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Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/ODD-SECTOR-REFIELDING-AND-NAHT.txt
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000001 | UR.SEAM.MIXED_CURRENT_ORIGIN.01 -- continuation 22 October 2026
000002 | Exact odd-sector re-fielding, global source winding, and original locality
000003 | Verdict: PARTIAL. No microscopic P1 spinor-origin theorem.
000004 | 
000005 | 1. Question and inherited premises
000006 | 
000007 | Can the already given eightfold organization of the ONE chiral source be
000008 | re-fielded into the odd D8 spinor extension without changing the source
000009 | Hamiltonian, total charge or vacuum? If yes, does the actual source winding
000010 | become a local spinor field? The first question is answered positively at
000011 | the global Hilbert-space level; the second is answered negatively for this
000012 | unchanged local-CAR inclusion. An independently derived gauging/outer sector
000013 | construction is explicitly outside that negative statement.
000014 | 
000015 | The source, sea, eightfold organization and its local-net interpretation
000016 | remain inherited premises, not a derivation of those choices from P1.
000017 | Source: seam-joint-ward-state-20261014/PROOF.txt sections16,17,21; and
000018 | joint-native-algebra-selection-20261016/
000019 | ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt section2.
000020 | In particular section16 explicitly distinguishes SOURCE-circle winding from
000021 | the as-yet unidentified primitive P1 core circle. No argument here erases
000022 | that outstanding physical identification.
000023 | 
000024 | 2. The exact global unitary
000025 | 
000026 | Indices k=0,...,7. P exchanges k and k+4 modulo8. Let
000027 |   r=(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2),
000028 |   delta_k=(k+alpha)/8-1/2, alpha=1/4 or3/4,
000029 |   K=Nosc+p.p/2+delta.p, H_source=8K, Q=sum p_k.
000030 | Exactly
000031 |   P^2=I, Pr=-r, Pdelta=delta+r,
000032 |   r.r/2+delta.r=0, sum r=0.
000033 | Consequently rho(p)=Pp+r is an involution preserving Q and K when P also
000034 | permutes the current oscillators. This is the old rho, reused directly.
000035 | 
000036 | The original integer lattice is Z8=D8 union v, with D8 the even-sum lattice
000037 | and v its odd-sum vector coset. Define Phi on an orthonormal charge and
000038 | Heisenberg-oscillator basis by
000039 |   Phi=identity on D8 and its entire oscillator module;
000040 |   Phi: |p;osc> -> |Pp+r;Gamma(P)osc> on v.
000041 | The target is L_c=D8 union c, where
000042 |   c={m+(1/2)^8: m in Z8, sum m odd}.
000043 | Indeed (Pp+r)-(1/2)^8=Pp+(0,0,0,0,-1,-1,-1,-1), whose parity is that of p.
000044 | Conversely rho maps c bijectively back to v. Thus Phi is a complex-linear
000045 | unitary between the specified full Hilbert spaces, fixes the full even
000046 | source module and vacuum, and intertwines Q and H_source at all energies.
000047 | The equality is analytic, not an inference from theta coefficients.
000048 | 
000049 | The common-time source vacuum is unique in this odd extension. The even
000050 | spinor coset has the different extra-zero-state problem previously proved;
000051 | that objection must not be transferred to this odd construction.
000052 | The exact charge/parity-refined finite theta control through physical E32
000053 | agrees in both quarters (352 nonzero energy/parity/charge keys each).
000054 | 
000055 | L_c is the usual even E8 lattice in an odd-spinor convention. Naming that
000056 | known target does not make its local fields original microscopic fields.
000057 | No new Fock copy, register or energy counterterm has been introduced here.
000058 | 
000059 | 3. A bounded witness against the unchanged local inclusion
000060 | 
000061 | The old D8 currents agree on the source even sector, but not on the full
000062 | source representation. For beta=e0-e4, the target Cartan pulls back as
000063 |   H_beta on source-even states, and 1-H_beta on source-odd states.
000064 | For a real smooth, mean-zero beta-current profile in a proper folded region
000065 | I, the constant affine term vanishes and the corresponding bounded Weyl
000066 | operator pulls back to
000067 |   Wprime=Pi_even W+Pi_odd W*,     Pbeta=-beta.
000068 | The original source support is contained in the eight preimage arcs of I;
000069 | their complement is nonempty. Choose a normalized odd CAR Majorana unitary
000070 | B in this complement. Original even locality gives [W,B]=0, whereas
000071 | parity B=-B parity. Direct multiplication then gives
000072 |   [Wprime,B]=parity B(W-W*) !=0
000073 | for a nontrivial profile with W!=W*. Therefore Wprime is not an even local
000074 | operator of the unchanged original CAR algebra with that support.
000075 | 
000076 | This is a test of an EVEN TARGET OBSERVABLE's pullback, not the definition
000077 | of a DHR morphism or charged endpoint. It does not exclude an outer local
000078 | extension of the even observable net, nor a physical gauging which changes
000079 | the assignment of local algebras/statistics. Such a change must be derived.
000080 | An exact three-mode CAR control has [W,B]=0, unitary Wprime and
000081 | norm[Wprime,B]=2. The analytic commutator, not this finite example, proves
000082 | the stated support obstruction.
000083 | 
000084 | 4. Explicit transport of SOURCE winding one
000085 | 
000086 | Let R be the cyclic permutation (Rp)_k=p_(k-1 mod8), so the actual source
000087 | winding has T1(p)=Rp+e0. Set
000088 |   A=PR=R^5, lambda=r+e4=(1/2)^5(-1/2)^3.
000089 | Exactly Rr+e0=Pe0+r=lambda. Conjugation by the explicit Phi therefore gives
000090 |   q -> A q+lambda
000091 | on BOTH target cosets, with oscillator permutation A on both. Define the
000092 | full global unitary by W1_tilde=Phi W1 Phi^-1. Any source cocycle/central
000093 | phases are inherited through this operator definition; the affine formula
000094 | alone is not a new cocycle calculation or a vertex-operator construction.
000095 | 
000096 | Exactly lambda.lambda=2, Q(lambda)=1, and for arbitrary q,
000097 |   K(Aq+lambda)-K(q)=(Q(q)+alpha)/8.
000098 | Hence on the transported finite-energy core
000099 |   [Q,W1_tilde]=W1_tilde,
000100 |   [K,W1_tilde]=W1_tilde (Q+alpha)/8.
000101 | Also A^8=I and sum_(j=0..7) A^j lambda=(1)^8, so eight steps yield
000102 |   q -> q+(1)^8, Q -> Q+8.
000103 | Unit spectral flow is retained; it is not divided by8 or16.
000104 | 
000105 | The source winding is nevertheless not a single localized spinor vertex:
000106 | it also permutes neutral Cartans and all oscillators by A. Moreover it is
000107 | odd under the inherited CAR parity. The bosonic target treats that sign as
000108 | an internal Z2. Deriving this statistics/local-net change and the SOURCE to
000109 | P1-core circle map remains essential. Global Hilbert transport is not it.
000110 | 
000111 | 5. Both boundaries
000112 | 
000113 | The original finite cylinder has both chiralities. These calculations on
000114 | one chiral source do not eliminate the opposite edge. The conjugate quarter
000115 | has the same rho/Phi identities; source reversal sends
000116 |   alpha ->1-alpha, Q->-Q, n->-n
000117 | and preserves the exact winding energy increment
000118 |   c_n(Q)=nQ/8+n(n-1)/16+alpha n/8.
000119 | A tensor product of the two global transports therefore keeps both edges;
000120 | it does not select one or construct a right-boundary absorber. The actual
000121 | v1033 register uses one common strip flux, not independent edge fluxes.
000122 | 
000123 | 6. Finite Arf identification check
000124 | 
000125 | The actual v774 parity lift is
000126 |   iota(f1,f2,f3,a)=(f1,f2,f3,a,f1+f2+f3+a).
000127 | Its finite chi_NSR=a is the fourth occupation bit on Lambda_even C5.
000128 | The sign C=(-1)^n4 has eight positive and eight negative eigenvalues.
000129 | The actual D5 root E45=epsilon4 iota5 acts inside this 16-dimensional
000130 | half-spinor and changes n4. Exactly
000131 |   rank[C,E45]=4, ||[C,E45]||_HS^2=16, ||[C,E45]||=2.
000132 | A uniform boundary spin-structure label would commute with the fixed
000133 | internal Spin(10) action. Thus this finite bit is not literally that central
000134 | NS/R label with the unchanged D5 action. The finite compiler theorem is
000135 | not refuted; a different geometric Arf dictionary would require its map.
000136 | 
000137 | [OMITTED: non-scientific conversation or execution metadata]
000138 | 
000139 | [OMITTED: non-scientific conversation or execution metadata]
000140 | [OMITTED: non-scientific conversation or execution metadata]
000141 | [OMITTED: non-scientific conversation or execution metadata]
000142 | [OMITTED: non-scientific conversation or execution metadata]
000143 | [OMITTED: non-scientific conversation or execution metadata]
000144 | [OMITTED: non-scientific conversation or execution metadata]
000145 | 
000146 | Even when J preserves parity, Rsel=Pi_even+Pi_odd J is real-linear, neither
000147 | complex-linear nor globally antilinear. In the favorable J=conjugation test,
000148 |   Ad_Rsel(iI)=i parity !=iI.
000149 | It cannot be a complex-linear observable-algebra automorphism. The original
000150 | spectral grading is in addition not automatically manybody CAR parity.
000151 | Likewise J-real vectors (v,Jv) form a real subspace, not the missing complex
000152 | Hilbert sector sum. Quarter determinant values 1-i and1+i have product2;
000153 | their relative phase is not an Arf sign or a gluing prescription.
000154 | 
000155 | [OMITTED: non-scientific conversation or execution metadata]
000156 | [OMITTED: non-scientific conversation or execution metadata]
000157 | 
000158 | 8. Verdict and reproducibility
000159 | 
000160 | Positive: an exact all-energy odd re-fielding and a unit-winding global
000161 | operator bridge are now explicit in the given folded source. The decisive
000162 | remaining condition is original microscopic locality/statistics/sector
000163 | generation, not its H,Q spectrum. No local P1 spinor has been constructed.
000164 | 
000165 | Run check_odd_refielding.py with python3 -B -OO. Its JSON records the exact
000166 | polynomial, theta, bounded-CAR and finite Arf controls. This experiment does
000167 | not change a verification module, physical gate, paper or ledger claim.