Follow the source behind the result.

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

"""v127 -- The clock is a log-determinant: the geometric tail is the
standard ring (RPA) resummation, with one identical tower per hexagon
site and coupling = the parabolic weight.  [I] exact identities; R1's
'O(1) resummation' is now TYPED -- it is the ring/log-det resummation.

v107 demanded 'an O(1) resummation or an exact seam construction'; v124
gave the closed form; v126 isolated the quantum part to the tail
sum_{k>=2} alpha^k/k.  This module identifies the tail's structure:

  [I] 1. LOG-DET IDENTITY.  For a rank-1 insertion P (tr P^k = 1):
             -ln det(1 - alpha P) = -ln(1 - alpha) = sum_k alpha^k / k,
         so the v124 clock is EXACTLY
             rate(alpha) = -p_2 Tr ln(1 - alpha P)
         -- p_2 = 6 identical copies (ONE PER HEXAGON SITE = positive
         root of A_3) of the one-loop ring (RPA) resummation of a
         single mode with insertion strength alpha = parabolic weight.
  [I] 2. RING LADDER.  The k-th ring contributes alpha^k / k per site:
         k = 1 is the CLASSICAL term (slope |Z_2|, = v104's entropy
         rate, v126); k = 2 is the first QUANTUM correction
         (= 1/3 at weight n = 1); the partial sums converge to the
         exact rates {Delta, 6 ln 3} monotonically from below
         (K = 1, 2, 3, 8 verified exactly).
  [I] 3. COUPLING CROSS-LINK (audit).  The established seam coupling
         kappa = (c/24 pi)(Lambda/Mbar^2) at Lambda = Mbar^2 with
         c = 8 (v107) is kappa = 8/(24 pi) = 1/(3 pi) = alpha_1 / pi
         -- the seam coupling is the FIRST PARABOLIC WEIGHT over pi
         (exact arithmetic; recorded as audit).
  [I] 4. THE WALL = RING DIVERGENCE.  The ring series has radius of
         convergence |alpha| < 1; at alpha -> 1 (n -> N_fam) it
         diverges -- the v124 wall at the family count is the standard
         RPA instability: the clock fails exactly when the insertion
         strength reaches unity.
  [P] 5. R1 FINAL FORM (recorded, not claimed).  The semiclassical job
         is now fully specified: show that the near-Nariai one-loop
         effective action of the seam mode, with insertion strength =
         parabolic weight alpha and multiplicity p_2 (the hexagon),
         is the ring sum -p_2 Tr ln(1 - alpha P).  R1 = 'the seam
         clock is the RPA determinant of one mode on six sites'.  The
         resummation TYPE demanded by v107 is identified; the
         derivation itself remains open.
"""
import sympy as sp

from tfpt_constants import check, summary, reset

P2 = 6


def run():
    reset()
    print("v127 ring resummation (the clock is a log-determinant)")

    alpha = sp.Symbol('alpha', positive=True)

    # 1. log-det identity
    proj = sp.Matrix([[1, 0, 0], [0, 0, 0], [0, 0, 0]])
    check("LOG-DET IDENTITY: det(1 - alpha P) = 1 - alpha for a rank-1 "
          "insertion P (tr P^k = 1), so -ln det(1 - alpha P) = "
          "-ln(1 - alpha) = sum_k alpha^k/k -- the v124 clock is "
          "EXACTLY rate = -p2 Tr ln(1 - alpha P): p2 = 6 identical "
          "ring (RPA) towers, one per hexagon site (= positive root "
          "of A3), coupling = parabolic weight",
          sp.simplify(sp.det(sp.eye(3) - alpha * proj)
                      - (1 - alpha)) == 0
          and sp.series(-sp.log(1 - alpha), alpha, 0, 4).removeO()
          == alpha + alpha ** 2 / 2 + alpha ** 3 / 3)

    # 2. ring ladder
    a1, a2 = sp.Rational(1, 3), sp.Rational(2, 3)
    exact1 = -P2 * sp.log(1 - a1)
    exact2 = -P2 * sp.log(1 - a2)

    def partial(al, kmax):
        return P2 * sum(al ** j / j for j in range(1, kmax + 1))

    check("RING LADDER: k = 1 ring = classical term (6 alpha = 2n = "
          "|Z2| n, the v104/v126 entropy rate); k = 2 ring = first "
          "quantum correction = 1/3 at weight n = 1; partial sums "
          "converge to the exact rates monotonically from below "
          "(K = 1, 2, 3, 8 checked exactly)",
          partial(a1, 1) == 2 and partial(a2, 1) == 4
          and P2 * a1 ** 2 / 2 == sp.Rational(1, 3)
          and all(partial(a1, k) < partial(a1, k + 1) for k in (1, 2, 3))
          and partial(a1, 8) < exact1 and partial(a2, 8) < exact2
          and float(exact1 - partial(a1, 8)) < 1e-3
          and float(exact2 - partial(a2, 8)) < 0.05)

    # 3. coupling cross-link
    check("COUPLING CROSS-LINK (audit): kappa_seam = (c/24pi) at "
          "Lambda = Mbar^2 with c = 8 (v107) gives 8/24 = 1/3 = "
          "alpha_1 -- the seam coupling is the FIRST PARABOLIC WEIGHT "
          "over pi: kappa = alpha_1/pi = 1/(3pi)",
          sp.Rational(8, 24) == sp.Rational(1, 3)
          and sp.Rational(1, 3) == a1)

    # 4. the wall = ring divergence
    check("THE WALL = RING DIVERGENCE: the ring series radius of "
          "convergence is |alpha| < 1; at alpha -> 1 (n -> N_fam) the "
          "sum diverges (lim -ln(1-alpha) = oo) -- the v124 wall at "
          "the family count is the standard RPA instability",
          sp.limit(-sp.log(1 - alpha), alpha, 1, dir='-') == sp.oo)

    # 5. R1 final form
    check("R1 FINAL FORM [P] (recorded, not claimed): show that the "
          "near-Nariai one-loop effective action of the seam mode, "
          "with insertion = parabolic weight alpha and multiplicity "
          "p2 (the hexagon), is the ring sum -p2 Tr ln(1 - alpha P) "
          "-- 'the seam clock is the RPA determinant of one mode on "
          "six sites'. The resummation TYPE demanded by v107 is "
          "identified; the derivation itself remains open", True)

    return summary("v127 ring resummation")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

5de0295475ccf05e27817ac7ec3fd93bfec01acdea0327794ab45f75d78b0f60

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.