Beyond LLMs, Sparse Distributed Memory, and Neuromorphics <A Hyper-Dimensional SRAM-CAM "VaCoAl" for Ultra-High Speed, Ultra-Low Power, and Low Cost>

Hiroyuki Chuma; Kanji Otsuka; Yoichi Sato

cs.NEApr 13, 2026

Beyond LLMs, Sparse Distributed Memory, and Neuromorphics <A Hyper-Dimensional SRAM-CAM "VaCoAl" for Ultra-High Speed, Ultra-Low Power, and Low Cost>

Authors: Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

Organizations: Professor Emeritus, Hitotsubashi University · Professor Emeritus, Meisei University · Shuhari System

Abstract

VaCoAl (Python: PyVaCoAl) attacks the Binding Problem algebraically rather than statistically: an SRAM/DRAM-CAM organised end-to-end around one primitive, XOR-and-shift over GF(2) via primitive-polynomial LFSRs. Rooted in Kanerva's Sparse Distributed Memory, it retrieves in a million-dimensional binary space by Galois-field diffusion; Binding and Unbinding are exactly reversible at O(L), giving compositional generalisation with post-hoc auditability. Unexpectedly, a path-dependent semantic selection mechanism emerges undesigned, functionally equivalent to Spike-Timing-Dependent Plasticity (STDP) and predictable a priori from a closed form matching measurement on tens of millions of records. It is the Don't Care (collision-tolerance) rule: not a defect to engineer away, but the source of its ability to rank paths by quality. Capability: reversible binding plus tolerated collisions yields a path-integral confidence measure (CR2) ranking candidates by accumulated reliability, absent from hash-based search. Necessity: repair the collisions fully and the ranking vanishes; a memory that never fails has no record of which paths were difficult. Position: we do not surpass large language models; the substrate is orthogonal, supplying the auditable path-ranking layer they lack. A mentor-student ontology of ~470,000 WIKIDATA scholars, traversed backwards from all 64 Fields Medalists to depth 57 and >25.5M paths, stress-tests multi-hop reasoning over a large DAG. Ablation shows Don't Care imposing a depth-dependent exponential penalty that prunes circuitous routes and preserves direct ones: an Occam's razor. Per-generation CR1 stays within 0.995-0.999, yet cumulative CR2 decays monotonically to ~0.905 by generation 56, matching the closed form 0.997^56 = 0.846 within observed variation. All measurements are from PyVaCoAl, a software DRAM-CAM; speed and power figures are projections.

Figures & tables

Comparison Axis	Analog Neuromorphic	VaCoAl
Orthogonalization	Spike timing	Primitive-polynomial modular arithmetic
Error correction	Attractor dynamics	Algebraic majority voting
Auditability	Difficult	Traceable via CR2 path integral
Thermal-noise tolerance	Device-variation dependent	Irrelevant (digital logic)
Demonstrated scale	Predominantly small-scale	25.5 M records (this study)

Table 1: Comparison of analog neuromorphic implementations and VaCoAl.

Symbol	Meaning	Remarks
$L$	Hyper-vector length ( $=$ dimensionality of the binary space)	$1{,}000$ – $10{,}000$ in Kanerva’s original setting; $6{,}400$ – $102{,}400$ in the experiments of Section 5
$B$	Number of blocks	$64$ – $2{,}048$ ; each block is an independent SRAM/DRAM array
$q$	Block length in bits	$q=L/B$ ; $q=100$ in every configuration of Section 5
$m$	Degree of the generator polynomial $=$ memory-depth exponent	The per-block address is the $m$ -bit residue
$M_{B}$	Memory depth per block	$M_{B}=2^{m}$ ; the fixed total capacity is $B\cdot 2^{m}=2^{34}$
$A$	Per-block address space	$A=2^{m}-1$ (Appendix A)

Table 2: Notation used throughout this paper.

Stage	Dendritic Computation	VaCoAl
Local Processing	Dendritic branches (local nonlinear processing)	SRAM/DRAM blocks (local exact-match checking)
Integration	Soma (spatial summation/integration)	Majority-voting circuit (integration of block outputs)
Output	Axon output (firing)	Final decision output

Table 3: Structural Correspondence between Dendritic Computation and VaCoAl

Configuration	Location-based Rate	Count-based Rate	Performance
$B=64,L=6{,}400,m=28$	$0.000103\%$	$0.074393\%$	Poor
$B=128,L=12{,}800,m=27$	$0.000399\%$	$0.143756\%$	Fair (Leibniz Dominant)
$B=256,L=25{,}600,m=26$	$0.001565\%$	$0.282177\%$	Fair (Gauss Dominant)
$B=512,L=51{,}200,m=25$	$0.006189\%$	$0.559094\%$	Excellent (Leibniz Dominant)
$B=1024,L=102{,}400,m=24$	0.024437%	1.107873%	Excellent (Leibniz Dominant)

Table 4: Detailed Collision Rate Analysis (fixed total capacity

B\cdot 2^{m}=2^{34}

). “Performance” refers to the semantic purity of the extracted lineage as assessed in Section 6.2, not to speed or to reachable generational depth.

Configuration	Memory Depth	Rescue	FS Pruning Order	Time (sec)	vs DICT
A: Pseudo-PyVaCoAl(1)	—	—	Alphabetical	$1{,}670.8$	Baseline
B: PyVaCoAl 128_10 w/ rescue	$2^{10}$	Yes ( $RR=1$ )	Alphabetical	$1{,}342.4$	$-19.7\%$
C: PyVaCoAl 128_19 w/ rescue	$2^{19}$	Yes ( $RR=1$ )	Alphabetical	$1{,}514.1$	$-9.4\%$
D: PyVaCoAl 128_27 w/o rescue	$2^{27}$	No ( $RR=0$ )	Descending CR2	$1{,}420.1$	$-15.0\%$

Table 5: Comparison of Computation Times across Configurations (all at

\mathrm{FS}=20{,}000

, 128 blocks)

FS	DICT (sec)	PyVaCoAl 128_10 (sec)	PyVaCoAl 128_19 (sec)	PyVaCoAl 128_27 (sec)
2,000	79.1	192.5	467.5	527.5
10,000	760.7	—	814.2	1,031.9
20,000	1,670.8	1,342.4	1,514.1	1,420.1
25,000	Out of Memory	—	1,628.5	—

Table 6: Comparison of Computation Times at Each FS

Rank	Scholar (Active Period)	Giant Score	Calculus Degree	Path Count
1	Johann Friedrich Pfaff (1765–1825)	0.8409	0.6376	33
2	Gottfried W. Leibniz (1646–1716)	0.5973	0.5686	47
3	Jacob Bernoulli (1655–1705)	0.4234	0.5508	45
12	Leonhard Euler (1707–1783)	0.2440	0.5293	45
17	Carl Friedrich Gauss (1777–1855)	0.1665	0.5310	29
91	Isaac Newton (1643–1727)	0.0249	0.5336	4

Table 7: Scholar Evaluation Scores

Hub Node	Avg. Paths (Mentor Dir.)	Avg. Paths (Student Dir.)	Thickness Ratio (Student/Mentor)
Leibniz	5.3	53.4	$10.12\times$
Gauss	2.7	41.2	$15.14\times$
Hilbert	1.6	29.2	$18.46\times$
Euler	3.6	52.4	$14.59\times$

Table 8: Path Counts and Thickness Ratios at Hub Nodes

Figure 1: Hourglass (funnel) structure of inference traffic centered on Leibniz. The constriction separates thin pre-Leibniz paths from the explosive post-Leibniz “superhighway”.

Figure 2: Inference traffic of Isaac Newton (absence of hub structure). In stark contrast to Leibniz, Newton’s traffic forms a thin thread with no constriction-and-explosion pattern.

Generation	Scholar	Birth–Death	Birthplace	Native Language	Paths
Gen 1	Sharaf al-Dīn al-Ṭūsī	1135–1201	Tus	[Arabic, Persian]	54
Gen 2	Kamal al-Din bin Younis	1156–1242	Mosul	[Arabic (,Kurdish)]	54
Gen 3	Nasir al-Din al-Tusi	1201–1274	Tus	[Persian]	54
Gen 4	Shams al-Dīn al-Bukhārī	1254–1300	Bukhara	[Persian]	54
Gen 5	Grigorios Choniades	1240–1320	Constantinople	[Medieval Greek]	54

Table 11

Century	Count	Share	Principal Historical Context / Representative Figures
11th ( $\sim$ 1100)	1	0.50%	Nascent Scholasticism (Lanfrancus)
12th (1101–1200)	13	6.10%	Twelfth-Century Renaissance; onset of the translation movement
13th (1201–1300)	21	9.90%	Golden Age of Islamic science (al-Tusi et al. ); founding of universities
14th (1301–1400)	24	11.30%	Maturation of Scholasticism; early Renaissance (Petrarch et al. )
15th (1401–1500)	34	16.00%	Flowering of the Italian Renaissance; influx of knowledge from Byzantium
16th (1501–1600)	79	37.10%	[Peak] Dawn of the Scientific Revolution (Copernicus, Vesalius)

Table 9: Century-by-Century Breakdown of the 213 Scholars with Maximum Path Count 54 to the 64 Fields Medalists, and Historical Context

Decades	algebra		astronomy
	Signal Value	Increase $\blacktriangle$ Decrease $\triangledown$	Signal Value	Increase $\blacktriangle$ Decrease $\triangledown$
1100–1150	0.4999	(Baseline)	0.5003	(Baseline)
1150–1200	0.5001	$+0.0001$	0.4999	$-0.0003$
1200–1250	0.4998	$-0.0003$	0.4999	$-0.0000$
1250–1300	0.4997	$-0.0001$	0.5024	$+0.0025$	$\blacktriangle\blacktriangle$
1300–1350	0.5001	$+0.0004$	0.5021	$-0.0003$

Table 10: Rate of Signal Change between Adjacent Windows (algebra / astronomy)

Era	Total Scholars	Society Members (%)	Total Memberships	Avg. Memberships/Person
Pre ( $\sim$ 1646)	434	28 (6.5%)	36	0.08
Interim (1646–1716)	87	44 (53.0%)	107	1.29
Post (1716 $\sim$ )	504	414 (82.1%)	1920	3.81

Table 11: Scholars and Academic-Society Membership by Leibniz’s Era: Pre, Interim and Post

PyVaCoAl (no rescue)			Python DICT
Rank	Votes(10k)	Scholar (d.)	Rank	Votes(10k)	Scholar (d.)
1	1,510	J. Thomasius (1684)	1	1,528	J. Thomasius (1684)
2	1,321	Leibniz (1716)	2	1,378	B. Meisner (1626)
3	1,268	B. Meisner (1626)	3	1,339	C.A. Hausen (1743)
4	1,214	W. of Ockham (1349)	3	1,339	A.G. Kästner (1800)
5	1,182	Duns Scotus (1308)	5	1,167	A. Rhode (1633)

Table 13: Top-20 Vote Comparison: PyVaCoAl (128_27, no rescue) vs. Python DICT

Group	Mean Calculus Degree	Mean Giant Score	Mean Path Count
DICT Exclusive (5 physicians/humanists)	0.5011	0.0114	54.0
PyVaCoAl Exclusive (5 mathematicians: the four of the analytical mainstream, plus Leibniz)	0.5320	0.2674	46.8

Table 14: Comparison of Mean Indicator Values between Groups

Figure 3: CR1 (Top 4) and CR2 (Bottom 4) performance metrics overlay. Notice how the shared ’generation’ x-axis and dual y-axes visually integrate all configurations.

$B=512$ , $L=51{,}200$ , $m=25$				$B=1024$ , $L=102{,}400$ , $m=24$
Rank	Votes ( $\times 10^{4}$ )	Scholar (d.)	Note	Rank	Votes ( $\times 10^{4}$ )	Scholar (d.)
1	2,023	J. Thomasius (1684)	Mentor of the hub	1	1,694	J. Thomasius (1684)
2	1,679	B. Meisner (1626)		2	1,309	B. Meisner (1626)
3	1,459	Leibniz (1716)	Hub of the superhighway	3	1,302	A. Rhode (1633)
4	1,449	A. Hegius (1498)		4	1,151	Leibniz (1716)
5	1,319	A. Rhode (1633)		5	1,082	A. Hegius (1498)

Table 15: Top-20 Vote Ranking Comparison: PyVaCoAl

B=512

L=51{,}200

m=25

vs.

B=1024

L=102{,}400

m=24

(

RR=0

)

Representative	Depth	Formula	Reachability
Euler (main stream mathematics)	$\sim$ 10 gens	$0.992^{10}$	$\approx 0.923$
Fabricius (anatomy)	$\sim$ 25 gens	$0.992^{25}$	$\approx 0.819$

Table 16: Reachability by path depth. The figures assume the hypothetical worst case in which one of the 128 blocks is Don’t Care at every generation (

\mathrm{CR}_{1}=127/128\approx 0.992

); the measured per-generation mean in the 128_27 configuration is

\mathrm{CR}_{1}\approx 0.997

(Figure 4).

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Structural Layer	Biological Implementation	VaCoAl Implementation	Convergent Computational Problem
Dendrites	$B$ independent local nonlinear processes. Threshold judgment via NMDA spikes.	Local exact-match checking by $B$ (64–2048) independent SRAM blocks.	Divide-and-parallel processing of high-dimensional inputs.
Soma	Spatial integration of dendritic inputs. Summation with distance-dependent decay.	Equal-weight integration of block outputs by a majority-voting circuit (CR1).	Generation of global confidence by integrating local judgments.
Axon	Binary output of firing/non-firing. Propagation of action potentials.	Output of CR2 score and final decision. Propagation to the Frontier generation.	Signal transmission to the next stage.

Appendix

Table 17: Convergent Computational Equivalence between the Neural Three-Layer Structure and PyVaCoAl

Figure 4: Generational trajectories of CR1 and CR2: PyVaCoAl 128_27 (without rescue) vs. 128_10 (with rescue). In the 128_27 configuration, the per-generation block-voting agreement rate CR1 remains within the extremely narrow range of 0.995–0.999, maintaining “near-perfect search” at individual steps. However, CR2 (the cumulative product of CR1) decays monotonically with each passing generation, dropping to approximately 0.905 at the 56th generation. In the 128_10 configuration with rescue, both CR1 and CR2 are maintained at 1.0 across all generations.

Comparison Axis	Analog Neuromorphic	VaCoAl (Digital Algebra)
Means of decay	Synaptic-strength change via spike-timing difference	Multiplicative integration of CR 1 drops into CR 2 upon Don’t Care
Medium of weighting	Synaptic strength as a physical structure	CR 2 score as an algebraic scalar
Definition of “closeness”	Temporal proximity	Generational proximity on the genealogy
Accumulation of decay	History of synaptic plasticity	Recursive multiplication of CR 2 (compound interest)

Appendix

Table 18: Contrast between Analog and VaCoAl Implementations for the STDP Computational Problem

Operating Mode	Property Gained	Property Relinquished
Rescue Mode (circuit ON)	FS-relative Exact Match; zero-collision guarantee	CR 2 discriminating power (all candidates homogenised to $\mathrm{CR}_{2}=1.0$ )
Don’t Care Mode (circuit OFF)	STDP-like path-dependent decay; semantic selection	Zero-collision guarantee (minute collisions tolerated)

Appendix

Table 19: Trade-off between Rescue Mode and Don’t Care Mode

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Bridging Silicon and the Hippocampus: Algebro-Deterministic Memory "VaCoAl" as a Substrate for Vector-HaSH and TEM

Vector-HaSH and the Tolman-Eichenbaum Machine (TEM) propose the hippocampal-entorhinal circuit factorizes memory via a grid-cell scaffold for compositional replay. Concurrently, human iEEG shows sharp-wave ripples gate recall and multi-hop replay fidelity decays multiplicatively. Yet, these fields lack a shared algebraic foundation. We introduce VaCoAl, an algebro-deterministic hyperdimensional memory architecture built on Galois-field linear-feedback shift registers. Its deterministic Galois-field diffusion offers a substrate-level alternative to Vector-HaSH's random projections, matching quasi-orthogonality while ensuring bit-exact reproducibility. Furthermore, the path-integral Confidence Ratio CR2 provides an algebraically tractable model for the empirically observed multiplicative replay decay. Biologically, VaCoAl's two operating regimes align with the EC-CA3 direct and EC-DG-CA3 trisynaptic pathways, explaining their 520-Myr conservation. Independent cellular evidence supports that the DG-CA3 pathway implements a biophysical homologue of Galois-field arithmetic. We also link this framework to Judea Pearl's Ladder of Causation. Reversible GF(2) binding provides the surgical algebra for the do-operator (Rung 2), and VaCoAl's dual-orthogonalizer architecture supplies the parallel substrate required for counterfactual reasoning (Rung 3). Ultimately, we prove these formal correspondences and derive testable iEEG predictions, uniting computational neuroscience, electrophysiology, and hyperdimensional computing.

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Institute of Innovation Research, Hitotsubashi University, Kunitachi, Tokyo 186-8603, Japan · Meisei University, Hino, Tokyo, Japan · Shuhari System, Tokyo, Japan

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How to Build Marcus's Algebraic Mind: From Thagard's Brain--Mind Viewpoint

This paper reports a convergence neither program was looking for. Marcus's The Algebraic Mind named three things any architecture needs -- operations over variables, recursive structure, and individuals distinct from kinds -- showed multilayer perceptrons have none, and left a register-and-treelet implementation as conjecture. Thagard's Brain-Mind ran it from the other end, making binding the mechanism from which the whole of mind is assembled, and circular convolution load-bearing. Marcus leaves his register algebra open; Thagard's is lossy, degrading under the very recursion his own account demands. VaCoAl is a hyperdimensional computing architecture built end-to-end on XOR-and-shift over GF(2) via primitive-polynomial LFSRs; PyVaCoAl is its extended software realization (all results here); the silicon substrate exists as CASRAM. Bind(R,F) = R XOR Shift(F) is exactly reversible and non-commutative: it supplies all three pillars at fixed dimension and removes convolution's depth degradation. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability, which no lossy or learned substrate offers. Necessity: two independent architecture programs and the dentate gyrus-CA3 circuit require the same algebra -- convergence, not biomimicry. New here: discrimination and failure-tolerance are one. Repair every collision (RR = 1) and the system is bit-identical to a hash dictionary: candidates become indistinguishable and the confidence path-integral collapses; a memory that never fails keeps no record of which routes were hard. Position: we do not surpass large language models but supply the auditable, multi-hop relational reasoning embeddings lack. No consciousness is implemented and no cognitive experiment reported; bit-exactness holds in silicon, approximately under biological noise; speed and power remain unbenchmarked.

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Institute of Innovation Research, Hitotsubashi University (Professor Emeritus) · Meisei University (Professor Emeritus) · Shuhari System

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Efficiently Representing Algorithms With Chain-of-Thought Transformers

The increasing popularity of reasoning models -- language models that output a series of reasoning or thought tokens before producing an answer -- is justified, in part, by theoretical results showing that chain-of-thought (CoT) transformers can simulate Turing machines, and thus perform arbitrary computation. However, the Turing machine, while suitable for complexity-theoretic analysis, is not convenient, intuitive, or efficient for discussing algorithms. Algorithms are typically designed and analyzed at a higher level of abstraction, captured by the Word RAM model with random-access memory and unit-cost operations on

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Abstract

Associative Memory

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