Intelligence Inertia: Physical Isomorphism and Applications

Jipeng Han

cs.AIMar 22, 2026

Intelligence Inertia: Physical Isomorphism and Applications

Authors: Jipeng Han

Organizations: OpenImmortal Technology Co., Ltd. Beijing 101100, China

Abstract

Classical frameworks like Fisher Information approximate the cost of neural adaptation only in low-density regimes, failing to explain the explosive computational overhead incurred during deep structural reconfiguration. To address this, we introduce Intelligence Inertia, a property derived from the fundamental non-commutativity between rules and states ( $[\hat{S}, \hat{R}] = i\mathcal{D}$ ). Rather than claiming a new fundamental physical law, we establish a heuristic mathematical isomorphism between deep learning dynamics and Minkowski spacetime. Acting as an effective theory for high-dimensional tensor evolution, we derive a non-linear cost formula mirroring the Lorentz factor, predicting a relativistic $J$ -shaped inflation curve -- a computational wall where classical approximations fail. We validate this framework via three experiments: (1) adjudicating the $J$ -curve divergence under high-entropy noise, (2) mapping the optimal geodesic for architecture evolution, and (3) deploying an inertia-aware scheduler wrapper that prevents catastrophic forgetting. Adopting this isomorphism yields an exact quantitative metric for structural resistance, advancing the stability and efficiency of intelligent agents.

Figures & tables

Figure 1: Geometric Partition of Logical Action. A particle collision with total action

l

is decomposed by a microscopic slant. The component

l_{R}=l\sin\theta

is absorbed by the adiabatic rule-manifold, while the normal component

l_{S}

governs state-expression. Heat emission is only registered when the cumulative normal action matches a full vertical collision relative to the system’s local energy level , naturally inducing the relationship

l^{2}=l_{S}^{2}+l_{R}^{2}

Dynamical Model	Reference Frame Treatment	Fit Error (RMSE)	Performance
Classical FIM	Absolute Coordinates	36.0	Failed : Incapable of capturing the non-linear “J-Curve” divergence.
Classical FIM	Galilean Shift ( $v-v_{0}$ )	30.0	Limited : Fits the low-speed regime but severely underestimates the wall effect.
Hybrid FIM	Lorentz Transformation	25.5	Suboptimal : Corrects for velocity addition but lacks mass expansion logic.
Relativistic Mass (Intelligence Inertia)	Universal Covariance	18.5 – 19.6	Optimal : Precisely fits the asymptotic divergence and exhibits frame independence.

Table 1: Dynamical Model Fitting Performance across Coordinate Systems. The Intelligence Inertia Theory, utilizing relativistic mass expansion, consistently outperforms classical Fisher Information approximations, particularly as the system approaches the informational speed limit.

Figure 2: Comparative Adjudication of Reference Frame Sensitivity and Model Robustness. Arena 1 (Left) demonstrates the sensitivity of the classical FIM model to coordinate shifts; the model fails to track the data trend in absolute coordinates. Arena 2 (Right) illustrates the Intelligence Inertia model, where the relativistic curves remain covariant and highly accurate regardless of the reference frame, maintaining an RMSE between 18.5 and 19.6.

Figure 3: Ablation Analysis of Relativistic Velocity Addition vs. Mass Expansion. Arena 3 (Left) contrasts the Galilean-shifted FIM against the relativistic mass model, showing the clear failure of the quadratic assumption at high speeds; Arena 4 (Right) introduces a “Hybrid FIM” model, which applies the relativistic Lorentz velocity transformation but retains the classical quadratic cost formula, highlighting that velocity correction alone is insufficient to explain the “J-Curve” inflation.

Arch. Name	Rule Reconfig. ( $dS_{R}$ Axis)	Ext. Gain ( $dS_{ext}$ Axis)	Vel. ( $v$ )	Vel. Dev. $\|v-0.5\|$	Reach. Limit ( $\mathcal{L}_{min}$ )	Work (TFLOPs)
MLP (Origin)	Baseline (R0)	Baseline (S0)	0.502	0.002	2.302	0.6
BN	Normalized (R1)	Baseline (S0)	0.522	0.022	1.389	68.7
Res	Residual (R2)	Baseline (S0)	0.818	0.318	1.322	117.0
MLP-CNN	Baseline (R0)	Locality (S1)	0.179	0.321	1.485	830.0
MLP-MCNN	Baseline (R0)	Multi-scale (S2)	0.194	0.306	1.339	3670.0
BN-CNN	Normalized (R1)	Locality (S1)	0.532	0.032	0.647	664.0

Table 2: Comparative Analysis of Dynamical Parameters and Reachability Limits across Neural Architectures. Abbreviations for architectures: MLP (Multi-Layer Perceptron), BN (Batch Normalization), Res (Residual), CNN (Convolutional Neural Network), MCNN (Multi-scale Convolutional Neural Network). Column Abbreviations: Rule Reconfig. (Internal Rule-Reconfiguration), Ext. Gain (External State Gain), Vel. (Velocity), Vel. Dev. (Velocity Deviation from 0.5), Reach. Limit (Reachability Limit

\mathcal{L}_{min}

), Work (Computational Effort in TFLOPs). The data shows that the lowest reachability limit is attained by architectures like Res-MCNN, which achieve optimal balance between rule effort and state feedback, minimizing the deviation from the

v\approx 0.5

golden axis.

Figure 4: 3D Reachability Topography and the Zig-Zag Evolutionary Geodesic. This 3D surface plots the Reachability Limit (

\mathcal{L}_{min}

) as a function of improvements in internal manifold smoothing (

dS_{R}

-axis) and external structural bias (

dS_{ext}

-axis). The red nodes represent measured architectures from our matrix, while the blue line illustrates the “Zig-Zag Geodesic,” representing the path of maximum efficiency. The topography exhibits a distinct saddle shape, where the steepest descent occurs along the diagonal of balanced development.

Figure 5: Velocity Deviation Topography and the Dynamical Riverbed. This 3D surface maps the absolute deviation of system velocity from the

0.5

golden axis. The resulting “V-shaped” valley—the Dynamical Riverbed—approximately aligns with the optimal evolutionary path, identifying the minimization of velocity deviation as the physical objective of architectural progress.

Sched.	Cfg.	Loss@30	Prog.@30	Accel.	Reach. $\mathcal{L}_{min}$	Gain	Best Ep.
Pure Inertia	Wrapper Only	1.3392	94.31%	N/A	1.2811	N/A	96
Cosine Ann.	Base / Enh.	1.419 / 1.395	82.2% / 83.7%	+1.74%	1.229 / 1.218	+0.87%	98 / 97
OneCycle	Base / Enh.	1.462 / 1.441	79.1% / 80.1%	+1.46%	1.240 / 1.227	+1.10%	96 / 97
Multi-Step	Base / Enh.	1.431 / 1.396	82.6% / 85.5%	+2.45%	1.247 / 1.242	+0.40%	91 / 92
Cyclic LR	Base / Enh.	1.431 / 1.396	84.3% / 87.4%	+2.46%	1.268 / 1.265	+0.21%	99 / 99
Polynomial	Base / Enh.	1.396 / 1.381	83.9% / 84.9%	+1.08%	1.222 / 1.217	+0.43%	96 / 98

Table 3: Comparative Statistical Analysis of Inertial Enhancement Across Schedulers. The table provides a quantitative leap in training efficiency and reachability. Sched.: PyTorch [ 68 ] base scheduler, including Cosine Annealing and Cosine Restart [ 47 ] , OneCycle Policy [ 69 ] , Cyclic LR [ 70 ] , Multi-Step, Polynomial, Exponential, and ReduceLROnPlateau; Cfg.: Operation mode; Prog.@30: Convergence percentage at Epoch 30; Accel.: Improvement in Progress; Gain: Reduction in

\mathcal{L}_{min}

Figure 6: Universal Enhancement of Learning Dynamics via the Inertia-Aware Scheduler Wrapper. The panels contrast validation loss (Left) and learning rate behavior (Right) across eight schedulers. Solid lines indicate enhanced systems, while dashed lines indicate baselines; the vertical red dotted line marks the critical Epoch 30 checkpoint where dynamical acceleration is most visible.

Group	Phase	Avg. Loss	Avg. Vel.	Avg. LR	Brake Ratio
Baseline (Exponential)	Pre-Shock	1.4878	0.6505	0.041532	N/A
	Clean Pulse	1.4142	0.6723	0.009068	0.9x (None)
	Noise Pulse	2.1845	0.6926	0.010367	-
Pure Inertia (Wrapper Only)	Pre-Shock	1.4379	0.7525	0.015874	N/A
	Clean Pulse	1.3541	0.8111	0.001659	1.2x ( Active )
	Noise Pulse	2.0841	0.8634	0.001420	-

Table 4: Comparative Dynamical Metrics under Periodic High-Entropy (100% Noise) Shocks. This table contrasts the response of a traditional exponential scheduler against our standalone inertia-aware wrapper during intermittent noise injection, revealing that the regulated system autonomously applies protective braking while the baseline remains blind to the noise-induced entropy surge.

Figure 7: Logical Resilience and Relativistic Braking under Noise Shock. (Left) Validation loss across epochs; the baseline (dashed yellow) exhibits convergence of clean and noisy loss curves, indicating disruption of rules and assimilation by noise, while the regulated system (solid blue) maintains a distinct dual-track separation. (Right) Relativistic Braking response; the Scheduler wrapper autonomously suppresses its step length during noise steps to preserve rule-purity.

Figure 8: Impact of Pulsed Noise on Velocity. This figure illustrates the velocity response (

v

) across training epochs. (Left) Experimental group (Regulated) and (Right) Control group (Baseline). Both groups show that the onset of label noise causes a distinct phase separation, where velocity spikes from a steady-state toward the informational limit, confirming that system velocity is a reliable physical indicator of informational shocks.

Metric	Exponential	Wrapper	Improvement
Pre-trans. Loss	0.9253	0.8879	-
Old Task Final Loss	9.1505	7.8801	13.88% Forgetting Reduction
Retention Deficit ( $\Delta\mathcal{L}$ )	8.2626	6.9922	15.38% Rule Retention
Full Task Final Loss	4.9093	4.3232	11.94% Comprehensive Synergy
Inst. Breaking Ratio	1.08x	2.92x	Autonomous Braking

Table 5: Comparison of Dynamical Stability Indicators in a Replay-Free Continual Learning Scenario. This table contrasts the baseline group against the regulated system during an abrupt task switch. The results indicate that a higher awareness of inertia leads to significantly lower forgetting and better overall task reachability by enforcing an autonomous collision brake at the moment of transition. Certain metrics are abbreviated: “Pre-trans. Loss” refers to Pre-transition Loss (Ep. 19) and “Inst. Breaking Ratio” refers to the Instantaneous Inertial Breaking Ratio.

Figure 9: Inertial Barrier during Abrupt Task Transitions. This figure contrasts the behavioral strategies during the task switch at Epoch 20. (Left) Validation loss trajectories where the regulated group suppresses the surge of Old Task Loss; (Right) Step-length dynamics, where the Scheduler wrapper executes an immediate “Protective Braking” maneuver in response to the high-velocity logic collision.

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Preserving Plasticity in Continual Learning via Dynamical Isometry

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Continual training of deep neural networks under non-stationarity often leads to a progressive loss of plasticity, eventually limiting further learning. We relate plasticity to the empirical Neural Tangent Kernel, and identify dynamical isometry (the condition that layer-wise Jacobian singular values remain close to one) as a key mechanism for preserving plasticity in continual learning. We revisit a class of networks that are almost-everywhere isometric while remaining universal Lipschitz function approximators, demonstrating that near-dynamical isometry is compatible with expressive nonlinear representations. For general architectures, we propose an efficient isometry-promoting regularization scheme and identify a novel mechanism by which it can reactivate dormant ReLU units. Building on this, we introduce AdamO, an Adam-style adaptive optimizer that decouples isometry regularization from gradient updates, analogous to AdamW. We further reinterpret prior plasticity-preserving approaches through the lens of dynamical isometry, showing that they target only a partial measure of isometry. Across supervised and reinforcement-learning continual-learning benchmarks designed to induce plasticity loss, our methods consistently match or outperform existing approaches.

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