TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-P1-VARIATION-GAP-AND-EVENT-SELECTION.txt
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Inclusive original lines: 135-185
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000135 | 4. Smallest actual3+2 test of the source-selected gap component
000136 | 
000137 | The original chi singles out a SPECTRAL COMPONENT, not a physical event.
000138 | In the actual quarter source the unique closest eigenvalue is
000139 |  alpha1/4: E0=+1/4, mode r0=0;
000140 |  alpha3/4: E0=-1/4, mode r0=-1.
000141 | These choices follow by integer ordering of the whole spectrum, no scan
000142 | or choice by desired rank. The gap projector has rank1 on the BOUNDARY
000143 | ONE-PARTICLE space. This rank is not an internal rank or particle count.
000144 | 
000145 | Given the already stipulated original CAR representation, its canonical
000146 | sea-normal-ordered spectral component is
000147 |  N_gap=sign(E0):a_r0^* a_r0:
000148 |       =a_0^*a_0 in the first quarter,
000149 |       =1-a_-1^*a_-1 in the second quarter.
000150 | It is a genuine bounded projector on the full original Fock space. There
000151 | is no newly chosen coupling or measurement postulate in defining this
000152 | existing CAR spectral component. Identifying it as THE physical event
000153 | would, however, be a further claim which the action has not supplied.
000154 | 
000155 | Decisive question: does even this directly source-derived component
000156 | literally supply the desired sharp mark on the five actual full native
000157 | inputs psi_i=R_(b,4)f_i? Retain the actual input metric B_ij=<psi_i,psi_j>
000158 | and all exterior output amplitudes. Compute
000159 |  H_ij=<psi_i,N_gap psi_j>=<N_gap psi_i,N_gap psi_j>.
000160 | Both original quarters, exactly overQ(i): H>0 and B-H>0, each rank5.
000161 | Hence the response E=B^-1/2 H B^-1/2 satisfies0<E<I. Its actual metric
000162 | similar form A=B^-1H is nonscalar and NOT idempotent. It is not a rank2
000163 | projector with rank3 complement, nor the opposite3+2 assignment.
000164 | 
000165 | The full exterior defect is
000166 |  Delta=H-H B^-1 H>0, rank5 in BOTH quarters.
000167 | If I:C5->H_source is the actual column map, then
000168 |  Delta=I^* N_gap(1-I B^-1 I^*)N_gap I.
000169 | This is an identity about the actual full output, not an executed
000170 | projection, postselection or a demand that every physical process close
000171 | onC5. It shows that the proposed literal restriction to a sharp mark
000172 | fails; unsharp observer responses are not ruled out as physical processes.
000173 | No eigenvalue grouping, extra normalization or mode weighting is performed.
000174 | 
000175 | N_gap is also a GLOBAL spectral occupation, not an insertion in a proper
000176 | local interval: [a_r^*a_r,psi(f)]=-<e_r,f>a_r, which is nonzero for suitable
000177 | f supported in any complementary interval. The source's local-net
000178 | definition cannot be satisfied by calling this global mode a local pulse.
000179 | This localization observation does not require the native5 to beH_adm.
000180 | 
000181 | This test excludes ONLY direct use of that canonical gap-occupation
000182 | component as the sharp3+2 input mark. It is not an all-P1/no-3+2 theorem.
000183 | The source gap covector, its CAR spectral component and a physical event
000184 | functional remain three correctly related but distinct objects.
000185 |