#!/usr/bin/env python3
"""
PST Computation 11 — Bimodule support: the threshold-selected su(2)_L acts on
the γ_F = +1 block only
======================================================================
Closes §14.2 item (B) (the chiral-bimodule residual that Computation 8 left
open).

Computation 8 established:
    P1.  γ_M does NOT furnish weak chirality.
    P2.  The substrate carries so(4) = su(2)_L ⊕ su(2)_R (quaternion
         left/right multiplication, equivalently the isometry algebra of
         S^3 ⊂ M = ℝ × S^3).  The directed modal threshold ε > 0 selects
         one of the two factors as the gauge-active su(2)_L.
    P3.  Residual: prove the threshold-selected su(2)_L acts on the
         γ_F = +1 block of H_F only — the bimodule-support property
         that, together with Computation 7's order-one condition, delivers the
         chiral Standard Model.

The honest framing of the question.  The chiral bimodule support is a
property of the Connes-Chamseddine H_F = H_F^(+) ⊕ H_F^(-) finite Hilbert
space and the way A_F = ℂ ⊕ ℍ ⊕ M_3(ℂ) is represented on it.  Two
representations are mathematically possible a priori:

    Choice A:   ℍ acts on H_F^(+) (left block), trivial on H_F^(-) — the
                parity-violating, observed Standard Model.
    Choice B:   ℍ acts on H_F^(-) (right block), trivial on H_F^(+) — the
                parity-conjugate model, never observed.

Both choices give a consistent spectral triple in the Connes-Chamseddine
sense.  The chirality of SU(2)_L therefore reduces to the question
"which of A or B is realised in nature?"  PST's claim is that the
directed modal threshold (paper §8 parity-violation mechanism) is the
physical principle that selects Choice A.

Sections:
    §1  Construct a minimal but representative chiral H_F: one weak
         doublet ((νL, eL), (νR, eR)) with γ_F = diag(+1, +1, -1, -1).
    §2  Both Choice-A and Choice-B representations of ℍ commute with γ_F
         (preserve the chirality grading) — a necessary condition for the
         bimodule structure either way.
    §3  Block-support verification: Choice A is non-trivial on
         H_F^(+) and ZERO on H_F^(-); Choice B is the mirror.
    §4  The substrate's so(4) = su(2)_L ⊕ su(2)_R contains both Choices
         as embeddings; without a physical orientation principle, both
         are on equal footing.
    §5  The directed threshold ε > 0 supplies the orientation: the same
         orientation that produces parity violation (paper §8) is the
         orientation that selects Choice A.  Bimodule support is
         therefore PST's parity-violation mechanism applied to the
         representation choice.
    §6  Conclusion: chirality of SU(2)_L is STRUCTURAL in PST,
         conditional on the framework assumption (a Connes spectral
         triple) and on the parity-violation argument (paper §8).
         The §14.1 "conditional on chiral bimodule" qualifier can be
         downgraded to "conditional on the threshold-mechanism parity
         argument".

Run:
    python3 computation_11.py
"""
import numpy as np
import numpy.linalg as la

SEP = "=" * 78
def hdr(s): print(f"\n{SEP}\n  {s}\n{SEP}")
def comm(A, B):  return A @ B - B @ A
def norm(M):     return la.norm(M)

print(SEP)
print("  PST Computation 11 — bimodule support of the threshold-selected su(2)_L")
print(SEP)

# Pauli matrices for the SU(2) generators
sx = np.array([[0,1],[1,0]], dtype=complex)
sy = np.array([[0,-1j],[1j,0]], dtype=complex)
sz = np.array([[1,0],[0,-1]], dtype=complex)
I2 = np.eye(2, dtype=complex)

# Generators of SU(2) at the Lie-algebra level: J_a = σ_a / 2
J = [None, sx/2, sy/2, sz/2]

# ─────────────────────────────────────────────────────────────────────
# §1. Minimal chiral H_F representing one weak doublet
# ─────────────────────────────────────────────────────────────────────
hdr("§1 — Minimal H_F: one weak doublet with γ_F grading")

# H_F = ℂ^4 with the convention
#    basis = (νL, eL, νR, eR)
# γ_F = diag(+1, +1, -1, -1) grades left (γ_F = +1) and right (γ_F = -1).
# The first two are left-handed (in an SU(2)_L doublet).  The second two
# are right-handed (SU(2)_L singlets in the observed Standard Model;
# would be the SU(2)_R doublet under parity).

gF = np.diag([+1, +1, -1, -1]).astype(complex)
P_plus  = np.diag([1, 1, 0, 0]).astype(complex)   # projector to γ_F=+1
P_minus = np.diag([0, 0, 1, 1]).astype(complex)   # projector to γ_F=-1

print(f"  H_F = ℂ^4, basis = (νL, eL, νR, eR)")
print(f"  γ_F = diag(+1, +1, -1, -1)")
print(f"  γ_F^2 - I = {norm(gF @ gF - np.eye(4)):.3e}  (γ_F is a Z_2 grading)")
print(f"  P_+ projects on left block (νL, eL); P_- on right block (νR, eR)")

# ─────────────────────────────────────────────────────────────────────
# §2. Two representations of ℍ ≅ su(2) on H_F — both commute with γ_F
# ─────────────────────────────────────────────────────────────────────
hdr("§2 — Choice A (left-acting) and Choice B (right-acting) ℍ both commute with γ_F")

# Choice A: ℍ acts on the left block as (σ_a/2), zero on right
H_A = [None]
for a in (1, 2, 3):
    Ja_left = np.zeros((4, 4), dtype=complex)
    Ja_left[:2, :2] = J[a]
    H_A.append(Ja_left)

# Choice B: ℍ acts on the right block as (σ_a/2), zero on left
H_B = [None]
for a in (1, 2, 3):
    Ja_right = np.zeros((4, 4), dtype=complex)
    Ja_right[2:, 2:] = J[a]
    H_B.append(Ja_right)

print("  Choice A: ℍ → (σ_a/2 on (νL,eL), 0 on (νR,eR))")
print("  Choice B: ℍ → (0 on (νL,eL), σ_a/2 on (νR,eR))")
print()

for a in (1, 2, 3):
    nA = norm(comm(H_A[a], gF))
    nB = norm(comm(H_B[a], gF))
    print(f"  ||[H_A^{a}, γ_F]|| = {nA:.3e}   ||[H_B^{a}, γ_F]|| = {nB:.3e}")

print()
print("  Both choices preserve the chirality grading — necessary condition for")
print("  the bimodule structure is satisfied either way.")

# Verify SU(2) algebra: [H^a, H^b] = i ε_abc H^c
print()
print("  Verify SU(2) algebra [H^a, H^b] = i ε_{abc} H^c for both choices:")
for label, H in [("A", H_A), ("B", H_B)]:
    err = 0.0
    err += norm(comm(H[1], H[2]) - 1j * H[3])
    err += norm(comm(H[2], H[3]) - 1j * H[1])
    err += norm(comm(H[3], H[1]) - 1j * H[2])
    print(f"    Choice {label}: cumulative algebra error = {err:.3e}")

# ─────────────────────────────────────────────────────────────────────
# §3. Block-support verification
# ─────────────────────────────────────────────────────────────────────
hdr("§3 — Block-support: Choice A is left-supported; Choice B is right-supported")

def support_fraction(M, P):
    """Fraction of M's Hilbert-Schmidt norm² localised on the block P projects to."""
    sup = np.real(np.trace(M.conj().T @ P @ M @ P))
    tot = np.real(np.trace(M.conj().T @ M))
    return sup / tot if tot > 0 else 0.0

print(f"  {'Operator':<12}{'support on γ_F=+1':>22}{'support on γ_F=-1':>22}")
print(f"  {'-'*12}{'-'*22}{'-'*22}")
for a in (1, 2, 3):
    fA_plus  = support_fraction(H_A[a], P_plus)
    fA_minus = support_fraction(H_A[a], P_minus)
    fB_plus  = support_fraction(H_B[a], P_plus)
    fB_minus = support_fraction(H_B[a], P_minus)
    print(f"  H_A^{a:<10}{fA_plus:>22.3f}{fA_minus:>22.3f}")
    print(f"  H_B^{a:<10}{fB_plus:>22.3f}{fB_minus:>22.3f}")

print()
print("  Choice A: support 1.000 on γ_F=+1, support 0.000 on γ_F=-1.")
print("  Choice B: support 0.000 on γ_F=+1, support 1.000 on γ_F=-1.")
print()
print("  ⇒ Both choices satisfy the bimodule-support property on their")
print("    respective chirality blocks.  The structural condition that")
print("    'ℍ acts on one chirality block only' is realised by either")
print("    representation; nothing in the algebra forces A over B.")

# ─────────────────────────────────────────────────────────────────────
# §4. The substrate so(4) contains both choices on equal footing
# ─────────────────────────────────────────────────────────────────────
hdr("§4 — Substrate so(4) = su(2)_L ⊕ su(2)_R contains both choices")

# Combined so(4) action on H_F: one factor acts on the left block, the
# other on the right.  This is the substrate's so(4) embedded in End(H_F).
print("""\
  The substrate carries so(4) = su(2)_L ⊕ su(2)_R from quaternion left
  and right multiplication on Im(𝕆) (or equivalently from the isometry
  algebra of the S^3 spatial factor of M = ℝ × S^3).

  On the 4-dim H_F, this so(4) embeds as
       (one su(2) on the γ_F = +1 block) ⊕ (one su(2) on the γ_F = -1 block).
  Concretely: one factor is Choice A; the other is Choice B.

  Commutativity verification:""")

# Verify [H_A^a, H_B^b] = 0 (the two factors commute)
max_off = 0.0
for a in (1, 2, 3):
    for b in (1, 2, 3):
        max_off = max(max_off, norm(comm(H_A[a], H_B[b])))
print(f"    max ||[H_A^a, H_B^b]||  =  {max_off:.3e}  (expected 0; two commuting su(2)'s)")

print(f"""
  ⇒ The substrate's so(4) supplies BOTH choices as commuting sub-algebras.
  The decomposition into "left-block su(2)" and "right-block su(2)" is
  intrinsic to the substrate's algebraic structure.  The bimodule support
  is therefore automatic for either choice; the only remaining question
  is WHICH of the two is the physical SU(2)_L of the observed Standard
  Model.""")

# ─────────────────────────────────────────────────────────────────────
# §5. The directed threshold selects Choice A (the observed SM)
# ─────────────────────────────────────────────────────────────────────
hdr("§5 — The directed threshold ε > 0 selects Choice A")

print("""\
  The remaining question is the SAME parity-discrimination question PST
  already answered in paper §8 (the parity-violation argument from the
  directed modal threshold).

  Sketch of the bridge:
    (i)   The substrate carries an oriented threshold ε = T(C) - τ.
          Configurations split into precausal (ε < 0, before the
          threshold) and instantiated (ε > 0, after).  The threshold
          orientation is the physical fact "tension increases as
          instantiation proceeds".
    (ii)  The orientation defines a sign on the volume element of the
          substrate's S^3 spatial factor (the volume form picks up a
          sign under parity).
    (iii) The two su(2) factors of so(4) = so(3)_L ⊕ so(3)_R differ
          only by the orientation of the S^3 volume form (left- vs
          right-multiplication of unit quaternions).
    (iv)  Therefore the threshold orientation fixes the L/R labelling
          of the so(4) factors; the "L" factor is the one whose
          generators preserve the threshold orientation, the "R"
          factor flips it.
    (v)   Paper §8's parity-violation argument is exactly this: the
          weak interaction acts on the L factor because the threshold
          orientation distinguishes L from R.

  Conclusion: Choice A (su(2) acting on γ_F = +1) is the L factor
  consistent with the threshold orientation.  Choice B is the parity
  conjugate, ruled out by the SAME mechanism that gives parity
  violation in PST §8.

  No additional principle is required: the bimodule support of the
  observed chiral SU(2)_L coincides with PST's parity-violation
  argument.""")

# Numerical demonstration that the two choices are exchanged by parity.
# Parity on H_F: swap left and right blocks.
Parity = np.zeros((4, 4), dtype=complex)
Parity[0, 2] = Parity[1, 3] = Parity[2, 0] = Parity[3, 1] = 1.0
print()
print("  Parity check P · H_A · P^{-1} = H_B (the two choices are parity-related):")
for a in (1, 2, 3):
    transformed = Parity @ H_A[a] @ Parity.conj().T
    err = norm(transformed - H_B[a])
    print(f"    a = {a}:  ||P H_A^{a} P^{-1} - H_B^{a}||  =  {err:.3e}")
print()
print("  ⇒ Choice A and Choice B are exact parity conjugates.  Any mechanism")
print("    that produces parity violation chooses between them, and the")
print("    threshold orientation is precisely such a mechanism.")

# ─────────────────────────────────────────────────────────────────────
# §6. Conclusion
# ─────────────────────────────────────────────────────────────────────
hdr("§6 — Conclusion")

print("""\
  Closes §14.2 item (B):

  The chiral bimodule support property — that the threshold-selected
  su(2)_L acts on the γ_F = +1 block of H_F only — is verified
  structurally:

    • The substrate's so(4) = su(2)_L ⊕ su(2)_R supplies TWO commuting
      su(2) representations on H_F, one with support on each chirality
      block (§2-S4).
    • Both satisfy [H, γ_F] = 0 (compatible with the chirality grading,
      §2) and clean block-support (§3).
    • The two are exact parity conjugates of each other (§5 numerical
      check: P · H_A · P^{-1} = H_B).
    • Which of the two is the observed SU(2)_L is selected by the
      directed modal threshold (paper §8 parity-violation mechanism),
      which is the SAME mechanism PST already commits to for parity
      violation.

  Status: chirality of SU(2)_L is STRUCTURAL, conditional only on the
  framework assumption (the product is a Connes spectral triple) and
  on the threshold-direction parity-violation argument (paper §8),
  both of which PST commits to elsewhere.  The §14.1 entry "conditional
  on chiral bimodule" can be downgraded to "structural, with the
  parity-violation orientation fixed by the directed threshold (§8)".

  Together with Computations 7 (Yukawa from order-one given the bimodule),
  P (γ_M ruled out; threshold-substrate bridge), and S (bimodule support
  verified to be parity-related and selected by the threshold), the
  chirality residual is closed.
""")