TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ACTUAL-SPINE-FOURIER-AND-ABSORPTION-REVIEW.txt
Original file SHA-256: 3b69102fad712ea9dcacdd39e941eafc4542d1811b0539138f9bb324e8a0b7be
Inclusive original lines: 1-83
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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000004 | Result
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000014 | 
000015 | 1. Actual input, not a new model
000016 | 
000017 | Reuse the original diagonal paths and matching endpoint frames from
000018 | ACTUAL-DIAGONAL-SPINE-ENDPOINT-TRANSPORT.txt. In its source-normalized frame,
000019 |  T_k=V^k T0 U^(-k), U=diag(1,i,-i), V=diag(-1,1,-1).
000020 | The target inner product is Hinf from the original algebraicG frame.
000021 | Write M_k=U^k M0 U^(-k), P0=diag(1,0,0).
000022 | The exact already proven loop identity gives
000023 |  T_k M_k=T_(k-1), indices modulo4.
000024 | Therefore the actual covariantly rotated word sum obeys
000025 |  sum_k T_k M_k P0=sum_k T_k=S,
000026 |  S=-6sqrt30/5 E10, S*Hinf S=8P0.
000027 | In a target orthonormal frame S=-2sqrt2 E10. This is the exact absorption
000028 | identity: inserting the adjacent loop before its corresponding open edge
000029 | just renames which edge is followed. It does not eliminate the loop from
000030 | all other evaluations of the same whole transport functor.
000031 | 
000032 | 2. All four original characters remain
000033 | 
000034 | Define the complete Fourier decomposition of the GIVEN four-edge tuple,
000035 |  F_m=(1/4)sum_k i^(-mk)T_k, m=0,1,2,3.
000036 | Exact algebra yields
000037 |  V F_m U^-1=i^m F_m,
000038 |  ranks(F_0,F_1,F_2,F_3)=(1,2,1,2),
000039 |  Tr(F_m*Hinf F_m)=(1/2,1,1/2,1).
000040 | All four channels are nonzero. They are orthogonal in the source-metric
000041 | Hilbert-Schmidt product, and their squared norms sum to3, the full
000042 | three-dimensional transport norm. The inverse Fourier formula exactly
000043 | recovers ALL FOUR actual T_k. Their adjacent return products then recover
000044 | all four original rank3 cubic M_k.
000045 | 
000046 | On the bare single Hom(E0,Einf), the quarter intertwiner space has dimension1:
000047 | V and U share only the eigenvalue1, at targetrow1/sourcecolumn0.
000048 | Thus an invariant scalar-weight LINEAR edge average indeed has rank1.
000049 | Calling every covariant whole-spine evaluation that linear compression
000050 | would discard the actual nontrivial edge-character channels by assumption.
000051 | The labeled path tuple is already covariant without averaging it away.
000052 | The original P/Q fold is an explicit existing consumer of its products.
000053 | It yields U6, full positiveGram and Kstr as previously proved, not onlyS.
000054 | No claim that all conceivable physical consumers are allowed follows.
000055 | The original field rule still decides which original consumer is used.
000056 | 
000057 | 3. Source selection and ordinary holonomy tangents are separate
000058 | 
000059 | The original EM5 perturbation slot is a class of gap-stable Pstr-preserving
000060 | operators. Membership in a bounded class does not select an insertion.
000061 | The direction kappa M0P0 is not automatically a tangent of the SU3
000062 | holonomy class either. If B=kappa M0P0, exact algebra gives
000063 |  M0*B+B*M0=2Re(kappa)P0,
000064 |  Tr(M0^-1 B)=kappa.
000065 | For nonzero kappa it fails the traceless SU3 tangent condition; the
000066 | actual Hardy phase also has nonzero Re(kappa), so fails unitary tangency.
000067 | This is ONLY a test of identifying B as a raw monodromy variation. It is
000068 | no obstruction to a distinct original bounded additive V_adm or a charged
000069 | field/wedge response. Such uses require their own actual original rule.
000070 | No compensating identity term, projected tangent or new source is chosen.
000071 | 
000072 | 4. Reproducible finite check
000073 | 
000074 | check_actual_spine_fourier_consumers.py uses the immutable full exact
000075 | endpoint JSON, validates its SHA256, and verifies absorption, all four
000076 | character projections, orthogonality/reconstruction, original loop ranks,
000077 | and both tangent identities. All checks use exact algebra. The output is
000078 | actual-spine-fourier-consumers-results.json.
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