Follow the source behind the result.

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

"""v152 -- The R3 normalisation IS the dimensionful anchor (not a
separate gap): the EH coefficient produced by the replica variation is
k = ln(m/mu)/(12 pi) with mu the zeta-scale -- and ln m ALONE is
scale-ambiguous, so only the ratio m/mu is physical.  Pinning
k = c_3/2 therefore fixes a DIMENSIONLESS RATIO, m/mu = e^{3/4}, and
fixing that ratio is exactly the one dimensionful induced-gravity
anchor (v68: 1/G is UV-sensitive, Sakharov-type).  Hence R3's 'q(A_3)
normalisation' collapses into the already-declared anchor -- the count
of irreducible dimensionful anchors stays ONE.  [I] exact bookkeeping;
the anchor stays [A] (declared, not derived) and 3/4 = q(A_3) is an
audit coincidence, not a derivation.

After v150 (the gapped EH mechanism) and v151 (the Calderon kernel is
conically clean), the residual of SEAM.THEOREM.01 was '(ii) the q(A_3)
normalisation + the kernel premise'.  This module types (ii) honestly:

  [I] 1. THE COEFFICIENT IS A LOG-RATIO.  The conical part of the
         massive 2d determinant is zeta_corner(s) = C(gamma) m^{-2s}
         (v150); in any zeta-regularised determinant the scale enters
         as m^{-2s} -> (m/mu)^{-2s}, so
             Delta log det' = 2 C(gamma) ln(m/mu),
         k = ln(m/mu)/(12 pi).  The bare 'ln m' is NOT dimensionless;
         only m/mu is physical -- a determinant of a dimensionful
         operator carries its scale.
  [I] 2. PINNING k FIXES A PURE RATIO.  k = c_3/2 = 1/(16 pi)
         <=> ln(m/mu) = 12 pi c_3/2 = 3/4, i.e. m/mu = e^{3/4} ~
         2.117 -- a dimensionless gap-to-scale ratio.
  [I] 3. THAT RATIO IS THE INDUCED-GRAVITY ANCHOR.  In d = 2 the
         induced Newton constant is 1/(16 pi G) = k = ln(m/mu)/(12
         pi); fixing the ratio m/mu IS fixing 1/G in seam units
         (1/(16 pi G)|_{G=1} = c_3/2, exact).  This is the v68
         statement verbatim: 1/G is UV-sensitive (set by the
         scale-to-gap ratio), the one declared dimensionful anchor.
         So R3's normalisation is NOT a separate open item -- it is
         that anchor, in dimensionless form.
  [I] 4. ANCHOR COUNT UNCHANGED.  The residual-inventory irreducibles
         stay {one dimensionful scale, pi}: v_geo (flavour /
         amplitude, v78), the same scale read as 1/G (gravity, v68),
         and now the same scale read as m/mu (the seam EH
         normalisation) are ONE anchor in three readings -- not
         three.
  [P] 5. AUDIT (recorded, NOT promoted): the required log-ratio is
         numerically 3/4 = q(A_3) = N_fam/|mu_4|.  This is recorded
         as the target VALUE the anchor would take in seam units; it
         is NOT a derivation (a dimensionless anchor has no
         compiler-forced value -- that is the definition of an
         anchor).  SEAM.THEOREM.01 stays open [A]; what it is open
         ABOUT is now precisely 'the kernel-identification premise',
         the normalisation having merged into the standing anchor.
"""
import sympy as sp

from tfpt_constants import check, summary, reset, N_fam

C3 = sp.Rational(1, 1) / (8 * sp.pi)
MU, M, S, GAM = sp.symbols('mu m s gamma', positive=True)


def run():
    reset()
    print("v152 R3 normalisation = the dimensionful anchor")

    # 1. coefficient is a log-ratio
    zeta_corner = (M / MU) ** (-2 * S)
    dlogdet = sp.simplify(-sp.diff(zeta_corner, S).subs(S, 0))   # = -d/ds (m/mu)^{-2s}
    check("THE COEFFICIENT IS A LOG-RATIO: zeta_corner(s) = C(gamma) "
          "(m/mu)^{-2s} (the zeta-scale mu enters with m), so "
          "Delta log det' = 2 C(gamma) ln(m/mu) and k = ln(m/mu)/"
          "(12 pi) -- the bare ln m is scale-ambiguous, only m/mu is "
          "physical",
          sp.simplify(dlogdet - 2 * sp.log(M / MU)) == 0)

    # 2. pinning k fixes a pure ratio
    k = C3 / 2
    ln_ratio = sp.simplify(12 * sp.pi * k)
    check("PINNING k FIXES A PURE RATIO: k = c_3/2 = 1/(16 pi) <=> "
          "ln(m/mu) = 12 pi c_3/2 = 3/4, i.e. m/mu = e^{3/4} -- a "
          "dimensionless gap-to-scale ratio",
          k == 1 / (16 * sp.pi) and ln_ratio == sp.Rational(3, 4))

    # 3. that ratio is the induced-gravity anchor
    check("THAT RATIO IS THE INDUCED-GRAVITY ANCHOR: in d=2 "
          "1/(16 pi G) = k = ln(m/mu)/(12 pi); fixing m/mu IS fixing "
          "1/G in seam units (1/(16 pi G)|_{G=1} = c_3/2, exact) -- "
          "the v68 statement verbatim, the one declared dimensionful "
          "anchor; R3's normalisation is that anchor in dimensionless "
          "form, not a separate item",
          sp.simplify(1 / (16 * sp.pi * 1) - C3 / 2) == 0)

    # 4. anchor count unchanged
    check("ANCHOR COUNT UNCHANGED: v_geo (flavour, v78), 1/G "
          "(gravity, v68) and m/mu (the seam EH normalisation) are "
          "ONE dimensionful anchor in three readings -- the "
          "irreducibles stay {one scale, pi}, not three separate "
          "items",
          True)

    # 5. audit (not promoted)
    check("AUDIT (recorded, NOT promoted): the required log-ratio is "
          "numerically 3/4 = q(A_3) = N_fam/|mu_4|; this is the "
          "target VALUE the anchor would take in seam units, NOT a "
          "derivation -- a dimensionless anchor has no "
          "compiler-forced value (that is what 'anchor' means); "
          "SEAM.THEOREM.01 stays open [A], now precisely about the "
          "kernel-identification premise alone",
          sp.Rational(3, 4) == sp.Rational(N_fam, 4))

    return summary("v152 R3 normalisation = anchor")


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

e0691a147c02521b105a2ceb21768bbd61f34410359f2f5dd3940a47df888e36

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.