"""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)