ChronaQ

A research programme on global quantum consistency and emergent spacetime.

Description

ChronaQ is a theoretical research programme investigating the hypothesis that physical reality is governed by a globally constrained, time-symmetric quantum structure, from which classical spacetime, probability, and dynamics emerge as effective descriptions.

The programme explores how two-boundary consistency conditions, entanglement structure, and projection geometry jointly determine observable physics without postulating fundamental time evolution.

Structure

The ChronaQ programme is organised as a sequence of technical papers to be published:

  • Foundations I — Branch-volume entropy and emergent temporal direction

  • Foundations II — Entanglement geometry and emergent spacetime

  • Foundations III — Radon–ABL reconstruction of consistent histories

  • Foundations IV — Local consistency geometry and effective field laws

  • Foundations V — Experimental and observational implications

To date, 24 internal physics research reports have been completed.

ψ-Mathematics (psi-m, ψ-M)

A mathematical programme for projection-based quantum structure.

Description

ψ-mathematics (psi-m) is a foundational mathematical research programme developed to support the ChronaQ framework. It investigates the hypothesis that existing mathematical formalisms - particularly tensor calculus, smooth manifolds, and local differential geometry - are insufficient to describe globally constrained, projection-defined quantum structures.

The programme develops alternative mathematical objects and transformation rules capable of representing nonlocal consistency, observer-relative projections, branched decoherence structure, and time-symmetric constraints. Rather than assuming a fixed background manifold or fundamental dynamics, psi-m treats geometry, entropy, and probability as emergent properties of projection structure on the ψ-net.

ψ-mathematics is not a physical theory in itself. It is a supporting mathematical language designed to formalise and extend the structures required by ChronaQ in regimes where standard tools fail or become ill-defined.

Structure

The ψ-mathematics programme is organised as a sequence of numbered internal technical briefs:

  • ψ-M0 — A manifesto for psi-mathematics
    (Scope, motivation, and breakdown of tensor-based formalisms)

  • ψ-M1–ψ-M5 — Projection geometry and observer patch structure
    (Non-manifold geometry, patch gluing, projection-defined spaces)

  • ψ-M6–ψ-M10 — Topology and consistency on the ψ-net
    (Sheaf-like structures, non-smooth intersections, global closure)

  • ψ-M11–ψ-M15 — Entropy, curvature, and variational structure
    (Entropy Hessians, minimal render surfaces, geometric constraints)

  • ψ-M16+ — Hybrid and recovery limits
    (Connections to standard geometry, local tensor limits, compatibility)

Each brief addresses a specific mathematical deficiency encountered in projection-based quantum frameworks, with the long-term aim of providing a coherent algebraic–geometric foundation for ChronaQ.

To date, 39 internal psi-m research reports have been completed, providing the mathematical groundwork required to support and extend the ChronaQ foundations papers.

Signed Probability (SP) & the Signed Probability Axiom (SPA)

A research programme extending probability theory beyond non-negativity

Description

The Signed Probability programme investigates a minimal extension of classical probability theory in which probabilities are permitted to take negative values on unobservable or intermediate events, while remaining non-negative on all empirically observable outcomes.

At its core is the Signed Probability Axiom (SPA), a generalisation of Kolmogorov’s axioms that relaxes global non-negativity while enforcing a strict reality constraint on observable events. This framework provides a rigorous foundation for quasi-probabilities, interference effects, and cancellation phenomena that arise naturally in quantum mechanics, postselected systems, and globally constrained models such as ChronaQ.

The programme develops the measure-theoretic, informational, and computational consequences of SPA, demonstrating that classical probability theory is recovered as a special case while enabling new tools for reasoning in contexts where standard probability breaks down.

Structure

The Signed Probability programme is organised as a sequence of research briefs:

  • SP0 — Programme charter and roadmap
    (Motivation, scope, and compatibility with ChronaQ and ψ-mathematics)

  • SP1 — Foundations of the Signed Probability Axiom
    (Axioms, consistency proofs, and classical limit)

  • SP2–SP5 — Measure theory and information
    (Hahn–Jordan decomposition, signed entropy, divergences)

  • SP6–SP10 — Dynamics, inference, and computation
    (Signed stochastic processes, inference rules, algorithms)

  • SP11+ — Geometry, operators, and applications
    (Signed information geometry, operator measures, ψ-net links)

Signed Probability serves both as a standalone extension of probability theory and as the probabilistic backbone for projection-based and two-boundary frameworks, where intermediate signed contributions arise without violating observable statistics.

As the ψ-m programme developed, it became clear that projection-based and two-boundary structures also require a generalisation of classical probability. This motivated the parallel development of the Signed Probability (SP). To date, 28 internal SP research reports have been completed.