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Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/HIGGS_VARIATION_PROOF.txt
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000052 | 3. Derive every local one-form in the retained quarter-circle realization
000053 | 
000054 | The existing source_reconstruction/PROOF.txt, sections4-5, reconstructs under
000055 | its explicit Wold/operator hypotheses
000056 |   H=L2(S1), B_alpha=N+alpha, N=-i partial_theta,
000057 |   alpha=1/4 or3/4, U=M_exp(i theta), C=1_(n>=0),
000058 | in periodic trivialization. The following calculation uses its local coordinate
000059 | algebra A_loc=C_infty(S1), acting by pointwise multiplication on the charged line.
000060 | This is a scoped local realization, not a claim that all allowed P1/P2 algebras
000061 | are commutative or that the full physical insertion net has been selected.
000062 | 
000063 | For every smooth periodic b and every vector in the smooth common core,
000064 |   [B_alpha,M_b]=-i M_(partial_theta b).
000065 | Thus every represented one-form is a multiplication operator M_v. Conversely,
000066 |   U* [B_alpha,U]=I,
000067 | so for any smooth h, M_h=M_(h)U* [B_alpha,U] is a represented one-form.
000068 | Consequently
000069 |   Omega1_(B_alpha)(A_loc)=M_(C_infty(S1)),
000070 | and its selfadjoint elements are EXACTLY real-valued smooth potentials v.
000071 | The dependence derived here, without choosing a Yukawa matrix, is
000072 |   B_alpha,v=-i partial_theta+alpha+v(theta).
000073 | 
000074 | This is a connection component on the given charged line. The pointwise
000075 | unitary algebra is U(1); an inhomogeneously transforming connection is not a
000076 | new fundamental weak SU(2) doublet. The representation and line multiplicity
000077 | are unchanged. Adding the opposite-charge conjugate source changes no such
000078 | conclusion: completion with its given J transports the same multiplication
000079 | operators, possibly cancelling or pairing them, and creates no new nonabelian
000080 | carrier module. No physical anomaly cancellation is inferred from this statement.
000081 | 
000082 | 4. Complete smooth-potential classification, including fixed quarter holonomy
000083 | 
000084 | Put vbar=(2pi)^-1 integral v. Let lambda be a real primitive of v-vbar; it is
000085 | periodic and smooth. Multiplication by exp(i lambda) is unitary on L2 and
000086 | preserves the H1 domain. Direct differentiation gives
000087 |   exp(-i lambda) B_(alpha+vbar) exp(i lambda)=B_alpha,v.
000088 | The same statement works on the actual twisted boundary domain in its
000089 | equivalent nonperiodic trivialization. In particular
000090 |   exp(2pi i B_alpha,v)=exp(2pi i(alpha+vbar)) I.
000091 | Keeping the original quarter transition therefore requires vbar in Z.
000092 | For vbar=k in Z the additional periodic large multiplication exp(i k theta)
000093 | relates B_(alpha+k) to B_alpha. A half-integer mean flips the quarter branch
000094 | and is not an allowed same-branch solution.
000095 | 
000096 | This proves a one-particle operator classification, not physical gauge
000097 | equivalence of fixed vacua, determinant-line triviality, cancellation of a
000098 | chiral anomaly, or a new physical time identification. A transported spectral
000099 | covariance is conjugated with the same operator; holding the old reference sea
000100 | fixed is a different question. Large gauge maps can carry charge. Likewise a
000101 | four-dimensional interpretation as gauge-Higgs unification would require an
000102 | actual spacetime/representation map and is not established by a circle label.
000103 | 
000104 | Any additional original reflection/state/admissibility constraints restrict
000105 | this local one-form space; they do not turn its unchanged U(1) representation
000106 | into a new SU(3)xSU(2) fundamental carrier. This exclusion does not cover a
000107 | larger independently derived local algebra or a different original source.