TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/NS-COVER-FULL-DIRAC-RP.txt
Original file SHA-256: a18c7589ac87b0ea875af7a2a46790989c80c836369b9ee001b0417dda6ab154
Inclusive original lines: 134-186
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000134 | removes that obstruction is2, which gives q^2=-1. This is NOT the unrelated
000135 | global relative C12 cover problem.
000136 | 
000137 | 5. The state and time bridge is still substantive
000138 | 
000139 | The cover Green construction quantizes x as Euclidean time. Its one
000140 | particle space is L2([-1,1],C2), its spatial operator is H, and its thermal
000141 | covariance is C=(I+exp(2H))^-1. In particular C v_b=v_b/2.
000142 | 
000143 | The earlier D sea is a projection1_(D<0) on L2 of BOTH x and w. The
000144 | original quarter current source instead uses the Hardy projection on
000145 | the longitudinal circle and its original circle energy. Neither projection
000146 | is C. These operators act on different spaces and perform different jobs.
000147 | No unitary conjugation can turn a projection into C while preserving a
000148 | one particle covariance, since C has eigenvalue1/2. An enlarged purification
000149 | can be considered only with its actual locality and time dictionary; it
000150 | does not silently identify the original representations.
000151 | 
000152 | Thus the newly proved full cover positivity supplies a valid free geometric
000153 | realization and explains the failure of the single quarter attempt. It does
000154 | not identify that realization with the old W8 Hardy source or with P1.
000155 | The next decisive original question is the SAME source map preserving
000156 | field algebra, support, time, state, and original collar/reference data.
000157 | Changing which coordinate is treated as time must appear in that map.
000158 | Original SEAM.THERMAL.KMS.01/v526 already starts from the given NS chiral
000159 | kernel and pinned free half algebra; it must be reused in that scope, not
000160 | treated as a derivation of the present two dimensional kernel from P1.
000161 | 
000162 | 6. Verification and sources
000163 | 
000164 | check_ns_cover_dirac_rp.py checks the exact energy dependent Green jump,
000165 | antiperiodicity, reflected Gram factorization, both Pin projector identities,
000166 | linear versus antilinear reflection action, both signs b, transverse
000167 | eigenfunctions and their endpoint lines, and the zero mode kernel pairing.
000168 | Exact rational exponential coordinates test positive Gram determinants at
000169 | several times without numerical rank tolerances. The all energy and all
000170 | mode claims are the spectral calculus proof above, not those finite checks.
000171 | 
000172 | Primary geometry: A.Chakraborty, Pin Equivariance and Local Dirac Boundary
000173 | Problems on the Flat Moebius Band, arXiv:2610.03745v1, sections6 and8.
000174 | https://arxiv.org/html/2610.03745v1
000175 | General Clifford Green reflection comparison: A.Jaffe and G.Ritter,
000176 | Reflection Positivity and Monotonicity, arXiv:0705.0712v2, section4.2.
000177 | https://arxiv.org/html/0705.0712v2
000178 | Original operational seed and separated geometric reflection roles:
000179 | archive--tfpt-45 boundary_kernel_source, sections001 and002, as located
000180 | precisely in P1-LONGITUDINAL-REFLECTION-REVIEW.txt.
000181 | 
000182 | Verdicts: PROVED for the given full Gaussian NS cover Green field;
000183 | EXCLUDED for its restriction as a positive temporal Green field in one
000184 | fixed Pinminus sector; OPEN for its original P1 identification and for a
000185 | same state, same time map to the previously constructed quarter W8 source.
000186 | physical_gates_closed=[]; complete_TFPT_solution=false.