This document provides the formal mathematical architecture for the alpha-less-than-one frameworkintroduced in SIP-PHY-01a. We derive the Lindbladian master equation governing open quantumsystems under dynamic imbalance, formalize the Modified Tsirelson Bound as a falsifiable prediction,and establish the spectral gap condition as the mathematical expression of the General Theory ofRegulated Stability (GTRS). The framework maps the Coherence-Decoherence-Recoherence (CDR)cycle onto non-equilibrium thermodynamics and connects to the Five Principles architecture.Speculative extensions (Genesis, topology specification) are explicitly bracketed as annexes pendingempirical verification.
This document provides the formal mathematical architecture for the alpha-less-than-one frameworkintroduced in SIP-PHY-01a. We derive the Lindbladian master equation governing open quantumsystems under dynamic imbalance, formalize the Modified Tsirelson Bound as a falsifiable prediction,and establish the spectral gap condition as the mathematical expression of the General Theory ofRegulated Stability (GTRS). The framework maps the Coherence-Decoherence-Recoherence (CDR)cycle onto non-equilibrium thermodynamics and connects to the Five Principles architecture.Speculative extensions (Genesis, topology specification) are explicitly bracketed as annexes pendingempirical verification.
The SIP-PHY Forks Register, v2.1 A consolidated register of eight registered foundational forks — well-formed open questions, each with named branches, traced consequences, and resolution criteria — in the quantum-foundations layer of the GTRS / coherence–decoherence–recoherence (CDR) framework. A registered fork turns a hidden theoretical tension into transparent research infrastructure; the register records what is open and why, and asserts no resolutions. A working connection map (explicitly a hypothesis for examination, not established structure) proposes that the eight forks may cluster around two deeper roots — an orientability question on the state manifold and an observer/selection question. Version 2.1 propagates the decidability finding of SIP-PHY-NOTE-04 (24 May 2026) into the topological branch of FORK-02, FORK-04, and FORK-06, and sharpens §3.1 accordingly. The topological reading of the "Klein cluster" is reclassified from open, attack via the toy model to open, but untestable on the single-oscillator model — a statement about the reach of the instrument, not a refutation — with the decisive test relocated to a holonomy check on a parameter manifold. No fork is closed and no new physics is introduced; the change is record-keeping. All eight forks remain open. Part of the SIP corpus, SymbioMind Research Practice. Released under CC BY-NC-SA 4.0 as a pedagogical boundary: use to teach, not to certify.
AbstractThe General Theory of Regulated Stability (GTRS), developed as a cross-domain diagnosticframework for recoverability in complex systems, has been applied to biological, ecological, andclinical domains where thresholds are observable but imprecise. This paper argues thatquantum error correction (QEC) provides the hardest possible falsification arena for GTRSbecause the fault-tolerance threshold, error correction cycles, and decoherence rates areexperimentally operationalised with high precision. We map the GTRS six-regime taxonomy andα-chain diagnostic decomposition onto quantum states and error correction operations, identifystructural correspondences (fault-tolerance threshold as GTRS threshold, Zeno/anti-Zenoeffects as protective/iatrogenic decoherence, decoherence-free subspaces as Regime IIpartition), and specify five falsification tests the mapping must pass. Adversarial mathematicalanalysis establishes that the α chain is a diagnostic bottleneck identifier — it identifies whichstage of recovery is the binding constraint — not a mathematical factorisation of the logical errorrate. Architectural analysis identifies boundary conditions where the mapping breaks: leakagefeedback creates temporal violations of the α chain's sequential independence assumption, andstray interactions create spatial coupling that the single-site model does not capture. TheGoogle Willow processor (2024) is identified as the primary test case, with cosmic ray burstevents providing the strongest candidate for non-trivial diagnostic application. This paperestablishes the research programme; execution of the falsification tests against the publishedZenodo data (DOI: 10.5281/zenodo.13273331) is scoped as the next phase, ideally incollaboration with quantum information researchers. Keywords: GTRS, quantum error correction, fault-tolerance threshold, surface code, α-chaindecomposition, decoherence, regime taxonomy, structural invariance, falsification
Description The SIP-PHY Forks Register is a living document recording identified-but-unresolved foundational questions in the SymbioMind SIP-PHY corpus. Each registered fork is a well-formed open question with named branches, traced consequences along each branch, a robustness assessment indicating which existing papers are structured to remain valid across branches, and criteria for what would resolve the fork. Naming forks explicitly turns hidden tensions into transparent research infrastructure. The practice is consistent with the GTRS principle that mature systems hold open tensions productively rather than collapsing prematurely to a single state. Authors of SIP-PHY papers reference forks by ID and indicate whether their argument is robust across branches or commits to a specific branch. Version 1 registers four foundational forks: FORK-01 (the ontological status of time within GTRS — emergent from CDR, coextensive with CDR, or fundamental substrate); FORK-02 (the dimensional realisation of the Klein-surface-like rulebook in physical or state-space contexts); FORK-03 (the status of the observer — internal, external, or co-constitutive); and FORK-04 (whether coherence is a regulatory phase co-equal with decoherence and recoherence, or an observer-operator that collapses a D/R superposition).
Description This working note preserves a foundational reframing proposal made during the 19 May 2026 SymbioMind research conversation. The proposal suggests that within the Coherence–Decoherence–Recoherence (CDR) triplet, coherence (C) may not be a third regulatory phase co-equal with decoherence (D) and recoherence (R), but rather an observer-operator that collapses a D/R superposition into a particular realisation. On this reading, D and R are bonded as the two outcomes of a quantum-state-like structure, with C functioning analogously to the slit-choice in a double-slit experiment. If the proposal holds, it restructures the foundations of CDR by making C and D/R different kinds of mathematical object (C as operator, D and R as states), introduces the observer at the foundations of the framework rather than as a late addition, opens a fourth branch on the registered fork concerning the ontological status of time, and gives the Klein-surface-like recoherence topology a sharper role as the natural geometry of D/R bondedness. The note is parked rather than developed because the proposal admits three possible versions — formal (importing the full quantum-mechanical apparatus), structural-analogy (using the quantum framing without the formalism), and heuristic (using the double-slit as a vivid picture only). Development requires deciding which version is intended. The note records the proposal cleanly so that future work can begin from an intact starting point. It is not for publication as a stand-alone paper at this stage. The note identifies five conditions that may trigger development into a formal paper: a clear decision on which version is intended; availability of a mathematical collaborator; completion of the η(Φ) thermodynamic foundation; literature search confirming originality of the configuration; or emergence of a test case that distinguishes the proposal from the alternative.
Abstract Classical coherence C_cl is introduced as the normalised mutual information C_cl(A,B) ≜ I(A;B)/H(A,B) for finite discrete biological observables. The measure is bounded in [0,1], satisfies five elementary information-theoretic properties, and admits a conditional geometric interpretation as the correlational component of the Fisher–Rao metric under equal-frequency discretisation (local second-order equivalence near the independence manifold). A complete measurement protocol (equi-marginal discretisation, KPSS + Pettitt stationarity checks, change-point handling) is supplied together with sensitivity analysis across copulas, sample sizes, and noise. Thresholds are derived exclusively from system-specific calibration against an independent gold-standard (e.g., carbon flux). This yields the falsifiable Prediction P1 for homeostatic stability in mycorrhizal–host systems. Network superadditivity is left as an open problem. Version 11 deepens the operator-theoretic foundations shared with Fisher information geometry, integrates C_cl into the Architecture of Regulatory Coherence (ARC) and Coherence–Decoherence–Recoherence (CDR) cycle, and explicitly links the framework to the General Theory of Regulated Stability (GTRS) across quantum and cosmological scales. The framework supplies a rigorous classical benchmark for the Quantum Homeostasis Programme.
This technical supplement establishes the rigorous mathematical foundation for the coupling parameter alpha and the off-diagonal winding-basis operator C-hat within the Epsilon (epsilon) Framework. By replacing the unphysical zero-point void with an irreducible ground-state plenum governed by the foundational axiom not-exists-zero, the framework eliminates adjustable fitting parameters. We detail the five-level nested Hilbert space architecture, derive the exact analytical expression for alpha via the trace of operator norms and the excess mass-charge coupling ratio eta_mass, formalize the epsilon-deformed position-momentum commutation relation, and outline a concrete, falsifiable dual-species Bose-Einstein Condensate (BEC) Feshbach tuning protocol to detect the predicted residual energy floor. This document serves as a methodological companion to the core preprint series on Zenodo.