{"schemaVersion":1,"publication":{"id":"cscf","title":"The Canonical Spectral-Curvature Field: A Unified Operator Origin for Coherence, Dissipation, and Spectral Budgets","subtitle":"A Unified Operator Origin for Coherence, Dissipation, and Spectral Budgets","description":"A positive selfadjoint carrier generates coherent evolution and dissipative relaxation through its functional calculus. The modular spectral measure identifies precisely what the ordered ratio readout retains.","doi":"10.5281/zenodo.23899458","date":"2026-11-03","url":"/source-readout/cscf","pdfUrl":"/publications/spectral/cscf/paper.pdf","markdownUrl":"/publications/spectral/cscf/complete.md","manifestUrl":"/source-readout/cscf/manifest.json","paperSlug":"cscf","headline":"One carrier. 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The Canonical Spectral-Curvature Generator $K$"},{"id":"section-4.1","title":"4.1 Admissible monotone curvature systems"},{"id":"section-4.2","title":"4.2 The canonical generator"},{"id":"section-4.3","title":"4.3 Functional dependence and base data"}],"anchors":["sec-k","section-4.1","def-admmono","section-4.2","thm-k","section-4.3","lemma-4.3","lem-directsum"],"words":417},{"paperId":"cscf","slug":"04-lambda-face","title":"The dissipative face and exact recovery","description":"Contractions, Fejér transmission, exact carrier recovery, spectral budgets and Lambda profiles.","order":4,"url":"/source-readout/cscf/04-lambda-face","sourcePath":"content/paper-integrations/cscf/04-lambda-face.mdx","sections":[{"id":"sec-diss","title":"5. The Dissipative / Lambda Face"},{"id":"section-5.1","title":"5.1 Contraction semigroup and Fejér transmission"},{"id":"section-5.2","title":"5.2 Exact recovery of the generator from the Lambda face"},{"id":"section-5.3","title":"5.3 The dissipative budget"},{"id":"section-5.4","title":"5.4 Asymptotic expansions"},{"id":"section-5.5","title":"5.5 Structural properties"},{"id":"section-5.6","title":"5.6 The lambda profile"},{"id":"sec-ldist","title":"5.7 Empirical spectral measures and Lambda distances"}],"anchors":["sec-diss","section-5.1","definition-5.1","def-lambda","prop-lambda","section-5.2","thm-lambda-exact-generator-recovery","cor-coherence-from-lambda","section-5.3","def-dbudget","lem-traceclass-lambda-det","section-5.4","prop-smallt","prop-larget","section-5.5","prop-dprops","section-5.6","sec-ldist","thm-profile"],"words":1807},{"paperId":"cscf","slug":"05-coherent-face","title":"The coherent face and its budget","description":"Reversible flow, resonant compression and the stability of declared coherence symbols.","order":5,"url":"/source-readout/cscf/05-coherent-face","sourcePath":"content/paper-integrations/cscf/05-coherent-face.mdx","sections":[{"id":"sec-coh","title":"6. The Coherent / Reversible Face"},{"id":"section-6.1","title":"6.1 Admissible coherence symbols"},{"id":"section-6.2","title":"6.2 The coherence generator and unitary flow"},{"id":"section-6.3","title":"6.3 The reversible Fejér compression"},{"id":"section-6.4","title":"6.4 The coherence budget"},{"id":"sec-symbol-stability","title":"6.5 Symbol-class stability"},{"id":"section-6.6","title":"6.6 Properties shared by the two faces"}],"anchors":["sec-coh","section-6.1","def-hadm","ex-default","section-6.2","def-c","def-ut","proposition-6.5","section-6.3","def-gamma","def-ct","section-6.4","def-bbudget","rem-resonant","prop-bprops","sec-symbol-stability","prop-symbol-stability","lem-finite-rank-budget-stability","cor-small-symbol-perturbation","remark-6.14","section-6.6","lem-commute"],"words":1320},{"paperId":"cscf","slug":"06-unified-field","title":"The unified field theorem","description":"The joint operator calculus and recovery of spectral algebras from readouts.","order":6,"url":"/source-readout/cscf/06-unified-field","sourcePath":"content/paper-integrations/cscf/06-unified-field.mdx","sections":[{"id":"sec-unified","title":"7. The Unified Field Theorem"},{"id":"sec-algebra-recovery","title":"7.1 Recovery of the spectral algebra from readouts"},{"id":"section-7.2","title":"7.2 The common spectral calculus"}],"anchors":["sec-unified","def-cscf","thm-unified","cor-combined","sec-algebra-recovery","thm-spectral-algebra-recovery","cor-cscf-algebra-recovery","section-7.2"],"words":1132},{"paperId":"cscf","slug":"07-classification","title":"Classification under explicit axioms","description":"The dissipative uniqueness result and the limits of reversible symbol selection.","order":7,"url":"/source-readout/cscf/07-classification","sourcePath":"content/paper-integrations/cscf/07-classification.mdx","sections":[{"id":"sec-class","title":"8. Classification and Uniqueness"},{"id":"section-8.1","title":"8.1 Dissipative classification"},{"id":"section-8.2","title":"8.2 Coherent classification"},{"id":"section-8.3","title":"8.3 Unified classification"}],"anchors":["sec-class","section-8.1","ax-diss","thm-dclass","section-8.2","ax-coh","thm-cclass","corollary-8.5","section-8.3","thm-unifclass"],"words":762},{"paperId":"cscf","slug":"08-modular-ratios","title":"The modular carrier of spectral ratios","description":"The exact finite-window observable, conjugation flow and what its transformations retain.","order":8,"url":"/source-readout/cscf/08-modular-ratios","sourcePath":"content/paper-integrations/cscf/08-modular-ratios.mdx","sections":[{"id":"sec-ratios","title":"9. Spectral ratios as coherent and modular readouts"},{"id":"section-9.1","title":"9.1 The exact operator representation"},{"id":"section-9.2","title":"9.2 Phase flow and the form factor"},{"id":"section-9.3","title":"9.3 Connection to the CSCF coherence symbol"},{"id":"section-9.4","title":"9.4 What the transformations preserve"}],"anchors":["sec-ratios","eq-v","section-9.1","thm-modular","eq-matrixunits","eq-eta","eq-nu","cor-windows","eq-native","section-9.2","prop-formfactor","eq-conjugation","eq-characteristic","section-9.3","eq-cohrecovery","eq-adapted","section-9.4"],"words":1060},{"paperId":"cscf","slug":"09-spectral-evidence","title":"What the measured profiles show","description":"Hydrogen, Kerr and image-graph recurrence with its conditional controls and diagnostic limits.","order":9,"url":"/source-readout/cscf/09-spectral-evidence","sourcePath":"content/paper-integrations/cscf/09-spectral-evidence.mdx","sections":[{"id":"sec-empirical","title":"10. 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Physical realizations, reciprocal memory and response"},{"id":"section-12.1","title":"12.1 Mobility converts curvature into a relaxation carrier"},{"id":"section-12.2","title":"12.2 Reciprocal storage, the full Hessian and exact memory"},{"id":"section-12.3","title":"12.3 A positive carrier with hidden cross-response"},{"id":"section-12.4","title":"12.4 Detailed balance and stochastic-matrix-form carriers"},{"id":"section-12.5","title":"12.5 An aligned complex-gradient realization"},{"id":"section-12.6","title":"12.6 Quantum and experimental response anchors"}],"anchors":["sec-physical-realizations","section-12.1","eq-physical-gradient","lem-mobility-whitened","eq-mobility-carrier","section-12.2","eq-reciprocal-potential","eq-reciprocal-dynamics","eq-reciprocal-dissipation","prop-storage-hessian","eq-storage-effective-potential","eq-storage-full-hessian","eq-storage-schur","eq-storage-full-carrier","eq-storage-square","equation-26","eq-exact-storage-memory","equation-28","equation-29","eq-storage-susceptibility","section-12.3","eq-three-coordinate-potential","eq-three-coordinate-h","equation-33","eq-hidden-cross-response","equation-35","section-12.4","eq-rk-carrier","eq-rk-dirichlet","equation-38","equation-39","equation-40","section-12.5","eq-aligned-gradient","prop-complex-alignment","eq-complex-aligned-factorization","section-12.6","equation-43"],"words":1800},{"paperId":"cscf","slug":"12-boundary-and-tests","title":"Boundary carriers and discriminating tests","description":"The collar realization, physical observables, selection obligations and the resulting account.","order":12,"url":"/source-readout/cscf/12-boundary-and-tests","sourcePath":"content/paper-integrations/cscf/12-boundary-and-tests.mdx","sections":[{"id":"sec-realization","title":"13. Boundary realization and physical discrimination"},{"id":"section-13.1","title":"13.1 An explicit curvature-to-boundary carrier"},{"id":"section-13.2","title":"13.2 What a joint physical test measures"},{"id":"sec-future","title":"14. Future work: selection of profiles by physical realization laws"},{"id":"section-15","title":"15. Conclusion"}],"anchors":["sec-realization","section-13.1","prop-collar","section-13.2","sec-future","section-15"],"words":897},{"paperId":"cscf","slug":"13-functional-calculus","title":"Appendix A: Functional calculus details","description":"Projection-valued measure and the common functional calculus.","order":13,"url":"/source-readout/cscf/13-functional-calculus","sourcePath":"content/paper-integrations/cscf/13-functional-calculus.mdx","sections":[{"id":"app-fc","title":"A. Functional Calculus Details"}],"anchors":["app-fc"],"words":92},{"paperId":"cscf","slug":"14-profile-injectivity","title":"Appendix B: When the Lambda profile is injective","description":"The compact-positive-support proof and finite-rank corollary.","order":14,"url":"/source-readout/cscf/14-profile-injectivity","sourcePath":"content/paper-integrations/cscf/14-profile-injectivity.mdx","sections":[{"id":"app-profile-inj","title":"B. Lambda-Profile Injectivity on Compact Positive Supports"}],"anchors":["app-profile-inj","thm-lambda-profile-injectivity","cor-finite-rank-profile"],"words":545},{"paperId":"cscf","slug":"15-determinant-scope","title":"Appendix C: Determinant scope","description":"Trace class and finite spectral regularization.","order":15,"url":"/source-readout/cscf/15-determinant-scope","sourcePath":"content/paper-integrations/cscf/15-determinant-scope.mdx","sections":[{"id":"app-det","title":"C. 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