TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/BOUNDED-RELATIVE-TRACE-REVIEW.txt Original file SHA-256: a9759fce892acfb585be64e624a77c7411dc8cc2ff0cfb8e4d20e0e090f7afbf Inclusive original lines: 1-142 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 | BOUNDED DIFFERENCE OPERATORS AND THE LITERAL ORDINARY SPECTRAL TRACE 000002 | 29 October 2026. Independent generalization within the existing source class. 000003 | No new model, profile selection, reflection choice, run or Atlas mutation. 000004 | 000005 | Result and deduplication 000006 | Read the registered existing finding mass-only-deck-relative-heat-action- 000007 | fails-free-infinite-trace and DECK-REFERENCE-AND-ACTION.txt:1-197. Its section4 000008 | proves the free identity contribution and a constant finite matrix repeated 000009 | over all modes. It scopes the combined bounded-D_rel/ordinary-trace contract. 000010 | The short additional result below proves the entire BOUNDED selfadjoint 000011 | relative-operator class, including varying zero-order backgrounds and arbitrary 000012 | infinite-rank admissibility projections that need not commute with them. 000013 | It is a general corollary of that same obstruction, not a new source model. 000014 | 000015 | 1. Exact operator/profile premises 000016 | Let H be the original infinite-dimensional source Hilbert space. Let D_ref 000017 | be selfadjoint elliptic on a compact geometric realization with compact 000018 | resolvent and its actual closed domain. Let V=V* be BOUNDED on H, and P an 000019 | orthogonal projection with infinite-dimensional RanP. Absorb the fixed positive 000020 | chi into D_ref,V. For the original even profile notation this means 000021 | f(D_rel/chi)=F((D_rel/chi)^2), V=D_rel/chi. 000022 | 000023 | Take a nonzero positive Laplace profile 000024 | F(x)=integral_[0,infinity) exp(-s x) dmu(s), mu>=0, 000025 | finite on spec(V^2), so that the ordinary bounded operator F(V^2) is defined. 000026 | For example every finite nonzero positive measure gives such a profile. 000027 | Assume separately that K=F(D_ref^2) is trace class. Compact resolvent alone 000028 | does not imply this for every possible positive Laplace mixture; the 000029 | trace-class condition is an explicit premise, satisfied by ordinary heat 000030 | profiles of a compact elliptic realization. 000031 | 000032 | 2. Whole bounded-operator proof 000033 | Put M=||V||. Since spec(V^2) is contained in[0,M^2], positivity of the measure 000034 | gives monotonicity and the strict finite lower bound 000035 | F(x)>=F(M^2)=c>0, 0<=x<=M^2. 000036 | Strict positivity follows because exp(-s M^2)>0 at every finite s and mu 000037 | is a nonzero positive measure. Functional calculus therefore gives 000038 | F(V^2)>=c I, 000039 | P F(V^2)P>=c P. 000040 | No commutation of P and V, or of P and F(V^2), is used in this inequality. 000041 | 000042 | Choose any infinite orthonormal sequence e_j in RanP. Then 000043 | >=c. 000044 | A compact bounded operator sends such a sequence to norm0. Thus the 000045 | compressed operator is not compact, and its positive ordinary trace on 000046 | RanP is infinite. Since P K P is trace class, hence compact, 000047 | P[F(V^2)-F(D_ref^2)]P 000048 | cannot be compact either: otherwise adding P K P would make P F(V^2)P 000049 | compact. In particular the displayed operator difference is not trace class. 000050 | Combining the terms before tracing therefore does not repair this contract. 000051 | 000052 | The result does not require V to be constant, finite rank, multiplication by 000053 | a scalar, or diagonal in the original Fourier modes. Even an extremely large 000054 | finite bound M only makes c small; it never removes infinitely many diagonal 000055 | contributions of at least c. The retained rank, rather than the geometric 000056 | compactness of the base, is decisive. 000057 | 000058 | If a proposed profile is singular at part of spec(V^2), its actual unbounded 000059 | functional-calculus/form domain must instead be specified. An undefined or 000060 | unbounded term is not a finite ordinary trace. The present bounded-operator 000061 | argument must not silently assume a new extension for such a singular term. 000062 | Whenever that positive form is well defined on the retained sector and has 000063 | the same strict lower bound, it likewise cannot have a finite positive trace. 000064 | 000065 | 3. When the geometric relative operator is actually bounded 000066 | For a LOCAL geometric reflection with its actual bundle/spin lift, preserving 000067 | the common closed domain and the full principal symbol of D, the conjugate 000068 | R D R^-1 is another first-order differential expression with the same 000069 | principal symbol. Their difference is order0. Smooth coefficients on a 000070 | compact base are bounded; its selfadjoint closure/extension supplies precisely 000071 | the V above for D_rel=(D-R D R^-1)/2. This is an explicitly CONDITIONAL 000072 | geometric contract and must be checked for the original lift and domain. 000073 | 000074 | No such conclusion follows just because R is an involution. A Pin lift which 000075 | reverses the principal symbol can leave a first-order D_rel. A nonlocal 000076 | Calderon/spectral reflection is not automatically in this local differential 000077 | argument. Neither case is excluded by this result. No reflection is freely 000078 | constructed from the Cauchy projector or substituted for the physical deck. 000079 | 000080 | 4. Original scope and the distinct pair-function contract 000081 | Original hash-validated passages reread: 000082 | boundary source001:302-316, registered research-source:b54c34c09388e6044a7dd009; 000083 | source003:1-24, registered research-source:28adffa8f2c767cca19b7efa. 000084 | The former permits any DECLARED spectral test profile; the latter explicitly 000085 | allows a specified relative PAIR as a convention for D_rel. Accordingly the 000086 | result excludes only the combined bounded relative operator, positive 000087 | Laplace profile, infinite-rank retained space and literal ordinary-trace 000088 | prescription. It is not an all-profile or all-P1 obstruction. 000089 | 000090 | A genuine relative pair function such as 000091 | F(D^2)-F(D_ref^2) 000092 | has different operator arguments from F((D-D_ref)^2)-F(D_ref^2). Its actual 000093 | trace-class/relative-trace conditions must be proved for the supplied pair. 000094 | This is a distinct possible contract, not an algebraic rewriting or a uniquely 000095 | forced correction of the original printed action. Finite cutoffs, trace 000096 | densities, supertraces or counterterms would also be different actual trace 000097 | prescriptions, none introduced or selected here. 000098 | 000099 | 5. Constructive whole heat reader for the actual bounded spectral pair 000100 | The positive alternative can be proved on exactly the same data. Let 000101 | D=D_ref+V, V=V* bounded, DomD=DomD_ref. 000102 | The already recorded bounded zero-order domain repair in 000103 | e-series-source-reconstruction-20261024/ANALYTIC-DOMAIN.txt and the original 000104 | domain-counterexample finding are reused. The bounded perturbation theorem 000105 | makes D selfadjoint on this common domain; its compact resolvent follows 000106 | from that of D_ref by the resolvent identity. No new field is introduced. 000107 | 000108 | The closed square forms have the SAME form domain DomD_ref and satisfy 000109 | ||D psi||^2>= (1/2)||D_ref psi||^2-||V||^2||psi||^2. 000110 | For example this is immediate from the scalar norm inequality 000111 | ||a+b||^2>=(1/2)||a||^2-||b||^2. The min-max principle therefore gives, with 000112 | eigenvalues of the nonnegative squares listed with multiplicity, 000113 | lambda_k(D^2)>=(1/2)lambda_k(D_ref^2)-||V||^2. 000114 | Consequently for EVERY t>0, 000115 | Tr exp(-t D^2) 000116 | <=exp(t||V||^2) Tr exp(-(t/2)D_ref^2)