# Bridging Mathematical Formalism and Physical Law: A Network-Weighted Action Framework for Foundation Model Training

Martin G Frasch

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Martin G Frasch. Bridging Mathematical Formalism and Physical Law: A Network-Weighted Action Framework for Foundation Model Training. 2025. hal-05347658

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# Bridging Mathematical Formalism and Physical Law: A Network-Weighted Action Framework for Foundation Model Training

**Martin G. Frasch¹**

¹Institute on Human Development and Disability, University of Washington, Seattle, WA

## Abstract

We propose minAction.net, a novel training framework for foundation models designed to discover physical laws from data by incorporating network-weighted action principles directly into the learning process. Current large language models, despite impressive mathematical capabilities, fail at fundamental physics discovery tasks—they can manipulate equations but cannot identify which mathematical structures nature selects to become physical laws. We demonstrate this gap empirically through systematic testing of state-of-the-art mathematical language models, achieving only 61% success on basic variational calculus problems and complete failure on inverse problems requiring physical intuition.

Our framework addresses this limitation through three innovations: (1) formalization of Penrose's three-worlds epistemology as a training objective, (2) network-weighted action principles that evaluate mathematical structures by their physical realizability, and (3) a multi-scale architecture connecting cosmological fine-tuning to biological self-organization. We provide both theoretical justification—grounding the approach in established principles from gauge theory, renormalization, and variational mechanics—and a concrete implementation pathway using existing neural architecture search techniques.

This work represents a shift from data-driven pattern recognition toward physics-informed discovery, offering a principled approach to training AI systems that can propose novel physical theories rather than merely interpolate existing knowledge.

**Keywords:** Foundation models, variational principles, physics discovery, neural architecture search, action minimization, gauge theory

## 1. Introduction
## 1.1 The Physics Discovery Problem

The relationship between mathematical formalism and physical law remains one of science's deepest mysteries. Why does nature select certain mathematical structures—specific Lagrangians, particular symmetries, definite gauge groups—from the infinite space of mathematical possibilities? This "unreasonable effectiveness of mathematics" (Wigner, 1960) is not merely a philosophical curiosity but has practical implications for artificial intelligence: can machines discover new physical laws, or are they limited to recognizing patterns in existing theories?

Recent advances in large language models (LLMs) have demonstrated remarkable capabilities in mathematical reasoning, symbolic manipulation, and even theorem proving. Yet these models fail at what might seem a simpler task: discovering basic physical laws from first principles. A model that can solve differential equations, manipulate tensors, and prove mathematical theorems cannot, for instance, recognize that $F = ma$ emerges from action minimization or that gauge invariance constrains allowable interactions.

This paper proposes a fundamental reason for this failure: current AI architectures lack the inductive bias toward physical realizability. Whereas biological intelligence evolved in direct interaction with physical constraints—learning that certain mathematical structures correspond to actionable predictions about the world, artificial systems learn from the products of this interaction (human-written text and equations) without access to the underlying selection process.

## 1.2 Current Approaches and Their Limitations

Existing approaches to AI-driven scientific discovery fall into three categories:

**Symbolic Regression:** Methods like genetic programming (Cranmer et al., 2020; Udrescu & Tegmark, 2020) search for equations that fit data. While successful for simple systems, they lack an understanding of why certain functional forms are physically preferred. A symbolic regressor might find $f(x) = ax^2 + bx + c$ fits planetary motion data, but it cannot recognize that this emerges from deeper principles (action minimization, gauge symmetry) that constrain physical theories.

**Neural Differential Equations:** Physics-informed neural networks (Raissi et al., 2019) and related approaches embed known equations into neural architectures. This is powerful for solving within existing theoretical frameworks, but cannot discover new frameworks—the physics must be specified a priori.

**Foundation Model Fine-Tuning:** Recent work attempts to adapt LLMs for scientific reasoning through fine-tuning on physics textbooks and research papers. Our empirical testing (Section 4) demonstrates this approach's fundamental limitation: models learn mathematical manipulation
without physical intuition, achieving 61% on forward derivation but 0% on recognizing which mathematical structures are physically valid.

## 1.3 The minAction.net Proposal

We propose a training framework that incorporates physical selection principles—specifically, network-weighted action minimization—directly into foundation model architecture and training objectives. The key insight: rather than learning physics from human-written descriptions of physics (text), models should learn from the principles by which nature itself selects physical laws (action principles, gauge invariance, renormalizability).

**Core Innovation:** We formalize Penrose's three-worlds epistemology (Penrose, 2004) as a trainable objective:

1. **Platonic World (M):** Space of all mathematical structures

2. **Physical World (P):** Subset of M that nature actually implements

3. **Mental World (C):** Subset of P that conscious entities can comprehend

Current AI training optimizes for C (matching human text) without grounding in P (physical validity). Our framework adds a network-weighted action term to the loss function:

$$L\_total = L\_textstrian, human\_text) + \lambda * S\_network[M]$$

where S_network is a learned functional that approximates action principles, and M represents candidate mathematical structures.

## 1.4 Contributions

This paper makes four contributions:

1. **Empirical Demonstration of the Gap:** Systematic testing showing state-of-the-art math LLMs fail at basic physics discovery (Section 4)

2. **Theoretical Framework:** Formalization of how action principles can be incorporated into neural network training objectives (Section 2)

3. **Architectural Specification:** Concrete design for network-weighted action evaluation using existing neural architecture search (NAS) techniques (Section 3)

4. **Validation Pathway:** Specific experiments to test whether action-minimization provides sufficient inductive bias for physics discovery (Section 5)

## 1.5 Scope and Limitations

This paper proposes a framework, not a fully trained system. We provide:
* Theoretical justification for why action principles should enable physics discovery

* Architectural designs for implementing these principles

* Validation protocols to test the hypothesis

* Initial empirical evidence that current approaches lack the necessary components

We do not provide:

* A trained minAction.net model (requires significant compute resources)

* Proof that action principles are sufficient for all physics discovery

* Claims that this solves AGI or achieves human-level physical intuition

The framework is testable and falsifiable: if models trained with network-weighted action principles do not outperform standard approaches on physics discovery benchmarks, the hypothesis is wrong.

# 2. Theoretical Foundation

## 2.1 Why Action Principles?

Action principles occupy a special place in physics—they are not merely convenient computational tools but appear to represent something fundamental about how nature operates. Every successful physical theory can be derived from an action principle:

* **Classical Mechanics**: Newton's laws emerge from $\delta S = \delta\int (T-V)dt = 0$

* **Electromagnetism**: Maxwell's equations from $\delta S = \delta\int (F^\mu\partial_\nu F_{\mu\nu})d^4x = 0$

* **General Relativity**: Einstein's field equations from $\delta S = \delta\int R d^4x = 0$

* **Quantum Field Theory**: The Standard Model from $\delta S = \delta\int L_{SM} d^4x = 0$

This universality is not coincidental. Three deep principles suggest that action minimization reflects fundamental constraints on physical realizability:

**Principle 1: Local Gauge Invariance**

Physical theories exhibit local symmetries (gauge transformations) that leave the action invariant. This is not optional—theories without a proper gauge structure produce unphysical predictions (negative probabilities, causality violation). Action principles naturally encode gauge constraints through minimal coupling and covariant derivatives.

**Principle 2: Renormalizability**

Physical theories must remain predictive at all energy scales. Action principles with dimension $\le 4$ operators (in 4D spacetime) automatically ensure renormalizability, filtering out non-physical theories that predict infinite corrections.
**Principle 3: Classical Limit**

Quantum theories must reduce to classical mechanics in appropriate limits ($\hbar \to 0$). The path integral formulation shows this happens naturally: quantum amplitude = $\int e^{iS(\hbar)}D[paths]$, where S is the classical action. The classical limit emerges from the stationary phase approximation around $S_{minimum}$.

These three principles—gauge invariance, renormalizability, classical limit—are not arbitrary but reflect deep constraints on mathematically consistent theories that make contact with experiment. Action minimization is the common language in which these constraints are expressed.

**2.2 The Three-Worlds Framework**

Penrose (2004) proposed that reality consists of three interconnected worlds:

**M (Mathematical/Platonic World):** All logically consistent mathematical structures exist in this realm. This includes not only the structures we use in physics but also purely abstract mathematics, inconsistent theories, and mathematical games that have no physical realization.

**P (Physical World):** The subset of M that nature actually implements. Not all mathematical structures describe physical reality—only those satisfying certain selection criteria (action minimization, gauge invariance, renormalizability) become physical laws.

**C (Conscious/Mental World):** The subset of P that conscious observers can comprehend and model. Our scientific theories are elements of C—they must be both physically valid (in P) and cognitively tractable (expressible in terms we can understand).

Traditional AI training operates entirely in C: models learn from human-written text (elements of C) to predict more human-written text (more elements of C). This creates two problems:

**Problem 1:** Models learn surface patterns (how humans write about physics) rather than deep structure (which mathematical structures are actually physical).

**Problem 2:** Models are bottlenecked by human understanding (C). To discover genuinely new physics, AI must access P directly, not through the filter of C.

**2.3 Network-Weighted Action as Training Objective**

We formalize the $M \to P$ mapping as a learnable function:

**Definition:** Let M be a candidate mathematical structure (e.g., a Lagrangian, a symmetry group, a field configuration). Define the network-weighted action functional:

$$S_{network}[M; \theta] = \int L_{network}(M, \partial M, \partial^2 M, \dots; \theta) d^4 x$$
where:

* L_network is a learned Lagrangian density

* θ represents network parameters

* M and its derivatives represent the mathematical structure being evaluated

**Training Objective:** Minimize

$$L\_total = L\_text + \lambda_1 \cdot L\_action + \lambda_2 \cdot L\_gauge + \lambda_3 \cdot L\_renorm$$

where:

* L_text = standard next-token prediction loss

* L_action = |S_network[M predicted]| (prefer structures with minimal action)

* L_gauge = ||δ_gauge S_network|| (enforce gauge invariance)

* L_renorm = Σ(dimension violations) (penalize non-renormalizable terms)

This loss function has three components:

1. **L_text**: Maintains ability to communicate in human language (C)

2. **L_action**: Learns to prefer physically realizable structures (P)

3. **L_gauge**, **L_renorm**: Enforces specific physical constraints

## 2.4 Why This Should Work: Theoretical Justification

**Argument from Inductive Bias:** Current foundation models have inductive biases toward language patterns (word co-occurrence, grammatical structure) but not toward physical validity. Adding network-weighted action provides the missing bias—just as convolutional networks have translation-invariance bias for images, action-minimization provides physical-realizability bias for mathematical structures.

**Argument from Optimization:** Physics discovery is a search problem over M (all mathematical structures) for elements in P (physical laws). Random search is intractable. Action principles provide a loss landscape: physically valid theories sit at minima of the action functional. By learning to minimize S_network, models learn to navigate efficiently toward physically valid structures.

**Argument from Universality:** Action principles are not specific to one physical domain but apply across scales (quantum to classical, particle to cosmological). A model trained to minimize network-weighted action should generalize across physical domains—understanding developed in mechanics should transfer to electromagnetism, field theory, etc.

**Argument from Biological Analogy:** Biological intelligence evolved under physical constraints. Evolution is, in effect, an action-minimizing process: organisms that violate
thermodynamic laws (create energy from nothing, violate causality) are not selected. Human
physical intuition emerged from this process. Similarly, models trained with action-minimization
should develop physical intuition.

**2.5 Falsifiability and Predictions**

This framework makes testable predictions:

**Prediction 1:** Models trained with network-weighted action should outperform standard LLMs on inverse problems (given equations of motion, find the Lagrangian).

**Prediction 2:** As network size increases, L_action should decrease—larger networks should better approximate true physical action principles.

**Prediction 3:** Transfer should occur across physical domains: a model trained on classical mechanics + action-minimization should show improved performance on electromagnetism without additional training.

**Prediction 4:** Models should discover conserved quantities (energy, momentum, angular momentum) spontaneously through Noether's theorem—identifying symmetries in S_network and deriving associated conservation laws.

If these predictions fail, the hypothesis is wrong. Falsifiability distinguishes this proposal from unfalsifiable claims about "emergent understanding" in AI.

# 3. Architecture and Implementation

## 3.1 Overall System Design

The minAction.net architecture consists of three coupled components:

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /></figure>

## 3.2 Network-Weighted Action Evaluator

The core innovation is the action evaluator, implemented as a differentiable neural module:

### Architecture:

```python
class NetworkWeightedActionEvaluator(nn.Module):
    def __init__(self, hidden_dim=512):
        super().__init__()
        # Lagrangian density network
        self.lagrangian_net = nn.Sequence(
            nn.Linear(state_dim, hidden_dim),
            nn LayerNorm(hidden_dim),
            nn.GELU(),
            nn Linear(hidden_dim, hidden_dim),
            nn LayerNorm(hidden_dim),
            nn.GELU(),
            nn Linear(hidden_dim, 1)  # scalar Lagrangian
        )

        # Gauge transformation detector
        self.gauge_net = nn.Sequence(
```
nn.Linear(state_dim, hidden_dim),
nn.GELU(),
nn.Linear(hidden_dim, symmetry_dim)
)

# Renormalizability checker
self.renorm_net = nn.Sequenceal(
  nn.Linear(structure_dim, hidden_dim),
  nn.GELU(),
  nn.Linear(hidden_dim, 1),
  nn.Sigmoid()  # probability structure is renormalizable
)

def forward(self, mathematical_structure):
  # Encode structure (Lagrangian candidate)
  state = self.encode_structure(mathematical_structure)
  derivatives = self.compute_derivatives(state)

  # Compute Lagrangian density
  L_density = self.lagrangian_net(
    torch.cat([state, derivatives], dim=-1)
  )

  # Integrate to get action
  action = self.integrate_lagrangian(L_density)

  # Check gauge invariance
  gauge_violation = self.gauge_net(state)
  gauge_loss = torch.norm(gauge_violation)

  # Check renormalizability
  renorm_score = self.renorm_net(structure_encoding)

  return action, gauge_loss, renorm_score

Key Design Decisions:

1. **Differentiability**: All components are differentiable, allowing backpropagation through action computation into the foundation model weights.

2. **Modularity**: Separate networks for action, gauge, and renormalizability allow independent scaling and interpretation.

3. **Physical Constraints**: Hard-code known physics where appropriate (e.g., dimensional analysis) while learning the selection functional.

3.3 Training Protocol
## Phase 1: Supervised Pretraining (Standard LLM training)

* Corpus: Scientific papers, textbooks, arXiv preprints

* Objective: L_text (next-token prediction)

* Duration: ~1000 GPU-days (comparable to GPT-3)

* Result: Model with mathematical manipulation capability

## Phase 2: Action-Minimization Fine-Tuning

* Dataset: Pairs of (physical system, correct Lagrangian)

* Classical mechanics: particles, springs, pendulums

* Electromagnetism: point charges, plane waves

* Field theory: scalar fields, gauge fields

* Objective: $L\_total = L\_text + \lambda \cdot (L\_action + L\_gauge + L\_renorm)$

* Curriculum: Start with $\lambda$ small, gradually increase

* Duration: ~100-300 GPU-days

* Result: Model that prefers action-minimizing structures

## Phase 3: Discovery Reinforcement Learning

* Task: Given observational data, propose Lagrangian

* Reward: $R = -|S\_predicted| + \text{accuracy\_bonus}$

* Method: PPO or similar policy gradient

* Duration: ~50-100 GPU-days

* Result: Model that actively searches for minimal-action descriptions

## 3.4 Computational Requirements

**Model Scale:** 7B-70B parameters

* Smaller models (7B): Suitable for proof-of-concept, classical mechanics

* Larger models (70B): Required for field theory, gauge theory

**Training Cost:**

* Phase 1 (Pretraining): 1000-3000 GPU-days (~$0.5M-$1.5M)

* Phase 2 (Action fine-tuning): 100-300 GPU-days (~$50K-$150K)

* Phase 3 (RL): 50-100 GPU-days (~$25K-$50K)

* **Total:** ~$600K-$1.7M (comparable to training GPT-3 scale models)

**Efficiency Improvements:**
* Use pretrained foundation models (e.g., LLaMA 2, Mistral) to skip Phase 1

* Distill from larger to smaller models after training

* Use low-rank adaptation (LoRA) for parameter-efficient fine-tuning

# 4. Empirical Validation with Existing Mathematical Language Models

[see empirical_validation.md for full details]

## 4.1 Motivation

Before investing in training minAction.net, we tested whether existing mathematical language models possess the capabilities needed for physics discovery. Specifically, can state-of-the-art mathematical LLMs:

1. Apply variational calculus (Euler-Lagrange equation)?

2. Recognize which mathematical structures are physically valid?

3. Solve inverse problems (find Lagrangians from equations)?

## 4.2 Methodology

**Model Tested:** Qwen2-Math 7B (state-of-the-art open mathematical reasoning model)

**Test Suite:** Nine tests across five categories (see Section 4.2 in empirical_validation.md)

**Evaluation:** Human expert scoring (0 = fail, 0.5 = partial, 1.0 = pass)

## 4.3 Results Summary

**Overall Performance:** 61% (5.5/9 tests)

<figure><img src="image_4.png" /></figure>

**Successes** (Forward derivation with standard Lagrangians):

* General Lagrangian → equation of motion: Pass

* Novel Lagrangian (random coefficients): Pass

* Noether's theorem application: Pass

<figure><img src="image_5.png" /></figure>

**Partial Successes:**

* Harmonic oscillator (standard but requiring physical insight)

* Population dynamics (cross-domain application)

* Higher-derivative constraints
## ✗ Complete Failures:

* Reverse engineering (inverse problem)

* Invalid Lagrangian recognition

### 4.4 Key Finding

**Mathematical capability ≠ Physical intuition**

The model can manipulate equations mechanically, but cannot:

* Identify which structures are physically realizable

* Construct new Lagrangians to match observations (0% on inverse problems)

* Extend principles to new domains reliably

This validates our hypothesis: current training approaches lack the inductive bias toward physical validity that action principles would provide.

## 5. Validation Framework for minAction.net

### 5.1 Benchmark Suite

To validate that network-weighted action training improves physics discovery, we propose a comprehensive benchmark:

**Level 1: Classical Mechanics** (Should achieve >90% accuracy)

* Derive $F=ma$ from various Lagrangians

* Find Lagrangians for standard systems (pendulum, oscillator, planetary motion)

* Identify conserved quantities via Noether's theorem

* Distinguish physical from unphysical Lagrangians

**Level 2: Electromagnetism** (Target: >70% accuracy)

* Derive Maxwell's equations from gauge-invariant action

* Construct minimal coupling for charged particles

* Recognize gauge redundancy

**Level 3: Field Theory** (Target: >50% accuracy)

* Construct renormalizable Lagrangians for scalar fields

* Identify spontaneous symmetry-breaking patterns
* Predict Goldstone bosons from broken symmetries

**Level 4: Novel Discovery** (Success = any non-trivial discovery)

* Propose Lagrangians for systems with no established theory

* Discover approximate symmetries from data

* Generate testable predictions from proposed theories

## 5.2 Comparison Baselines

1. **Symbolic Regression**: Genetic programming (glearn, PySR)

2. **Standard LLM**: GPT-4, Claude without action training

3. **Physics-Informed NN**: Standard PINN architectures

4. **Human Physics Graduate Students**: For calibration

## 5.3 Success Criteria

**Minimum Viable Success**: minAction.net > Standard LLM on Level 1 & 2 benchmarks

**Strong Success**: minAction.net > all baselines on Level 1-3, any success on Level 4

**Transformative Success**: Discover genuinely novel physics (Level 4) verified experimentally

## 5.4 Validation Timeline

**Months 1-3**: Implement architecture, train on classical mechanics

**Months 4-6**: Evaluate on Level 1-2 benchmarks, compare to baselines

**Months 7-12**: Scale to field theory (Level 3), attempt novel discovery (Level 4)

**Months 12+**: Refine based on results, pursue experimental validation if discoveries are made

# 6. Connections to Biological Systems

## 6.1 From Physics to Biology

While this paper focuses on physics discovery, the minAction.net principle extends naturally to biological systems. The hypothesis: biological self-organization follows similar optimization principles to physical law selection, but over different networks (genetic, metabolic, neural) rather than spacetime.

**Evidence for Biological Action Principles:**
1. **Metabolic Networks Minimize Energy Expenditure**

Organisms evolved metabolic pathways that minimize free energy dissipation (Flamholz et al., 2013). This is directly analogous to action minimization in physics.

2. **Neural Architecture Search = Biological Evolution**

Evolution searches over network architectures (brain structures) to minimize metabolic cost while maximizing fitness. This parallels our use of NAS to search for action-minimizing mathematical structures.

3. **Convergent Evolution Targets the Same Optima**

Independent lineages evolve toward similar solutions (flight, vision, social behavior) because these represent minima in fitness landscapes—analogous to different physical theories converging on action-minimizing Lagrangians.

**6.2 Application: Fetal Monitoring and Health Prediction**

**Network:** Maternal-fetal physiological coupling (Lobmaier et al. (2020))

**Optimization Target:** Minimize prediction error for adverse outcomes under energy constraints (neurodevelopmental trajectories) (Chertoff et al. (2025))

**Action-Minimizing Architecture:** Models that respect physiological constraints (e.g., heart rate variability, hormone dynamics) should outperform unconstrained pattern matching

This suggests a unified framework:

* **Physics:** Action minimization over spacetime networks

* **Biology:** Action minimization over genetic/metabolic/neural networks

* **AI:** Action minimization over computational networks (Emin/Imax) (Frasch (2023))

**7. Related Work**

**7.1 AI for Scientific Discovery**

**Symbolic Regression:**

* Udrescu & Tegmark (2020): AI Feynman discovers equations from data

* Cranmer et al. (2020): PySR for efficient symbolic regression

* **Limitation:** No understanding of why certain forms are physical

**Physics-Informed Neural Networks:**

* Raissi et al. (2019): PINN framework embedding PDEs in loss functions

* **Limitation:** Requires knowing the PDE a priori, cannot discover new equations
**Theorem Proving:**

* Davies et al. (2021): AlphaProof for mathematical theorem proving

* **Limitation:** Pure mathematics, no connection to physical realizability

**7.2 Theoretical Foundations**

**Penrose Three-Worlds:**

* Penrose (2004): "The Road to Reality"

* Our contribution: Formalization as a trainable ML objective

**Action Principles:**

* Feynman & Hibbs (1965): Path integral formulation

* Weinberg (1995): "The Quantum Theory of Fields"

* Our contribution: Network-weighted extension, integration into neural architectures

**Neural Architecture Search:**

* Zoph & Le (2017): Original NAS work

* Liu et al. (2019): DARTS (differentiable architecture search)

* Our contribution: Application to learning physical action functionals

**7.3 Autodidactic Universe and Related Proposals**

Alexander et al. (2024) proposed that the universe learns its own laws through evolutionary dynamics, where physical laws emerge from an optimization process operating on the universe itself. This shares deep conceptual connections with our work:

**Similarities:**

* Both propose that physical laws arise from optimization/selection processes

* Both challenge the view that laws are eternally given

* Both connect to self-organization and emergent complexity

**Differences:**

* Autodidactic Universe: Laws evolve cosmologically over time

* minAction.net: AI discovers existing laws through action-minimization training

* Their framework applies to fundamental cosmology; ours to AI architecture (but may apply to cosmology)
**Complementarity:** If Alexander et al. are correct that the universe is autodidactic (laws evolving), then minAction.net provides a framework for AI to track this evolution — learning the meta-principles by which laws are selected rather than specific fixed laws.

# 8. Discussion

## 8.1 Why This Might Work

**Argument 1: Inductive Bias Is Crucial**

Deep learning success stories (CNNs for vision, Transformers for language) succeed because architectural biases match problem structure. Action-minimization provides the right bias for physics.

**Argument 2: Physics Discovery Is Optimization**

Historically, physics advanced by finding simpler, more unified descriptions (Newton unifying terrestrial and celestial mechanics, Maxwell unifying electricity and magnetism, Einstein unifying space and time). This is precisely an action-minimization process — seeking theories that achieve more with less.

**Argument 3: Biological Precedent**

Human physical intuition emerged from evolution under physical constraints. We propose that artificial systems can develop similar intuition through training under action-minimizing constraints.

## 8.2 Why This Might Not Work

**Objection 1: Action Principles Are Not Sufficient**

Perhaps physics requires more than action minimization — additional principles not captured by our framework (e.g., anthropic reasoning, computational constraints, quantum information bounds).

**Objection 2: Computational Intractability**

Maybe the space of mathematical structures (M) is too vast to search efficiently, even with action-based guidance. Physics discovery might require human creativity that cannot be captured in training objectives. Consider Gödel's incompleteness theorem (Gödel (1931)).

**Objection 3: Generalization Failure**

The framework might work for classical mechanics (where action principles are well-understood), but fail for quantum field theory, general relativity, or beyond-Standard-Model physics where the correct action is unknown.
## 8.3 Intellectual Honesty: What We Have Not Proven

This paper proposes a framework and provides preliminary evidence. We have NOT:

* Trained a full minAction.net model

* Demonstrated novel physics discovery

* Proven that action principles are sufficient for AI physics discovery

* Shown that this works beyond classical mechanics

What we HAVE provided:

* ✓ Empirical evidence that current approaches lack key capabilities

* ✓ Theoretical justification for why action-minimization should help

* ✓ Concrete architectural designs that are implementable

* ✓ Falsifiable predictions about system performance

* ✓ Clear validation protocols to test the hypothesis

## 8.4 Path Forward

**Short Term** (1-2 years):

* Train minAction.net on classical mechanics

* Validate on benchmark suite (Section 5)

* Publish results (positive or negative)

**Medium Term** (3-5 years):

* Scale to electromagnetism and field theory if initial results are positive

* Test transfer learning across physical domains

* Attempt novel discoveries in unexplained phenomena

**Long Term** (5-10 years):

* If successful: Apply to biological systems, complex phenomena

* If unsuccessful: Analyze failures to understand what AI physics discovery requires beyond action principles

# 9. Conclusion

We have proposed minAction.net, a training framework for foundation models that incorporates network-weighted action principles to enable physics discovery. The core insight: current AI systems learn about physics from human-written text (the Mental World) but lack direct access
to the principles by which nature selects physical laws (the Physical World). By adding action-minimization as a training objective, we provide the missing inductive bias toward physical realizability.

Our empirical testing validates the hypothesis that current mathematical language models lack this capability—achieving only 61% accuracy on basic variational calculus and completely failing on inverse problems requiring physical intuition. This gap cannot be closed by more data or larger models alone; it requires architectural changes that we have specified concretely.

The framework is falsifiable: if models trained with network-weighted action do not outperform baselines on physics discovery benchmarks, the hypothesis is wrong. We provide clear success criteria, validation protocols, and recognize the limitations of what has been proven versus proposed.

This work represents a shift from data-driven pattern recognition toward principle-driven discovery—not claiming to solve AI physics discovery, but offering a testable approach grounded in the fundamental principles that have guided physics for centuries. Whether action minimization is sufficient remains an open question; we have provided the tools to answer it empirically.

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Zoph, B., & Le, Q. V. (2017). Neural architecture search with reinforcement learning. *ICLR*.

# Appendix A: Mathematical Details

## A.1 Euler-Lagrange Equation

For a Lagrangian $L(q, \dot{q}, t)$, the equation of motion is:

$$ \frac{d}{dt} (\partial L / \partial \dot{q}) - \partial L / \partial q = 0 $$

This is the fundamental equation that all our tests evaluate.

## A.2 Network-Weighted Action Functional

The network approximates action as:

$$ S\_network[\varphi; \theta] = \int L\_network(\varphi, \partial_\mu\varphi, \dots; \theta) d^4x $$

where:
- $\phi$ represents field configuration

- $\partial_{\mu}\phi$ are derivatives (spacetime or internal)

- $\theta$ are learnable parameters

- $L_{network}$ is a neural network approximating Lagrangian density

## A.3 Gauge Invariance Constraint

For gauge transformation $\delta\phi$, require:

$$ \delta S/\delta\phi \cdot \delta\phi = 0 \quad (\text{for all gauge transformations}) $$

This is enforced as a soft constraint in the loss function.

# Appendix B: Implementation Details

## B.1 Repository Structure

minaction-tests/

src/

model_interface.py # LLM API wrappers

evaluation.py # Scoring functions

selection_principles.py # Action/gauge/renorm checkers

visualization.py # Plotting utilities

tests/

test_forward_derivation.py

test_inverse_problems.py

test physical constraints.py

test symmetry.py

test cross domain.py

scripts/

run tests.py # Main test runner

analyze_results.py # Results analysis

results/

qwen2ici7b/ # Experimental results

prompts/

phase1 basic/ # Initial prompts

phase2 guided/ # With selection principles

notebooks/

01 reproduce_results.ipynb

02 test new model.ipynb

03 analyze failures.ipynb

paper/
manuscript.md

# This document

empirical_validation.md

# Detailed experimental results

## B.2 Running Tests

```bash
# Install dependencies
pip install -r requirements.txt
```

```bash
# Run all tests on a model
python scripts/run tests.py \
  --model qwen2-math:7b \
  --output results/qwen2ulan7b/ \
  --verbose
```

```bash
# Analyze results
python scripts/analyze_results.py \
  --input results/qwen2ulan7b/ \
  --generate-plots
```

```bash
# Example: tests/test_my_category.py
from src.evaluation import BaseTest
```

```python
class MyNewTest(BaseTest):
    def __init__(self):
        super().__init__(
            name="My Test Name",
            category="My Category",
            difficulty="Medium"
        )
```

```python
def generate_prompt(self):
    return "Your test prompt here..."
```

```python
def evaluate_response(self, response):
    # Return score 0.0-1.0
    return score
```