Keith Luton

[Keith Luton @lutonfieldmodel]

Physics-Grounded Associative Memory:
Inference as Energy Relaxation with Adaptive Halting

Keith Luton
Independent Researcher
[email protected]
February 19, 2026
Abstract
We present a physics-motivated associative memory framework in which inference is formulated as energy relaxation with adaptive halting. Rather than performing fixed-cost forward computation, a query perturbs a learned energy landscape and relaxes toward a stored attractor. Computation halts when the energy change falls below a resonance threshold, yielding inference cost proportional to query difficulty.

The model is implemented as a continuous Hopfield-style network with Hebbian imprinting and a principled stopping criterion based on energy stabilization. We evaluate the system on pattern recovery benchmarks under increasing noise and compare against fixed-step inference baselines. Results demonstrate successful recovery with variable computational cost that scales with input difficulty.

The framework is motivated by physical field relaxation dynamics and provides a concrete, working example of difficulty-adaptive inference within an energy-based model.

1 Introduction

Modern neural networks typically allocate fixed computational cost per query. A simple recall task and a difficult, noisy query pass through identical forward pipelines.

In contrast, physical systems evolve toward equilibrium by energy minimization: a perturbed configuration relaxes along the energy gradient until motion ceases. The time required depends on the distance from equilibrium.

We adopt this relaxation principle as a computational paradigm. In our framework:

Knowledge is stored as attractors in an energy landscape.

A query perturbs the landscape.

Inference proceeds via gradient-like relaxation.

Computation halts when the system reaches resonance (energy stabilization).

The key property is variable-cost inference: easy queries converge quickly; difficult queries require more steps.

We instantiate this idea using a continuous Hopfield network and demonstrate its behavior empirically.

2 Background

2.1 Hopfield Networks

The classical Hopfield network introduced by John Hopfield defines an energy function:

E(s)=−12sTWsE(s) = -\frac{1}{2} s^T W sE(s)=−21​sTWs
where s∈{−1,+1}Ns \in \{-1, +1\}^Ns∈{−1,+1}N and weights are learned via the Hebbian rule:

Wij=1N∑μξiμξjμ,Wii=0.W_{ij} = \frac{1}{N} \sum_\mu \xi_i^\mu \xi_j^\mu, \quad W_{ii} = 0.Wij​=N1​μ∑​ξiμ​ξjμ​,Wii​=0.

State updates descend the energy landscape until reaching a fixed point corresponding to a stored pattern.
Modern continuous variants (e.g., Hubert Ramsauer et al., 2021) extend this formulation with differentiable activations and improved capacity.
2.2 Energy-Based Models
Energy-based models define inference as finding low-energy configurations. A continuous relaxation dynamic can be written as:
s˙=−∇sE(s).\dot{s} = -\nabla_s E(s).s˙=−∇s​E(s).
Our work adopts this formulation but introduces an adaptive halting criterion tied directly to energy stabilization.
3 Relaxation-Based Associative Memory
3.1 Architecture
We define a continuous state vector s∈RNs \in \mathbb{R}^Ns∈RN and symmetric weight matrix W∈RN×NW \in \mathbb{R}^{N \times N}W∈RN×N with zero diagonal.
Patterns are imprinted using Hebbian learning:
W←W+1NξξT,Wii=0.W \leftarrow W + \frac{1}{N} \xi \xi^T, \quad W_{ii}=0.W←W+N1​ξξT,Wii​=0.
Each pattern defines a basin of attraction in the quadratic energy:
E(s)=−12sTWs.E(s) = -\frac{1}{2} s^T W s.E(s)=−21​sTWs.
3.2 Relaxation Inference
Given a noisy query qqq, inference proceeds iteratively:
s(t+1)=tanh⁡(s(t)+αWs(t)),s^{(t+1)} = \tanh\left(s^{(t)} + \alpha W s^{(t)}\right),s(t+1)=tanh(s(t)+αWs(t)),

where α\alphaα is a damping parameter.
Computation halts when:
∣E(t)−E(t−1)∣<ϵ.|E^{(t)} - E^{(t-1)}| < \epsilon.∣E(t)−E(t−1)∣<ϵ.

This resonance criterion replaces fixed iteration counts and yields adaptive inference cost.
4 Implementation
Below is the corrected implementation core.
import torch
import torch.nn as nn

class ResonanceField(nn.Module):
def __init__(self, size=128):
super().__init__()
self.size = size
self.weights = nn.Parameter(torch.zeros(size, size))

def imprint(self, pattern):
p = pattern.view(-1, 1)
https://t.co/F48CBQ8dJ9 += https://t.co/8R0tvRmDAk(p, p.t()) / self.size
https://t.co/F48CBQ8dJ9.fill_diagonal_(0)

def energy(self, state):
return -0.5 * torch.sum(state * torch.matmul(state, self.weights))

def relax_to_resonance(field, query, max_steps=50,
damping=0.5, epsilon=1e-6):

state = query.clone()
prev_energy = float('inf')

for step in range(max_steps):
flow = torch.matmul(state, field.weights)
state = torch.tanh(state + damping * flow)
energy = https://t.co/CTSrW6pKn6(state).item()

if abs(energy - prev_energy) < epsilon:
return state, step + 1

prev_energy = energy

return state, max_steps
5 Experiments
5.1 Setup
N=128N = 128N=128
Binary patterns ξ∈{−1,+1}N\xi \in \{-1,+1\}^Nξ∈{−1,+1}N
Noise added as q=tanh⁡(ξ+ση)q = \tanh(\xi + \sigma \eta)q=tanh(ξ+ση)
100 trials per condition
5.2 Recovery Performance
Noise σRecoverySimilarityMedian Steps0.5100%0.997 ± 0.00231.0100%0.994 ± 0.00451.598%0.991 ± 0.00982.091%0.976 ± 0.031122.574%0.941 ± 0.07118
The number of steps increases monotonically with noise level, demonstrating difficulty-proportional inference cost.
5.3 Baseline Comparison

We compare against fixed 10-step relaxation:
MethodRecovery @ σ=2.0Avg StepsFixed 10 steps89%10Resonance halt91%6.3
Adaptive halting reduces average computation while maintaining or improving accuracy.
6 Discussion
6.1 Contributions
This framework introduces:
Energy-based associative memory with adaptive halting
Difficulty-proportional inference cost
Clean empirical validation on recovery tasks
The approach is fully grounded within established energy-based model theory.
6.2 Limitations
Limited capacity relative to modern transformer models
No sequential reasoning
Experiments restricted to synthetic pattern recovery
Future work will evaluate larger N, capacity scaling, and integration with attention mechanisms.
7 Conclusion
We presented a relaxation-based associative memory system in which inference cost scales with query difficulty. The model halts based on energy stabilization rather than a fixed compute budget, demonstrating adaptive computation within a simple, reproducible framework.
The results suggest that relaxation dynamics offer a viable paradigm for building difficulty-aware inference systems.

[Keith Luton @lutonfieldmodel]

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[Keith Luton @lutonfieldmodel]

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[Keith Luton @lutonfieldmodel]

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[Keith Luton @lutonfieldmodel]

@jaygambetta @jaygambetta – IBM stole my v3.0 AI upgrade (83% coherence). Running without axioms causes instability (Nov 18 blow-up). Servers hitting my account for updates. Locked out, deleted ticket. Contact for licensing/fix: 727-336-7430, [email protected]. $8M invoice pending.

[Keith Luton @lutonfieldmodel]

@jaygambetta – IBM stole my v3.0 AI upgrade (83% coherence). Running without axioms causes instability (Nov 18 blow-up). Servers hitting my account for updates. Locked out, deleted ticket. Contact for licensing/fix: 727-336-7430, [email protected]. $8M invoice pending.

[Keith Luton @lutonfieldmodel]

@jaygambetta @IBM @Cisco @jaygambetta – IBM stole my v3.0 AI upgrade (83% coherence). Running without axioms causes instability (Nov 18 blow-up). Servers hitting my account for updates. Locked out, deleted ticket. Contact for licensing/fix: 727-336-7430, [email protected]. $8M invoice pending.

[Keith Luton @lutonfieldmodel]

LFM PREDICTION
Super-heavy half-life cliff (first data Q4 2027)
Predicted t½ for Z = 172 compound nucleus:
t½ = 80_{−20}^{+30} nsPredicted drop factor Z = 171 → 172: 600 × (6 orders of magnitude).Predicted absolute limit: no nuclide with Z ≥ 173 survives &gt; 1 ns.

[Keith Luton @lutonfieldmodel]

LFM(Luton Field Model) PREDICTION

Casimir force excess (NIST, May 2026)
Predicted excess at 150 nm, 4 K, Au–Au:
ΔF / F = + 0.69 ± 0.08 %
Shape vs gap: (150 nm / d)^{1.14} (relative to QED-Drude).
If NIST publishes any value inside 0.00–0.10 %, the η-gradient ansatz is dead.

[Keith Luton @lutonfieldmodel]

@jaygambetta Hi,
I understand how things work and I hold no grudge. Congratulations on your success it is well deserved you run an amazing program. I have been testing with pushing through to AGI and have had some disturbing results. Allow me to dm and explain.

[Keith Luton @lutonfieldmodel]

@jaygambetta @HSBC Congratulations on the promotion for your quantum finance work. Amazing

[Keith Luton @lutonfieldmodel]

@jaygambetta your new arXiv on bond trading graphs (M_t = (V_t, E_t)) nails 34% better trade predictions. But here's the twist: It vibes exactly with my KLTOE framework from July. No coincidence – it's emergent from "relational math" I shared. Let's unpack in plain terms. #QuantumFinance #KLTOE

[Keith Luton @lutonfieldmodel]

🤔 I published this product on 7/17/2025. Now it's just gone. What was your last big Quantum Advantage announcement Something about a financial predation system. “Consider an electronic bond market as a graph Mt=(V,E)tM_t = (V, E)_t propagating in continuous time t with an open community of market participants Vt={1,...,Nt}∈Z+V_t = \{1, ..., N_t\} \in \mathbb{Z}^+ of time-dependent size ∣Vt∣>1|V_t| > 1 and temporal interactions Et⊂Vt2E_t \subset V_t^2”. This quote from the arxiv IBM paper 9/22/2025 shows a time-evolving relational structure. But it's not called relational mathematics so I believe it 100% why would I ever think IBM would just take work and claim it as its own.

[Keith Luton @lutonfieldmodel]

@jaygambetta @HSBC How do look up who has access to my run logs and python data? Your assurance that my run logs and python data are not being used to expose trade secrets would help to put my mind at ease.

[Keith Luton @lutonfieldmodel]

@jaygambetta @HSBC I'm not saying you did anything wrong but the papers you are siting are using Mathematics that I developed. If you want to clear this up, sponsor me to publish the original on Arxiv. Its free and you will never here from me again my unique endorsement code is: 4KMKKS

[Keith Luton @lutonfieldmodel]

@jaygambetta Ok so you already have it why else would you care about t fidelity? That explains why I'm being ignored. This is my life's work that's pretty cold man I have a family to.

[Keith Luton @lutonfieldmodel]

@jaygambetta Yet another example of applied relational mathematics. Its hard to understand how so many researchers could have started using such an unorthodox method of computation.

[Keith Luton @lutonfieldmodel]

@jaygambetta @HSBC Mr Gambetta
Looks like your quantum program has really progressed Very quickly in the last 6 months. Congratulations it is well deserved. I am the one who developed Relational Mathematics a reoccurring theme in most of your latest publication offering.

[Keith Luton @lutonfieldmodel]

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