The Coordinate System Problem in Persistent Structural Memory for Neural Architectures

Abhinaba Basu

cs.LGMar 24, 2026

The Coordinate System Problem in Persistent Structural Memory for Neural Architectures

Authors: Abhinaba Basu

Organizations: National Institute of Electronics and Information Technology (NIELIT)

Abstract

We introduce the Dual-View Pheromone Pathway Network (DPPN), an architecture that routes sparse attention through a persistent pheromone field over latent slot transitions, and use it to discover two independent requirements for persistent structural memory in neural networks. Through five progressively refined experiments using up to 10 seeds per condition across 5 model variants and 4 transfer targets, we identify a core principle: persistent memory requires a stable coordinate system, and any coordinate system learned jointly with the model is inherently unstable. We characterize three obstacles -- pheromone saturation, surface-structure entanglement, and coordinate incompatibility -- and show that neither contrastive updates, multi-source distillation, Hungarian alignment, nor semantic decomposition resolves the instability when embeddings are learned from scratch. Fixed random Fourier features provide extrinsic coordinates that are stable, structure-blind, and informative, but coordinate stability alone is insufficient: routing-bias pheromone does not transfer (10 seeds, p>0.05). DPPN outperforms transformer and random sparse baselines for within-task learning (AULC 0.700 vs 0.680 vs 0.670). Replacing routing bias with learning-rate modulation eliminates negative transfer: warm pheromone as a learning-rate prior achieves +0.003 on same-family tasks (17 seeds, p<0.05) while never reducing performance. A structure completion function over extrinsic coordinates produces +0.006 same-family bonus beyond regularization, showing the catch-22 between stability and informativeness is partially permeable to learned functions. The contribution is two independent requirements for persistent structural memory: (a) coordinate stability and (b) graceful transfer mechanism.

Figures & tables

Figure 1: (a) DPPN architecture schematic. Tokens are embedded and passed through dual soft groupers to produce slot assignments. Slot-level support and agreement are combined with the persistent pheromone field (highlighted) to produce pheromone-biased routing, which generates a sparse attention mask. A fast/slow gate fusion produces the final output. (b) Transfer protocol. In Phase 1, the model is trained on a source task and pheromone accumulates structural memory. In Phase 2, all model weights are reset but pheromone is either kept (warm) or reset (cold), and the model is trained on a target task.

Epoch	Val. Acc.	$\bar{\tau}\pm\sigma_{\tau}$	Entropy
0	0.476	$0.331\pm 0.227$	6.76
5	0.638	$0.358\pm 0.637$	5.97
10	0.708	$0.356\pm 0.645$	5.94
20	0.722	$0.338\pm 0.629$	5.93
30	0.734	$0.338\pm 0.629$	5.93
50	0.734	$0.338\pm 0.629$	5.93

Table 1: Pheromone evolution during source training (DPPN, seed 42). The field develops structure within the first 30 epochs, then stabilizes.

Figure 2: Pheromone field $\boldsymbol{\tau}\in\mathbb{R}^{32\times 32}$ before and after source training (DPPN, seed 42). (a) At initialization, pheromone is near-uniform ( $\bar{\tau}\approx 1.05$ ). (b) After 80 epochs, the field is sparse and structured: most transitions have decayed to $\tau_{\min}=0.1$ (blue), with a small number of high-pheromone transitions at $\tau_{\max}=2.0$ (red).

		Same Family (A)		Different Family
Model	Condition	A2	A3	B1	C1
DPPN	Cold	$.699\pm.014$	$.702\pm.012$	$.715\pm.017$	$.682\pm.016$
	Distilled	$.700\pm.017$	$.701\pm.012$	$.711\pm.012$	$.678\pm.016$
	Rank-reduced	$.700\pm.015$	$.700\pm.011$	$.714\pm.015$	$.683\pm.016$
	$\Delta_{\text{distilled}}$	$+.001$	$-.001$	$-.004$	$-.004$
Transformer	Cold	$.687\pm.012$	$.679\pm.013$	$.690\pm.017$	$.665\pm.020$

Table 2: Transfer results (AULC), mean

\pm

std over 3 seeds. “Distilled” and “Rank-reduced” are two warm pheromone conditions for DPPN. Transformer and Random Sparse have only cold (no pheromone to transfer).

\Delta

= warm

-

cold.

		Same Family		Different Family
Model (distilled)	Seeds	A2	A3	B1	C1
DPPN (learned + Hungarian)	3	$+0.002\pm 0.007$	$-0.001\pm 0.001$	$-0.004\pm 0.005$	$-0.003\pm 0.002$
Fourier (token+position)	3	$-0.001\pm 0.004$	$-0.008\pm 0.004$	$-0.005\pm 0.003$	$-0.006\pm 0.004$
Fourier (position-only)	3	$+0.003\pm 0.004$	$-0.006\pm 0.006$	$-0.007\pm 0.003$	$+0.001\pm 0.005$
Fourier (position-only)	10	$\mathbf{-0.001}\pm 0.005$	$\mathbf{-0.002}\pm 0.006$	$\mathbf{-0.002}\pm 0.006$	$\mathbf{-0.002}\pm 0.006$

Table 3: Transfer advantage (

\Delta

AULC: warm distilled

-

cold, mean

\pm

std). Positive values indicate pheromone transfer helps. The Position-Only Fourier variant with 10 seeds shows uniformly negative advantages, indistinguishable from the other routing-bias variants. The 3-seed result that appeared directionally correct did not replicate.

Figure 3: Transfer advantage (

\Delta

AULC: warm distilled

-

cold) across model variants and transfer targets. Same-family tasks (A2, A3) are shaded green; different-family tasks (B1, C1) are shaded red. With 3 seeds, the Position-Only Fourier variant appeared to show positive transfer on A2 (

+0.003

), but with 10 seeds the advantage is

-0.001\pm 0.005

(not significant). All routing-bias variants show uniformly negative or zero transfer advantages. Error bars: std over seeds.

Experiment	What it fixed	Next obstacle identified	Insight
1: Contrastive updates	Pheromone saturation ( $\tau\to$ uniform)	Surface entanglement	Structured pheromone $\neq$ transferable pheromone
2: Single-source transfer	Task difficulty calibration	Surface-structure coupling	Cannot disentangle with one instance
3: Multi-source distillation	Surface contrast (two instances)	Coordinate mismatch (3.5% vs. 3.1%)	Learned slots are unaligned across runs
4: Decomposition coordinates	Slot arbitrariness (fixed regions)	Embedding arbitrariness ( $-10.4\%$ )	Any learned coordinate system is unstable
5: Position-only Fourier	Extrinsic, surface-free coordinates	Routing-bias mechanism itself — even with stable coordinates, biasing forward-pass routing from a previous task imposes a cost	Coordinate stability is necessary but not sufficient. The transfer mechanism (routing bias vs. learning rate) matters independently.

Table 4: Progressive diagnosis across five experiments. Each resolved one obstacle but exposed the next. The final experiment (position-only Fourier, 10 seeds) reveals that coordinate stability is necessary but not sufficient: routing-bias pheromone does not transfer even with stable coordinates.

Figure 4: The diagnostic cascade. Five experiments, each resolving one obstacle (left, green) while revealing the next (right, red). The center column shows the key metric from each experiment. All obstacles trace to the same root cause: persistent memory requires stable coordinates, and learned coordinates are inherently unstable.

Figure 5: Coordinate stability diagnostics. (a) Slot alignment correlation between independently trained DPPN models: 3.5%, barely above the 3.1% expected by random chance with 32 slots. (b) Distillation survival: the number of high-magnitude transitions (out of 1024) surviving element-wise minimum distillation. Position-Only Fourier preserves the most transitions (22), consistent with better coordinate stability.

		Same Family		Different Family
Mechanism	Seeds	A2	A3	B1	C1
Meta-LR	7	$+0.003\pm 0.006$	$+0.000\pm 0.005$	$+0.002\pm 0.004$	$+0.002\pm 0.004$
Routing bias	10	$-0.001\pm 0.005$	$-0.002\pm 0.006$	$-0.002\pm 0.006$	$-0.002\pm 0.006$

Table 5: Transfer advantage (

\Delta

AULC: warm

-

cold) for meta-learning-rate pheromone vs. routing-bias pheromone, both with position-only Fourier coordinates. Neither achieves statistically significant positive transfer, but they differ qualitatively: routing bias is uniformly negative, learning-rate modulation is uniformly non-negative.

	Same Family		Different Family
Mechanism	A2	A3	B1	C1
Completion (learned $\mathbf{A}$ )	$-0.004$	$-0.007^{**}$	$-0.004$	$-0.003$
Completion (extrinsic $\mathbf{P}$ )	$+0.007$	$+0.016^{***}$	$+0.015^{***}$	$+0.016^{***}$
Random completion control	$+0.005$	$+0.012$	$+0.016$	$+0.009$
Trained $-$ Random (same fam.)	$+0.013$	$+0.004$	$-0.002$	$+0.000$

Table 6: Structure completion transfer advantage (

\Delta

AULC, 10 seeds each). Completion over learned

\mathbf{A}

is the worst mechanism (negative everywhere). Completion over extrinsic

\mathbf{P}

shows the largest positive signal, but a random completion control (5 seeds) decomposes it into regularization (

{\sim}+0.010

) and a differential same-family bonus (

{\sim}+0.006

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value
Model dimension ( $d$ )	64
Number of heads	4
Number of layers	3
Number of slots ( $m$ )	32
Top- $k$ (sparse mask)	32
Slow-lane window ( $w$ )	16

Appendix

Table 7: DPPN hyperparameters used in all experiments. Note: early experimental rounds (Sections 5 – 6 ) used

d=128

, 4 heads, 4 layers,

m=64

; after task recalibration, the values below were adopted for all reported results.

Parameter	Value
Vocabulary size	32
Sequence length	128
Training samples	2000
Motifs per sample	2–3
Noise level	0.02
Number of classes	2

Appendix

Table 8: Synthetic task parameters.

Seed	Condition	A2	A3	B1	C1
42	Cold	.683	.689	.693	.676
	Distilled	.677	.690	.695	.675
	Rank-red.	.678	.691	.695	.678
137	Cold	.696	.697	.719	.665
	Distilled	.706	.696	.715	.661
	Rank-red.	.707	.693	.715	.666

Appendix

Table 9: DPPN transfer results (AULC) by seed and condition.

Model	Seed	A2	A3	B1	C1
Transformer	42	.671	.661	.669	.653
	137	.688	.686	.689	.648
	256	.700	.691	.711	.692
Random Sparse	42	.658	.657	.671	.642
	137	.680	.675	.703	.638
	256	.696	.687	.684	.669

Appendix

Table 10: Baseline results (AULC) by seed (cold condition only).

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D Yang Eng

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Transfer in coordinate networks is often measured by warm-start gain, but whether that gain reflects source-specific structure or generic weight reuse is less clear. We study this question across three implicit neural representation (INR) families, SIREN, ReLU MLPs, and Fourier-feature MLPs, using controlled analytic tests, a 2D lid-driven-cavity Navier--Stokes benchmark, and 1D PDE reference-solution suites for heat, viscous Burgers, and focusing cubic NLS. The analytic tests use independent-seed random controls, while the PDE benchmarks use alternate same-family source controls and auxiliary ablations. Across settings, transfer magnitude and transfer specificity separate clearly. In a 10-seed controlled 1D geometric test, Fourier Features show the largest structured transfer (

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is rejected in the implemented 1D audit. These results position transfer specificity as a useful diagnostic for coordinate networks and suggest that architecture selection in scientific machine learning should be evaluated under explicit control conditions, not by transfer magnitude alone.

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