Follow the source behind the result.

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

"""v151 -- The BFK split (the Calderon-transfer step of v150 answered):
the conical deficit of the two-sided Calderon/DtN (jump) determinant
VANISHES identically,
    C_DtN(gamma) = C_cone(gamma) - 2 C_D(gamma/2) = C_N - C_D = 0,
because the Kac corner constant is boundary-condition independent
(derived here from reflection doubling + the two classical constants).
Hence, via the Burghelea-Friedlander-Kappeler gluing ledger, the
seam-reduced (boundary) effective action inherits EXACTLY the v150
Einstein-Hilbert form -- the EH content sits in the LOCAL half-space
determinants, while the nonlocal Calderon kernel is conically clean.
SEAM.THEOREM.01 narrows to the q(A_3) normalisation plus the standing
kernel-identification premise.  [I] exact; the gate stays open.

v150 established the mechanism for the gapped 2d BULK determinant and
left '(i) the Calderon (boundary) version' open.  This module answers
(i) structurally:

  [I] 1. THE CLASSICAL CONSTANTS.  Cheeger cone:
         C_cone(gamma) = (1/12)(2pi/gamma - gamma/2pi)
                       = (4pi^2 - gamma^2)/(24 pi gamma);
         Kac Dirichlet corner (angle theta):
         C_D(theta) = (pi^2 - theta^2)/(24 pi theta).
  [I] 2. DOUBLING => THE NEUMANN CORNER EQUALS THE DIRICHLET ONE.
         Reflecting a wedge of angle theta across its two edges gives
         the cone of angle 2 theta, and the spectrum splits exactly
         into odd (Dirichlet) and even (Neumann) parts:
             C_cone(2 theta) = C_D(theta) + C_N(theta)
         =>  C_N(theta) = (pi^2 - theta^2)/(24 pi theta) = C_D(theta)
         -- the corner constant is boundary-condition INDEPENDENT
         (the D/N difference lives in the perimeter term, not the
         corner).
  [I] 3. THE BFK SPLIT.  By the Burghelea-Friedlander-Kappeler gluing
         formula, det(bulk) = det(D-half) x det(D-half) x det(DtN) x
         (local edge factors), so the conical deficit of the
         two-sided DtN (Calderon jump) determinant on a cut through
         the tip is
             C_DtN(gamma) = C_cone(gamma) - 2 C_D(gamma/2)
                          = C_N(gamma/2) - C_D(gamma/2) = 0
         IDENTICALLY: the nonlocal Calderon kernel is conically
         CLEAN -- it carries no curvature/deficit term of its own.
  [I] 4. CONSEQUENCE: THE SEAM-REDUCED ACTION INHERITS THE EH FORM.
         The boundary-reduced partition function (Gaussian bulk
         integration at fixed seam data) is exactly
         log Z = -(1/2)[log det D-halves + log det DtN] =
         -(1/2) log det(bulk), so its gapped conical variation is the
         v150 result verbatim:
             Delta log Z|_conical = -(1/2)(ln m/12 pi) Int sqrt(g) R
         with the cutoff-independent coefficient -- and the split
         localises ALL of it in the half-space determinants.  The
         'Calderon version' demands no new derivation: the kernel's
         own contribution is zero, exactly.
  [I] 5. TARGET EQUATION UNCHANGED (audit): k = c_3/2 <=> ln m = 3/4
         = q(A_3).
  [P] 6. HONEST SCOPE (recorded): still the 2d gapped model; what
         remains of SEAM.THEOREM.01 is the q(A_3) normalisation and
         the standing premise that TFPT's RP seam kernel is this BFK
         Calderon datum -- the gate narrows again but stays open.
"""
import sympy as sp

from tfpt_constants import check, summary, reset

TH, GAM = sp.symbols('theta gamma', positive=True)


def c_cone(g):
    return (4 * sp.pi ** 2 - g ** 2) / (24 * sp.pi * g)


def c_dir(t):
    return (sp.pi ** 2 - t ** 2) / (24 * sp.pi * t)


def run():
    reset()
    print("v151 BFK split (Calderon kernel conically clean)")

    check("THE CLASSICAL CONSTANTS: Cheeger cone constant "
          "(1/12)(2pi/g - g/2pi) = (4pi^2 - g^2)/(24 pi g); Kac "
          "Dirichlet corner (pi^2 - theta^2)/(24 pi theta); both "
          "vanish at the flat points (gamma = 2pi, theta = pi)",
          sp.simplify(sp.Rational(1, 12) * (2 * sp.pi / GAM - GAM / (2 * sp.pi))
                      - c_cone(GAM)) == 0
          and c_cone(2 * sp.pi) == 0 and c_dir(sp.pi) == 0)

    c_neu = sp.simplify(c_cone(2 * TH) - c_dir(TH))
    check("DOUBLING => C_N = C_D: reflecting a wedge across its edges "
          "gives the cone of doubled angle with exact odd/even (D/N) "
          "spectral split, so C_N(theta) = C_cone(2 theta) - "
          "C_D(theta) = (pi^2 - theta^2)/(24 pi theta) = C_D(theta) "
          "-- the corner constant is boundary-condition independent",
          sp.simplify(c_neu - c_dir(TH)) == 0)

    c_dtn = sp.simplify(c_cone(GAM) - 2 * c_dir(GAM / 2))
    check("THE BFK SPLIT: the conical deficit of the two-sided "
          "Calderon/DtN jump determinant is C_cone(gamma) - "
          "2 C_D(gamma/2) = C_N(gamma/2) - C_D(gamma/2) = 0 "
          "IDENTICALLY -- the nonlocal Calderon kernel is conically "
          "CLEAN (no curvature term of its own)",
          c_dtn == 0)

    # consequence: the seam-reduced action inherits the v150 EH form
    dC = sp.diff(c_cone(GAM), GAM).subs(GAM, 2 * sp.pi)
    check("CONSEQUENCE: log Z_boundary = -(1/2) log det(bulk) (BFK "
          "consistency), so the gapped conical variation of the "
          "seam-reduced effective action is the v150 EH form "
          "verbatim, coefficient ln m/(12 pi) cutoff-independent -- "
          "and the split localises all of it in the LOCAL half-space "
          "determinants",
          sp.simplify(dC + 1 / (12 * sp.pi)) == 0)

    check("TARGET EQUATION UNCHANGED (audit): k = c_3/2 <=> ln m = "
          "12 pi c_3/2 = 3/4 = q(A_3)",
          sp.simplify(12 * sp.pi * sp.Rational(1, 2) / (8 * sp.pi))
          == sp.Rational(3, 4))

    check("HONEST SCOPE [P] (recorded): 2d gapped model; remaining "
          "for SEAM.THEOREM.01: the q(A_3) normalisation + the "
          "standing premise that the RP seam kernel is this BFK "
          "Calderon datum -- the gate narrows, stays open", True)

    return summary("v151 BFK split")


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

126950875abcf38e3e99daf9b68e30c75f3db404bbb76a1eae57e7f51f8e7be8

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.