{"schemaVersion":1,"publication":{"id":"sharp-uniqueness","title":"Sharp Uniqueness and Bistability in Driven Adaptive Spectral Carriers","subtitle":"A sharp curvature threshold gives uniqueness even at equality. Below it, the three-node realization exhibits bistability, with a complete stationary count in the stated transverse-convexity regime.","description":"A sharp curvature threshold gives uniqueness even at equality. Below it, the three-node realization exhibits bistability, with a complete stationary count in the stated transverse-convexity regime.","doi":"10.5281/zenodo.23932350","date":"2026-11-03","url":"/source-readout/sharp-uniqueness","pdfUrl":"/publications/spectral/sharp-uniqueness/paper.pdf","markdownUrl":"/publications/spectral/sharp-uniqueness/complete.md","manifestUrl":"/source-readout/sharp-uniqueness/manifest.json","paperSlug":"sharp-uniqueness","headline":"One input can sustain two stable states.","author":"Jeremy Rodgers","programme":"source-readout","supplementUrl":"/publications/spectral/sharp-uniqueness/verification.zip"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"sharp-uniqueness","slug":"01-complete-driven-source","title":"The Same Drive Can Sustain Two Stable States","description":"The complete reciprocal source, its physical rate port, and the strict tilted-energy argument.","order":1,"url":"/source-readout/sharp-uniqueness/01-complete-driven-source","sourcePath":"content/paper-integrations/sharp-uniqueness/01-complete-driven-source.mdx","sections":[{"id":"section-1","title":"1. The uniqueness boundary"},{"id":"section-2","title":"2. The complete source and external energy contract"},{"id":"section-2-1","title":"2.1 Constitutive equations and units"},{"id":"section-2-2","title":"2.2 Power and the constant-input tilt"}],"anchors":["abstract","section-1","section-2","section-2-1","eq-1","eq-2","eq-3","eq-4","section-2-2","eq-5","eq-6","eq-7","eq-8","proposition-2-1","eq-9"],"words":858},{"paperId":"sharp-uniqueness","slug":"02-endpoint-and-stability","title":"Sharp Curvature and Complete Dynamical Stability","description":"Stationary memory elimination yields the endpoint; the mobility–inertia theorem returns all amplitude and memory directions.","order":2,"url":"/source-readout/sharp-uniqueness/02-endpoint-and-stability","sourcePath":"content/paper-integrations/sharp-uniqueness/02-endpoint-and-stability.mdx","sections":[{"id":"section-3","title":"3. Stationary reduction and endpoint uniqueness"},{"id":"section-3-1","title":"3.1 Exact memory elimination and curvature"},{"id":"section-4","title":"4. Full stability from stationary curvature"},{"id":"section-4-1","title":"4.1 Return the eliminated memory"},{"id":"section-4-2","title":"4.2 Varying mobility at fixed state"}],"anchors":["section-3","section-3-1","eq-10","eq-11","eq-12","eq-13","eq-14","eq-15","theorem-3-1","eq-16","eq-17","eq-18","section-4","section-4-1","eq-19","eq-20","eq-21","lemma-4-1","eq-22","eq-23","eq-24","eq-25","section-4-2"],"words":752},{"paperId":"sharp-uniqueness","slug":"03-sharp-path-threshold","title":"An Exact Bistability Threshold on Three Nodes","description":"A transverse convexity argument counts the complete stationary set and exposes a misleading sector-stable candidate.","order":3,"url":"/source-readout/sharp-uniqueness/03-sharp-path-threshold","sourcePath":"content/paper-integrations/sharp-uniqueness/03-sharp-path-threshold.mdx","sections":[{"id":"section-5","title":"5. Three-node path: exact count and threshold"},{"id":"section-5-1","title":"5.1 Modes and scalar stationary law"},{"id":"section-5-2","title":"5.2 Sector stability can miss a full instability"},{"id":"section-5-3","title":"5.3 Complete count for highest-mode preparation"}],"anchors":["section-5","section-5-1","eq-26","eq-27","eq-28","eq-29","eq-30","eq-31","eq-32","eq-33","section-5-2","section-5-3","theorem-5-1","eq-34","corollary-5-2","eq-35"],"words":764},{"paperId":"sharp-uniqueness","slug":"04-complete-folds","title":"Both Folds Are Full Dynamical Saddle-Nodes","description":"The complete left and right null vectors establish nondegeneracy and two independent cubics retain all eight modes.","order":4,"url":"/source-readout/sharp-uniqueness/04-complete-folds","sourcePath":"content/paper-integrations/sharp-uniqueness/04-complete-folds.mdx","sections":[{"id":"section-6","title":"6. Saddle-nodes and complete modal cubics"},{"id":"section-6-1","title":"6.1 Full dynamical nondegeneracy"},{"id":"section-6-2","title":"6.2 Independent all-coordinate blocks"}],"anchors":["section-6","section-6-1","eq-36","eq-37","eq-38","eq-39","eq-40","eq-41","section-6-2","eq-42","eq-43","eq-44","eq-45","eq-46","eq-47"],"words":544},{"paperId":"sharp-uniqueness","slug":"05-basin-and-observation","title":"Full-State Basins, Powered Switches and Lost Readout Information","description":"A tilted barrier controls basin membership while two equally powered sources retain distinct full-feedback responses.","order":5,"url":"/source-readout/sharp-uniqueness/05-basin-and-observation","sourcePath":"content/paper-integrations/sharp-uniqueness/05-basin-and-observation.mdx","sections":[{"id":"section-7","title":"7. Full-state basin regions and powered switching"},{"id":"section-7-1","title":"7.1 A cylinder barrier"},{"id":"section-7-2","title":"7.2 Finite pulse experiments and resources"},{"id":"section-8","title":"8. Observable response and source-to-readout information"},{"id":"section-8-1","title":"8.1 Conditional versus free memory"},{"id":"section-8-2","title":"8.2 What the coarse summary loses"}],"anchors":["section-7","section-7-1","theorem-7-1","eq-48","eq-49","eq-50","eq-51","section-7-2","table-1","eq-52","eq-53","section-8","section-8-1","eq-54","eq-55","eq-56","eq-57","eq-58","section-8-2","eq-59","eq-60"],"words":923},{"paperId":"sharp-uniqueness","slug":"06-reproducible-path-example","title":"The Three-Node Example in Numbers","description":"Exact scalar brackets, complete stationary spectra, basin preparations, finite switching trajectories and replay.","order":6,"url":"/source-readout/sharp-uniqueness/06-reproducible-path-example","sourcePath":"content/paper-integrations/sharp-uniqueness/06-reproducible-path-example.mdx","sections":[{"id":"section-9","title":"9. Fixed example and numerical evidence"},{"id":"section-9-1","title":"9.1 Analytic design and stationary values"},{"id":"section-9-2","title":"9.2 Full-state preparations and controls"},{"id":"section-9-3","title":"9.3 Physical ledgers for the two switches"},{"id":"section-9-4","title":"9.4 Methods and reproduction"}],"anchors":["section-9","section-9-1","eq-61","eq-62","table-2","table-3","table-4","figure-1","section-9-2","eq-63","table-5","table-6","section-9-3","table-7","figure-2","section-9-4","eq-64"],"words":1346},{"paperId":"sharp-uniqueness","slug":"07-scope-and-references","title":"What the Threshold and Readout Obstruction Establish","description":"The exact scope of the counted regime, physical realization requirements, and bibliography.","order":7,"url":"/source-readout/sharp-uniqueness/07-scope-and-references","sourcePath":"content/paper-integrations/sharp-uniqueness/07-scope-and-references.mdx","sections":[{"id":"section-10","title":"10. Discussion and limits"},{"id":"references","title":"References"}],"anchors":["section-10","references","reference-1","reference-2","reference-3","reference-4","reference-5","reference-6","reference-7","reference-8","reference-9"],"words":510}],"completeMarkdown":"/publications/spectral/sharp-uniqueness/complete.md","originalPdf":"/publications/spectral/sharp-uniqueness/paper.pdf"}