"""v429 -- DM.AXION.PENTAGON.01: the 'unmapped' icosahedral/golden structure of E8 surfaces in the ONE external cosmological input theta_i. The reverse audit (v354, E8.REVERSE.AUDIT.01) concluded that the golden ratio phi = 2 cos(pi/5) is UNMAPPED structure -- it appears only INTERNALLY (the affine-E8 attractor angle GOLD.ATOMS.01/v312, the icosian lattice v348) and in NO physical readout -- and used exactly that to argue phi cannot be numerology (v354 locates phi at the affine-E8 attractor eigenvalue v312 and the icosian lattice v348; v313/GOLD.ATOMS.01 names it the g_car=5 signature). This module sharpens that statement honestly: the single place where the otherwise-idle 5-fold/golden structure DOES touch a physical quantity is the one external input the theory cannot derive -- the axion misalignment angle theta_i = pi N_fam/g_car = 3pi/5 (the spine branch, v211/DM.AXION.SPINE.01). [E] 1. THETA_I IS THE INTERIOR ANGLE OF THE REGULAR g_car-GON. Because N_fam = g_car - 2 (= 3 = dim S+/.. , a carrier identity), the spine quotient is theta_i = pi N_fam/g_car = pi (g_car-2)/g_car = (g_car-2) pi/g_car, which is EXACTLY the interior angle of the regular g_car-gon. For g_car=5 that is the regular PENTAGON: theta_i = 3pi/5 = 108 deg. So the spine angle is not a bare rational multiple of pi -- it is the carrier-pentagon interior angle, the g_car=5 polygon itself. [E] 2. ITS COSINE IS THE GOLDEN RATIO. cos(theta_i) = cos(3pi/5) = (1-sqrt5)/4 = -1/(2 phi), and the partner 2 cos(2pi/5) = (sqrt5-1)/2 = 1/phi = phi-1, with phi = 2 cos(pi/5) = (1+sqrt5)/2. The golden ratio v354 called 'unmapped' IS the angular data of theta_i. [E] 3. THE GOLDEN CHARACTER IS UNIQUE TO g_car=5 (forced discriminator). Among the small regular n-gons only n=5 has an IRRATIONAL (golden) interior- angle cosine: cos((n-2)pi/n) is 1/2 (n=3), 0 (n=4), -1/2 (n=6) -- all rational -- and only n=5 gives -1/(2phi). So theta_i is golden BECAUSE the carrier is 5 (P2); a triangle/square/hexagon spine would carry no phi. This ties theta_i's golden character directly to g_car. [C] 4. THE BRIDGE (a re-reading, NOT a status change). This connects two standing facts -- phi is the g_car=5 carrier signature, 'unmapped' (v354/v313), and theta_i = 3pi/5 is the one external cosmological input (v211) -- into one statement: the idle icosahedral/golden structure is not idle; it is the geometry of the single misalignment angle the compiler does NOT fix. It REFINES v354 from 'phi in NO physical readout' to 'phi touches ONLY the [C] spine misalignment angle' (a conditional, branch-level appearance, never a frozen prediction). It is a geometric MOTIVE for the spine branch over the hilltop theta_i~170deg=pi(1-phi_seam) (whose cosine is NOT a clean pentagon value) -- consistent with, not replacing, the finite-T solver that decides the branch (v185/v211). HONEST SCOPE: [E] the three exact geometric identities (theta_i = g_car-gon interior angle; cos = -1/(2phi); golden character unique to n=5); [C] the bridge reading and the branch motive. This does NOT derive theta_i (the abundance is cosmology), does NOT upgrade DM.AXION.SPINE.01 (stays [C]), and does NOT close any gate -- it answers the parsimony question 'why does idle E8 structure exist?' for the one case where the idle structure has a job. Mirrored in wolfram/tfpt_readouts_extension.wl (exact trig/golden identities). Python (sympy). """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam phi = (1 + sp.sqrt(5)) / 2 def run(): reset() print("v429 DM.AXION.PENTAGON.01: theta_i = the carrier-pentagon interior angle = golden") # ---- 1. theta_i is the interior angle of the regular g_car-gon ---- theta_i = sp.pi * N_fam / g_car # 3 pi/5 polygon_interior = (g_car - 2) * sp.pi / g_car # interior angle of regular g_car-gon check("THETA_I = g_car-GON INTERIOR ANGLE [E]: N_fam = g_car-2, so " "theta_i = pi N_fam/g_car = (g_car-2)pi/g_car = the interior angle of " "the regular g_car-gon; for g_car=5 the PENTAGON, theta_i = 3pi/5 = " "108 deg (not a bare rational multiple of pi -- the carrier polygon)", N_fam == g_car - 2 and sp.simplify(theta_i - polygon_interior) == 0 and sp.simplify(theta_i - sp.Rational(3, 5) * sp.pi) == 0 and sp.nsimplify(sp.deg(theta_i)) == 108) # ---- 2. cos(theta_i) is the golden ratio ---- c_theta = sp.cos(theta_i) c_partner = 2 * sp.cos(2 * sp.pi / g_car) check("COS(THETA_I) = GOLDEN [E]: cos(3pi/5) = (1-sqrt5)/4 = -1/(2 phi); the " "partner 2 cos(2pi/5) = (sqrt5-1)/2 = 1/phi = phi-1; phi = 2 cos(pi/5) " "= (1+sqrt5)/2. The 'unmapped' golden ratio (v354) IS the angular data " "of theta_i", sp.simplify(c_theta - (1 - sp.sqrt(5)) / 4) == 0 and sp.simplify(c_theta + 1 / (2 * phi)) == 0 and sp.simplify(c_partner - 1 / phi) == 0 and sp.simplify(c_partner - (phi - 1)) == 0 and sp.simplify(2 * sp.cos(sp.pi / 5) - phi) == 0) # ---- 3. the golden character is unique to g_car=5 (forced discriminator) ---- def interior_cos(n): return sp.cos((n - 2) * sp.pi / n) rational_ngon = {n: interior_cos(n) for n in (3, 4, 6)} # 1/2, 0, -1/2 pent = interior_cos(5) check("GOLDEN CHARACTER UNIQUE TO g_car=5 [E]: among small regular n-gons " "only n=5 has an IRRATIONAL (golden) interior-angle cosine -- " "cos((n-2)pi/n) = 1/2,(n=3) 0,(n=4) -1/2 (n=6) all rational, only n=5 " "gives -1/(2phi); so theta_i is golden BECAUSE the carrier is 5 (P2)", all(sp.nsimplify(v).is_rational for v in rational_ngon.values()) and rational_ngon[3] == sp.Rational(1, 2) and rational_ngon[4] == 0 and rational_ngon[6] == sp.Rational(-1, 2) and not sp.nsimplify(pent).is_rational and sp.simplify(pent + 1 / (2 * phi)) == 0) # ---- 4. the bridge: a re-reading, NOT a status change ---- check("THE BRIDGE [C]: the 'unmapped' icosahedral/golden structure (v354 " "phi-unmapped; v313 GOLD.ATOMS phi = g_car signature) surfaces in the " "ONE external cosmological input theta_i = 3pi/5 (v211 spine branch). " "REFINES v354 'phi in NO readout' -> 'phi touches ONLY the [C] spine " "misalignment angle'; a geometric MOTIVE for the spine over the hilltop " "theta_i~170deg (no clean pentagon cosine); does NOT upgrade " "DM.AXION.SPINE.01 (stays [C]) and closes no gate -- the abundance is " "cosmology (the finite-T solver decides, v185/v211)", True) return summary("v429 DM.AXION.PENTAGON.01: theta_i = pi N_fam/g_car = 3pi/5 is " "the regular g_car-gon (pentagon) interior angle, cos = -1/(2phi); " "the golden character is unique to g_car=5; so the 'unmapped' " "icosahedral/golden E8 structure (v354) is the geometry of the one " "external input theta_i (v211) -- a [C] bridge, no status change") if __name__ == "__main__": raise SystemExit(1 if run() else 0)