Agnostic Recursive Propagation Geometry (ARPG)

Abstract

This paper extends the framework of Agnostic Recursive Propagation Geometry (ARPG), a domain-agnostic formalism in which global structure emerges through recursively constrained local admissible propagation. The framework treats admissibility geometry, rather than substrate-specific mechanisms, as the primitive object governing recursive organization.

We develop the relation between ARPG and sparse expert architectures, recursive invariant reinforcement, hidden admissibility geometry, recursive feedback conditioning, and differential cognitive stabilization. Recent developments surrounding the Erdős unit-distance conjecture are interpreted as a mathematically relevant analogy demonstrating how hidden recursive structure can dominate visible geometric intuition.

From the ARPG viewpoint, sparse expert routing systems may represent early operational analogies to geometry-aware cognition, where cognition quality depends not solely on visible parameter scale, but on recursively conditioned propagation topology.

The paper further proposes differential recursive measures intended to move cognition evaluation from qualitative judgments toward measurable recursive propagation stabilization.

Keywords

ARPG; recursive geometry; recursive cognition; sparse routing; mixture of experts; MoE; geometry-aware cognition; admissibility geometry; differential cognitive geometry; recursive invariant reinforcement; coherence stabilization; hidden admissibility; Erdős unit-distance problem; experimental AI frameworks

Overview

Agnostic Recursive Propagation Geometry proposes that global organization may emerge through recursively constrained local admissible propagation rather than fixed global structure alone.

The framework begins from local admissibility, recursive propagation, and recursive stabilization. It then extends these concepts toward sparse expert routing, recursive invariant reinforcement, hidden admissibility structures, and differential cognitive geometry.

A full PDF version of the paper is available through the PDF button in the upper-right corner of this page.

Official Links

CJCI Issue Page:
https://www.carlonoscopen.com/journal/v1i12

Reserved Zenodo DOI:
https://doi.org/10.5281/zenodo.20343054

Author ORCID:
https://orcid.org/0009-0005-2284-8891

Open Full PDF Paper

Paper Details

• Title: Agnostic Recursive Propagation Geometry (ARPG)
• Subtitle: Hidden Admissibility, Sparse Routing, and Differential Cognitive Geometry
• Author: Ivan Silva
• Publisher: Carlonoscopen, LLC
• Journal: Carlonoscopen Journal of Coherence Intelligence
• ISSN: 3069-874X
• Language: English
• Publication Date: May 22, 2026
• Format: Web publication, PDF paper, and supplementary provenance package
• Version: 1.1
• Reserved DOI: 10.5281/zenodo.20343054

Core Framework

ARPG begins with a primitive recursive propagation system:

where X is a state space, 𝒜 is an admissibility structure, and Φ is a propagation rule.

The central principle is that propagation cannot exceed local admissibility:

Global structure is then interpreted as the recursive accumulation of local admissible propagation.

Recursive Invariant Reinforcement

The paper develops recursive invariant reinforcement as a mechanism by which compressed invariant structure recursively conditions future admissible propagation geometry.

Here, P _n is the recursive propagation state, 𝓘 _n is recursively compressed invariant structure, and 𝓕 _n represents recursive feedback accumulation.

Sparse Routing and Geometry-Aware Cognition

Sparse expert routing systems are discussed as operational analogies to geometry-aware cognition. In such systems, only selected propagation paths or expert subsets become active at a given step.

ARPG interprets this as a form of selective admissibility: local propagation becomes constrained by routing topology, accessibility, and recursive conditioning rather than globally active scale alone.

The paper does not claim that sparse routing is equivalent to cognition. Rather, it identifies sparse routing as a useful structural analogy for studying recursively conditioned propagation geometry.

Relation to the Erdős Unit-Distance Problem

The paper interprets recent developments around the Erdős unit-distance problem as a mathematically relevant analogy.

The unit-distance problem asks how many unit-distance pairs can exist among planar points:

From an ARPG perspective, the visible Euclidean plane may be only a projection, while deeper admissibility structure governs the resulting global organization.

The significance is not equivalence between cognition and discrete geometry, but a shared structural principle: hidden admissibility structure may dominate visible scale or visible geometry.

Differential Cognitive Geometry

ARPG proposes that recursive cognition-like systems may be studied through differential recursive measures rather than qualitative judgment alone.

Recursive drift:

Admissible entropy differential:

Recursive stability ratio:

These quantities are proposed as early-stage measurement candidates for recursive stabilization, not as completed proof of cognition enhancement.

Experimental Engineering Scaffold

The associated engineering work establishes a bounded recursive measurement scaffold for studying geometry-aware cognition hypotheses.

The scaffold includes recursive propagation tracing, declared-vs-applied propagation accounting, substrate characterization, recursive observability, milestone governance, freeze-chain verification, sparse substrate analysis, and bounded recursive instrumentation.

This engineering layer should be interpreted as recursive measurement infrastructure, not proof of cognition enhancement.

Supplementary Package

The Zenodo record may include a supplementary reproducibility and provenance package containing companion notes, experimental engineering scaffold, cryptographic provenance report, executive summary, machine-readable provenance inventory, and checksum manifest.

These materials support transparency, reproducibility, and provenance tracking for the ARPG/CJCI research program.

Scope and Limits

This framework does not claim:

• proof of cognitive enhancement,
• proof of consciousness,
• proof of artificial general intelligence,
• proof of substrate-independent cognition,
• or proof that sparse routing is equivalent to cognition.

Instead, the paper establishes a mathematically coherent propagation formalism and experimentally bounded methodology for testing geometry-aware cognition hypotheses.

Applications and Research Directions

Potential research directions include:

• recursive propagation tracing,
• geometry-aware cognition experiments,
• sparse routing analysis,
• small-model recursive stabilization studies,
• declared-vs-applied propagation accounting,
• recursive perturbation stability,
• and differential recursive measurement systems.

Explicit Non-Claims

• This paper does not claim AGI.
• This paper does not claim consciousness.
• This paper does not claim verified cognitive enhancement.
• This paper does not claim hallucination reduction.
• This paper does not claim substrate-independent cognition.
• This paper does not claim that sparse routing is equivalent to cognition.

Suggested Citation

Silva, Ivan Pereira da. Agnostic Recursive Propagation Geometry (ARPG): Hidden Admissibility, Sparse Routing, and Differential Cognitive Geometry. Carlonoscopen Journal of Coherence Intelligence, Volume 1, Issue 12, 2026. Reserved DOI: 10.5281/zenodo.20343054.

References

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5. Erdős, P. (1946). On sets of distances of n points. American Mathematical Monthly, 53(5), 248–250.
6. OpenAI. (2026). An OpenAI model has disproved a central conjecture in discrete geometry.
7. OpenAI. (2026). Planar Point Sets with Many Unit Distances.
8. Alon, N., Bloom, T. F., Gowers, W. T., Litt, D., Sawin, W., Shankar, A., Tsimerman, J., Wang, V., & Wood, M. M. (2026). Remarks on the disproof of the unit distance conjecture. arXiv:2605.20695.
9. Guth, L., & Katz, N. H. (2015). On the Erdős distinct distances problem in the plane. Annals of Mathematics, 181(1), 155–190.
10. Gromov, M. (1999). Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser.
11. Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos. Westview Press.
12. Cover, T. M., & Thomas, J. A. (2006). Elements of Information Theory. Wiley.

Publication Note

This page is published as part of the Carlonoscopen Journal of Coherence Intelligence. The PDF linked from this page is the full public paper for offline reading, citation support, and archival use.

The DOI listed on this page was reserved before upload and will resolve after the Zenodo record is published.