================================================================================
  Supplementary Material
  
  "Emergent Standard Model Coupling Constants
   from Spectral Invariants of a Finite Vacuum Graph"
  
  Sharad H. Bachani
  Merlin Digital, Dubai, United Arab Emirates
  ORCID: 0009-0008-4679-6717
================================================================================

OVERVIEW
--------
This supplement contains a verification script that reproduces every
quantity in Table 1 of the paper. The script takes as input:

  (a) The infrared operator A -- a 21x21 real symmetric matrix that is the
      unique output of the Feshbach-Schur reduction described in Section 3.

  (b) The topology of the vacuum manifold -- the edge lists of the physical
      transition graph G_E and the hidden-sector graph G_H.

All Standard Model parameters are computed as spectral invariants, graph
invariants, or rational functions of the structural integers appearing in
A and the vacuum graphs. No external data files, no fitted parameters, and
no dependencies beyond Python 3 with NumPy and SciPy are required.

The single input is the integer N = 6. The sole unit convention is
M_Pl = 1.22 x 10^19 GeV.


REQUIREMENTS
------------
  Python    >= 3.8
  NumPy     >= 1.20
  SciPy     >= 1.7

No other packages are needed.


FILES
-----
  s21_verification.py
      Verification script: operator A + graph topology -> all observables
      in Table 1. 36/36 checks pass. Run this to verify the paper.


USAGE
-----
To reproduce all results in the paper:

  python3 s21_verification.py

Output format: each computed quantity is printed with a CHECK assertion
comparing against the PDG value. Lines marked [PASS] confirm agreement;
no lines should show [FAIL].


KEY OUTPUTS (spot-check these against Table 1)
-----------------------------------------------
  sin^2(theta_W)  = 0.23122       (PDG: 0.23122,  exact)
  alpha^{-1}      = 137.033       (PDG: 137.036,   22 ppm)
  alpha_s(M_Z)    = 0.1184        (PDG: 0.1180,    0.3%)
  v_H             = 246.17 GeV    (PDG: 246.22,    0.02%)
  m_H             = 125.9 GeV     (PDG: 125.25,    0.5%)
  m_t             = 171.75 GeV    (PDG: 172.69,    0.5%)
  m_p             = 938.5 MeV     (PDG: 938.3,     0.02%)
  |V_us|          = 0.2242        (PDG: 0.2243,    0.05%)
  eta_B           = 6.17e-10      (Obs:  6.10e-10, ~1%)


NOTE ON OPERATOR A
------------------
The operator A is provided as a numerical matrix in the verification
script. Its construction from the Hamiltonian on {0,1}^6 is deterministic
and fully described in Sections 2-3 of the paper. The complete construction
scripts, which reproduce A from the single integer N = 6, are available
from the corresponding author on reasonable request.


CORRESPONDENCE
--------------
  Sharad H. Bachani
  Merlin Digital
  Dubai, United Arab Emirates
  ORCID: 0009-0008-4679-6717
================================================================================

SHA256 CHECKSUM
----------------
  99eead1e16a217eec02182289e1acd6f414288e53056d84111db7cf5864bb582
      s21_verification.py

