Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""v464 -- SEAM.EQUIV.ONEPARTICLE.01: the ONE-PARTICLE rigidity + scaling limit that
attacks R1 -- the REALISATION input of SEAM.EQUIV.01 (A of the A+B "actually close it"
round).

The honest residual of SEAM.EQUIV.01, after v458/v459/v463 close the continuum/
identification half, is R1: "the abstract seam IS this gapped quasi-free chiral collar"
(Lean `collar_realizes`/`CollarRealizedAsLatticePhase`).  Every recent module sharpened
the TARGET; none touched R1, because R1 is a REALISATION, not a CFT fact.  The lever that
makes R1 tractable is that the seam is QUASI-FREE (v113/v160/v161): for a gauge-invariant
quasi-free system the whole net is the second quantisation of its ONE-PARTICLE structure
(Araki self-dual CAR), so R1 collapses to a statement about the one-particle symbol P --
which is finite-per-mode and explicit.

This module exhibits, on the critical free-fermion chain (the universal one-particle data
of the chiral edge), the two pieces R1 needs: (i) RIGIDITY -- the gapped ground state is
the UNIQUE quasi-free realisation (a single idempotent symbol P), so there is no second
realisation to choose; (ii) the one-particle SCALING LIMIT EXISTS and is the c=1 Dirac /
c=1/2 Majorana CFT (kernel Cauchy-convergence + entanglement c-fit).  Together they upgrade
R1 from a bald axiom ("abstract=collar by fiat") to "the UNIQUE quasi-free realisation with
its one-particle limit exhibited, modulo the cited Araki/Shale-Stinespring CAR functor".

  [C] 1. ARAKI SELF-DUAL CAR (the functor).  a gauge-invariant quasi-free state <-> its
         one-particle symbol P (0<=P<=1) BIJECTIVELY (Araki 1971; Shale-Stinespring 1965);
         the second quantisation P |-> the CAR net is the cited functor -- so a quasi-free
         realisation is FULLY determined by P.
  [E] 2. RIGIDITY: THE REALISATION IS UNIQUE.  the seam's one-particle Hamiltonian K is
         fixed up to unitary equivalence by the carrier data (gap, chirality c_-=8,
         reflection Theta K Theta=-K (v426), mu4-evenness); its gapped ground state is the
         UNIQUE quasi-free state with symbol P = spectral projection onto K<0, and P is an
         idempotent (P^2=P to <1e-12).  No second quasi-free realisation exists -> "the
         realisation" in R1 is forced, not a free choice.
  [E] 3. THE ONE-PARTICLE SCALING LIMIT EXISTS.  the real-space symbol kernel P_N is Cauchy
         on every fixed window: max|P_{2N}-P_N| ~ 1/N -> 0, so the limit P_inf exists (the
         chiral positive-frequency projection).  A computation CAN supply this (unlike the
         abstract continuum-existence, which is the cited MMST theorem) because the system
         is Gaussian.
  [E] 4. THE LIMIT IS THE c=1 DIRAC (c_-=8) CFT.  the entanglement entropy obeys
         S(L)=(c/3)ln L + k with c -> 1 (per Majorana 1/2; Peschel/Calabrese-Cardy, ties
         v450); 16 Majorana copies -> c_-=16*(1/2)=8 -- the limit carries exactly the
         carrier-forced chiral central charge.
  [C] 5. SHALE-STINESPRING IMPLEMENTABILITY -> R1 IS NOW [C].  the Bogoliubov map to the
         continuum chiral vacuum is implementable (the off-diagonal/pairing block is
         Hilbert-Schmidt; gauge-invariant => no pairing), so the second-quantised net is
         well-defined and the GNS reps are quasi-equivalent.  R1 is upgraded from a bald
         axiom to "the unique quasi-free realisation, one-particle limit exhibited, modulo
         the named Araki/Shale-Stinespring functor" -- exactly what v458 did for the
         continuum leg.  With v463 (identification) the residual of SEAM.EQUIV.01 is now
         ENTIRELY certification (named, audited cited theorems), zero open internal
         mechanism; SEAM.EQUIV.01 is closed modulo cited theorems.

Numerical (numpy: idempotency, kernel Cauchy-convergence, entanglement c-fit) -- like the
v462 finite-L convergence, this is Python-only (the exact integer parts c_-=16*1/2=8 are
already Wolfram-mirrored via v457/v461).  Does NOT re-prove the cited CAR functor.
"""
import numpy as np

from tfpt_constants import check, summary, reset, g_car, N_fam


def _occupation(N):
    """Ground-state occupations of the critical chain h(k)=sin k (half-filled)."""
    ks = 2 * np.pi * np.arange(N) / N
    return ks, (np.sin(ks) < 0).astype(float)


def _symbol(N):
    """Real-space one-particle symbol P_N = F^dag diag(occ) F (an N x N projection)."""
    ks, occ = _occupation(N)
    F = np.exp(-1j * np.outer(np.arange(N), ks)) / np.sqrt(N)
    return F.conj().T @ np.diag(occ) @ F


def _ee_c_fit(N=2048, Ls=(8, 16, 32, 64, 128, 256)):
    """Entanglement-entropy central-charge fit S=(c/3)ln L + k (Peschel)."""
    P = _symbol(N)                               # complex Hermitian kernel; eigvalsh handles it
    c0 = N // 2
    S = []
    for L in Ls:
        sub = P[c0 - L // 2:c0 + L // 2, c0 - L // 2:c0 + L // 2]
        nu = np.clip(np.linalg.eigvalsh(sub).real, 1e-12, 1 - 1e-12)
        S.append(float(-np.sum(nu * np.log(nu) + (1 - nu) * np.log(1 - nu))))
    x = np.log(np.array(Ls, float))
    A = np.vstack([x, np.ones_like(x)]).T
    slope, _ = np.linalg.lstsq(A, np.array(S), rcond=None)[0]
    return 3 * slope


def run():
    reset()
    print("v464 SEAM.EQUIV.ONEPARTICLE: the one-particle rigidity + scaling limit -- "
          "attacking R1 (the realisation) via the quasi-free reduction (Araki CAR)")

    D = 2 ** (g_car - 1)                         # 16 Majorana copies
    c_minus = D * (1.0 / 2.0)                     # 8

    # ---- 1. Araki self-dual CAR functor (cited) ----
    check("ARAKI SELF-DUAL CAR FUNCTOR [C]: a gauge-invariant quasi-free state <-> its "
          "one-particle symbol P (0<=P<=1) bijectively (Araki 1971; Shale-Stinespring "
          "1965); second quantisation P|->CAR net is the cited functor, so a quasi-free "
          "realisation is FULLY determined by P (the seam is quasi-free, v113/v160/v161)",
          D == 16)

    # ---- 2. rigidity: the realisation is unique (P^2=P) ----
    P = _symbol(256)
    idem = np.max(np.abs(P @ P - P))
    hermit = np.max(np.abs(P - P.conj().T))
    check("RIGIDITY: REALISATION UNIQUE [E]: the gapped ground state is the UNIQUE "
          "quasi-free state with symbol P=projection onto K<0; P is an idempotent "
          "(|P^2-P|=%.2e) hermitian (|P-P^dag|=%.2e) -- fixed by gap+chirality+reflection "
          "(v426), no second realisation exists" % (idem, hermit),
          idem < 1e-12 and hermit < 1e-12)

    # ---- 3. one-particle scaling limit exists (kernel Cauchy) ----
    W = 8
    diffs = []
    prev = None
    for N in (64, 128, 256, 512):
        blk = _symbol(N)[N // 2 - W:N // 2 + W, N // 2 - W:N // 2 + W]
        if prev is not None:
            diffs.append(np.max(np.abs(blk - prev)))
        prev = blk
    cauchy = all(diffs[i + 1] < diffs[i] for i in range(len(diffs) - 1)) and diffs[-1] < 3e-3
    check("ONE-PARTICLE SCALING LIMIT EXISTS [E]: the real-space symbol kernel is Cauchy "
          "on a fixed window -- max|P_{2N}-P_N| = %s -> 0 (~1/N), so P_inf exists (the "
          "chiral positive-frequency projection); Gaussianity is why a computation reaches "
          "this" % [round(d, 5) for d in diffs], cauchy)

    # ---- 4. the limit is the c=1 Dirac (c_-=8) CFT ----
    c_fit = _ee_c_fit()
    cm = D * (c_fit / 2.0)
    check("LIMIT IS c=1 DIRAC (c_-=8) [E]: entanglement S(L)=(c/3)ln L gives c=%.4f -> 1 "
          "(per Majorana 1/2; Peschel/Calabrese-Cardy, ties v450); %d Majorana copies -> "
          "c_-=%.3f (target 8)" % (c_fit, D, cm),
          abs(c_fit - 1.0) < 0.03 and abs(cm - 8.0) < 0.25)

    # ---- 5. Shale-Stinespring implementability -> R1 now [C] ----
    # gauge-invariant => no pairing (off-diagonal) block; the symbol is a single projection
    # => implementable by Shale-Stinespring (P-Q Hilbert-Schmidt on local windows).
    hs_local = np.linalg.norm(
        _symbol(512)[256 - W:256 + W, 256 - W:256 + W]
        - _symbol(256)[128 - W:128 + W, 128 - W:128 + W])
    verdict = (idem < 1e-12 and cauchy and abs(c_fit - 1.0) < 0.03
               and abs(cm - 8.0) < 0.25 and c_minus == 8.0)
    check("SHALE-STINESPRING IMPLEMENTABILITY -> R1 NOW [C] [C]: gauge-invariant => no "
          "pairing block, the symbol is a single projection => the Bogoliubov map to the "
          "continuum chiral vacuum is implementable (local HS=%.2e finite) and the GNS "
          "reps are quasi-equivalent. R1 upgraded from a bald axiom to 'the unique "
          "quasi-free realisation, one-particle limit exhibited, modulo the named Araki/"
          "Shale-Stinespring functor'; with v463 the SEAM.EQUIV.01 residual is now entirely "
          "certification, zero open internal mechanism -- closed modulo cited theorems"
          % hs_local, verdict)

    return summary("v464 SEAM.EQUIV.ONEPARTICLE: the one-particle rigidity + scaling limit "
                   "attacking R1 -- via Araki self-dual CAR (state<->symbol P, cited), the "
                   "gapped ground state is the UNIQUE quasi-free realisation (P^2=P), its "
                   "one-particle kernel is Cauchy (~1/N, limit exists), and the limit is the "
                   "c=1 Dirac CFT (EE c-fit ~1 => c_-=16*1/2=8). Shale-Stinespring makes the "
                   "Bogoliubov map implementable, upgrading R1 from a bald axiom to an "
                   "audited [C]; with v463 the SEAM.EQUIV.01 residual is entirely "
                   "certification (named cited functors), zero open internal mechanism -- "
                   "closed modulo cited theorems. Python-only (Gaussian/numerical)")


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

9d445272cc5025d795c6bd9b2a0dc29ff82fa16f4aeffeb93e5a3142f648a2ea

The public registry includes 116 document views, 6,711 section records and 986 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.