ULTRAMETRIC PHYSICS
ULTRAMETRIC PHYSICS
RESEARCH PLAN
Author: Rowan Brad Quni-Gudzinas
Contact: rowan.quni@outlook.com
ORCID: 0009-0002-4317-5604
ISNI: 0000000526456062
Date: 2026-04-06 Version: 2.0
EXECUTIVE
SUMMARY: THE RATIO-CENTRIC PARADIGM
This research plan presents a framework for ultrametric physics based
on fundamental scaling ratios rather than
anthropocentric number representations. The core insight: physical
reality is described by dimensionless scaling ratios
(π, φ, e, α, etc.) that operate on hierarchical discrete structures,
independent of any base representation (decimal, binary, etc.) or
continuum assumptions.
Key Innovations: 1. **Ratio-Based
Valuations**: Scaling bases q ∈ ℝ⁺ as pure ratios, not decimal
numbers 2. Base-Invariant Formulations: All mathematics
independent of representation (decimal/binary/etc.) 3.
Hierarchical Discrete Reality: Bruhat-Tits trees as
fundamental substrate 4. Unified Framework: Addressing
quantum computation, quantum gravity, and measurement theory
simultaneously 5. Constrained Methodology: 0.5 FTE PI,
LLM-assisted, open dissemination via Zenodo/ResearchGate
RESEARCH
CONSTRAINTS AND METHODOLOGY
Resources:
Personnel: 0.5 FTE Principal Investigator (20
hrs/week)
Tools: Large Language Models (DeepSeek, Google
Gemini)
Computation: Python execution within LLM chat
threads
Output Format: Markdown with MathJax → Obsidian →
Dissemination: Zenodo (DOI), ResearchGate, social
media
Excluded: Experimental facilities, arXiv, LaTeX,
formal peer review
Research Protocol:
Module Execution: Each module as single LLM chat
thread
Document Generation: Markdown with MathJax,
base-independent formulations
Code Implementation: Python for simulations/toy
models (executable in chat)
Archiving: Zenodo deposit with DOI, ResearchGate
upload
Dissemination: Social media with Zenodo links
RESEARCH MODULES (12
MODULES)
*Prioritized for consistency and quality control based on existing
progress and monograph “Ultrametric Physics: From Discrete Hierarchical
Geometry to Intrinsic Fault Tolerance and Quantum Gravity”
(0.10.md)*
FOUNDATIONAL
MATHEMATICS (Modules 1-4)
Module 1:
Ratio-Based Valuation Theory
Core Concept: Scaling ratios q as fundamental, not
integer primes Research Questions: 1. How to formalize
valuations with arbitrary scaling ratios q ∈ ℝ⁺? 2. What algebraic
structures emerge when q is a geometric ratio (π, φ, e)? 3. How do
scaling ratios generate continuous appearances from discrete
structures?
Expected Output: 8-12 page Markdown document
defining ratio-based valuation framework, with theorems/proofs for q =
π, φ, e, and Python implementations.
Key Insight: q treated as pure scaling operator,
never written in decimal form; base-invariant formulations; physical
interpretation as fundamental scaling ratio.
Module
2: Bruhat-Tits Trees with Scaling Ratios
Core Concept: Trees with edge weight log q,
independent of q’s decimal representation **Research
Questions:** 1. How to construct trees parameterized by scaling
ratio q and residue field size N? 2. What tree properties (growth,
boundary dimension) depend only on ratios log N/log q? 3. How do tree
automorphisms relate to scaling transformations?
Expected Output: 6-10 page document with tree
constructions for q = π, φ, e, Python code for tree generation/analysis,
visualizations.
Key Insight: Edge weight = log q (not decimal
expansion); boundary dimension = log N/log q (pure ratio); physical
interpretation: q as scale separation ratio.
Module
3: Vladimirov Operator and Ratio-Based Calculus
Core Concept: Pseudodifferential operator with
scaling ratio q Research Questions: 1. How to define
D_q^α for arbitrary scaling ratio q? 2. What is the spectrum in terms of
q^(nα) (powers of scaling ratio)? 3. How does tree Laplacian approximate
D_q^α in continuum limit?
Expected Output: 5-8 page document with ratio-based
calculus, Python implementations for eigenvalue calculations, connection
to tree Laplacians.
Key Insight: Normalization constant C_q(α) expressed
in terms of q; eigenvalues as powers of scaling ratio; base-independent
formulations.
Module
4: Adelic Theory for General Number Fields
Core Concept: Extend adelic framework beyond ℚ to
fields with fundamental scaling ratios **Research
Questions:** 1. How to construct adeles for fields with scaling
ratios π, φ, e? 2. What physical interpretations emerge from different
completions? 3. How do product formulas generalize with scaling
ratios?
Expected Output: 7-10 page document extending adelic
theory, physical interpretation of different completions, mathematical
consistency proofs.
Key Insight: Treat all completions democratically;
scaling ratios as fundamental parameters; connection to multiple scale
hierarchies in physics.
QUANTUM
COMPUTATION THEORY (Modules 5-7)
Module 5:
Ratio-Based Quantum Error Correction
Core Concept: Error suppression εL ∼ q^(-d) εP
with scaling ratio q Research Questions: 1. How does
scaling ratio q determine error suppression efficiency? 2. What is the
optimal q for given resource constraints? 3. How does this compare to
surface codes and other approaches?
Expected Output: 8-12 page document deriving error
suppression formulas, resource optimization with scaling ratio q,
comparison tables.
Key Insight: q as fundamental scaling ratio of
architecture; optimization over q space; physical implementation
considerations.
Module 6:
Quantum Gate Theory on Ratio-Based Trees
Core Concept: Universal gate sets from tree
automorphisms with scaling ratio q Research Questions:
- What gate sets are universal for computation on trees with scaling
ratio q? 2. How to compile arbitrary unitaries into scaling
transformations? 3. What are complexity implications of ratio-based
computation?
Expected Output: 7-10 page document with gate
classification, compilation algorithms, complexity analysis, Python
implementations.
Key Insight: Gates as scaling transformations;
compilation as path finding in ratio space; complexity dependent on log
q.
Module 7:
Thermodynamic Limits with Scaling Ratios
Core Concept: Minimum energy requirements scale with
q^d Research Questions: 1. What are fundamental
thermodynamic limits for ratio-based computation? 2. How does
temperature affect error suppression with scaling ratio q? 3. What are
Landauer limits for ratio-based operations?
Expected Output: 6-9 page document with
thermodynamic derivations, temperature limits analysis, comparison to
conventional architectures.
Key Insight: Energy barriers scale as q^d;
temperature limits depend on log q; thermodynamic advantages of
ratio-based structures.
QUANTUM
GRAVITY AND COSMOLOGY (Modules 8-10)
Module
8: Wheeler-DeWitt Equation on Ratio-Based Trees
Core Concept: Discrete WdW equation with kinetic
term from scaling ratio q Research Questions: 1. How to
formulate WdW equation on trees with scaling ratio q? 2. What solutions
correspond to cosmological histories? 3. How does time emerge from tree
navigation with scaling ratio q?
Expected Output: 10-15 page document with discrete
WdW formulation, solution methods, time emergence mechanism, Python
implementations.
Key Insight: Time as navigation in ratio-scaled
tree; WdW operator depends on q; connection to cosmic evolution.
Module
9: Emergent Lorentz Symmetry from Scaling Ratios
Core Concept: Lorentz group emerges from
automorphisms with scaling ratio q Research Questions:
- How does Lorentz symmetry emerge from tree automorphisms in limit? 2.
What is the relation between scaling ratio q and speed of light? 3. How
do Lorentz violations appear at small scales?
Expected Output: 8-12 page document deriving Lorentz
group emergence, q-c relations, Lorentz violation predictions.
Key Insight: Speed of light related to log q;
Lorentz violations scale with q^(-d); testable predictions.
Module
10: Cosmological Dynamics from Ratio-Based Navigation
Core Concept: Scale factor a(t) from tree branching
statistics with scaling ratio q Research Questions: 1.
How does a(t) emerge from branching with scaling ratio q? 2. What tree
parameters correspond to (Ωm, ΩΛ, H₀)? 3. How does inflation arise
from ratio-based dynamics?
Expected Output: 9-13 page document deriving
cosmological dynamics, parameter mapping, inflation mechanism, Python
simulations.
Key Insight: Cosmological parameters as ratios;
inflation as accelerated branching; testable CMB predictions.
CONSCIOUSNESS
AND SYNTHESIS (Modules 11-12)
Module
11: Monna Map as Ratio-Based Consciousness Interface
Core Concept: M: K → ℝ as projection from discrete
ratio-based states to continuous experience **Research
Questions:** 1. How does Monna map transform ratio-based quantum
states to qualia? 2. What neural implementations are possible? 3. How
does this resolve the hard problem?
Expected Output: 8-12 page document with Monna map
model, neural implementation hypotheses, hard problem resolution.
Key Insight: Consciousness as ratio-based
information processing; qualia as specific ratio patterns; measurement
as projection.
Module
12: Synthesis: Adelic Ontology and Ratio-Based Unification
Core Concept: Comprehensive synthesis based on
monograph “Ultrametric Physics” Research Questions: 1.
How do ratio-based frameworks unify quantum computation and quantum
gravity? 2. What is the adelic ontology of discrete hierarchical
reality? 3. What are testable predictions and philosophical
implications?
Expected Output: 15-20 page synthesis document
integrating all modules, developing adelic ontology, identifying
testable predictions.
Key Insight: The adelic universe A = ℝ × ∏q Kq as
comprehensive mathematical arena; each scaling ratio q as a “layer” of
reality; fundamental discreteness with emergent continuity.
EXECUTION TIMELINE
AND MILESTONES
Phase 1:
Foundational Completion (Weeks 1-4)
Complete Modules 1-4: Mathematical foundations
Deliverables: 4 Zenodo deposits (Modules 1-4)
Phase 2:
Core Theory Development (Weeks 5-12)
Develop Modules 5-10: Quantum computation and gravity
Implement computational simulations
Deliverables: 6 Zenodo deposits (Modules 5-10)
Phase 3:
Synthesis and Completion (Weeks 13-16)
Develop Modules 11-12: Consciousness and synthesis
Cross-module coherence verification
Complete framework documentation
Deliverables: 3 Zenodo deposits (Modules 11-12,
synthesis)
QUALITY METRICS
AND SUCCESS CRITERIA
Mathematical Rigor:
All theorems with proofs or proof sketches
Internal consistency across modules
Base-invariance verification
Computational
Soundness:
Executable Python code in chat environment
Code documentation and examples
Symbolic verification of analytical results
Physical
Plausibility:
Connection to known physics where required
Testable predictions identified
Consistency with empirical constraints
Interdisciplinary
Coherence:
Connections between modules clearly articulated
Synthesis across quantum computation, gravity, consciousness
Philosophical consistency
Dissemination
Impact:
Zenodo downloads and views
ResearchGate engagement
Social media discussion and sharing
Citation in related work
RISK MITIGATION
STRATEGIES
Theoretical Risks:
Mathematical inconsistencies: Regular cross-module
consistency checks
Over-reliance on LLMs: Critical verification of all
derivations
Isolation from community: Active social media
engagement
Computational Risks:
Code execution limitations: Design for constrained
chat environment
Numerical accuracy: Analytical verification where
possible
Scalability issues: Focus on small-scale
demonstrators
Dissemination Risks:
Low visibility: Strategic social media
approach
Misunderstanding: Clear explanations, responsive
communication
Platform changes: Primary archiving on Zenodo
EXPECTED
CONTRIBUTIONS AND IMPACT
Theoretical
Contributions:
Complete ratio-based ultrametric physics framework
Generalized valuation theory with scaling ratios
Discrete Wheeler-DeWitt formulation on trees
Ratio-based consciousness model via Monna map
Unification of computation, gravity, and measurement
Methodological
Innovations:
LLM-assisted theoretical physics methodology
Constrained research protocols for foundational work
Open science practices for theoretical physics
Base-invariant mathematical formulations
Practical
Applications:
Quantum computing architectures with geometric protection
Cosmological predictions testable with CMB data
Consciousness criteria for artificial systems
Hierarchical material design principles
CONCLUSION: THE
RATIO-CENTRIC PARADIGM
This research plan presents a focused, executable program for
developing ultrametric physics based on fundamental scaling ratios. By
rejecting anthropocentric mathematical conventions (integer primes, real
continuum, base-10 decimals) and focusing on ratio-based hierarchical
structures, the framework offers a unified approach to quantum
computation, quantum gravity, and consciousness.
The constrained methodology (0.5 FTE PI, LLM-assisted, open
dissemination) demonstrates that significant foundational progress is
possible without traditional resources. The 12-module structure ensures
consistency and quality control while covering the essential components
of the framework.
Ultimate Goal: A paradigm shift
from number-based to ratio-based physics, developed with strict
empirical discipline and rigorous mathematical foundations.