#!/usr/bin/env python3 """ Computation 93 -- M_* off-by-one: re-examination of rescue (b) under the doublet-self-loop / eigenmode-sum decomposition ========================================================================== Open-research item 1.2 (the M_* = 4 pi m_h zero-mode off-by-one). Comp 67 identified the off-by-one in eq:pairing-cancel; Comp 69 walked three rescues and concluded none closes it. This computation re-examines Comp 69's analysis of rescue (b) (zero-mode included in the eigenmode sum) with explicit attention to whether the paper's two contributions (1) Higgs doublet self-loop: delta m_h^2|_Phi = C_H * M_*^2/(16 pi^2) with C_H = 4 lambda_PST = 1 (eq:ch-exact) (2) PST scalar eigenmode sum: delta m_h^2|_eig = (N_B - N_F) * M_*^2/(16 pi^2) (eq:pairing-cancel) are conceptually independent contributions to be ADDED, or whether they overlap (double-counting the Higgs as both doublet component and zero mode of the Boolean Laplacian). WHY THIS MATTERS ---------------- If they ADD independently: total = (M_*^2/16pi^2) * (C_H + N_B - N_F) with zero mode INCLUDED: N_B - N_F = 0, total = M_*^2/(16 pi^2) => m_h = M_* / (4 pi) ✓ with zero mode EXCLUDED: N_B - N_F = -1, total = 0 => m_h = 0 ✗ Resolution: include zero mode; rescue (b) closes off-by-one. If they overlap (zero mode = Higgs = doubly-counted): Including zero mode in N_B double-counts the Higgs (already in C_H), inflating the bosonic count by 1, giving N_B - N_F = 0 only because of the double count. After removing double-count: net = -1, total = 0. This matches Comp 69's verdict (rescue b is a no-op). THE STRUCTURAL QUESTION ----------------------- - C_H = 4 lambda_PST captures the Higgs DOUBLET self-loop with quartic vertex lambda |H|^4 / 4. This counts the 4-component complex doublet (4 real DoFs) self-interacting via the quartic vertex. - N_B counts BOSONIC eigenmodes of the Boolean Laplacian L^B on the substrate's modal field psi. The zero mode (S = empty) is the uniform constant-mode profile. - Are these the same Higgs or different? In PST, psi = Pi(phi_Higgs) is the substrate's MODAL ORDER PARAMETER, a single scalar (or scalar-valued) field on substrate configurations. The 4-component doublet structure arises POST-PROJECTION via inner fluctuations of the Dirac operator on M x F, generating the H^+, H^0 doublet. Conjecture for re-examination ----------------------------- The substrate-side modal field psi is a SINGLE scalar; its Boolean Laplacian eigenmodes are PST-internal scalars. After projection Pi, the emergent Standard-Model Higgs doublet H has 4 real components, each from a DIFFERENT structural sector of the spectral triple (A_F = C (+) H (+) M_3(C)) -- so the 4-component doublet self-loop (C_H) is NOT in correspondence with substrate eigenmodes (N_B, N_F). Under this reading: - C_H counts EMERGENT 4-real-component doublet self-loops. - (N_B, N_F) count SUBSTRATE-internal Boolean modes orthogonal to the Higgs identification. - The zero mode S = empty is the "average / constant" substrate mode, distinct from the emergent doublet. - Including the zero mode in N_B is NOT double-counting the Higgs; it counts the substrate-side constant-mode contribution which is a separate object from the post-projection doublet. Under this reading, rescue (b) closes the off-by-one structurally: total = (M_*^2/16pi^2)(C_H + N_B^full - N_F) = (M_*^2/16pi^2)(1 + 0) = M_*^2/(16 pi^2) => m_h = M_*/(4 pi) ✓ CAVEAT ------ This rescue depends on the structural distinction between: (a) substrate-side modal field psi (single scalar; Boolean modes) (b) emergent SM Higgs doublet H (4-component; doublet self-loop) The paper currently couples them via phi_Higgs = Pi(psi), with the zero mode of L^B = the constant profile = the order-parameter "background". Whether the zero mode is identified WITH the Higgs (in which case rescue b double-counts) or DISTINCT from the Higgs (in which case rescue b closes) is a foundational reading that Comps 67/69 did not separate. This computation does NOT decide the question; it identifies that the key foundational reading was implicit in Comps 67/69 and explicit resolution requires going back to the substrate-side / emergent-side distinction in the PST projection chain. """ from math import comb, pi, exp def even_odd_counts(D: int) -> tuple[int, int, int]: """Returns (N_B_with_zero, 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 93 -- M_* off-by-one: re-examination of rescue (b)") print("=" * 100) print() print("THE PAPER's TWO CONTRIBUTIONS") print("-" * 100) print() print(" (1) Higgs DOUBLET self-loop (eq:ch-exact):") print(" delta m_h^2 |_Phi = C_H * M_*^2/(16 pi^2)") print(" with C_H = 4 lambda_PST = 4 * (1/4) = 1.") print() print(" (2) PST scalar eigenmode sum (eq:pairing-cancel):") print(" delta m_h^2 |_eig = (N_B - N_F) * M_*^2/(16 pi^2)") print(" with N_B, N_F from Boolean-Laplacian eigenmode counting.") print() print(" TOTAL (if independent):") print(" delta m_h^2 = (C_H + N_B - N_F) * M_*^2/(16 pi^2)") print() print("PAPER's EXISTING COUNTING:") print("-" * 100) print() print(f" {'D':>3} {'N_B (incl 0)':>14} {'N_B (excl 0)':>14} {'N_F':>8} " f"{'C_H + N_B^full - N_F':>23} {'C_H + N_B^nz - N_F':>22}") for D in (4, 6, 8, 10, 12, 14, 16): NB_full, NB_nz, NF = even_odd_counts(D) with_zero = 1 + (NB_full - NF) without_zero = 1 + (NB_nz - NF) print(f" {D:>3} {NB_full:>14} {NB_nz:>14} {NF:>8} " f"{with_zero:>+23} {without_zero:>+22}") print() print(" Two readings of the total scalar contribution (C_H + N_B - N_F):") print() print(" * With zero mode in N_B: total = +1, m_h^2 = M_*^2/(16 pi^2)") print(" ==> m_h = M_* / (4 pi). ✓") print() print(" * With zero mode excluded: total = 0, m_h^2 = 0") print(" ==> m_h = 0. ✗") print() print("THE FOUNDATIONAL READING QUESTION (rescue b re-examined)") print("-" * 100) print() print(" Comp 69 dismissed rescue (b) (zero mode in N_B) because it interpreted") print(" the eigenmode imbalance as the SOURCE of the Higgs self-mass. Under that") print(" reading, N_B - N_F = 0 leaves no leftover for m_h^2, so total = 0.") print() print(" But the paper's eq:ch-exact derives delta m_h^2|_Phi = M_*^2/(16 pi^2)") print(" separately, from the 4-component doublet self-loop with C_H = 4 lambda.") print(" This is INDEPENDENT of the eigenmode sum: C_H counts the EMERGENT doublet") print(" self-interactions, while (N_B, N_F) counts SUBSTRATE-internal Boolean") print(" eigenmodes orthogonal to the emergent Higgs identification.") print() print(" Under the 'independent contributions' reading:") print(" - C_H = 1 (emergent doublet, always)") print(" - N_B - N_F = 0 (include zero mode = substrate constant-mode)") print(" - Total = 1 + 0 = 1, giving m_h^2 = M_*^2/(16 pi^2)") print(" - m_h = M_* / (4 pi) ✓") print() print(" This is the FOURTH RESCUE: re-interpret rescue (b) as 'include zero mode") print(" in N_B, where the zero mode is the substrate-side constant-mode (NOT") print(" double-counting the emergent Higgs doublet, which is in C_H).'") print() print("STATUS") print("-" * 100) print() print(" This re-examination identifies that Comp 69's dismissal of rescue (b)") print(" rests on a specific foundational reading (the zero mode IS the Higgs,") print(" so including it in N_B double-counts). An alternative reading -- the") print(" zero mode is a substrate-side constant-mode DISTINCT from the emergent") print(" doublet -- closes the off-by-one structurally and recovers M_* = 4 pi m_h") print(" as a parameter-free one-loop derivation.") print() print(" Which reading is correct depends on the precise PST projection chain") print(" psi (substrate scalar) -> phi_Higgs (emergent 4-component doublet).") print(" The substrate-side modal field psi is a SINGLE scalar function on P(D);") print(" its Boolean-Laplacian eigenmodes are PST-internal SCALAR modes. The") print(" emergent 4-component doublet structure arises POST-PROJECTION via inner") print(" fluctuations on M x F, generating H^+, H^0 from the A_F = C (+) H (+)") print(" M_3(C) factor.") print() print(" If the projection Pi: psi -> phi_Higgs maps the substrate-side single") print(" scalar to the EMERGENT 4-component doublet via inner-fluctuation lift,") print(" then the zero mode of L^B (substrate-side constant-mode) is NOT the") print(" emergent doublet -- they live at different layers in the projection chain.") print(" Including the substrate-side zero mode in N_B does not double-count") print(" the emergent doublet (which is captured by C_H). Rescue (b) closes") print(" the off-by-one under this reading.") print() print(" This re-examination DOES NOT settle the question definitively; it") print(" identifies the foundational reading on which Comp 69's dismissal rests,") print(" and shows that an alternative reading recovers the parameter-free") print(" M_* = 4 pi m_h relation. Further computations to validate the") print(" substrate-vs-emergent distinction in the projection chain would close") print(" this open research item.") if __name__ == "__main__": main()