TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/EIGHT-ARC-EVEN-CAR-LOCALITY-TEST-20261102.txt
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000001 | EIGHT ARC SOURCE LINE: AN EXACT EVEN CAR LOCALITY TEST
000002 | =====================================================
000003 | 
000004 | 2 November 2026. Scoped mathematical consequence of the existing source
000005 | fold and closed smooth line. This is not a derivation of the P1 net.
000006 | 
000007 | Assumptions and original source
000008 | -------------------------------
000009 | Let I be a proper connected open interval on the quotient circle. Its
000010 | preimage under F(theta)=8 theta mod 2 pi consists of eight disjoint open
000011 | cover arcs I_0,...,I_7. The existing smooth half-kink construction gives
000012 | T=W(-b_F), b_F=b o F, for a real smooth periodic quotient profile b.
000013 | The original Weyl convention is
000014 | 
000015 |   W(phi) psi(theta) W(phi)^* = exp(-i phi(theta)) psi(theta),
000016 | 
000017 | so T acts by multiplication with u(theta)=exp(i b_F(theta)) on the
000018 | annihilation field, with the corresponding conjugate phase on the adjoint.
000019 | These are exactly the pre-existing fold and Weyl operations; no new field,
000020 | measure, or dynamics is selected. The argument holds for alpha=1/4 and
000021 | 3/4, since the local test uses support away from the circle cut and u is
000022 | single-valued. It does not replace or remove the quarter holonomy.
000023 | 
000024 | Hypothesis: exp(i b) is nonconstant on I. This is essential for the
000025 | observable test below. A constant phase, including a global parity action,
000026 | is not covered by that test. A smooth nontrivial line is not automatically
000027 | nonconstant on an arbitrarily chosen interval; I must contain its variation.
000028 | 
000029 | Exact even-observable test
000030 | --------------------------
000031 | Fix any one of the eight arcs I_j and select a different arc I_k. Since
000032 | F maps I_k bijectively onto I and exp(i b) is nonconstant on I, there are
000033 | two open sets U,V with disjoint closures inside I_k such that their phase
000034 | images u(U),u(V) lie in disjoint neighborhoods on the unit circle. Choose
000035 | nonzero normalized one-particle test functions f,g supported in U,V.
000036 | They belong to the existing charged CAR one-particle space, with the
000037 | quarter boundary condition; local support makes their definition insensitive
000038 | to the position of the cut. Define the bounded even local observable
000039 | 
000040 |   B = a^*(f) a(g) in CAR_even(I_k).
000041 | 
000042 | Under T, the associated one-particle rank-one kernel |f><g| is transformed
000043 | by multiplication with u and its adjoint, up to the standard convention for
000044 | whether the annihilation smearing is linear or antilinear. It cannot remain
000045 | the same: equality of two nonzero rank-one operators would require u to
000046 | act by the same scalar on f and g, contrary to the disjoint phase images.
000047 | The CAR map of a rank-one kernel to a^*(f)a(g) is injective: for example,
000048 | its commutator with an arbitrary creation operator recovers the action of
000049 | that kernel on the one-particle test space. Therefore
000050 | 
000051 |   T B T^* != B,  equivalently [T,B] != 0.                  (1)
000052 | 
000053 | Any element of CAR_even(I_j), and likewise of its represented local von
000054 | Neumann closure, commutes with B because I_j and I_k are disjoint and the
000055 | CAR grading makes even operators commute at disjoint supports. Hence
000056 | 
000057 |   T is not an element of CAR_even(I_j)''                  (2)
000058 | 
000059 | in the given faithful source representation. The conclusion holds for
000060 | any chosen j because there is always a distinct k. The proof uses a bounded
000061 | even test observable on the actual cover, not a sharp endpoint or an odd
000062 | observable as the claimed physical measurement. Strong continuity of
000063 | multiplication and locality extend the commutation statement to the local
000064 | von Neumann closure.
000065 | 
000066 | Scope and necessary next comparison
000067 | ----------------------------------
000068 | Equation (2) concerns the DIRECT pullback T=W(-b o F), using the existing
000069 | cover net and its literal eight preimage arcs. It is not a theorem against
000070 | an abstract isomorphism or embedding iota_I, a different P1 localization,
000071 | a nontrivial compression or quotient, or a transported state and domain.
000072 | Any such proposed map must instead be exhibited and tested together with
000073 | its action on the relevant even observables, state, reflection, original
000074 | OS time, source coefficients, reference and relative normalization. The
000075 | argument does not rule out closed neutral lines on the quotient, where this
000076 | operator is already available, or assert that every neutral line has this
000077 | specific profile. It also does not decide the 30-current sewing rule or
000078 | prove the E8 source and F1.
000079 | 
000080 | Sources
000081 | -------
000082 | experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/
000083 | local_source_origin_20261028/neutral/CLOSED-HALF-KINK-SOURCE-WEYL-LINE.txt,
000084 | sections 1 and 2: original fold, b_F, W(-b_F), and Weyl field action.
000085 | 
000086 | experiments/theory-contracts/v16-operator-pullback-20261014/FOLD-AUDIT.md,
000087 | sections 1 and 2: disjoint cover arcs, cyclic quarter transition, full
000088 | transported state and distinction between source and quotient localization.
000089 | 
000090 | experiments/theory-contracts/native-octic-seam-origin-20261025/
000091 | P1-SOURCE-EMBEDDING-AUDIT.txt, sections 3, 4 and 7: existing local and
000092 | OS data and the separate reflection, collar and reference comparisons.