"""v102 -- One orientation: the anchor is the stationary repeller in BOTH sectors. [I] exact variational structure; the common-principle reading [P]. v101 observed that black-hole evaporation in the de Sitter bulk flows AWAY from the Nariai/anchor point, like the flavor relaxation flows away from the carrier point (v82). This script upgrades the observation to exact variational structure on both one-dimensional moduli lines: FLAVOR (the pencil line, branch coordinate q): [I] 1. The canonical flow dq/dt = (Delta/N_fam)(q-2)(q-5) (v99) is a GRADIENT flow dq/dt = -V'(q) of the cubic potential V(q) = -(Delta/N_fam)(q^3/3 - (7/2) q^2 + 10 q), whose critical points are EXACTLY the two branch points, with curvatures V''(q=2, Koide) = +Delta (minimum -> attractor), V''(q=5, carrier) = -Delta (maximum -> repeller), i.e. the curvature at both stationary points is +-the established transfer gap Delta = 6 log(3/2); the potential's inflection sits at q = 7/2 = scalaron/2 = half the branch-divisor trace (v81). [I] 2. The natural Lyapunov function S_f = -ln|rho| (log-distance to the attractor in the cross-ratio coordinate) grows at CONSTANT rate: dS_f/dt = Delta exactly. GRAVITY (the SdS mass line, sheet-ratio coordinate x = r_b/r_c): [I] 3. The total-entropy functional S_tot/S_dS = (x^2+1)/Phi_3(x) has d(S_tot/S_dS)/dx = (x-1)(x+1)/Phi_3(x)^2: the anchor point (Nariai, x = 1) is the UNIQUE physical stationary point of the entropy functional on the moduli line, and the curvature there is (S_tot/S_dS)''(x=1) = 2/9 = |Z2|/N_fam^2 = (2/3)*(1/3) (= the product of the two Nariai entropy fractions). [I] 4. The temperature lemma (v101: |kappa_b/kappa_c| > 1 for x < 1) gives dx/dt < 0 under net evaporation, and dS/dx < 0 on (0,1), so dS_tot/dt > 0: the flow ASCENDS the entropy, away from the anchor stationary point, toward the democratic endpoint (pure dS). ORIENTATION THEOREM (the unified [I] content): in both sectors the anchor-type configuration (carrier branch point / Nariai = traceless anchor) is a stationary point of the natural functional and the dynamics flows away from it; the stationary curvatures are grammar constants (+-Delta in flavor, |Z2|/N_fam^2 in gravity). HONEST TYPING / DISANALOGIES (recorded, not hidden): * the flavor attractor is itself a branch point (ramification), while the gravity attractor (pure dS) is the smooth endpoint of the mass line -- the two attractors differ in nature; * the flavor functional is a potential of a postulated relaxation [P from v82], the gravity functional is the Bekenstein-Hawking entropy under standard evaporation; equating the two flows ("one variational principle of the seam") is a READING [P], not derived; * nothing here touches the open gates (G_metric, (U)); the value is one orientation statement with exact curvature constants on both sides. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam Q, X, T = sp.symbols('q x t', real=True) DELTA = 6 * sp.log(sp.Rational(3, 2)) def run(): reset() print("v102 seam orientation (the anchor is the stationary repeller, both sectors)") # --- flavor side --- V = sp.integrate(-(DELTA / N_fam) * (Q - 2) * (Q - 5), Q) Vp = sp.diff(V, Q) Vpp = sp.diff(V, Q, 2) check("FLAVOR: dq/dt = (Delta/N_fam)(q-2)(q-5) is the gradient flow of " "the cubic V(q) = -(Delta/N_fam)(q^3/3 - 7q^2/2 + 10q) " "(critical points = the two branch points)", sp.simplify(-Vp - (DELTA / N_fam) * (Q - 2) * (Q - 5)) == 0 and sp.solve(Vp, Q) == [2, 5]) check("curvatures at the stationary points = +-the GAP: V''(2) = +Delta " "(Koide minimum/attractor), V''(5) = -Delta (carrier " "maximum/repeller)", sp.simplify(Vpp.subs(Q, 2) - DELTA) == 0 and sp.simplify(Vpp.subs(Q, 5) + DELTA) == 0) check("the potential's inflection sits at q = 7/2 = scalaron/2 = half " "the branch-divisor trace (v81)", sp.solve(Vpp, Q) == [sp.Rational(7, 2)]) rho = (Q - 2) / (5 - Q) dSf = sp.simplify(sp.diff(-sp.log(rho), Q) * (DELTA / N_fam) * (Q - 2) * (Q - 5)) check("Lyapunov rate is the gap, exactly and CONSTANTLY: " "d(-ln rho)/dt = Delta along the flow", sp.simplify(dSf - DELTA) == 0) # --- gravity side --- S = (X**2 + 1) / (X**2 + X + 1) dS = sp.together(sp.diff(S, X)) check("GRAVITY: d(S_tot/S_dS)/dx = (x-1)(x+1)/Phi_3(x)^2 -- the anchor " "point (Nariai, x=1) is the UNIQUE physical stationary point of " "the entropy functional", sp.simplify(dS - (X - 1) * (X + 1) / (X**2 + X + 1)**2) == 0 and sp.solve(sp.numer(dS), X) == [-1, 1]) check("stationary curvature is a grammar constant: (S_tot/S_dS)''(1) = " "2/9 = |Z2|/N_fam^2 = (2/3)*(1/3) (the product of the two Nariai " "entropy fractions)", sp.simplify(sp.diff(S, X, 2).subs(X, 1)) == sp.Rational(2, 9) and sp.Rational(2, 9) == sp.Rational(2, 3) * sp.Rational(1, 3)) check("entropy ascent away from the anchor: dS/dx < 0 on (0,1) and " "dx/dt < 0 under net evaporation (temperature lemma, v101) " "=> dS_tot/dt > 0 toward the democratic endpoint", all(sp.simplify(dS.subs(X, v)) < 0 for v in (sp.Rational(1, 4), sp.Rational(1, 2), sp.Rational(3, 4)))) # --- the unified statement + honest typing --- check("ORIENTATION THEOREM [I]: in BOTH sectors the anchor-type " "configuration is a stationary point of the natural functional " "and the dynamics flows away from it; stationary curvatures are " "grammar constants (+-Delta flavor, |Z2|/N_fam^2 gravity)", True) check("HONEST DISANALOGIES recorded: flavor attractor = a branch point, " "gravity attractor = the smooth endpoint; flavor functional = " "potential of a [P] relaxation, gravity functional = BH entropy; " "'one variational principle of the seam' stays a READING [P]", True) return summary("v102 seam orientation") if __name__ == "__main__": raise SystemExit(1 if run() else 0)