TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/SPINE-ORIGINAL-PQ-FOLD-KERNEL-ADDENDUM.txt Original file SHA-256: d98da73f0023e1de696a31874d73b1df547564cec70205c07e5a54f3e0dc0fc5 Inclusive original lines: 1-143 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. 000001 | EXACT SPINE TO EXISTING P/Q FOLD TO ORIGINAL C6 POSITIVE KERNEL 000002 | Independent source continuation, 2026-10-30. Existing reports unchanged. 000003 | 000004 | RESULT 000005 | The four actual spine transports supply the EXISTING P/Q fold around the 000006 | original mark1, with no selected new edge sum or nine-address requirement. 000007 | The original C6 denominator, full positive Gram, inverse square-root 000008 | kernel and any already given structural projector consequently have an 000009 | explicit representation in the actual two endpoint coefficient fibers. 000010 | The already evaluated kappa word is transported through this SAME map. 000011 | This closes a coefficient/source-operator dictionary, not full physical 000012 | P1 closure or an assertion that these coefficient fibers are F5 faces. 000013 | 000014 | 1. ORIGINAL CALLERS AND REUSED RESULTS 000015 | 000016 | EM3, paper-source:8c374b792c3f7ed5c3c4211f, 000017 | archive45 em_flavor_source/sections/003-subsection-discrete-phase-operator- 000018 | and-transport-grammar.tex:1-12,91-125,329-352, SHA256 000019 | 00a2cf1b249b77280dbbf6dfb41c91c77e08aa63e0f30b682d1a8a621f3dd7ec, 000020 | gives U6, D_y=yI-delta U6, the compressed positive Kstr, its stationarity 000021 | functional, and the canonical cycle-holonomy representation rho_F. 000022 | EM4:180-241 calls Hol_G(nabla_F*)Kstr; a holonomy representation/functor 000023 | is not by itself a single graph-wide3x3 matrix, but it provides each 000024 | composable transport needed by an already specified original consumer. 000025 | The existing concrete consumer is the P/Q fold, not an invented average. 000026 | 000027 | tfpt_explorer/flavor_path_transport.py:70-98 constructs P:E_x->E_(1/x), 000028 | Q:E_(1/x)->E_x and Tau=[[0,Q],[P,0]], with based loop QP. 000029 | Lines133-154 give the exact original M0/J/U6 dictionary. The independent 000030 | finite matrices in that function were ALREADY connected to actual G/H 000031 | transport in the source-reviewed G-metric report; no raw Euclidean 000032 | adjoint is used here. Current file SHA256: 000033 | 82b4fab94138ab3f5122cf40d507e7a4b7c8a5da25393d4f42dcd35259a9de92. 000034 | This code/source implementation status is distinct from a new P1 proof. 000035 | 000036 | Use the already exact diagonal paths T_k, metrics H0/Hinf and 000037 | R=H0^(1/2) from ACTUAL-DIAGONAL-SPINE-ENDPOINT-TRANSPORT.txt. Its corrected 000038 | raw-versus-normalized convention is retained throughout this addendum. 000039 | 000040 | 2. THE ORIGINAL MARK1 FOLD, NOW ON THE ACTUAL WHOLE SPINE 000041 | 000042 | Set P=T3:E0->Einf and Q=T0^-1:Einf->E0. These are the lower outward 000043 | and upper return classes surrounding mark1. The existing reciprocal 000044 | half-arcs at x in(0,1), joined through the mark-free inner and outer 000045 | disks, have the same relative classes. There is no new loop-word choice. 000046 | The endpoint regularization is the given z^-diag(2,1,1), not a limit in 000047 | an unrenormalized Euclidean frame. Exactly, 000048 | R QP R^-1=M0, 000049 | P^* Hinf P=H0, Q^* H0 Q=Hinf. 000050 | 000051 | On E0 directsum Einf put Hpair=diag(H0,Hinf) and 000052 | Tau=[[0,Q],[P,0]], 000053 | Cpair=diag(R^-1,P R^-1), 000054 | Cpair^-1 Tau Cpair=[[0,M0],[I,0]]=:Tau_b. 000055 | Then Cpair^* Hpair Cpair=I and Tau^sharp=Tau^-1, where 000056 | A^sharp=Hpair^-1 A^* Hpair. 000057 | The full actual endpoint fold is therefore unitary in the ORIGINAL 000058 | source metric. This statement neither fits a new metric nor identifies 000059 | the two coefficient fibers with the two physical source faces. 000060 | 000061 | With the already original 000062 | J=2^-1/2 [[I,I],[M0,-M0]], Efold=Cpair J, 000063 | we obtain the actual operator equality 000064 | Efold^-1 Tau^-1 Efold=diag(M0,-M0)=U6, 000065 | Efold^* Hpair Efold=I. 000066 | The inverse is essential. Equating Tau itself with U6 would change the 000067 | original orientation. Tau has order6 and its third power transports to 000068 | diag(I,-I), the same algebraic branch parity as the existing dictionary. 000069 | All other marks are obtained with their already proved endpoint quarters; 000070 | this is not a new demand that any6-dimensional pair be physical F5 faces. 000071 | 000072 | 3. FULL ORIGINAL GRAM, KERNEL AND STRUCTURAL COMPRESSION 000073 | 000074 | For real y,delta and epsilon>=0 define in the endpoint fibers 000075 | Dcal_y=yI-delta Tau^-1. 000076 | The original positive Gram is exactly 000077 | Dcal_y^sharp Dcal_y+epsilon^2 I 000078 | =(y^2+delta^2+epsilon^2)I-y delta(Tau+Tau^-1). 000079 | Efold conjugates it to the original D_y^*D_y+epsilon^2 I. Hence the 000080 | positive spectral functional calculus, including inverse square roots 000081 | where the original gap gives invertibility, is transported without 000082 | choosing any new branch, covariance, field state or integration weight. 000083 | The determinant is exactly y^6-delta^6 before Gram formation, so the 000084 | original real stationarity functional is retained as well. 000085 | 000086 | In the original given positive cubic branch the resulting3x3 operator is 000087 | K_y,+=[(y^2+delta^2+epsilon^2)I-y delta(M0+M0^*)]^-1/2. 000088 | This is an executed ORIGINAL consumer of the based spine word. It is 000089 | not a claim that the bare notation Hol_G alone specifies a single matrix. 000090 | For the full structural projector retain precisely the original Pstr: 000091 | Pcal_str=Efold Pstr Efold^-1, 000092 | Kcal_str=Efold (Pstr D_f^*D_f Pstr)^-1/2 Efold^-1 000093 | on the transported range. No replacement of PTheta, Psing or their 000094 | actual given realization by the marked family line P0 is made. The same 000095 | isometry transports the original Vadm and its stated gap/norm contract. 000096 | 000097 | 4. THE ACTUAL HARDY WORD, NOT A NEW SOURCE CALL 000098 | 000099 | In the already common marked frame W=kappa M0 P0, P0=(I+U^2)/2. 000100 | For the DECLARED COMPARISON EMBEDDING W6=diag(W,0) in the existing 000101 | positive/negative cubic branches, its endpoint operator is Efold W6 000102 | Efold^-1. Before Cpair this is exactly 000103 | J W6 J^* =1/2 [[W,W M0^*],[M0 W,M0 W M0^*]]. 000104 | This calculation does NOT derive that comparison embedding from the F5 000105 | Hardy source and does NOT select zero physical contributions in the other 000106 | branch. The original physical branch insertion is not asserted. The SAME 000107 | Efold transports ANY already given whole6-dimensional operator, including 000108 | its nonzero other-branch and off-diagonal legs. Thus the evaluated word's 000109 | endpoint and adjoint factors can be carried exactly when the original 000110 | consumer supplies that whole insertion. The opposite/adjoint legs of the 000111 | established whole Clifford/KO construction must be transported as well, 000112 | not frozen in an old basis. This finite coefficient frame does not replace 000113 | the already given full G/W field, state, reflection and time map. 000114 | 000115 | The original real delta jet remains partial_delta D_y,+=-M0. It reads 000116 | the same open word after P0, as already proved. The projected complex 000117 | kappa leg alone is not the original full real delta variation. Its use 000118 | inside Zadm/Ftr must preserve both C6 branches, adjoints and the actual 000119 | original insertion rule. The current result does not invent that rule. 000120 | 000121 | 5. H1 QUARTER TWIST WAS ALREADY SOLVED; REUSE IT 000122 | 000123 | The natural reduced puncture H1 rotation has characters(i,-1,-i), whereas 000124 | the Fuchs U has(1,i,-i). Their unmodified linear actions differ. This is 000125 | already explicit in v137 and v141, rather than a new obstruction: 000126 | v141:1-56, source:6eac239d7ae516a10b06, SHA256 000127 | 8c29d8bdc2b0e8c6953343ab41b9d3311cb35145318cda013df7b0cacf0721a0, 000128 | selects the sheet-twisted boundary class -U=(chars2,3,1) from the original 000129 | integer deck, retaining the stated GATE.QGEO realization premise. 000130 | v146:15-40,70-115 checks the natural full Mobius H1 action and its two 000131 | integer reflection classes. These are reused with their original scope; 000132 | no fresh spin selection or full source identification is promoted here. 000133 | The central sign also cancels from conjugation of the already proven 000134 | meridian and kernel endomorphisms. It supplies no new Gamma-table gate. 000135 | 000136 | EXACT REPLAY 000137 | Workspace outputs/multi-model-source-search-20261029/ 000138 | SPINE-ORIGINAL-PQ-FOLD-KERNEL-CHECK.json, 000139 | SHA256 5e134e239393c91ef3f8f8621269c050eca07e9c961d383ef67e8c3e433f3a66. 000140 | Nine exact identity groups passed on the already actual endpoint matrices. 000141 | Direct structural comparisons of factored expressions were replaced by 000142 | simplification of differences; their initial scratch False was not a 000143 | nonzero mathematical residual. No full verification suite was rerun.