{"schemaVersion":1,"publication":{"id":"slow-passage","title":"Slow Passage and Prepared Hysteresis in Adaptive Spectral Carriers","subtitle":"A slow sweep delays switching past the static fold by a preparation-dependent amount. The prepared cycle has a nonzero geometric loop limit, while physical maintenance work grows as the sweep slows.","description":"A slow sweep delays switching past the static fold by a preparation-dependent amount. The prepared cycle has a nonzero geometric loop limit, while physical maintenance work grows as the sweep slows.","doi":"10.5281/zenodo.23935694","date":"2026-11-04","url":"/source-readout/slow-passage","pdfUrl":"/publications/spectral/slow-passage/paper.pdf","markdownUrl":"/publications/spectral/slow-passage/complete.md","manifestUrl":"/source-readout/slow-passage/manifest.json","paperSlug":"slow-passage","headline":"Switching remembers how the state was prepared.","author":"Jeremy Rodgers","programme":"source-readout","supplementUrl":"/publications/spectral/slow-passage/verification.zip"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"slow-passage","slug":"01-source-and-energy","title":"A Slow Sweep Passes the Stationary Fold","description":"The complete reciprocal source, uninterrupted triangular input and bounded-input energy estimates.","order":1,"url":"/source-readout/slow-passage/01-source-and-energy","sourcePath":"content/paper-integrations/slow-passage/01-source-and-energy.mdx","sections":[{"id":"section-1","title":"1. Where switching happens"},{"id":"section-2","title":"2. Time-dependent source, energy and boundedness"},{"id":"section-2-1","title":"2.1 Complete state and prescribed input"},{"id":"section-2-2","title":"2.2 Physical balance and the varying tilt"}],"anchors":["abstract","section-1","section-2","section-2-1","eq-1","eq-2","eq-3","eq-4","eq-5","eq-6","section-2-2","eq-7","eq-8","eq-9","eq-10","proposition-2-1","eq-11","eq-12","eq-13"],"words":891},{"paperId":"slow-passage","slug":"02-folds-and-critical-coordinate","title":"Complete Folds and the Mixed Critical Coordinate","description":"The stationary count, full saddle-node coefficients and a passing-trajectory estimate in the fixed-fold chart.","order":2,"url":"/source-readout/slow-passage/02-folds-and-critical-coordinate","sourcePath":"content/paper-integrations/slow-passage/02-folds-and-critical-coordinate.mdx","sections":[{"id":"section-3","title":"3. Stationary geometry and complete fold data"},{"id":"section-4","title":"4. Fixed-fold reduction with dynamic memory"},{"id":"section-4-1","title":"4.1 Projection and extended nilpotent structure"},{"id":"section-4-2","title":"4.2 Attraction during passage"}],"anchors":["section-3","eq-14","eq-15","eq-16","eq-17","lemma-3-1","eq-18","eq-19","eq-20","eq-21","eq-22","eq-23","eq-24","eq-25","eq-26","table-1","section-4","section-4-1","eq-27","eq-28","eq-29","eq-30","eq-31","eq-32","section-4-2","lemma-4-1","eq-33","eq-34"],"words":902},{"paperId":"slow-passage","slug":"03-airy-delay-and-observables","title":"Preparation Selects the Airy Delay","description":"The two-thirds-power law, its proof by matching and exit bounds, and the difference between local departure and complete arrival.","order":3,"url":"/source-readout/slow-passage/03-airy-delay-and-observables","sourcePath":"content/paper-integrations/slow-passage/03-airy-delay-and-observables.mdx","sections":[{"id":"section-5","title":"5. Inner scaling and preparation-dependent passage"},{"id":"section-5-1","title":"5.1 Orient the Riccati equation"},{"id":"section-5-2","title":"5.2 The incoming stable branch selects the solution"},{"id":"section-5-3","title":"5.3 Exact-fold and finite-inner preparations"},{"id":"section-6","title":"6. Which switching event is measured?"}],"anchors":["section-5","section-5-1","eq-35","eq-36","eq-37","eq-38","section-5-2","eq-39","eq-40","eq-41","theorem-5-1","eq-42","eq-43","eq-44","section-5-3","eq-45","eq-46","eq-47","section-6","eq-48","eq-49"],"words":971},{"paperId":"slow-passage","slug":"04-capture-hysteresis-and-work","title":"From Local Exit to a Complete Hysteresis Return","description":"Strict loss gives the outgoing connection; stable tracking gives the prepared return and separates geometric area from physical work.","order":4,"url":"/source-readout/slow-passage/04-capture-hysteresis-and-work","sourcePath":"content/paper-integrations/slow-passage/04-capture-hysteresis-and-work.mdx","sections":[{"id":"section-7","title":"7. Capture and the prepared quasistatic return"},{"id":"section-7-1","title":"7.1 The full outgoing connection"},{"id":"section-7-2","title":"7.2 Stable tracking through reversal"},{"id":"section-7-3","title":"7.3 The geometric limit"},{"id":"section-8","title":"8. The cost of deleting dynamic memory"},{"id":"section-9","title":"9. Work and the evolving carrier"}],"anchors":["section-7","section-7-1","lemma-7-1","section-7-2","eq-50","eq-51","eq-52","theorem-7-2","eq-53","eq-54","section-7-3","eq-55","eq-56","section-8","eq-57","eq-58","eq-59","table-2","section-9","eq-60","proposition-9-1","eq-61","eq-62","eq-63"],"words":1253},{"paperId":"slow-passage","slug":"05-executed-cycles","title":"Four complete cycles test delay, capture and work","description":"Compare event-localized switching with unfit Airy laws, test preparation controls, and separate the geometric loop from the energy ledger.","order":5,"url":"/source-readout/slow-passage/05-executed-cycles","sourcePath":"content/paper-integrations/slow-passage/05-executed-cycles.mdx","sections":[{"id":"section-10","title":"10 Executed numerical tests and reproducibility"},{"id":"section-10-1","title":"10.1 Frozen experiment and actual event locations"},{"id":"section-10-2","title":"10.2 Preparation, complete-state controls and independent formulation"},{"id":"section-10-3","title":"10.3 Continuous cycles, capture and work"},{"id":"section-10-4","title":"10.4 Verification and correction traceability"}],"anchors":["section-10","section-10-1","eq-64","eq-65","table-3","table-4","figure-1","section-10-2","table-5","eq-66","eq-67","eq-68","table-6","section-10-3","figure-2","table-7","table-8","figure-3","section-10-4"],"words":2009},{"paperId":"slow-passage","slug":"06-scope-residuals-and-references","title":"What the Prepared Return Proves","description":"The limits of the hysteresis result and the complete finite-preparation residual audit.","order":6,"url":"/source-readout/slow-passage/06-scope-residuals-and-references","sourcePath":"content/paper-integrations/slow-passage/06-scope-residuals-and-references.mdx","sections":[{"id":"section-11","title":"11. Discussion"},{"id":"appendix-a","title":"Appendix A. Finite-preparation residual audit"},{"id":"references","title":"References"}],"anchors":["section-11","appendix-a","eq-69","table-9","references","reference-1","reference-2","reference-3","reference-4","reference-5","reference-6","reference-7","reference-8","reference-9","reference-10","reference-11","reference-12"],"words":696}],"completeMarkdown":"/publications/spectral/slow-passage/complete.md","originalPdf":"/publications/spectral/slow-passage/paper.pdf"}