Follow the source behind the result.

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

"""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)

80873c1a8f85ce2486fd2e8683bee9371b9c1c84d611465bf04e0ae709402350

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.