Exploratory research at the edge of logic, geometry, computation, and epistemology. This repository is a sandbox for frontier ideas: formal results, speculative frameworks, and auditable constructions that sit between mathematics, theoretical physics, and the philosophy of knowledge.
- Formal core results (definitions, theorems, proofs),
- Model-level constructions (discrete kinematics, locality bounds),
- Speculative extensions clearly labeled as such.
Nothing here claims experimental validation unless explicitly stated.
Frontier Research (4 Fun) is not a product, not a framework, and not a manifesto. It is a collection of precise questions explored with rigorous tools:
- What can be encoded exactly in arithmetic?
- How do limits of knowledge acquire geometric structure?
- Can spacetime, causality, and agency be treated as discrete, auditable objects?
- Where do computational and epistemic barriers structurally arise?
The unifying theme is structure over interpretation.
Oscar Riveros
Summary Introduces the SAT Equation Theorem: any balanced CNF formula can be encoded as a single natural number whose binary expansion exactly marks unsatisfying assignments.
Main contribution
- Provides a lossless arithmetic representation of satisfiability for balanced CNFs.
- Makes explicit the correspondence between logical structure and integer arithmetic.
- Serves as a foundational "exact interface" between logic and number theory.
Summary A methods paper presenting a fully auditable pipeline from finite discrete physical constraints to SAT instances, including witness verification.
Main contribution
- Establishes SAT as a mechanical referee for finite discrete models.
- Demonstrates SAT/UNSAT instances for combinatorial gravity and locality constraints.
- Emphasizes verification, not performance or continuum claims.
Summary Defines epistemic curvature as a metric invariant measuring the irreducible gap between syntax and semantics in formal systems.
Main contribution
- Recasts incompleteness as a geometric obstruction, not a logical pathology.
- Introduces metric interfaces and a derivational refinement principle (DRP).
- Links Gödel incompleteness, prime irregularities, and P vs NP as manifestations of curvature.
Summary A unified exposition connecting formal results, discrete geometry, and speculative extensions toward cognition, society, and AI.
Main contribution
- Systematizes epistemic curvature across logic, arithmetic, and computation.
- Separates formal theorems from physical models and metaphysical interpretation.
- Provides a second-order framework for reasoning about the limits of knowledge itself.
Summary Introduces the Layered Metric Space (LMS): a discrete framework where geometry, time, and curvature emerge from evolving edge weights on a fixed graph.
Main contribution
- Defines inter-layer strain and curvature as purely kinematical quantities.
- Shows how temporal ordering emerges from monotonic deformation.
- Positions LMS alongside Regge calculus, CDT, and causal sets—without assuming a manifold.
Summary Extends LMS by adding quantum-like propagation and a materialization phase where transitions saturate.
Main contribution
- Separates metric kinematics from quantum-like dynamics cleanly.
- Introduces an orthogonal Procrustes construction for unitary evolution.
- Identifies a "materialized backbone" where effective curvature becomes operational.
Summary Formalizes agency as the operational ability to create distinguishability in remote regions under locality constraints.
Main contribution
- Derives soft causal cones using Lieb–Robinson–type bounds.
- Translates geometric suppression into information-theoretic capacity limits.
- Rules out "fantastical agency" without assuming relativistic spacetime.
Summary A unifying paper organizing the project into four modules: SAT interfaces, epistemic curvature, LMS, and operational agency.
Main contribution
- Shows how the modules interlock without collapsing into a single theory.
- Clearly demarcates formal core vs speculative extensions.
- Provides a roadmap for future discrete-physics research programs.
Summary Studies an exact SAT/#SAT algorithm based on maintaining a disjoint cover of forbidden subcubes in the Boolean hypercube.
Main contribution
- Proves tight exponential lower bounds via parity obstructions.
- Connects SAT solving to deterministic DNF (DSOP) knowledge compilation.
- Establishes conditional links to PH collapse under uniform compilation assumptions.
This repository follows three strict principles:
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No hidden metaphysics Formal results stand independently of interpretation.
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Auditability over elegance Finite, checkable constructions are preferred to asymptotic promises.
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Curiosity without closure Open problems are features, not bugs.
If you are looking for certainty, this repo is not for you. If you enjoy standing precisely at the edge of what can be said — welcome.
Unless otherwise specified in individual files: All rights reserved. The author explicitly permits reading, discussion, and critique.