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"""v199 -- QGEO.STATE.01 (work package A): attacking omega o rho = omega directly,
on the RAW seam, not on the mu4 normal form. The honest outcome is one more
localising reduction (not a closure): the residual is sharpened to "the bounded
sub-principal symbol of the raw DtN has no off-character matrix elements", and the
setup is confirmed NON-CIRCULAR (rho is a carrier-algebra automorphism, the
algebra is the raw CAR net -- neither imports the seam geometry).

  [E] 1. THE SETUP IS NON-CIRCULAR.  rho is the carrier clock = the Coxeter
        element of W(A3)=S4 (order 4, rho^4 = 1, v117), an automorphism of the
        RAW CAR algebra (16 Majoranas = the carrier net, v156/v175). Both rho and
        the algebra are defined from the CARRIER side, with no seam geometry --
        so 'rho preserves the seam state' is a genuine, non-circular question.
  [E] 2. GROUND-STATE REDUCTION.  For a quasi-free RP state the covariance is the
        positive-frequency projection C_Sigma = 1/2 (1 + sgn H_1) of the
        one-particle Hamiltonian H_1 (~ the DtN). So omega o rho = omega <=> rho
        preserves C_Sigma <=> [rho, H_1] = 0.
  [E] 3. CHARACTER-BLOCK-DIAGONAL CRITERION.  rho has order 4, so the one-particle
        space splits into the four mu4-character classes {n = r mod 4} (rho acts
        as i^n). [rho, H_1] = 0 <=> H_1 is BLOCK-DIAGONAL in these four classes
        (no off-character matrix elements) -- verified: a character-block-diagonal
        H commutes with rho, an off-character entry breaks it.
  [E] 4. THE PRINCIPAL SYMBOL ALREADY PASSES.  |k| = diag(|n|) is diagonal, hence
        trivially character-block-diagonal, so [rho, |k|] = 0 exactly (v198) --
        the leading order carries no off-character elements.
  [O] 5. THE SHARPENED RESIDUAL.  What remains is purely: the BOUNDED sub-principal
        symbol of the raw seam DtN has NO off-character (mod 4) matrix elements.
        This is the most-localised form of omega o rho = omega -- a bounded,
        finite-character-class statement, not a diffuse geometry. It holds on the
        finite H^1 block (the eigenforms are mu4-eigenforms, v177); the open part
        is the off-character entries on the continuum.

  VERDICT [O]: package A confirms the omega o rho = omega question is non-circular
  and reduces it to 'the bounded sub-principal symbol is mu4-character-block-
  diagonal'. NOT a closure (the foundational symmetry persists) -- but the
  residual is now a bounded, sharply-typed operator condition. Approaching the
  point where further reductions are restatements of the same postulate.

  Python-only (finite-mode linear algebra; the principal-symbol vanishing is exact
  and already Wolfram-mirrored at v198).
"""
import numpy as np

from tfpt_constants import check, summary, reset


def run():
    reset()
    print("v199 QGEO.STATE.01: omega o rho = omega reduced to a bounded character-block-diagonal residual (non-circular)")

    N = 8
    n = np.arange(-N, N + 1)
    d = len(n)
    rho = np.diag((1j) ** n)                          # carrier clock, order 4

    # 1. setup non-circular: rho^4 = I (carrier automorphism, no geometry)
    check("SETUP NON-CIRCULAR [E]: rho = the Coxeter element of W(A3)=S4 (v117), "
          "rho^4 = I (order 4) = an automorphism of the RAW CAR algebra (16 "
          "Majoranas, v156/v175); both defined carrier-side, NO seam geometry -- "
          "so 'rho preserves the seam state' is a genuine non-circular question",
          np.allclose(np.linalg.matrix_power(rho, 4), np.eye(d)))

    # 3. character-block-diagonal criterion: [rho,H]=0 <=> H block-diag in {n mod 4}
    rng = np.random.default_rng(0)
    Hbd = np.zeros((d, d), complex)
    for a in range(d):
        for b in range(d):
            if (n[a] - n[b]) % 4 == 0:                # same mu4 character class
                Hbd[a, b] = rng.standard_normal()
    Hbd = (Hbd + Hbd.conj().T) / 2
    Hoff = Hbd.copy()
    i0, i1 = int(np.where(n == 0)[0][0]), int(np.where(n == 1)[0][0])
    Hoff[i0, i1] += 0.7; Hoff[i1, i0] += 0.7          # mode 0 (char 1) <-> mode 1 (char i)
    bd_ok = np.allclose(rho @ Hbd - Hbd @ rho, 0)
    off_bad = not np.allclose(rho @ Hoff - Hoff @ rho, 0)
    check("GROUND-STATE + CHARACTER CRITERION [E]: for a quasi-free RP state, "
          "C_Sigma = 1/2(1+sgn H_1), so omega o rho=omega <=> [rho,H_1]=0 <=> H_1 "
          "is block-diagonal in the four mu4-character classes {n=r mod 4} -- "
          "verified: a character-block-diagonal H commutes (%s), an off-character "
          "entry breaks it (%s)" % (bd_ok, off_bad),
          bd_ok and off_bad)

    # 4. principal symbol passes automatically
    absK = np.diag(np.abs(n).astype(float))
    check("PRINCIPAL SYMBOL PASSES [E]: |k| = diag(|n|) is diagonal, hence "
          "character-block-diagonal, so [rho,|k|]=0 exactly (v198) -- the leading "
          "order carries no off-character elements",
          np.allclose(rho @ absK - absK @ rho, 0))

    check("SHARPENED RESIDUAL [O]: what remains is purely 'the BOUNDED sub-"
          "principal symbol of the raw seam DtN has NO off-character (mod 4) matrix "
          "elements' -- a bounded, finite-character-class condition (not a diffuse "
          "geometry); holds on the finite H^1 block (mu4-eigenforms, v177), open on "
          "the continuum. Non-circular and sharply typed; not a closure -- the "
          "foundational symmetry persists", True)

    return summary("v199 QGEO.STATE.01: omega o rho=omega -> bounded character-block-diagonal residual; non-circular [O]")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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