TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-FIVE-INPUT-STATE-AND-VACUUM-CLUSTER-20261102.txt Original file SHA-256: daa072468c8e6be3b35285710869833a88e50b35edecbe16792aea9ad28a7dc4 Inclusive original lines: 1-95 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 | R094: ORIGINAL FIVE INPUTS, SOURCE VACUUM AND EUCLIDEAN CLUSTERING 000002 | 2 November 2026 000003 | 000004 | The original quarter CAR source has vacuum Omega_alpha, K_alpha >= 0, 000005 | K_alpha Omega_alpha=0, and a positive gap. Its specified bounded-even 000006 | history construction gives vacuum reflection positivity and Euclidean time 000007 | clustering; see native-octic-seam-origin-20261025/ 000008 | P1-SOURCE-EMBEDDING-AUDIT.txt sections 2--4. P1 itself asks for clustering 000009 | on its admissible local algebra, without specifying the clustering direction 000010 | at the cited seed statement. The CAR K_alpha is not yet identified with the 000011 | actual P1 OS time, nor the eight-arc algebra with the P1 local algebra. 000012 | 000013 | 1. The original five inputs are excitations, not the source vacuum 000014 | 000015 | In joint-native-algebra-selection-20261016/ 000016 | ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt sections 2--4, each 000017 | of the five ACTUAL source inputs P_i^alpha has Q=0 and cover energy 32. 000018 | The original K_alpha=(L0-ground)/8 therefore gives K_alpha P_i=4 P_i. 000019 | Their original finite Schur expressions have 694,710,711,713,688 terms; 000020 | write the corresponding finite raising-current polynomial B_i on the 000021 | finite-energy domain. Then P_i=B_i Omega_alpha. This is a state/current 000022 | intertwiner, not a new assertion that B_i is a local primary field. 000023 | For the actual R093 line X=T(t)T*, the full already-derived result is 000024 | = . 000025 | This preserves the original vacuum and both original quarter coefficients; 000026 | it does not identify either with a P1-selected preparation or insertion. 000027 | 000028 | 2. A decisive source test for all five-dimensional inputs 000029 | 000030 | For both alpha=1/4 and 3/4, the original saved CAR occupation Gram 000031 | G_ij= is Hermitian with all five leading principal 000032 | minors strictly positive. Exact determinant: 000033 | det G=1607041858632363297/1073741824 > 0. 000034 | Thus J8 psi != 0 for every nonzero psi in span(P_0,...,P_4). The basis is 000035 | not orthonormal: normalized bounds require the input Gram B and the 000036 | generalized eigenvalues of (G,B), not eigenvalues of G alone. 000037 | 000038 | Take a real smooth quotient profile b, supported in an arbitrary nonempty 000039 | proper quotient interval, with b_1 != 0; such a bump exists. Its ACTUAL 000040 | cover profile b_F(theta)=b(8theta) has support in the corresponding eight 000041 | disjoint cover arcs. Let W_epsilon=exp(i epsilon J(b_F)). This is a bounded 000042 | even Weyl operator on that eight-arc source. On finite-energy vectors 000043 | the strongly differentiable Weyl action gives, with Pi_3 the K=3 projector, 000044 | Pi_3 W_epsilon psi = i epsilon conjugate(b_1) J8 psi + o(epsilon). 000045 | The other Fourier modes have different energy shifts. For every nonzero 000046 | psi in this five-space, the right side is nonzero for all sufficiently 000047 | small nonzero epsilon (the allowed epsilon can depend on psi). 000048 | 000049 | For normalized psi define the particular stationary excited-state response 000050 | C_psi(s)= 000051 | - ||^2, s>=0. 000052 | The middle operator is bounded for each s since K_alpha>=0. Its spectral 000053 | measure is nonnegative and contains the strictly positive contribution 000054 | exp(s)||Pi_3 W_epsilon psi||^2; the subtraction is constant in s. 000055 | Hence C_psi(s) diverges as s goes to infinity and is not time-clustering. 000056 | The same reasoning applies to any positive normalized density matrix 000057 | supported entirely in the five-space: trace(rho J_-8 J8)>0 and, for each 000058 | fixed rho, the energy-3 weight is positive for sufficiently small epsilon. 000059 | 000060 | For the first named input the original J32 response reaches the unique 000061 | charge-zero vacuum: J32 P0 is a scalar multiple of Omega_alpha. The 000062 | coefficient of exp(4s) for the same W_epsilon has expansion 000063 | epsilon^2 |b_4|^2 R32^alpha+o(epsilon^2), provided b_4 != 0, with 000064 | R32^(1/4)=1142672/328635469, 000065 | R32^(3/4)=1560464/331641421. 000066 | The normalized numbers use the actual Hall input norm, not a different 000067 | scaled Gram. The original current and the whole finite-energy state are 000068 | not replaced by a one-oscillator source. 000069 | 000070 | 3. Scope and consequence for the source bridge 000071 | 000072 | The result rules out treating any pure state or positive mixture solely 000073 | inside this five-input space as the clustering vacuum in THIS fixed CAR 000074 | time test, whenever this eight-arc Weyl word is admitted. The CAR vacuum 000075 | itself is not excluded, and the inputs remain valid excited insertions. 000076 | Other time maps, a physical algebra that does not contain the word, mixtures 000077 | with other supports, a finite-beta KMS trace, and other P1 realizations are 000078 | not ruled out. In particular the s->infinity statement is not applied to 000079 | the finite thermal interval at fixed beta. Reflection positivity, eta, 000080 | the full interacting Higgs source, E8 currents and P1 source selection 000081 | are not derived by this comparison. 000082 | 000083 | Next: use the unique original vacuum for the underlying source and retain 000084 | all five B_i insertions, both alpha states, the full dynamical source and 000085 | Z0/reference/Eta. Compare the actual P1 source/state, reflection, collar, 000086 | time and local-field map with these same vacuum-plus-insertion words; 000087 | do not replace them by the excited-state clustering test or assume that 000088 | the quotient localization was physically selected. 000089 | 000090 | Reproduction: python3 -B -OO aps_source_review/ 000091 | check_original_excitation_cluster_20261102.py from the unified source 000092 | contract directory, or run the full repository-relative path from repo root. 000093 | The exact checker verifies the stored input degrees, positive Gram minors, 000094 | determinants and R32 ratios. The spectral proof is the argument above; a 000095 | matrix computation alone is not a proof of the physical source embedding.