Established in scope
Whole bounded-difference class and finite genuine pair-heat reader
In the question: Automated construction of operator maps and sources
No all-profile/all-P1 exclusion. Newer originals allow any declared f and separately a relative pair. Pair reader is not algebraic rewriting of F((D-Dref)²),original selection of profile/normalization or physical completion. Singular profiles require actual functional calculus and integrable heat mixture. Local symbol-preserving reflection may yield bounded V; other Pin/anticommuting or nonlocal reflections are outside inference. Exponential bound compares eigenvalues/traces,not Loewner exponentials. Existing coupled sources and Seed time retained.
Read the exact result, assumptions and evidence
Result
For every bounded selfadjoint V and specified nonzero positive Laplace contract,F(V²)>=F(||V||²)I=cI,c>0. Thus PF(V²)P is noncompact on every infinite-rank retained subspace; no P/V commutation needed. Subtracting trace-class reference does not repair it. This extends prior free/constant counterexample to entire bounded class. Constructively,same supplied operators admit genuine pair D=Dref+V: common square-form domain gives D²>=.5Dref²-||V||² as form bound; minmax yields Tr exp(-tD²)<=exp(t||V||²)Tr exp(-(t/2)Dref²)<∞. Whole pair heat exp(-tD²)-exp(-tDref²) is trace class for every t>0,without heat factorization or fixed square domain.
Required assumptions
- Bounded relative operator,positive Laplace profile and compact source pair
Dref is actually selfadjoint on its fixed compact elliptic source domain,V=V* bounded and P any infinite-rank orthoprojection. Negative trace theorem assumes nonzero positive Laplace F(x)=∫e^(-sx)dμ,F(V²) a defined bounded operator and F(Dref²) trace class. Positive pair-heat theorem uses D=Dref+V on same domain and trace-class reference heat for every positive t. Whether actual original Drel is bounded zero order must be checked against full actual reflection lift and principal symbol; not generally assumed.
- Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Original evidence
- BOUNDED-RELATIVE-TRACE-REVIEW.txt lines 1–142
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/BOUNDED-RELATIVE-TRACE-REVIEW.txt
a9759fce892acfb585be64e624a77c7411dc8cc2ff0cfb8e4d20e0e090f7afbf
- DECK-REFERENCE-AND-ACTION.txt lines 1–197
Source checked at this snapshot
Original location and fingerprint
experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DECK-REFERENCE-AND-ACTION.txt
d392cfc2c0b3b50d39a041c6c1ee8bd3a1e173f80e767f35e05ba82344eaf0f1
- 001-section-primitive-core-and-carrier-interface-handoff.tex lines 302–316
Source checked at this snapshot
Original location and fingerprint
documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/001-section-primitive-core-and-carrier-interface-handoff.tex
7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79
- 003-subsection-relative-objects.tex lines 1–24
Source checked at this snapshot
Original location and fingerprint
documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/003-subsection-relative-objects.tex
d1136b33055198aedd6d1322912672977374f13f30e65dd073312ed272df0858
finding:bounded-relative-trace-obstruction-and-whole-genuine-pair-heat-reader