TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-ADMISSIBILITY-AND-TOTAL-CHIRAL-TRACE-PROOF.txt Original file SHA-256: 08fdf72e888a4ef511bdfb4d7db1e2919c72ee3dc193c27c16a5dfee72fb6d25 Inclusive original lines: 113-222 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000113 | that {D_H,Gamma_tot}=0. Define 000114 | Pi=I4 tensor(I96+GammaF)/2, 000115 | Pc=(I384+Gamma_tot)/2. (11) 000116 | Pi and Pc are distinct. Both have total rank192, which is not a physical 000117 | dictionary or a finite48-dimensional submodule identity. Pc does NOT 000118 | commute with the individual gamma^mu. The parallel spin-COMMUTING 000119 | projection assumption of R036 does not apply to Pc as an assumption about 000120 | the first-order Clifford generators, even though Pc reduces D_H^2. 000121 | Pc IS parallel for the given product connection. Directly for its square, 000122 | Pc E(I-Pc)=0, so the two heat readers coincide without discarding kinetics. 000123 | 000124 | In the full square, the derivative term Qder=i gamma^mu gamma5 tensor 000125 | D_mu Phi is spin odd AND internally odd. Hence 000126 | [Gamma_tot,Qder]=0, Pi Qder Pi=0, 000127 | Pc Qder Pc generally NONZERO, 000128 | (I-Pc) D_H^2 Pc=0. (12) 000129 | The total-chiral half retains the Higgs kinetics directly. The R036 000130 | internal-half kinetic difference must not be called a physical chiral 000131 | projection effect without an actual original identification. 000132 | 000133 | 5. Full spectral identity for EVERY supplied H, not just four heat terms 000134 | Suppose the SAME given self-adjoint K has Gamma-stable domain and is odd, 000135 | with trace-class heat/compact-resolvent conditions where traces are used. 000136 | It can be written K=[[0,T*],[T,0]] on the Gamma eigenspaces. Then 000137 | K^2=diag(T*T,TT*). 000138 | For every lambda>0, T/sqrt(lambda) identifies the two lambda eigenspaces 000139 | isometrically, with inverse T*/sqrt(lambda). Thus their positive spectra 000140 | and multiplicities agree. With P0=projection onto ker K, 000141 | Tr(Pc F(K^2)(I-P0))=1/2 Tr(F(K^2)(I-P0)) (13) 000142 | for EVERY allowed summable spectral function F, not a finite scan. 000143 | For heat INCLUDING zero modes, the full identity is 000144 | Tr(Pc exp(-tK^2))=1/2 Tr exp(-tK^2)+index(T)/2. (14) 000145 | The zero-mode term is essential. It is not legitimate to halve an absolute 000146 | heat trace while silently deleting an index imbalance. For H and H0 in 000147 | the same fixed-domain Fredholm homotopy, index(T_H)=index(T_H0), and their 000148 | whole heat differences in(14) are exactly half. With the explicitly 000149 | ZERO-EXCLUDED convention, the entire chiral zeta function is half of the 000150 | full one on its convergence domain and hence under the same meromorphic 000151 | continuation. This includes Res and FP, with their zero-mode convention. 000152 | 000153 | The combined reader is also fixed without another spectral search. If the 000154 | ORIGINAL positive operator really is L=K^2 for this same odd K, then 000155 | Pgap=1_[kappa,infinity)(L) commutes with Gamma. Pairing every retained 000156 | positive eigenvalue in(13) gives 000157 | Tr(Pc Pgap F(L))=1/2 Tr(Pgap F(L)). (13a) 000158 | There is NO index correction in(13a), since kappa>0 removes the whole zero 000159 | kernel. This holds for the entire allowed function class. It does NOT 000160 | supply the unproved original Ladm=Ksigma^2 map. The local index density 000161 | has not vanished: in a heat expansion its integral is compensated by the 000162 | nonlocal finite-rank removal of the unpaired kernel. 000163 | 000164 | Apply(13)-(14) to the existing D_H family. For scalar f commuting with 000165 | Gamma_tot, fD_Hf is still odd on its transported Gamma-stable domain. 000166 | This is the full inner producer from R034. The affine outside presentation 000167 | is the same family with its transported H0, action and sources; it does 000168 | not create a new chiral freedom. If a further Brel is really present, 000169 | its oddness must be checked and it must not be counted twice. 000170 | 000171 | For a bound reference R_tau(K)=(K+tau K tau^-1)/2, 000172 | tau Gamma tau^-1=+Gamma OR -Gamma is sufficient for R_tau(K) to remain odd. 000173 | The sign only determines whether tau preserves or swaps chiral halves; 000174 | either way its conjugate of an odd K is odd. GammaF itself COMMUTES with 000175 | GammaF, so multiplying a lift by GammaF does not make it internally odd. 000176 | This corrects Gemini's proposed additional internal flip. Domain stability 000177 | and the actual original lift/reference are still required. The same 000178 | argument also works for an antiunitary tau with the stated real grading. 000179 | 000180 | For two such original operators, the even relative spectral trace inherits 000181 | (13), and its heat includes half the relative index in(14). If the actual 000182 | reference is not odd, do not apply the theorem to it. The boundary eta 000183 | term in the original functional is NOT set to zero by an odd bulk spectrum. 000184 | No fermion Pfaffian, phase, full mixed source or RP/state selection follows 000185 | from an EVEN bosonic spectral-trace identity alone. All such data survive. 000186 | A factor one-half in an action is not an ignorable normalization of its 000187 | path integral. Do not infer equal normalized H moments or equal Z0 by 000188 | halving the spectral trace; the original action/measure contract decides. 000189 | 000190 | 6. Local H coefficients and the index density 000191 | In the fixed product connection, every H-dependent term in a2 and a4 000192 | has zero trace with Gamma_tot: scalar Phi^2/Phi^4/RPhi^2, Qder^2, 000193 | mixed curvature/Qder, and total derivatives all have zero corresponding 000194 | gamma5 spin trace. Thus, for u=|H|^2 and original A,B,C from R031, 000195 | a2H,Pc= -A/(8 pi^2) int u, 000196 | a4H,Pc= (1/(16 pi^2)) int[A|DH|^2+B u^2+4C u] 000197 | +A/(96 pi^2) int R u. (15) 000198 | The H-independent Majorana background is retained in the absolute source; 000199 | the formula uses the original H0=0 baseline of R031. For another H0, 000200 | subtract the corresponding full polynomial/background, as required by R034. 000201 | The local supertrace can contain gauge/gravity index densities. Those 000202 | must not be globally halved away, inferred to vanish from this H test, 000203 | or canceled against a different reference without its actual dictionary. 000204 | Their zero-mode correction is exactly accounted for by(13)-(14). 000205 | In particular coeff_a2(u)/coeff_a4(Ru) remains -12, not the printed -2. 000206 | Neither total-chiral halving nor the same finite low-spectrum removal 000207 | supplies an independent factor6 or relative Higgs potential sign repair. 000208 | 000209 | 7. Precise resulting original task 000210 | The actual Padm reduces Ladm by construction. The product total chirality 000211 | reduces the whole product square for all H and scalar scales. Both are now 000212 | fully evaluated spectral roles; R036 internal-half feedback is not a free 000213 | replacement for either of them. No new projection search is useful here. 000214 | The next actual map is the ORIGINAL realization of Dadm on Ran(Pstr), 000215 | its relation to Ksigma=fDgeo f+Brel, and the same relative trace/reference 000216 | on that realization. Compare the positive operators and transported 000217 | domains, not just matrices with the same size or their index. The original 000218 | Wedge->96H consumer, whole words/memory, state/time/RP and Flavor/Alpha, 000219 | E8 and clocks remain. This derives no missing normal metric or geometry. 000220 | The named source passages give the roles above; failure to find an 000221 | explicit identification there is not a corpus-wide absence proof. 000222 | F1 remains in progress; no new physical origin or Lean theorem is claimed.