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Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/PROOF.txt
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000001 | UR.NATIVE.ARCCHARGE.01
000002 | 2026-10-26. Exact source-algebra obstruction; physical selection remains open.
000003 | 
000004 | 1. Question and first discriminating test
000005 | The native five quartic fields are already constructed in the original
000006 | quarter CAR source. Can their original P1 insertion be generated using only
000007 | the additive scalar density/stress net on their eight proper preimage arcs?
000008 | Success for this route requires invariance under all phases constant separately
000009 | on these arcs. A nonzero native highest-degree commutator aborts this route.
000010 | This tests an original insertion algebra, not a proposed new dynamics.
000011 | 
000012 | 2. Retained data
000013 | Use the actual M, displayed J-equivariant O, positive Cartan embedding V,
000014 | and the five existing native quartic constructors. V*V=I4, V^t V=0.
000015 | No new metric, weighting, coupling or source state is chosen for the exact test.
000016 | The source folding remains eta_r(x)=psi((x+2pi r)/8)/sqrt(8),
000017 | chi=U_alpha* eta, U_alpha[r,k]=exp(2pi i(k+alpha)r/8)/sqrt(8),
000018 | with alpha=1/4 or 3/4. Retain the one-circle sea and its reference subtraction.
000019 | 
000020 | 3. Exact top-degree coefficient calculation
000021 | For each native polynomial f_i, expand f_i(V^t t) in four distinct commuting
000022 | nilpotent occupations t_k. Let C[K,i] be its coefficient, |K|=4.
000023 | C has rank5 and has 70 rows. This is coefficient bookkeeping for the highest
000024 | CAR degree, not a physical identification of the full source with 70 states.
000025 | The corresponding creator/annihilator tensor is diagonal on exterior^4 C8.
000026 | A phase on original arc r acts there by dGamma(E_r),
000027 | E_r=U_alpha* |r><r| U_alpha. If K,L differ by replacing one index,
000028 | the matrix element has modulus1/8. All other offdiagonal entries vanish.
000029 | Thus the exact Hilbert-Schmidt Gram of these principal commutators is
000030 | G_ij=(1/64) sum_{ordered K~L} conj(C[K,i]-C[L,i]) (C[K,j]-C[L,j]).
000031 | There are1120 directed edges. Direct wedge replacement signs independently
000032 | reproduce the same Gram, without floating eigendecomposition.
000033 | Result: rankG=5, detG=498159/262144.
000034 | With original norms(96,8,8,8,1), detG_normalized=166053/4294967296.
000035 | All leading principal minors are positive. This holds for each of eight arcs
000036 | and both quarter phases. Every nonzero native quartic responds to each
000037 | individual infinitesimal original arc phase.
000038 | Normal ordering contractions, b/z shifts and derivative counterterms have
000039 | lower CAR degree. They cannot cancel this nonzero degree8 tensor.
000040 | The total-density coefficient vector C[K]=1 has zero commutator, a control.
000041 | 
000042 | 4. Stronger relative-phase statement
000043 | Partition edges by replaced index difference +/-s modulo8.
000044 | For s=1,2,3,4 the edge counts are320,320,320,160 and each difference Gram
000045 | has rank5. For real lambda_r, U_alpha*diag(lambda)U_alpha has offdiagonal
000046 | entries given by the discrete Fourier coefficients at those differences.
000047 | Each nonconstant real lambda has a nonzero difference component.
000048 | The supports of distinct differences in the matrix tensor do not cancel.
000049 | Therefore every nonzero native quartic responds to every nonconstant real
000050 | infinitesimal arc-phase vector. The common constant phase remains trivial:
000051 | the fields are globally charge neutral. No claim about every finite phase
000052 | is made, since2pi phases and possible finite stabilizers require separate care.
000053 | 
000054 | 5. Precise algebraic consequence
000055 | Let E=union_r I_r have disjoint closures, and let
000056 | B_add(E)=join_r B(I_r), where B(I_r) is generated by the original scalar
000057 | density and its geometric stress supported in the individual original arc.
000058 | Choose one smooth source phase h constant on a neighborhood of the entire
000059 | closure of each I_r, with independent constants on the eight components and
000060 | transitions in the complementary gaps. Disjoint closures make this possible.
000061 | Its Weyl implementer fixes each local density/stress generator: derivatives
000062 | of h, including anomalous terms, vanish on these supports. It therefore
000063 | fixes the additive net pointwise. The degree8 calculation proves the native
000064 | quartic field distributions are not fixed, and hence are not generated by
000065 | that additive density/stress net. This is not inferred merely from support.
000066 | For a smoothly smeared closed operator realization on a common invariant
000067 | core, noninvariance also excludes affiliation with that additive von Neumann
000068 | algebra: a unitary fixing the algebra pointwise fixes its affiliated closed
000069 | operators and their domains. Existence/domain hypotheses for that closed
000070 | realization are explicit; the finite coefficient test alone proves no such
000071 | analytic theorem. The unconditional result here is the field-tensor statement.
000072 | The older odd-fermion support commutator establishes multilocal support;
000073 | the new even charge commutator establishes a different, stronger boundary.
000074 | 
000075 | 6. Positive source route retained
000076 | The full original CAR algebra on E already contains bilocal neutral products
000077 | eta_r* eta_s, r!=s. They transfer relative arc charge while preserving total
000078 | charge. They are not in general in the additive scalar density/stress subnet.
000079 | The actual native currents use A_a=U_alpha diag(V[:,a]) U_alpha*,
000080 | which has such offdiagonal entries. Their quartic products give the native
000081 | fields. This does not add particles, a new E8 extension or an interaction fit.
000082 | The offdiagonal Gram for positive native Cartan versus arc diagonal scalar
000083 | potentials is V*V - V* ones(8,8) V/8 and has rank4.
000084 | No nonzero positive native Cartan direction is an arc diagonal potential.
000085 | This classifies the failed scalar insertion route, not all original P1 maps.