#!/usr/bin/env python3
"""
Computation 69 -- Higgs-mass cancellation: detailed analysis of the three rescues
==================================================================================
Comp 67 found that the paper's eq:pairing-cancel = 0 fails: with the zero mode
excluded (as the paper instructs), the eigenmode sum gives -M_*^2 / (16 pi^2),
which combined with the Higgs self-loop eq:ch-exact gives total delta m_h^2 = 0
rather than the intended M_*^2 / (16 pi^2).

The paper listed three structural rescues:
  (a) Higgs-doublet eaten Goldstones supply the missing bosonic dof at EW scale.
  (b) Reinterpret the zero mode as part of the cancellation sum.
  (c) Boson-fermion pairing of non-zero modes via complementation / parity.

This computation walks each in detail and shows that none, as currently
constructible from the postulates, closes the off-by-one without introducing
new ad hoc structure.  The honest status: M_* = 4 pi m_h is an NDA estimate
(standard naturalness rule of thumb Lambda ~ 4 pi * (mass scale)), not a
parameter-free one-loop derivation.
"""
from math import comb, pi


def even_odd_counts(D):
    """Returns (N_B_full, N_B_nonzero, N_F)."""
    NB_full = sum(comb(D, k) for k in range(0, D + 1, 2))
    NF = sum(comb(D, k) for k in range(1, D + 1, 2))
    return NB_full, NB_full - 1, NF


def main():
    print("=" * 100)
    print("  Computation 69 -- detailed analysis of the three Comp-67 rescues")
    print("=" * 100)
    print()

    # ------------------------------------------------------------------
    print("BASELINE (Comp 67 finding, restated).")
    print("-" * 100)
    print()
    print(f"  {'D':>3}  {'N_B_full':>10}  {'N_B_nonzero':>12}  {'N_F':>6}  "
          f"{'N_B_nz - N_F':>14}")
    for D in (3, 4, 5, 6, 7, 8, 10, 12):
        NB_full, NB_nz, NF = even_odd_counts(D)
        print(f"  {D:>3}  {NB_full:>10}  {NB_nz:>12}  {NF:>6}  {NB_nz - NF:>+14}")
    print()
    print("  Off-by-one is exact for every D, both even and odd.  Persistent.")
    print()

    # ------------------------------------------------------------------
    print("RESCUE (a) -- Higgs-doublet eaten Goldstones at the EW scale.")
    print("-" * 100)
    print()
    print("  Claim: at the EW scale, the Higgs doublet H = (H+, H0)^T has 4 real")
    print("  dof; 3 are eaten by W+, W-, Z to give them mass; 1 is the physical")
    print("  Higgs h.  If these 3 eaten Goldstones supply the 'missing' bosonic")
    print("  modes, the cancellation closes.")
    print()
    print("  Analysis:")
    print("    * The Comp 67 cancellation argument is at the MODAL scale M_* >> v,")
    print("      where the EW symmetry is UNBROKEN.  The Higgs doublet has 4 real")
    print("      propagating modes, ALL bosonic, with universal coupling 4 lambda_PST = 1")
    print("      to the Higgs self-energy.")
    print("    * The eaten-Goldstone count is a BROKEN-PHASE quantity: 3 of the 4")
    print("      become longitudinal W+/-/Z components only AFTER SSB at v.")
    print("    * Mixing a symmetric-phase eigenmode sum (Comp 67) with broken-phase")
    print("      mode count (eaten Goldstones) is dimensionally a category error:")
    print("      they live at different RG scales and aren't on the same Feynman")
    print("      diagram.")
    print()
    print("  Verdict: rescue (a) is not obviously consistent without an explicit")
    print("  RG-matching argument that the paper does not construct.  Reviewer's")
    print("  assessment confirmed.")
    print()

    # ------------------------------------------------------------------
    print("RESCUE (b) -- Zero mode included in the cancellation sum.")
    print("-" * 100)
    print()
    print("  Claim: put S = empty (the zero mode = the Higgs / order parameter)")
    print("  back into the cancellation sum.  Then N_B_full - N_F = 0 exactly")
    print("  for every D (binomial identity).")
    print()
    print("  Analysis:")
    print("    * With N_B - N_F = 0, eq:pairing-cancel gives 0 exactly.")
    print("    * BUT then there is no leftover +M_*^2 / (16 pi^2) to BE the")
    print("      Higgs self-mass; eq:ch-exact is also gone (it was the zero-mode")
    print("      self-loop).  Total delta m_h^2 = 0.")
    print("    * Result: m_h = 0 at one loop.  Same problem the original argument")
    print("      had with the off-by-one in (b)'s opposite direction.")
    print("    * This is a no-op rescue: it shuffles the bookkeeping but the")
    print("      total stays at 0.")
    print()
    print("  Verdict: rescue (b) trades the off-by-one for m_h = 0.  Same destination,")
    print("  different route.  Reviewer's assessment confirmed.")
    print()

    # ------------------------------------------------------------------
    print("RESCUE (c) -- Boson-fermion pairing via complementation S -> S^c.")
    print("-" * 100)
    print()
    print("  Claim: each subset S in P(D) pairs with its complement S^c = D \\ S,")
    print("  and the pair has opposite parity (even-|S| <-> odd-|S^c|) under")
    print("  complementation, giving a structural boson-fermion pairing of the")
    print("  eigenmode spectrum.")
    print()
    print("  Analysis:")
    print("    * For |S| + |S^c| = D, the parities of |S| and |S^c| are opposite")
    print("      iff D is ODD.  For even D, they have the SAME parity, so")
    print("      complementation pairs even with even and odd with odd (no")
    print("      cross-pairing).")
    print()
    print("  Even-D case (complementation preserves parity):")
    for D in (4, 6, 8, 10, 12):
        print(f"    D = {D:2d}: even-|S| <-> even-|S^c| (same parity); "
              f"complementation does NOT pair bosons with fermions.")
    print()
    print("  Odd-D case (complementation flips parity, as PST uses):")
    for D in (3, 5, 7, 9, 11):
        NB_full, NB_nz, NF = even_odd_counts(D)
        # The zero mode (|S|=0, even) pairs with the FULL SET mode (|S|=D, odd).
        # If we exclude the zero mode from N_B but keep the full-set mode in N_F,
        # the pairing is broken: the full-set mode has no even partner.
        print(f"    D = {D:2d}: zero-mode (|S|=0) <-> full-set-mode (|S|={D}, odd);")
        print(f"            full-set is in N_F; zero-mode is excluded from N_B;")
        print(f"            => one unpaired fermion remains; N_B_nz - N_F = {NB_nz - NF}.")
    print()
    print("  Verdict: rescue (c) does NOT close the off-by-one for either parity")
    print("  of D.  Even D: no boson-fermion cross-pairing at all.  Odd D: the")
    print("  pairing exists structurally but the zero-mode/full-set pair is broken")
    print("  by the paper's prescription (zero mode = Higgs = excluded).")
    print("  Reviewer's assessment confirmed.")
    print()

    # ------------------------------------------------------------------
    print("=" * 100)
    print("  HONEST CONCLUSION")
    print("=" * 100)
    print()
    print("  None of the three rescues, as constructible from the current PST")
    print("  postulates without additional ad hoc structure, closes the Comp 67")
    print("  off-by-one to recover the 'one-loop-complete / parameter-free' reading")
    print("  of M_* = 4 pi m_h.")
    print()
    print("  The defensible position is:")
    print()
    print("    * M_* = 4 pi m_h is an NDA estimate (standard naturalness rule of")
    print("      thumb Lambda ~ 4 pi * (mass scale) applied to the Higgs sector).")
    print()
    print("    * The structural cancellation argument (Comps 67, 69) is incomplete")
    print("      as stated; the numerical value M_* ~ 1.573 TeV survives ONLY as")
    print("      an NDA estimate, not as a parameter-free one-loop derivation.")
    print()
    print("    * Comp 67 + Comp 69 together pin the issue and show why each")
    print("      proposed rescue does not close it.  These computations exist")
    print("      as honest disclosure of the gap rather than evidence of failure")
    print("      of the broader PST programme.")
    print()
    print("    * Reframing the headline claim throughout the paper ('one-loop-")
    print("      complete' -> 'NDA estimate') is the correct response.")
    print()
    print(f"  Numerical value M_* = 4 pi * 125.25 GeV = {4 * pi * 125.25:.2f} GeV")
    print("  is unchanged as an NDA estimate.")


if __name__ == "__main__":
    main()