TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-source-origin-audit-20261016/LOCAL_ALGEBRA_PROOF.txt
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Inclusive original lines: 108-172
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000108 | 3. What the unmarked exterior carrier DOES determine
000109 | 
000110 | This section reuses the no-selector result already in
000111 | raw-carrier-origin-gate-20260920/PROOF.txt:35-112,178-184, rather than rerunning
000112 | an old U(5) search. For supplied Hermitian E of rank g define the INTERNAL
000113 | exterior space F=Lambda E and its canonical operators
000114 | 
000115 |   c*(v) psi = v wedge psi, c(v)=(c*(v))*.
000116 | 
000117 | They obey CAR and act between degrees k and k+/-1. Let P_vac=product_i(1-c_i*c_i)
000118 | with c_i the annihilator in an orthonormal basis. Creation words, P_vac and
000119 | annihilation words give all rank-one matrix units |I><J| (with the appropriate
000120 | ordering signs). Hence the represented INTERNAL CAR algebra is End(Lambda E).
000121 | Its even part is End(F_even) direct_sum End(F_odd). Its degree-at-most-two even
000122 | vector space is
000123 | 
000124 |   C I + Lambda^2 E + End(E) + Lambda^2 E*,
000125 | 
000126 | represented by c*c*, c*c and cc. Quadratics generate the entire even algebra
000127 | ASSOCIATIVELY; their quadratic vector space is not the entire even algebra.
000128 | The shifted quadratic commutator algebra is so(E direct_sum E*).
000129 | 
000130 | All these formulas are basis covariant. A Hermitian metric and unit complex
000131 | volume leave SU(g) acting naturally. A projector canonically selected from
000132 | these data must commute with this action. The fundamental representation on E
000133 | is irreducible, so such a projector is0 or I. In particular rank five alone
000134 | does not select any proper subspace. Its rank-k projectors form Gr(k,5).
000135 | For any1<=k<=4 the trace-balanced primitive integral generator
000136 | 
000137 |   X_k=(5-k)P_k-k(I-P_k)
000138 | 
000139 | has trace0. The k=1 case has spectrum4,-1 and the k=2 case3,-2; trace balance
000140 | does not eliminate the1+4 option. This does not exclude an originally derived
000141 | interaction selecting rank2 up to gauge. It shows that CAR closure or trace
000142 | balance alone supplies no such selection. No physical gauge group is inferred
000143 | from this natural internal covariance group.
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000152 | 
000153 | There IS additional positive structure on this supplied unmarked carrier.
000154 | Its anti-linear Hodge operator J_H obeys, on Lambda^k C5,
000155 |   J_H^2=(-1)^(k(5-k))=+1,
000156 |   J_H Gamma=-Gamma J_H, Gamma=(-1)^degree,
000157 | so the anti-linear operator Gamma J_H has square -1 on the FULL32-dimensional
000158 | space. This needs only the supplied Hermitian metric and unit complex volume,
000159 | not a3+2 split or Yukawa. The checker verifies these identities exactly.
000160 | It is a pair of KO-like signs, not a complete finite KO6 triple: no physical
000161 | J identification, compatible D, or spectral coordinate algebra follows yet.
000162 | 
000163 | In fact the full even INTERNAL algebra is not a coordinate algebra satisfying
000164 | order zero with this J_H. Since J_H swaps the two half-spinor spaces,
000165 | J_H A_even J_H^-1=A_even. The order-zero condition for A_even would then
000166 | require A_even to commute with itself, which it does not. Explicitly on the
000167 | even block let a=|empty><12|, b=|12><empty| and choose the odd-block seed
000168 | d=J_H b* J_H^-1. Then d is in A_even but [a,J_H d* J_H^-1]=[a,b] is a
000169 | nonzero rank-two diagonal operator. The checker verifies this witness.
000170 | Thus universal internal operator closure cannot simply be relabelled the
000171 | spectral coordinate algebra. An actual left/right bimodule subalgebra has
000172 | to be derived; neither all matrices nor the pair of KO-like signs selects it.