Research — Digital Dynamics AI

These works define the formal frameworks that underlie KARIOS, PET integration, and structured computation.

PET formalizes computation through the lens of observer access, indistinguishability, and structured collapse. It provides a principled explanation of:

• Irreversibility as loss of operational distinguishability
• Learning as refinement of access maps
• Memory as preserved inferability
• Temporal inference within bounded access

These papers define accessible state, echo resources, and observer-relative computation in rigorous mathematical terms.

🔗 Phase–Echo Theory Core Specification

🔗 PET–A: Access and Learning

🔗 PET–E: Action Timing and Engineering

🔗 PET–RS: Resource Theory of Echo

🔗 PET–V/VE: Temporal Inference Bounds

Our systems succeed only where structure allows them to succeed. This body of work focuses on:

• Formal limits on generic solvers
• Structural conditions that permit efficient resolution
• Exact solver design for bounded structural classes

We prove general barriers for broad classes of optimization solvers when they rely on local moves or indiscriminate mixing. These results formalize circumstances under which commonly assumed convergence behaviors cannot hold.

🔗 Universal Washout Barrier

🔗 Structural Conflicts Between Expansion and Compression

Rather than propose universal solvers, these works identify structured islets of solvability — where polynomial-time algorithms exist due to explicit restrictions:

• Bounded treewidth regimes
• Spectrally constrained energy landscapes
• Locally convex regions under structural priors

🔗 Bounded Treewidth Exact Solvers

🔗 Spectral PL Convergence in Structured Regimes

This category includes theoretical explorations that extend beyond the core engineered pathways. These works are clearly labeled as exploratory and include:

• Hypotheses about unification of computation and cognition
• Formal investigation into observer-relative state spaces at general horizons
• Speculative frameworks with falsifiable predictions

Importantly: Exploratory research is not presumed in production systems like KARIOS; it serves to probe the boundaries of known frameworks while maintaining explicit assumptions.

🔗 Exploratory Access and Observer Hypotheses

🔗 Unified Formal Models of Observer Influence

To ensure clarity and rigorous interpretation, our research always adheres to the following:

Every assertion is accompanied by defined prerequisites and non-claims.

Proofs and specifications are provided in formats that subject matter experts can verify.

Negative results, complexity barriers, and structural impossibilities are documented with the same emphasis as positive findings.

Where research informs or is integrated into systems like KARIOS, this relationship is clearly stated.

Whether you are:

• A researcher evaluating formal frameworks,
• An engineer implementing structured systems,
• An enterprise partner assessing capability claims,

we recommend reading the papers with attention to:

• Scope statements
• Assumption sections
• Proof and construction details
• Validated limitations

and distinguishing between:

• Core foundational results
  and
• Exploratory hypotheses

This practice ensures correct interpretation and appropriate integration.

We encourage collaboration with:

• Academic institutions
• Government labs
• Open research initiatives
• Structured optimization communities

If you are interested in joint research, shared publications, or technical briefings, please contact us.

Below are curated categories of published work:

• Phase–Echo Theory Core
• PET–A, PET–E, PET–RS, PET–V/VE

• Washout Barriers
• Structural Solvability
• Exact Treewidth Solvers

• Unified Observer Formalisms

Each entry links to full research PDFs, with clear labeling of scope, assumptions, and conclusions.

At Digital Dynamics AI:

• We do not hide assumptions behind optimized code
• We do not anchor claims on proprietary black boxes
• We do not substitute benchmarks for proofs

Our research is a first-class artifact of our engineering.