TFPT scientific source excerpt
Original repository path: experiments/tfpt-discovery/spin10_det16_projector_probe.py
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000001 | #!/usr/bin/env python3
000002 | r"""spin10_det16_projector_probe -- EXPLORATION ONLY.
000003 | 
000004 | This probe tests the number-violating determinant vertex on the complete
000005 | 2^16-dimensional Fock space of one positive-chirality Spin(10) spinor.  The
000006 | 45 one-body generators are the verified Clifford/Weyl construction used by
000007 | ``spin10_mirror_projector_probe.py``:
000008 | 
000009 |     T_ab = (i/2 gamma_a gamma_b)|_16,  0 <= a < b < 10.
000010 | 
000011 | Let Omega^dagger = chi_1^dagger ... chi_16^dagger, with
000012 | |F> = Omega^dagger|0>, and Omega its adjoint.  In the occupation-basis
000013 | convention used here the exact partial-isometry algebra is
000014 | 
000015 |     Omega^dagger Omega = P_16,       Omega Omega^dagger = P_0,
000016 |     P_16 Omega^dagger = Omega^dagger = Omega^dagger P_0,
000017 |     P_0 Omega = Omega = Omega P_16,
000018 |     (Omega)^2 = (Omega^dagger)^2 = 0.
000019 | 
000020 | Consequently A_phi = P_0 + P_16 + exp(-i phi) Omega
000021 | + exp(i phi) Omega^dagger obeys A_phi^2 = 2 A_phi, and
000022 | P_phi = A_phi/2 is the rank-one projector onto
000023 | 
000024 |     |Omega_phi> = (|0> + exp(i phi)|F>)/sqrt(2).
000025 | 
000026 | Thus h_mir = 1-P_phi has exact spectrum {0^1, 1^(2^16-1)}.  For disjoint
000027 | clusters every h_x is even: P_0/P_16 preserve number and Omega contains 16
000028 | fermion operators.  Even operators with disjoint fermionic support commute,
000029 | so [h_x,h_y]=0 and sum_x h_x has the unique product ground state
000030 | tensor_x |Omega_phi_x> and exact, volume-independent gap 1.  The same
000031 | statement is checked explicitly for two and three four-mode determinant
000032 | clusters (dimensions 2^8 and 2^12); the four-operator vertex is also even.
000033 | 
000034 | Gauge invariance is checked infinitesimally.  Each chiral-16 generator is
000035 | traceless, so det(exp(i theta T_a)) = exp(i theta tr(T_a)) = 1.  Its second
000036 | quantization annihilates both |0> and |F>, hence commutes with P_phi.  This
000037 | finite-cluster identity does not supply dynamical gauge fields, a domain-wall
000038 | geometry, or a 4D thermodynamic/volume theorem.
000039 | 
000040 | CONTRACT: CHIRAL4D.MIRROR.DET16.01 remains [O].
000041 | 
000042 | VERDICT ENUM:
000043 |   DET16_PROJECTOR_EXACT(gap=1,hopping_stability=PASS_t<=0.2_toy)
000044 |   DET16_PROJECTOR_FAILS(where)
000045 | """