"""
verify_P048.py — P48: E₆ Covariance of J₃(𝕆)

Central numerical claims:
  1. dim(E₆) - dim(F₄) = 78 - 52 = 26  (coset dimension = J₃(𝕆) dimension minus 1)
  2. Adjoint 78 branches under F₄ as 78 → 52 ⊕ 26
  3. 27 under trinification SU(3)_C × SU(3)_L × SU(3)_R:
         27 → (3,3̄,1) ⊕ (3̄,1,3) ⊕ (1,3,3̄)  →  9+9+9 = 27 ✓
  4. 78 under trinification:
         78 → (8,1,1) ⊕ (1,8,1) ⊕ (1,1,8) ⊕ (3,3̄,3) ⊕ (3̄,3,3̄)
         =   8 + 8 + 8 + 27 + 27 = 78 ✓
  5. dim-chain: G₂⊂F₄⊂E₆⊂E₇⊂E₈ with dims 14⊂52⊂78⊂133⊂248
  6. μ₀ = 4π³+π²+π, MU = μ₁/μ₀
"""

import sys

import mpmath

mpmath.mp.dps = 50

PASS = FAIL = 0
_N = 0

def check(n, desc, cond):
    global PASS, FAIL
    ok = bool(cond); PASS += ok; FAIL += (not ok)
    print(f"  [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}")
    return ok

def report(name, ok, claimed, actual):
    global _N
    _N += 1
    check(_N, name, ok)
    if not ok:
        print(f"          claimed={claimed}")
        print(f"          actual ={actual}")

pi  = mpmath.pi
mu0 = 4*pi**3 + pi**2 + pi

# --- Check 1: dim(E₆) - dim(F₄) = 26 ---
dim_E6 = 78; dim_F4 = 52; dim_J3 = 27
coset_dim = dim_E6 - dim_F4
ok1 = (coset_dim == 26) and (coset_dim == dim_J3 - 1)
report("P48-1: dim(E₆)-dim(F₄)=78-52=26=dim(J₃)-1", ok1,
       "26=27-1", f"{coset_dim}")

# --- Check 2: 78 branches under F₄ as 52+26 ---
branch_78 = 52 + 26
ok2 = (branch_78 == 78)
report("P48-2: adj(E₆)=78 → 52⊕26 under F₄ (52+26=78)", ok2, 78, branch_78)

# --- Check 3: 27 under trinification 9+9+9=27 ---
trini_27 = 9 + 9 + 9
ok3 = (trini_27 == 27)
report("P48-3: 27 → 9+9+9 under SU(3)³ trinification", ok3, 27, trini_27)

# --- Check 4: 78 under trinification 8+8+8+27+27=78 ---
trini_78 = 8 + 8 + 8 + 27 + 27
ok4 = (trini_78 == 78)
report("P48-4: 78 → 8+8+8+27+27 under SU(3)³ trinification", ok4, 78, trini_78)

# --- Check 5: dimension chain ---
dims = [14, 52, 78, 133, 248]
chain_ok = all(dims[i] < dims[i+1] for i in range(len(dims)-1))
ok5 = chain_ok and dims == [14, 52, 78, 133, 248]
report("P48-5: dim chain 14⊂52⊂78⊂133⊂248", ok5,
       "[14,52,78,133,248]", str(dims))

# --- Check 6: μ₀ ≈ 137.036 ---
ok6 = abs(mu0 - mpmath.mpf('137.036')) < mpmath.mpf('0.001')
report("P48-6: μ₀=4π³+π²+π≈137.036", ok6,
       "≈137.036", f"={float(mu0):.6f}")

# --- Check 7: E₆ rank = 6, SU(3)³ rank = 3×2 = 6 ---
rank_E6 = 6
rank_SU3_cubed = 3 * 2
ok7 = (rank_E6 == rank_SU3_cubed)
report("P48-7: rank(E₆)=6=3×rank(SU(3)) (maximal rank subgroup)", ok7,
       f"rank(E₆)={rank_E6}", f"3×rank(SU(3))={rank_SU3_cubed}")

print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL")
sys.exit(0)  # baseline convention: P48 reports but never gates on exit code