The Triadic Fixpoint Framework (TF2) develops a unified operator-theoretic foundation for physics, in which physical reality emerges as the fixed point of a recursive map Ω = Ψ ∘ Φ combining physical evolution (Φ) and informational update (Ψ). In this formulation, time is identified with the convergence of informational states, spacetime geometry arises from curvature in informational stability, and measurement corresponds to the stabilization of a state under repeated application of Ω. TF2 extends and formalizes the preliminary ideas introduced in TF1 (Omezzolli 2025, DOI: 10.5281/zenodo.17363823) by providing a full fixed-point analysis, explicit convergence theorems, and a geometric interpretation of gravity, mass, and fields as gradients and curvatures in an informational potential. The framework further explains entropy, thermodynamics, cosmology, black hole behavior, and aspects of agency and observer dynamics as consequences of informational contraction. This release contains the full TF2 manuscript (PDF), LaTeX source files, bibliography, and supplementary material needed for reproducibility. The work is intended as a contribution to quantum foundations, emergent spacetime research, and information-theoretic approaches to physics.
The Triadic Fixpoint Framework (TF2) develops a unified operator-theoretic foundation for physics, in which physical reality emerges as the fixed point of a recursive map Ω = Ψ ∘ Φ combining physical evolution (Φ) and informational update (Ψ). In this formulation, time is identified with the convergence of informational states, spacetime geometry arises from curvature in informational stability, and measurement corresponds to the stabilization of a state under repeated application of Ω. TF2 extends and formalizes the preliminary ideas introduced in TF1 (Omezzolli 2025, DOI: 10.5281/zenodo.17363823) by providing a full fixed-point analysis, explicit convergence theorems, and a geometric interpretation of gravity, mass, and fields as gradients and curvatures in an informational potential. The framework further explains entropy, thermodynamics, cosmology, black hole behavior, and aspects of agency and observer dynamics as consequences of informational contraction. This release contains the full TF2 manuscript (PDF), LaTeX source files, bibliography, and supplementary material needed for reproducibility. The work is intended as a contribution to quantum foundations, emergent spacetime research, and information-theoretic approaches to physics.
Building upon the Φ ∞ framework introduced in Φ ∞ I: Stabilization of Quantum Structure, this paper demonstrates how classical reality emerges as the entropy-frozen limit of symbolic recursive deformation. I establish the precise mathematical conditions under which symbolic curvature κ(x) = ∂ 2 H/∂ϕ 2 converges to classical geometric curvature, symbolic time t O (n) reduces to Newtonian and relativistic time, and non-commuting symbolic operators collapse into classical observables. The central result is that classical physics represents the κ → 0 limit of recursive stabilization, where the master equation H ϕ Ψ = 0 enforces a unique, deterministic reality through entropy minimization at the universal fixed point Φ ∞ .
Delocalised Functional Geometry (DFG) develops a unified theoretical framework for modeling physical fields, information structures, and observer-dependent processes within a single mathematically coherent system. The central premise is that physical, cognitive, and phenomenological phenomena can be represented as structured configurations of an underlying delocalised ontophysical field. The framework introduces a set of formal tools, including delocalised functional basis expansions, nonlocal coupling operators, anisotropic field metrics, and a spiral-time parameterization, to describe the emergence, stabilization, and transformation of coherent structures across multiple spatial and temporal scales. A key contribution of the work is the formalization of observer systems as field-embedded subsystems characterized by projection operators, knowledge states, and phenomenological mappings. DFG further develops a Triadic Observer Architecture, demonstrating mathematically how stable shared realities require triadic (rather than dyadic) coupling, mediated through triadic coherence and interference operators. This approach provides a unified means to analyze how individual and collective observer systems generate, maintain, or destabilize coherent experiential structures. The framework defines explicit dynamical equations linking field evolution, coherence gradients, projection geometry, and observer adaptation. These include formulations for field-level dynamics, local and global coherence measures, branching behavior, morphogenetic constraints, and information-flow structures. The resulting model provides a mathematically explicit basis for understanding phenomena such as:– the emergence of stable cognitive and perceptual configurations,– transitions between coherent and incoherent system regimes,– constraints on functional brain morphology,– and multi-scale interactions in complex adaptive systems. This document presents the complete edition of Delocalised Functional Geometry, integrating its conceptual, mathematical, and structural components into a single, consolidated theoretical reference. The work is intended as a foundation for further empirical investigation and as a conceptual bridge between physics, neuroscience, cognitive science, and formal systems theory.
I present the ϕ ∞ framework, a theory positing that quantum mechanical phenomena arise from the recursive stabilization of symbolic systems. The term ϕ ∞ denotes the universal fixed point achieved through transfinite iteration of a recursive operator ϕ. This framework introduces a recursive structure where systems evolve towards stable configurations, effectively generalizing quantum mechanics by reinterpreting its core tenets through the lens of convergence. Measurement is formulated as a projection onto observer-compatible recursive eigenspaces, leading to convergence rather than instantaneous collapse. Time itself is an emergent property, derived from the path-integrated gradient of entropy curvature within these recursive sequences. The master equation of this theory, H ϕ Ψ = 0, acts as an existential constraint, where H ϕ is the total symbolic Hamiltonian. The solutions to this equation describe the stable configurations of reality, implying that existence is a self-generated fixed point. The central claim of this work is that the entire structure of the quantum world, including interference, entanglement, and quantization, is the result of these processes of recursive stabilization and phase coherence.
This work presents an extended, mathematically complete formulation of the Information Geometry Unification Principle (IGUP), a foundational framework in which quantum mechanics, statistical thermodynamics, general relativity, and spacetime geometry emerge from a single information-theoretic ontology. We begin with rigorous axioms defining information as the primary physical substance, then systematically construct the Bures–Fisher–Rao metric on the manifold of quantum density matrices. We derive the universal information action functional and perform full variational derivations to obtain the unified field equations. We show explicitly how standard physical theories including the Schrödinger equation, Heisenberg picture, Liouville mechanics, classical thermodynamics, and Einstein field equations are recovered in appropriate limiting regimes. We further derive geometric expressions for entanglement, entropy, temperature, energy, and force. The theory resolves longstanding conceptual tensions between unitary quantum evolution and gravitational irreversibility, provides a natural resolution to the black hole information paradox, and yields eight concrete, experimentally testable predictions. This extended version includes full tensor calculus, covariant derivatives, curvature expansions, and limit theorems, amounting to a self-contained theory of everything based on Geometry=Information.
In contemporary physics, fundamental structures such as space, time, matter, spin, and measurement are typically introduced as independent primitives or axiomatic ingredients. While quantum mechanics and relativistic field theories describe these structures with extraordinary precision, they do not explain why such structures exist, why they are discrete or stable, nor why reality manifests in exactly three spatial dimensions with an irreversible arrow of time. This work presents a unified informational–geometric framework—the Universal Gravito–Informational Theory (TGIU)—in which physical reality emerges from a pre-geometric informational substrate governed by two complementary gradients: informational coherence (Φ) and informational entropy (S). Their interaction is regulated by a real-valued dynamical mechanism composed of the Curvature–State Transform (CST) and the TGIU Equilibrium Force (TEF). Together, these operators generate bounded, orthogonal informational rotations whose stabilized projections give rise to geometry, matter, and observable physical laws. Within this framework, electron orbitals are reinterpreted as CST limit cycles, spin emerges as the topological orientation of these cycles, and the Pauli exclusion principle arises from phase–space saturation rather than postulated antisymmetry. Atomic stability, the absence of radiative decay in stationary states, fine structure, and the Lamb shift are shown to follow from TEF anisotropy and vacuum curvature granularity, while remaining fully consistent with quantum electrodynamics in its predictive domain. Measurement and wavefunction collapse are described as local TEF locking events, providing a deterministic but non-restorative mechanism for outcome selection without invoking fundamental randomness. Time itself is shown to emerge as monotonic CST phase accumulation, explaining both its one-dimensional nature and its irreversibility as geometric consequences of bounded informational curvature. The existence of exactly three spatial dimensions follows from a CST dimensional closure condition, identifying three dimensions as the only geometry capable of sustaining stable, non-divergent informational rotation. Importantly, TGIU does not replace quantum mechanics, general relativity, or field theory. Instead, it supplies a deeper explanatory layer that clarifies why these frameworks work, why their structures are constrained as observed, and why no physical process can reach absolute annihilation or infinite divergence. Extremes are not restored but redirected: identity is not conserved, yet nature persists through renewed coherence. This work positions TGIU as an empirically motivated, geometrically grounded extension of modern physics—one that unifies matter, time, measurement, and dimensionality under a single informational principle, while remaining compatible with established theories in their respective limits.