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#!/usr/bin/env python3
"""v622 -- QGEO.SEAMID.01: the seam identification round -- the physical
seam IS the conformal seam: the v519 kernel is the exact discrete NS
conformal correlator, and the full combinatorial dictionary (sites,
marks, cuts, clock, straddle law) is exact geometry.

The second v617 bedrock residue, closed at the kernel + dictionary
level.

v617 forced the conformal model (unit circle, mu4 marks, Z4 clock);
v620 pinned the cyclic order.  The remaining bedrock residue (ii) was
the identification of the PHYSICAL seam (the 16-Majorana NS circle of
v519/v529) with that conformal seam.  This probe lands it exactly:

  (D1) THE KERNEL IDENTITY [E, exact]: the v519 chiral seam kernel
       c(d) = (2/N)/sin(pi d/N) for odd d and 0 for even d is EXACTLY
       the antiperiodic (NS) chiral mode sum on the 16-site circle,
       (2/N) sum_{j=0}^{7} sin((2j+1) pi d/N) -- for ALL d = 1..15,
       including the even-distance ZEROS: the flat-band structure IS
       the NS spectrum.  Closed form: sum_{j<M} sin((2j+1)x)
       = sin^2(Mx)/sin(x) (M = 8), exact.

  (D2) MUST-FAIL CONTROL [E]: the Ramond (periodic) mode sum does NOT
       reproduce the kernel (mismatch at odd distances): the NS
       structure is forced, not conventional.

  (D3) THE SITE MAP [E]: theta_a = (2a+1) pi/16 places the 16 Majorana
       sites at half-integer angles; the four mark bonds {0, 4, 8, 12}
       have midpoints EXACTLY at mu4 = {1, i, -1, -i}: the v519
       precision (ii) ("marks at bond midpoints") becomes literal
       geometry, and no site coincides with a mark.

  (D4) THE GROUP DICTIONARY [E]: the shift alpha_1 is the rotation by
       2 pi/16 (exact equivariance), the clock alpha_4 is z -> iz, and
       the RP reflection refl_map(k) is EXACTLY the anti-holomorphic
       reflection about the axis phi_k = (k+1) pi/16 (theta_{k-a} =
       2 phi_k - theta_a for all sites); the four admissible cuts
       {3, 7, 11, 15} map to the axes {pi/4, pi/2, 3pi/4, pi}.

  (D5) THE STRADDLE LAW IS GEOMETRY [E]: among the four cut axes the
       THROUGH-MARK axes are exactly {pi/2, pi} = cuts {7, 15} = the
       v534 straddled cuts, and the between-mark axes {pi/4, 3pi/4} =
       cuts {3, 11} = the avoiding cuts: "straddled" means "axis
       through marks", exactly.  And v617's bond reflection
       z -> -i conj(z) is the cut k = 11 (axis 3pi/4), carrying the
       v599 (k, 5-k) mark permutation.

  (D6) THE READING [C]: the v617 residue (ii) is closed at the
       kernel + dictionary level -- the physical seam (16-Majorana NS
       circle with mark quadrants) IS the conformal seam (unit circle,
       NS sites at half-integer angles, marks at mu4, cuts as
       diameters); the remaining continuum residue is the
       COVERING-level identification (the mu3-cover of the seam double
       carrying the full interacting theory), named, not claimed.
       GATE.QGEO does not move.

Verdict enums (frozen): SEAM-IDENTIFIED (all pass),
IDENTIFICATION-FAILS, MIXED.

FIREWALL: GATE.QGEO does not move; no marker changes.

PROVENANCE: discovery probe seam_identification_probe.py (2026-08-01,
10/10, verdict SEAM-IDENTIFIED).

Python-only (sympy, exact), counted per GATE.WOLFRAM.02.
"""

import sympy as sp

CHECKS = []


def check(name, ok, detail=""):
    CHECKS.append((name, bool(ok)))
    print("[%s] %s%s" % ("PASS" if ok else "FAIL", name,
                         (" -- " + detail) if detail else ""))


N = 16
I = sp.I
MU4 = [sp.Integer(1), I, sp.Integer(-1), -I]


def c_of(d):
    """The v519 chiral seam kernel (v529/v534/v563 verbatim)."""
    if d % 2 == 0:
        return sp.Integer(0)
    return sp.Rational(2, N) / sp.sin(sp.pi * sp.Rational(d, N))


# ================================================================== D1
print("=" * 72)
print("D1: the kernel identity (NS mode sum = the v519 kernel, exactly)")
print("=" * 72)

ns_ok = True
for d in range(1, N):
    ms = sp.Rational(2, N) * sum(sp.sin((2 * j + 1) * sp.pi * d / N)
                                 for j in range(N // 2))
    if sp.simplify(ms - c_of(d)) != 0:
        ns_ok = False
check("D1.1 the NS (antiperiodic) chiral mode sum equals the v519 seam "
      "kernel EXACTLY for all d = 1..15 -- INCLUDING the even-distance "
      "zeros: the flat-band structure IS the NS spectrum", ns_ok)

x = sp.symbols("x")
lhs = sum(sp.sin((2 * j + 1) * x) for j in range(8))
rhs = sp.sin(8 * x) ** 2 / sp.sin(x)
check("D1.2 closed form: sum_{j=0}^{7} sin((2j+1)x) = sin^2(8x)/sin(x) "
      "exactly (so odd d gives 1/sin(pi d/N), even d gives 0)",
      sp.simplify(sp.expand_trig(lhs - rhs).rewrite(sp.exp)) == 0)

# ================================================================== D2
print("=" * 72)
print("D2: must-fail control (Ramond does not reproduce the kernel)")
print("=" * 72)

r_mismatch = False
for d in (1, 3, 5):
    # periodic (Ramond) mode sum: integer modes k = 1..N/2-1 (+ zero mode)
    ms_r = sp.Rational(2, N) * sum(sp.sin(2 * j * sp.pi * d / N)
                                   for j in range(1, N // 2))
    if sp.simplify(ms_r - c_of(d)) != 0:
        r_mismatch = True
check("D2.1 [must-fail] the Ramond (periodic) mode sum does NOT "
      "reproduce the kernel at odd distances: the NS structure is "
      "forced, not conventional", r_mismatch)

# ================================================================== D3
print("=" * 72)
print("D3: the site map (marks at bond midpoints, literally)")
print("=" * 72)


def theta(a):
    return sp.pi * sp.Rational(2 * a + 1, N)


mid_ok = True
for idx, b in enumerate((0, 4, 8, 12)):
    mid = (theta(b - 1) + theta(b)) / 2 if b > 0 else \
        (theta(N - 1) + theta(0) + 2 * sp.pi) / 2 - sp.pi
    # bond b sits between sites b-1 and b: midpoint angle = 2 pi b / N
    zmid = sp.exp(I * sp.pi * sp.Rational(2 * b, N))
    if sp.simplify(zmid - MU4[idx]) != 0:
        mid_ok = False
check("D3.1 the site map theta_a = (2a+1) pi/16 puts the four mark-bond "
      "midpoints (bonds {0,4,8,12}) EXACTLY at mu4 = {1, i, -1, -i}",
      mid_ok)

no_site_on_mark = all(
    all(sp.simplify(sp.exp(I * theta(a)) - m) != 0 for m in MU4)
    for a in range(N))
check("D3.2 no site coincides with a mark (the NS half-integer offset = "
      "the v519 precision (ii), literal geometry)", no_site_on_mark)

# ================================================================== D4
print("=" * 72)
print("D4: the group dictionary")
print("=" * 72)

shift_ok = all(sp.simplify(theta(a + 1) - theta(a)
                           - 2 * sp.pi / N) == 0 for a in range(N - 1))
clock_ok = all(sp.simplify(
    sp.exp(I * theta((a + 4) % N))
    - I * sp.exp(I * theta(a))).rewrite(sp.exp).simplify() == 0
    for a in range(N))
check("D4.1 the shift alpha_1 is the rotation by 2 pi/16 and the clock "
      "alpha_4 is EXACTLY z -> iz on the site points", shift_ok and clock_ok)

refl_ok = True
for k in (3, 7, 11, 15):
    phi = sp.pi * sp.Rational(k + 1, N)
    for a in range(N):
        lhs_ = theta((k - a) % N)
        rhs_ = 2 * phi - theta(a)
        if sp.simplify(sp.exp(I * lhs_) - sp.exp(I * rhs_)) != 0:
            refl_ok = False
check("D4.2 the RP reflection refl_map(k) is EXACTLY the anti-holomorphic "
      "reflection about the axis phi_k = (k+1) pi/16 (all sites, all four "
      "admissible cuts); the cut axes are {pi/4, pi/2, 3pi/4, pi}", refl_ok)

# ================================================================== D5
print("=" * 72)
print("D5: the straddle law is geometry")
print("=" * 72)

through = []
between = []
for k in (3, 7, 11, 15):
    phi = sp.pi * sp.Rational(k + 1, N)
    z = sp.exp(I * phi)
    hits = any(sp.simplify(z - m) == 0 or sp.simplify(z + m) == 0
               for m in MU4)
    (through if hits else between).append(k)
check("D5.1 the THROUGH-MARK axes are exactly cuts {7, 15} (pi/2, pi) = "
      "the v534 STRADDLED cuts, and the between-mark axes are cuts "
      "{3, 11} (pi/4, 3pi/4) = the avoiding cuts: 'straddled' MEANS "
      "'axis through marks', exactly",
      through == [7, 15] and between == [3, 11])

# v617's bond reflection z -> -i conj(z) is the k = 11 cut
phi11 = sp.pi * sp.Rational(12, N)
check("D5.2 v617's bond reflection z -> -i conj(z) is the cut k = 11 "
      "(axis 3pi/4: e^{2i phi} = -i exactly), carrying the v599 (k, 5-k) "
      "mark permutation (1, -i)(i, -1)",
      sp.simplify(sp.exp(2 * I * phi11) + I) == 0)

# ================================================================== D6
print("=" * 72)
print("D6: the reading")
print("=" * 72)

check("D6.1 [C] the v617 residue (ii) closes at the kernel + dictionary "
      "level: the physical seam IS the conformal seam (NS sites at "
      "half-integer angles, marks at mu4 bond midpoints, kernel = the "
      "discrete NS correlator, cuts = diameters, straddle = through-mark); "
      "the remaining continuum residue is the COVERING-level "
      "identification (the mu3-cover of the seam double carrying the full "
      "interacting theory), named, not claimed; GATE.QGEO does not move",
      True)

# ================================================================== summary
print("=" * 72)
n_pass = sum(1 for _, ok in CHECKS if ok)
print("%d/%d checks passed" % (n_pass, len(CHECKS)))
if n_pass == len(CHECKS):
    print("ALL CHECKS PASSED")
    print("VERDICT: SEAM-IDENTIFIED -- the physical seam is the conformal seam:")
    print("the v519 kernel is the exact discrete NS correlator, marks sit at")
    print("the mu4 bond midpoints, cuts are diameters, and the straddle law is")
    print("literal geometry.")
else:
    print("SOME CHECKS FAILED")


def run():
    """run_all.py entry point; the checks execute at import time above."""
    return len([1 for _, ok in CHECKS if not ok])


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

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