# Neuromorphic operator learning: mechanisms, evidence and limits

September 14, 2026. Bounded autonomous campaign; all four numerical studies
completed before the September 15, 18:00 CEST upper bound. This report closes
the tested panels, not the broader vision. No application-scale training,
neuromorphic hardware experiment or practical breakthrough is reported.

## Executive assessment

The vision is small, always-on machines that perceive, retain useful experience
and adapt under strict resource budgets. The prospective contribution is an
operational calculus for compact adaptive dynamics, not another activation
function. The research question is whether the existing toolbox supplies a
mechanism with a defensible advantage over the strongest relevant classical
implementation.

Two tempting automatic-advantage claims did not survive this campaign.
First, accurate event/cell integration of nonlinear spline learning statistics
was verified, but its frozen speed gate failed. A subsequent explicitly
post-exposure comparator audit removed the apparent long-quiet-interval
accuracy advantage using ordinary polynomial quadrature. Second, a continuous
synaptic-consolidation chain was verified equivalent to a bank of exponential
modes, but the modal realization produced more distortion under the frozen
equal-bit quantization comparison.

The positive deliverables are executable, tested mechanisms and clear
boundaries: accurate event/cell objective accumulation; a regularized strong
quadrature control; an input-output equivalence check for consolidation
dynamics; and reproducible finite-precision evidence. These are useful
scientific engineering results, not evidence of a new learning-capacity law,
superior robot, hardware energy advantage or recursive self-improvement.

## 1. Prior evidence and the opportunity landscape

The project-wide audit and Gram/SSP claim corrections constrain this work.
Continuous calculus applies to the represented function and the stated
observation contract. A fixed-feature Gram and right-hand side can reproduce
a quadratic estimator without replaying all samples; this does not preserve
arbitrary old-task accuracy or supply evidence for newly invented features.
Cardinal locality does not make every network Gram banded or circulant, and
exact held-input linear propagation is not an exact solver for arbitrary
coupled nonlinear neurons.

The nearest literature makes several attractive slogans insufficient as
novelty. NEST documents exact integration of suitable linear neuron models
and augmented synaptic dynamics.[^1] Stapmanns and colleagues already address
event-based updates and compressed history for voltage-dependent synaptic
learning; separable temporal factors can support shared accumulated
integrals.[^2] EventProp supplies event-based exact gradients under its model
and differentiability conditions.[^3] E-prop separates eligibility traces
from learning signals, with practical credit-assignment approximations.[^4]
Therefore neither "between-event calculus" nor "local trace learning" alone
is an unclaimed invention.

Compact continuous-time history representations also have strong precedents:
LMU and HiPPO use polynomial projection/state-space constructions.[^5][^6]
The Neural Intermediate Representation provides a common dynamical interface;
it does not imply that a new cubic feature/Gram-learning primitive is already
supported efficiently on a particular chip.[^7] NeuroBench distinguishes
algorithmic evaluation from system-level evaluation.[^8] CPU operation counts
and a sparse mathematical graph cannot substitute for measured hardware power,
communication, readout and preprocessing costs.

Biological consolidation is especially relevant but not unexplored. Benna and
Fusi's interacting fast/slow variables motivate a physical diffusion chain;
their bounded and stochastic memory theory is more than that chain's
unquantized linear core.[^9] Complex synapses have already been applied to
continual reinforcement learning.[^10] Loss of learning plasticity is also
distinct from forgetting old tasks.[^11] None of these distinctions disappear
by expressing dynamics in an exponential basis.

## 2. Frozen study 01: event/cell objective integration

### Observation and representation

The scalar nonnegative trace obeys `x(t+s)=x(t) exp(-s/tau)` between supplied
positive impulses. The time constant and events are observed, not learned.
The learned output is a linear combination of cardinal cubic features of
**trace amplitude**, with fixed spacing. This is not a new exponential
B-spline generator or a full spiking recurrent network. Polynomial pieces in
amplitude become exponential-polynomial pieces along the trace; their crossing
times are generally irregular.

The retained statistics are `G = integral phi phi^T dt` and
`b = integral phi y dt`. The teacher value is deliberately supplied at each
interval start and held constant until the next event. This synthetic
supervision contract must not be generalized to unobserved real sensor values.
The statistics determine the quadratic objective up to its additive
`integral y^2 dt` constant; they determine its gradient and ridge minimizer,
not its absolute loss without that constant.

Each method receives the same partition at actual amplitude-knot crossings.
The analytic route integrates polynomial products against `dt=-tau dx/x`.
Stable exponential moments handle the zero-containing cell; a bounded
convergent series evaluates nonzero-cell moments. This is numerically checked
analytic-function evaluation, not interval-certified symbolic computation.
Controls use fixed-order Gauss quadrature or event-aware midpoint stepping.
Crossing discovery and accumulation are included in timing. Gram storage in
this implementation is dense and fully counted; no packed-banded memory
advantage is claimed.

### Panel and all-arm results

The development seed is 914001. Confirmation uses six independent seeds,
915001–915006, each paired across four mean-gap/time-constant ratios and two
knot spacings: 48 cases of 256 intervals. These are six independent seed
groups, not 48 independent datasets. All arm definitions, tolerances, teacher,
ridge, seeds and continuation gates were frozen before evaluation. Three
randomized-order timing repeats are summarized by their medians.

| Method | Cases passing numerical criteria / 48 | Median accumulation time |
| --- | --- | --- |
| Analytic event/cell | 48 | 2.118 ms |
| Gauss 4 | 0 | 2.062 ms |
| Gauss 8 | 24 | 2.119 ms |
| Gauss 16 | 32 | 2.210 ms |
| Midpoint step/tau 0.1 | 0 | 12.194 ms |
| Midpoint step/tau 0.01 | 0 | 16.422 ms |
| Midpoint step/tau 0.001 | 5 | 55.658 ms |

The analytic maximum relative Gram error is `9.92e-16`; maximum prediction
RMS disagreement is `2.01e-14`. High-order quadrature references agree within
`5.97e-15` in the recorded reference check. Adaptive quadrature verifies small
development cases and selected confirmation intervals, not every full
confirmation trajectory.

The primary correctness gate passes. The primary computational gate fails:
there is no twofold advantage over a numerically qualified quadrature control
in at least three of four rate groups. For mean gaps/tau 0.02 and 0.2, paired
quadrature/analytic timing ratios are about 0.965 and 0.966. At ratio 2, only
eight cases have a qualified fixed-order control and their median is about
1.053. At ratio 20, none of the original quadrature orders qualifies, so no
matched-accuracy speed ratio is asserted. Midpoint comparisons at inadequate
accuracy are not substitutes for that gate.

Whole-process peak RSS is approximately 117.3 MiB including the independent
reference work. Dense sufficient-statistic arrays range from 160 to 49,296
bytes; these are not entire deployed network memory measurements. Timings
describe this shared-host implementation, not asymptotic or hardware-energy
superiority. The [stored summary](summary_01.json) includes per-seed records,
rate breakdowns and descriptive whole-seed bootstrap intervals.

## 3. Study 02: stronger comparator, not fresh confirmation

After seeing study 01, the zero-cell numerical difficulty was isolated in a
separately recorded comparator audit. The original panel and scores were not
changed. This is explicitly exposed-data diagnostic evidence.

Write the local cubic feature vector as `p(u)=a+u*d(u)`, where `d` is quadratic.
Then the zero-cell quadratic integral is the constant time contribution
`duration*a*a^T` plus an amplitude integral whose polynomial degree is five.
Three-point Gauss quadrature integrates that remainder exactly in real
arithmetic. This removes the apparent endpoint singularity without discarding
the long quiet tail. Other cells use the declared fixed Gauss-8 rule.

The regularized comparator passes all 48 cases. Its maximum relative Gram
error is `6.97e-13`, and maximum prediction RMS discrepancy is `1.47e-12`.
Median quadrature/analytic time ratios across the four rate groups are
0.980, 0.984, 1.001 and 0.993: approximately parity. Classical polynomial
integration therefore removes the apparent distinctive accuracy benefit on
these cases. There is no follow-up timing or quadrature-order rescue search.

This identity generalizes in the straightforward polynomial sense; it is not
presented as a new theorem. Its research value here is a stronger comparator
and a numerical primitive worth retaining.

## 4. Study 03: what biological consolidation contributes

A second, separately frozen mechanism audit considers the continuous linear
core of a grounded consolidation chain. With positive diagonal capacitance
matrix `C`, grounded symmetric conductance Laplacian `K`, and
`A=-C^{-1} K`, the similarity transform gives the symmetric generator
`S=-C^{-1/2} K C^{-1/2}`. Diagonalizing `S=Q Lambda Q^T` yields independent
exponential modes. For a co-located first-coordinate input and readout, the
impulse response is a sum of decaying exponentials with nonnegative residues.

This is ordinary state-space realization theory applied to the selected
consolidation core. It exposes an implementation connection to operator-based
exponential models; it does not invent biological consolidation or show more
memory capacity.

Nine cases use dimensions 4, 8 and 12 and three seeds, with 256 signed
impulses per case. Physical dense-matrix propagation and modal propagation
agree: maximum relative output discrepancy `1.54e-15`, maximum state
discrepancy `2.96e-15`. All nine identity checks pass. A coordinate-rounding
counterexample shows why this equivalence need not survive finite precision,
but is not itself a fair precision ranking.

The model omits key aspects of the original bounded stochastic synapses. No
memory-capacity scaling, useful recall, learning plasticity or task retention
has been demonstrated by the identity audit.

## 5. Study 04: equal-bit realization hypothesis rejected

The independent hypothesis was that modal coordinates offer lower distortion
for the same consolidation operator under a fixed persistent-state budget.
The frozen comparison uses an eight-state chain, unit-time exact transitions,
2,048 synapses, 2,048 independent signed-input steps, six new seeds and two
precisions. Coordinate scales are fixed from theoretical stationary variances
before sampling; both arms use stochastic rounding and explicit clipping.
There are no scale, spectrum, dimension or optimizer sweeps.

The primary metric is normalized RMS output distortion against the common
unquantized operator over the last 512 updates. The gate requires at least a
twofold reduction in five of six seeds at both precisions, with no seed worse
and clipping below 0.1%.

| Effective bits / coordinate | Median modal / physical distortion | Descriptive 95% seed-bootstrap interval |
| --- | --- | --- |
| 6 | 1.403 | 1.379–1.447 |
| 8 | 1.750 | 1.727–1.811 |

Every modal/physical ratio exceeds one. Median normalized distortions are
0.767 versus 1.084 at six bits and 0.257 versus 0.453 at eight bits, physical
versus modal respectively. Maximum clipping fraction at six bits is
`3.70e-6`; eight-bit clipping is zero. Thus clipping did not dominate the
observed ranking. These results reject the declared modal-precision advantage
on this panel; they do not establish that physical coordinates are universally
optimal.

Both persistent arrays actually use int8. The six-bit condition has 48 logical
bits per synapse but occupies 64 physical bits in this unpacked prototype;
the eight-bit condition uses 64 bits by both measures. Shared numeric arrays
require 640 bytes for the physical realization and 192 for the modal one,
plus shared RNG state. Float64 transient computation and the evaluation-only
ideal state are additional resources. Peak process RSS is about 56.5 MiB.
There is no low-power hardware demonstration.

All secondary ideal-observer overlaps, including negative ones, are retained.
They are not an executable recall task or a capacity theorem. The primary
panel is now closed: no pole, quantizer, bit-depth or scale rescue search.

## 6. Verification, provenance and resources

The directory retains protocols, implementation hashes, every seed/regime
record and precision-state packages. The artifact audit checks 16 source or
protocol hash entries and reconstructs all 12 precision final outputs and
final errors from stored codes and scales. A separate read-only seed replay
matches all 12 primary last-512-step scores and saved arrays; this is stronger
than reconstructing only the final state.

The combined suite passes 36 tests: 26 new tests and ten existing Gram/SSP
claim-boundary tests. One warning occurs in the short precision smoke test
because secondary overlap ages exceed its deliberately short horizon; those
undefined ages are not used in a primary score. All declared full-panel ages
exist and full-panel JSON metrics are finite. The frozen experiment source was
not changed merely to silence that warning.

All numerical work uses the existing environment, one CPU process and
single-threaded numerical libraries. MPS was not needed for these small
kernels. No reserved data, production service, external messaging or package
installation was used. Git checkpoints separate protocol freezes, completed
results and the final assessment. Other untracked research directories are
not implicitly included in this campaign's backup.

## 7. What remains credible, and what would admit another experiment

The grand vision remains a legitimate problem statement. These results remove
two proposed shortcuts, not all routes to it. A further experiment should
begin with an actual unavailable capability and its information budget, not
with another basis substitution.

One possible frontier is an adaptive physical device whose behavior must be
calibrated cheaply from limited measurements. Noise-aware digital-twin
training already has substantial prior art.[^12] Its public article identifies
processed figure data and makes raw experimental data available by contact;
this campaign did not qualify a public raw device-drift cohort. A listed
validation array is not evidence of measured hardware provenance. No external
request was sent. Without the necessary observation and evaluation contract,
this is a prospective opportunity rather than an admitted benchmark.

Another is useful bounded memory whose representation can change without
silently pretending old sufficient statistics constrain every new direction.
Any proposal must specify what evidence is retained, what new evidence is
allowed, which queries are protected, and how state and maintenance costs
grow. Repeatedly adding immutable models is not bounded unlimited memory.

The user's additional compression idea connects these opportunities. A
compact object that supports physical operations, streaming updates and
declared later-selected queries could be valuable. However scientific
compressed computation and error-controlled multilevel representations already
exist, and this project's earlier codec/inspection failures remain binding
evidence. The separate [compression vision assessment](COMPRESSION_VISION.md)
states the possible mechanism, strong comparators and limitations. No
compression experiment has been launched.

**Continuation decision:** retain and document the verified mechanisms; close
the two failed advantage panels. Do not admit a broad SNN/robotics training
run from these results. A new capability experiment requires a distinct
mechanism, qualified observations, full resource accounting and a strong
non-spline comparator. Nothing in this report establishes a practical or
SOTA breakthrough.

## Sources and reading scope

The [reading ledger](READING_LEDGER.md) distinguishes detailed method reading
from abstract screening and unqualified data access. This is a targeted
literature audit, not an exhaustive full-text reading of every neuromorphic
publication.

[^1]: NEST, [Exact integration documentation](https://nest-simulator.readthedocs.io/en/v3.5/neurons/exact-integration.html), describing the Rotter–Diesmann approach for eligible systems.
[^2]: Stapmanns et al. (2021), [Event-Based Update of Synapses in Voltage-Based Learning Rules](https://www.frontiersin.org/journals/neuroinformatics/articles/10.3389/fninf.2021.609147/full).
[^3]: Wunderlich and Pehle, [EventProp: exact gradients for spiking neural networks](https://arxiv.org/abs/2009.08378).
[^4]: Bellec et al. (2020), [A solution to the learning dilemma for recurrent networks of spiking neurons](https://www.nature.com/articles/s41467-020-17236-y).
[^5]: Voelker, Kajić and Eliasmith (2019), [Legendre Memory Units](https://proceedings.neurips.cc/paper/2019/hash/952285b9b7e7a1be5aa7849f32ffff05-Abstract.html).
[^6]: Gu et al. (2020), [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://arxiv.org/abs/2008.07669).
[^7]: Pedersen et al. (2024), [Neuromorphic Intermediate Representation](https://www.nature.com/articles/s41467-024-52259-9); [primitive definitions](https://neuroir.org/docs/primitives/).
[^8]: Yik et al. (2025), [NeuroBench](https://www.nature.com/articles/s41467-025-56739-4).
[^9]: Benna and Fusi (2016), [Computational principles of synaptic memory consolidation](https://www.nature.com/articles/nn.4401); [author-hosted PDF](https://www.gatsby.ucl.ac.uk/~pel/tnlectures/papers/benna_fusi.pdf).
[^10]: Kaplanis, Shanahan and Clopath (2018), [Continual Reinforcement Learning with Complex Synapses](https://proceedings.mlr.press/v80/kaplanis18a.html).
[^11]: Dohare et al. (2024), [Loss of plasticity in deep continual learning](https://www.nature.com/articles/s41586-024-07711-7).
[^12]: Manneschi et al. (2025), [Noise-aware training of neuromorphic dynamic device networks](https://www.nature.com/articles/s41467-025-64232-1); [released project](https://github.com/LucaManneschi/NoiseAwareTwins_Project).