TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/collar-current-source-bridge-20261021/DIRAC-LIFT-ADMISSIBILITY.md
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Inclusive original lines: 125-248
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000125 | original selfadjoint domain is selfadjoint. Conjugation by U_chi no longer
000126 | identifies the same-domain operators unless it preserves Bsa. This by itself
000127 | does not prove different spectra; a fixed-domain spectral/action response
000128 | has not been computed here. After conjugating back by U_chi*, its
000129 | boundary condition is `U_boundary* Bsa`. Thus any such change is a relative
000130 | boundary attachment, and must be specified as that physical context; one
000131 | cannot discard its variation in the reference, projector or action.
000132 | 
000133 | For the actual reference split, keep the geometric involution R distinct
000134 | from the Calderon polarization. At fixed R, writing `R_U=U* R U`, exactly
000135 | 
000136 | ```
000137 | D_ref(D_phi,R)=U (D+R_U D R_U)/2 U*,
000138 | D_rel(D_phi,R)=U (D-R_U D R_U)/2 U*.
000139 | ```
000140 | 
000141 | In particular, if U commutes with the fixed R the reference and relative
000142 | DIFFERENTIAL expressions are still conjugate; holding R fixed alone does
000143 | not make the gauge family a physical action variation. Domains remain part
000144 | of this statement. If `[U,R]!=0`, with `U_t=exp(itF)` and `K=[F,R]`,
000145 | 
000146 | ```
000147 | delta D_ref - i[F,D_ref] = -i (K D R + R D K)/2,
000148 | delta D_rel - i[F,D_rel] = +i (K D R + R D K)/2.
000149 | ```
000150 | 
000151 | These are the extra reference terms; deleting them is not a fixed-R
000152 | calculation. For a covariantly transported full admissibility complex its
000153 | Hodge projector instead obeys `P_t=U P_0 U*`. None of these identities
000154 | determines the original P1 physical F or the OS operator implementing it.
000155 | 
000156 | For the gapped APS sign subspace on a circle, a scalar multiplier and its
000157 | adjoint preserve the fixed subspace iff g is constant: preserving both Hardy
000158 | halves forces every nonzero Fourier coefficient of g to vanish. Infinitesimally,
000159 | `sin(4theta)` sends the negative mode -1 to positive mode 3 with coefficient
000160 | `1/(2i)`, and `1-cos(4theta)` does so with coefficient `-1/2`.
000161 | Fixing the four marks therefore does not preserve the original APS subspace.
000162 | This is NOT a failure of selfadjointness of the bounded perturbation on its
000163 | old domain. Nor does it rule out a physical boundary drive with that domain.
000164 | 
000165 | Two interior extensions chi with the same endpoints differ by a smooth unitary
000166 | equal to I on both boundary components. It preserves the fixed trace condition
000167 | and conjugates the two realizations. Therefore even the fixed-domain spectrum
000168 | depends on the endpoint attachment, not on an arbitrary interior interpolation.
000169 | 
000170 | There is a second, stronger trace-space distinction: the one-face endpoint
000171 | unitary `V=I direct_sum g` preserves the original scalar graph G only for g=I.
000172 | Indeed its off-diagonal block would require `c g*=c`, and c is injective.
000173 | This observation concerns the Cauchy trace projection. **It does not identify
000174 | that projection with Padm on the OS Hilbert space.**
000175 | 
000176 | The original physical net requires BOTH A and A* to preserve Ran(Padm),
000177 | equivalently `[A,Padm]=0` for bounded A on the full OS space. Neither a moving
000178 | Cauchy graph nor a sign-projector calculation supplies this missing physical
000179 | operator map. Compression by Padm is explicitly not the definition of the net.
000180 | 
000181 | ## 5. The chiral route survives, but its measure is an actual missing datum
000182 | 
000183 | The original Gamma already defines the chiral block D_plus. One need not
000184 | invent an oscillator or a new component label to formulate that route.
000185 | However replacing the full module by its chiral determinant/quantization is
000186 | not a change of notation: it selects which field measure and line connection
000187 | are physically used. That identification is the outstanding task.
000188 | 
000189 | The existing source integration result already prevents a shortcut here:
000190 | `finding:joint-fermionic-source-schur-functional` shows that a DIRECT
000191 | identification of a primitive trace profile with `-log det K` requires
000192 | `x f_prime(x)+1=0`. An arbitrary declared spectral profile does not provide
000193 | that fermionic measure. A separate fermionic action is possible, but its
000194 | actual kinetic operator, measure and source coupling must then be supplied
000195 | by the original construction. This is reused prior evidence, not a new test.
000196 | 
000197 | For the paired adapted B, Jn pairs every eigenvalue with its negative, so
000198 | `eta(B)=0`. For the scalar A in the original periodic quarter frame,
000199 | Hurwitz zeta continuation gives `eta(A)=1-2alpha`, namely +1/2 or -1/2.
000200 | The two quantities belong to DIFFERENT operators. Even the scalar eta is
000201 | constant along `g A g*`; its value is not by itself the curvature of the
000202 | chiral implementer line. There is no license to substitute one for the other.
000203 | 
000204 | Original source contracts explicitly require a chiral measure and line WITH
000205 | connection: `CHIRAL4D.NOMIRROR.01` C1-C6, and
000206 | `SEAM.DETLINE.UNIFICATION.01`. Their original current manuscript passages and
000207 | the live CSV ledger still state Open. The Atlas annotation of the former has
000208 | stale source references; its status was therefore checked directly in the
000209 | current ledger, not inferred from that annotation. No four-dimensional part
000210 | of those contracts is proved by this boundary calculation.
000211 | Their four-dimensional requirements are not imposed as prerequisites for
000212 | further one-dimensional boundary work. The immediate missing map is already
000213 | the boundary source/measure/connection identification in the original datum.
000214 | 
000215 | ## 6. Consequence for the solution search
000216 | 
000217 | The automatic route "full P1 Calderon graph -> same scalar one-face current
000218 | phase" fails in the stated complete Dirac/CAR realization. Restoring the
000219 | omitted graded component cancels its central phase. The previous scalar
000220 | bridge is correct on its stated scalar/chiral source, and remains conditional.
000221 | 
000222 | The next indispensable original map is now more specific: the P1 real/graded
000223 | source and its physical admissible measure must select the chiral block (or
000224 | supply a compensating bulk line/inflow) and identify its connection with the
000225 | original charged CAR/E8 source, while retaining the actual boundary attachment,
000226 | reference and A/A*-admissibility. A declaration of a new projector or fitted
000227 | coupling would not establish that map.
000228 | 
000229 | Native Hamming/E8/J, quartic4/5, erasure20, primary20 carry, all eight source
000230 | preimages, original time and winding are unchanged. No finite code-closure
000231 | condition is added. No T1-T8 goal, event-selection rule or full solution closes.
000232 | 
000233 | ## Sources and verification
000234 | 
000235 | - Original full product Dirac: `seam-full-collar-reconstruction-20261015/PROOF.txt:21-61`.
000236 | - Original P1 datum, primitive action and physical net:
000237 |   `documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex:65-80,302-343`.
000238 | - Original full selector: `documents/archive--tfpt-45--source_extracts--04_qft_source/sections/002-section-closure-theorems-from-the-admissibility-complex.tex:12-123`.
000239 | - Existing frame subtraction: `seam-collar-multifield-20261016/P1-COVARIANT-INSERTION.txt:26-146`.
000240 | - Existing joint source-measure identification test: `seam-collar-multifield-20261016/PROOF.txt:271-326`.
000241 | - Original open chiral and line contracts: `documents/tfpt_research_contracts/sections/015-source.tex:235-266`, `017-source.tex:239-285`.
000242 | - [Baer and Ballmann, arXiv:1307.3021v2](https://arxiv.org/abs/1307.3021), Lemma 3.2/Remark 3.3, Theorems 4.11-4.12: adapted operator, paired spectrum, boundary domains.
000243 | - [Langmann, JMP35 (1994), arXiv:hep-th/9305014](https://arxiv.org/abs/hep-th/9305014), equations 31-34: implementable CAR currents and trace-class cocycle.
000244 | 
000245 | `check_dirac_lift.py` reuses the existing graph constructor and checks the
000246 | normal-complement identity, full/component cocycles, symbol commutant and
000247 | differential lift exactly. Finite matrices are controls of those analytic
000248 | identities, not a numerical proof of an infinite field theory.