational(2, 3) and sp.simplify(NF.subs(PSI, sp.pi / 2)) == 1) check("deck involution psi -> -psi: m and sigma both even (= the v101 " "x -> 1/x deck); the Z3 trisection rotation = the triality deck " "(coker Q = Z/N_fam, v72) [structural cross-reference]", sp.simplify(NF.subs(PSI, -PSI) - NF) == 0 and sp.simplify(sp.cos(-PSI) - sp.cos(PSI)) == 0) # 3. canonical curvature curv = sp.diff(NF, PSI, 2).subs(PSI, 0) check("CANONICAL CURVATURE at the anchor: sigma''(0) = 8/27 = (2/3)^3 " "= (|Z2|/N_fam)^N_fam -- the Koide constant to the family power", curv == sp.Rational(8, 27) and sp.Rational(2, 3)**N_fam == sp.Rational(8, 27)) # 4. invariant base slope mpsi = sp.cos(PSI) / 3 slope = sp.simplify(curv / sp.diff(mpsi, PSI, 2).subs(PSI, 0)) check("INVARIANT BASE SLOPE: d sigma/dm |_Nariai = sigma''/m'' = -8/9 " "= -rank(E8)/N_fam^2 (cross-checked below in the x coordinate)", slope == sp.Rational(-8, 9)) gen = sp.simplify(sp.diff(NF, PSI) / sp.diff(mpsi, PSI)) check("generally d sigma/dm = -(4/3) sin(2 psi/3)/sin(psi); limit at " "the anchor = -8/9", sp.simplify(gen + sp.Rational(4, 3) * sp.sin(2 * PSI / 3) / sp.sin(PSI)) == 0 and sp.limit(gen, PSI, 0) == sp.Rational(-8, 9)) # cross-check in the v101 sheet coordinate x Phi3 = X**2 + X + 1 mx = sp.sqrt(3) / 2 * X * (1 + X) / Phi3**sp.Rational(3, 2) sx = (X**2 + 1) / Phi3 check("x-coordinate cross-check: m(x) deck-invariant, m'(1) = 0, " "m''(1) = -1/4 = -1/|mu4| (non-invariant atom footnote), and " "sigma''(1)/m''(1) = (2/9)/(-1/4) = -8/9 (the invariant slope)", sp.simplify(mx.subs(X, 1 / X) - mx) == 0 and sp.simplify(sp.diff(mx, X).subs(X, 1)) == 0 and sp.simplify(sp.diff(mx, X, 2).subs(X, 1)) == sp.Rational(-1, 4) and sp.simplify(sp.diff(sx, X, 2).subs(X, 1) / sp.diff(mx, X, 2).subs(X, 1)) == sp.Rational(-8, 9)) check("dimensionful: dS_tot/dM |_Nariai = -(8 pi/3) r_N = " "-r_N/(N_fam c3) (8 pi = 1/c3)", sp.simplify(sp.Rational(-8, 9) * (3 * sp.pi / L) * sp.sqrt(L) + 8 * sp.pi / (3 * sp.sqrt(L))) == 0) # 5. the bridge check("BRIDGE: flavor multiplier (2/3)^6 = (2/3)^(2 N_fam) (a RATE, " "eigenvalue -Delta) vs gravity curvature (2/3)^N_fam (a " "CURVATURE): same base 2/3 = |Z2|/N_fam, exponent ratio 2 = " "|Z2|; full identification needs the gravity-side CLOCK " "(near-Nariai evaporation generator) -- the same missing-object " "class as P2's transfer generator [P]: one clock, two known " "geometries", sp.Rational(2, 3)**6 == (sp.Rational(2, 3)**N_fam)**2 and 6 == 2 * N_fam) return summary("v103 trisection normal form") if __name__ == "__main__": raise SystemExit(1 if run() else 0)