TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/C58-THOULESS-AND-CAP-REVIEW.txt Original file SHA-256: 8539b8b4a901ae9cac751f299a2753ee06f75c25f95cdb74dc4a17b3b5f2a036 Inclusive original lines: 1-200 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 | C58: SHALE EQUIVALENCE, THOULESS PATCHES AND THE ACTUAL WEAK CAP 000002 | Independent targeted review, 30 October 2026. 000003 | Verdict: SAME_FOCK_VECTOR_SCOPED_TRUE; ORIGINAL_POSITIVE_TIME_OS_IDENTIFICATION_AND_UNIQUE_THOULESS_CLAIMS_FALSE. 000004 | No Atlas or original edit. 000005 | 000006 | 1. What Hilbert-Schmidt equivalence actually proves 000007 | For two pure CAR polarizations P0 and Ph, the Shale-Stinespring/Araki 000008 | criterion 000009 | 000010 | P0-Ph is Hilbert-Schmidt (1) 000011 | 000012 | puts their quasifree representations in the same Fock equivalence class. In a 000013 | fixed irreducible common representation the P0 quasifree vacuum ray therefore 000014 | exists and is unique up to phase. This is the precise positive part of C58. 000015 | 000016 | Equation(1) does not by itself identify a single Gaussian boundary insertion 000017 | F_cap in the original positive-time OS algebra. The OS Hilbert space is a 000018 | completion of equivalence classes of positive-time admissible observables. A 000019 | vector in that completion need not be represented by one bounded, local or 000020 | polynomial observable in the original OS algebra. Nor is a Gaussian 000021 | implementer automatically localized on the declared boundary interval or 000022 | preserved by P_adm. 000023 | 000024 | Reflection positivity says, once an admissible F has been supplied, 000025 | 000026 | ([F],T_ell[F])_OS>=0 000027 | 000028 | It does not establish the positive-time membership of a proposed insertion. 000029 | An original identification still needs an explicit map from the common CAR 000030 | Fock vector to the QFT8 positive-time admissible algebra. This is the precise 000031 | gap in C58's Gaussian boundary-insertion claim; no operator-positivity claim is 000032 | attributed to C58 here. 000033 | 000034 | 2. Smallest counterexample to the universal Thouless claim 000035 | Take a two-dimensional one-particle space and 000036 | 000037 | Ph=|e1>Ran Ph is invertible, the vacuum overlap is 000056 | nonzero, and a Hilbert-Schmidt graph kernel Z gives (2). 000057 | * Zero-overlap but index zero: extract finitely many orthogonal 000058 | particle-hole creators, then use a Thouless exponential on the transverse 000059 | complement. 000060 | * Nonzero relative index: additional unpaired finite creators or holes carry 000061 | the relative charge; a pair-only exponential cannot represent the state. 000062 | 000063 | The P0 vacuum ray is unique up to phase once the representation and zero-mode 000064 | occupation are fixed. The implementing unitary and an OS observable 000065 | representative are not unique: stabilizer factors, OS null vectors and dense 000066 | approximations remain. 000067 | 000068 | 3. Conditions for the actual weak family 000069 | For the already given constant finite coefficient and weak direction set 000070 | 000071 | h_p(t)=(pI+tQ)gamma_F-i gamma_F D_F, 000072 | p in Z+alpha, alpha=1/4 or3/4, 000073 | P0,p=1_(-infinity,0)(p gamma_F), 000074 | Ph,p(t)=1_(-infinity,0)(h_p(t)). (3) 000075 | 000076 | Here Q and D_F are fixed finite matrices. Write h_p(t)=p gamma_F+B(t), with 000077 | B(t) bounded independently of p. Since Q and D_F are constant along the 000078 | circle, h remains diagonal in the Fourier label p. What is lost for generic Q 000079 | is only the older reduction to independent anticommuting singular-value pairs, 000080 | not the mode-by-mode decomposition. For fixed finite t and sufficiently large 000081 | |p|, the gap of p gamma_F dominates B(t). Standard resolvent estimates give 000082 | 000083 | ||Ph,p(t)-P0,p||_HS=O(1/|p|). 000084 | 000085 | Since the internal dimension is finite and sum_p |p|^-2 converges, the direct 000086 | sum difference is Hilbert-Schmidt. This reuses the previously recorded 000087 | one-dimensional whole-sea mechanism; it is not a new source theorem. 000088 | 000089 | Only finitely many p can cross zero at fixed bounded t. Away from the critical 000090 | values satisfying 000091 | 000092 | 0 in spec h_p(t), (4) 000093 | 000094 | Ph(t) is continuous, its relative Fredholm index is constant, and the old sea 000095 | is a well-defined ray in the common Fock representation. Starting in the 000096 | known no-crossing scalar-quarter component gives index zero until the first 000097 | crossing. 000098 | 000099 | At a crossing, the coupled ground space is degenerate. The negative projection 000100 | alone does not fix the coupled vacuum ray or its relative charge. A spectral 000101 | section or the original modified-APS Lagrangian is needed only for that vacuum 000102 | and Thouless-chart description. The cap formula below is independent of the 000103 | occupation of ker h: both dGamma(h) and E0=Tr(h 1_(h<0)) are unchanged there. 000104 | Crossing may change the relative charge by a finite integer. This is a finite 000105 | event, not failure of Hilbert-Schmidt equivalence. 000106 | 000107 | Even without a zero-energy crossing, a principal angle can reach pi/2 and make 000108 | the old/new vacuum overlap zero. The big-cell Thouless formula then fails while 000109 | the Fock vector and finite-time cap remain. The precise nonzero-overlap 000110 | condition is 000111 | 000112 | ker(Ph P0|Ran P0)=0 000113 | 000114 | together with relative index zero. Under (1), absence of exact orthogonal 000115 | angles makes the corresponding Fredholm overlap product positive and supplies 000116 | the Hilbert-Schmidt graph patch. 000117 | 000118 | 4. The general cap survives every Thouless patch 000119 | For a finite set of modes, define Ph=1_(h<0) and the coupled ground energy 000120 | 000121 | E0=Tr(h Ph). 000122 | 000123 | For the fixed old Slater vector Omega_P0, exterior functoriality gives 000124 | 000125 | Z_cap(P0,h;ell) 000126 | = 000127 | =exp(ell Tr(hPh)) 000128 | det_RanP0(P0 exp(-ell h) P0). (5) 000129 | 000130 | Formula(5) remains valid when the ground-state overlap is zero. At finite 000131 | ell, exp(-ell h) is strictly positive and its compression to Ran P0 is 000132 | positive definite. A zero overlap only changes the large-ell leading sector. 000133 | 000134 | For the whole circle, the ground-energy exponential and determinant must be 000135 | combined before the limit. They need not exist as two separate absolute 000136 | factors. The previously proved fixed-cap Fredholm limit supplies this 000137 | combination in its scalar-quarter scope. For the weak family, (1), the fixed 000138 | old P0 and coupled normal ordering give more than a merely formal candidate. 000139 | Let 000140 | 000141 | Z_p(ell)=. 000142 | 000143 | Every finite-dimensional H_exc,p is nonnegative, hence 02||B(t)||, the projection estimate gives 000145 | 000146 | ||Ph,p-P0,p||_HS^2=O(p^-2). 000147 | 000148 | More explicitly, put g=|p|, v=||B(t)|| and let r=48 be the rank of the 000149 | opposite unperturbed chirality in the actual 96-dimensional block. For p>0, a 000150 | negative normalized eigenvector u of h_p with eigenvalue lambda has wrong 000151 | chirality component 000152 | 000153 | ||P_plus u|| <= v/(g-lambda) <= v/(2g-v). 000154 | 000155 | For p<0 the same estimate holds with the two chiralities exchanged. 000156 | 000157 | Consequently 000158 | 000159 | Tr(Ph,p P_plus) <= r v^2/(2g-v)^2, (6a) 000160 | 000161 | and the elementary product bound for the principal angles gives the same 000162 | upper bound for one minus the squared many-body ground overlap. 000163 | 000164 | The squared ground overlap is therefore 1-O(p^-2). Resolving Omega_P0,p into 000165 | the coupled ground component and its excited components yields uniformly in 000166 | ell>=0 000167 | 000168 | 0<=1-Z_p(ell) 000169 | <=1-||^2 000170 | =O(p^-2). (6) 000171 | 000172 | Thus sum_p(1-Z_p)0. Any zero overlap or 000174 | relative-charge change occurs in only finitely many low modes and contributes 000175 | a finite positive factor at finite ell. A zero mode does not require an 000176 | occupation choice for Z_p itself. This proves the 000177 | weak-family cap in the already given coupled-normal-energy convention without 000178 | using the old scalar C_p shortcut. 000179 | 000180 | This does not yet identify that product with the original relative partition 000181 | functional. If the original denominator subtracts the free reference energy 000182 | instead of the coupled ground energy, the result contains the additional 000183 | factor exp[-ell(E_h-E_ref)] with its prescribed regularization. Shale 000184 | equivalence does not choose or cancel that factor. The original reference and 000185 | domain caller therefore remain necessary for a Z_rel identity. 000186 | 000187 | The fixed-domain theorem in the original primitive source does provide a real 000188 | mathematical distinction: a bounded relative perturbation is realized on the 000189 | same modified-APS domain. In that family P0 remains the declared old boundary 000190 | datum and (5) is nontrivial. A moving choice P0=Ph makes the cap vacuum 000191 | trivial. This choice determines which polarizations are compared. It still 000192 | does not by itself prove that their Fock vector is one original QFT8 observable. 000193 | 000194 | 5. Constructive conclusion 000195 | For the actual weak h_p(t), no new source state needs to be selected. Use the 000196 | already fixed P0 and evaluate (5) on intervals of t between the finitely many 000197 | zero crossings. In a big-cell interval one may use the normalized Thouless 000198 | graph for efficient Wick calculations. At an overlap zero, switch charts by 000199 | extracting the finite particle-hole factor. At an h zero mode, retain the 000200 | original modified-APS zero-mode choice only when a definite coupled vacuum ray