TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/DET16-WHOLE-SOURCE-REDUCTION.txt
Original file SHA-256: fc43fcfef453bb278b8f5c28123078331a36d1e4350672b45c47a5d99c992897
Inclusive original lines: 133-178
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000133 | 5. ATTACHMENT TO THE WHOLE ORIGINAL HAMILTONIAN SOURCE
000134 | 
000135 | In the declared decoupled ambient table, for rest-only source legs,
000136 | 
000137 |  Z_Lambda^amb[J_R]=Z_R,Lambda[J_R] product_x Z_m.
000138 | 
000139 | This is exact partial trace, at every volume; the interacting H_R is NOT
000140 | solved or discarded. Its normalized rest words factor, while its actual
000141 | unnormalized source retains all mirror free-energy factors. The free-box
000142 | KMS subnet therefore has these same finite mirror thermal factors together
000143 | with the rest subnet state. This is not a unique-phase theorem for H_R.
000144 | 
000145 | The physical Gauss projection must not be replaced by this ambient product.
000146 | Suppose the ACTUAL supplied gauge action factors as
000147 |  U(g)=U_R(g) tensor product_x U_m,x(g_x), and use its SAME original
000148 | P_G=|G|^(-|V_int|)sum_g U(g). For gauge-compatible rest source legs,
000149 | 
000150 |  Z_phys[J_R]=|G|^(-|V_int|) sum_g
000151 |    Tr_R[U_R(g) exp(-beta H_R) original_ordered_source[J_R]]
000152 |    * product_x z_m,x(g_x),
000153 |  z_m(g)=a Tr U_m(g)+b<phi|U_m(g)|phi>.                  (7)
000154 | 
000155 | Proof: substitute the actual group projector into the full trace, retain
000156 | its order, then take the mirror partial traces using (1). This performs
000157 | exact reduction of the65536-dimensional factor. It does NOT replace the
000158 | sum by independent projected Hilbert factors or solve the256-dimensional
000159 | rest lattice. Nonfactorizing gauge actions require their actual map.
000160 | 
000161 | For the expressly given canonical second quantization U_m=Gamma(r),
000162 | with det r=1 and no extra implementer phase, both0 andF are invariant.
000163 | Exterior-algebra trace gives Tr Gamma(r)=det(I+r), hence
000164 | 
000165 |                   z_m(g)=a det(I+r(g))+b.              (8)
000166 | 
000167 | Actual Spin10 central q=1,i,-1,-i acts as q^N on this native spinor Fock
000168 | factor. It fixes phi, and its four weights are
000169 |  a(1+q)^16+b: a*65536+b, a*256+b, b, a*256+b.
000170 | The native code verifies them directly. This is not an identified v1013
000171 | gauge embedding or an equality to the P1 clock just because both have
000172 | order4. That actual mirror gauge representation was not printed in the
000173 | reviewed v1013 table.
000174 | 
000175 | With mirror source legs, replace z_m by the entire alternating heat/word
000176 | trace (4), with its actual U_m insertion. Keep the original graded tensor
000177 | ordering and gauge carriers for mixed even observables. The untwined
000178 | renewal routine does not pretend to implement these unknown gauge maps.