Clockless analogue MVP for physical EP research

Prepared: 2026-07-11
Starting point: COMPONENT_HW_MAP.md
Scope: academic prototype, not a product demonstrator; analogue learning core; no processor, ADC, DAC, FPGA, sample-and-hold bank, phase clock, or digital optimizer in the learning loop.


1. Executive decision

The current $5k–$20k Demo-0 is not an MVP. It is an architecture demonstrator that tries to validate, at once, an SRAM-CIM module, bidirectional transpose reads, RoPE mixers, QK/RMS normalization, analogue softmax, SwiGLU multipliers, analogue state handling, ADC/DAC boundaries, a phase sequencer, write programming, and digital supervisory logic. A failure would be difficult to attribute to one mechanism.

The lower-risk research question is narrower and more publishable:

Can a reciprocal nonlinear analogue network continuously learn by a local equilibrium contrast, with the free and nudged states physically present at the same time, without a clock or processor in the learning loop?

Build that first. The recommended core is a twin-equilibrium analogue learning tile:

Recommended first populated board: 8 sign-update edges plus one exact multiplier reference channel, approximately $170–$300 excluding instruments, tax, and shipping. This is about 17×–118× below the current $5k–$20k board class. A one-edge calibration rig is approximately $70–$130.


2. What “completely analogue and clockless” can honestly mean

A defensible claim is:

The state evolution, nudge, local learning rule, and weight storage are continuous-time analogue processes. No periodic control signal, processor, converter, sampled state memory, or digitally computed parameter update participates in learning.

Allowed outside the claim:

A full language-model demonstration cannot honestly be entirely analogue and clockless at the token interface: symbol lookup, presentation of a sequence, cross-entropy labels, and token sampling are discrete operations. The academic MVP should therefore validate the physical learning primitive, then add a very small reciprocal attention cell as a second experiment.


3. Core topology

                      analogue input x          analogue target y*
                             │                           │
                ┌────────────┴────────────┐              │
                │                         │              │
                ▼                         ▼              │
        FREE reciprocal network    NUDGED reciprocal network
          natural RC settling        natural RC settling
             output y0              output yβ + nudge current
                │                         ▲              │
                │                         │      OTA/error transconductor
                │                         └──────────────┘
                │
       for every trainable edge e:

          edge voltage Δv0,e               edge voltage Δvβ,e
                    │                              │
                    └──── local contrast cell ─────┘
                                  │
                         charge/discharge current
                                  │
                            shared weight Cw,e
                                  │
                     ┌────────────┴────────────┐
                     ▼                         ▼
             MOSFET edge in free net   MOSFET edge in nudged net

The two replicas must be laid out symmetrically and use matched transistor pairs where practical. The shared capacitor ensures both edge copies always use the same learned weight.

3.1 State dynamics

The node voltages are the states. Resistors/MOSFET conductances and node capacitances produce the relaxation automatically. Add small capacitors only where needed to set a reproducible pole and suppress oscillation; do not build a capacitor-plus-OTA integrator for every abstract model state.

Design target, not a theorem:

τ_{weight}/τ_state ≥ 10^2,     preferably 10^3.

The state should equilibrate much faster than the weight capacitors move. This time-scale separation replaces a settle detector and phase sequencer.

3.2 True EP nudge and the CL control

Populate both modes on the same board.

EP mode — current/force nudge. For voltage outputs, compute the output error and inject a proportional current at the nudged output. A practical small-nudge implementation is

I_β = -g_β (y_0-y^*)

with an LM13700-class OTA or a discrete transconductor. The polarity is chosen to push the nudged output toward the target. This crosses voltage error with current nudge and is the mode to use for the formal EP/gradient-flow claim.

CL mode — voltage constraint. Buffer a weighted voltage between the free output and the target and impose it on the second replica. This is easier and reproduces the demonstrated clockless Coupled Learning architecture, but it should not be called exact EP.

A two-position switch should select EP-current or CL-voltage nudge. That comparison is itself a useful experiment.

3.3 Exact local contrast

For a conductance-like parameter, the local energy derivative is proportional to the squared voltage drop. Use

C_{w,e} dV/dt_{w,e}
 = s_e k≤ft[(\Delta v_{β,e})^2-(\Delta v_{0,e})^2],

where s_e = ±1 accounts for whether increasing capacitor voltage increases or decreases effective conductance.

Do not square twice. Use

a^2-b^2=(a-b)(a+b),

so one four-quadrant multiplier can implement an exact reference channel. The established laboratory circuit used an AD633 with op-amp conditioning and a local capacitor.

3.4 Low-cost sign contrast

The exact multiplier dominates cost. A much cheaper local rule is

C_{w,e} dV/dt_{w,e}
 = s_e I_0 sgn≤ft(|\Delta v_{β,e}|-|\Delta v_{0,e}|)

outside a deadband δ, with zero current inside the deadband. Implement it with absolute-value/rectifier stages, a comparator with hysteresis, and two matched charge/discharge current sources.

This rule has precedent in simulated memristor EP hardware because it removes the analogue multiplier. The continuous capacitor implementation proposed here is an engineering adaptation, not an already demonstrated result. That is a legitimate research contribution, but it must be labeled correctly.

3.5 Freeze and retention

Use a mechanical toggle or relay to disconnect update current from every weight capacitor. Avoid a clocked switch matrix. Use film capacitors initially; characterize leakage and dielectric absorption. Volatile analogue weights are acceptable for an academic demonstrator, but not for a storage product.


4. Recommended hardware ladder

Rung A — one-edge metrology tile

Purpose: validate the physics before assembling a network.

Populate:

Estimated cost: $70–$130.

Required measurements:

  1. transfer surface I_update(Δv0, Δvβ);
  2. exact-channel gain and offset;
  3. sign-channel boundary, hysteresis, and deadband;
  4. zero-contrast drift at Δv0 = Δvβ;
  5. weight retention with learning frozen;
  6. temperature drift;
  7. nudge linearity versus β;
  8. time-scale ratio between node settling and weight motion.

Do not build the multi-edge board until this tile gives a stable null at zero contrast.

Rung B — two modular four-edge cards

Build a four-edge card and populate two cards for eight trainable edges. Each card contains:

Recommended population: eight sign channels plus one parallel exact AD633 channel on a selected edge. This gives a continuously measured exact-versus-sign comparison without buying eight multipliers.

Estimated cost: $170–$300.

Initial task:

Do not promise XOR at eight edges. The published nonlinear clockless network used 32 twin edges for XOR. Design the card so eight identical four-edge modules can be stacked later.

Rung C — 32-edge replication-class network

Use eight four-edge cards and the sign-update rule. This approaches the scale of the published nonlinear demonstration while avoiding 32 AD633 multipliers.

Estimated cost: $450–$900.

This is still below the low end of the original plan by approximately 5.5×–44× and is large enough for a serious robustness and nonlinear-learning study.

Rung D — reciprocal attention microcell

Only after Rungs A–C work, add a transformer-adjacent cell:

For two alternatives, softmax reduces to a logistic function of a score difference, so a differential pair can replace a general N-way entropic-resistor array. This is an attention-shaped energy cell, not an OLMo2 block.

Estimated total including the learning core: $300–$700.


5. Costed options

Budgetary single-quantity catalogue prices checked on 2026-07-11; prices exclude tax, shipping, instruments, assembly labor, and rework.

Item Planning role Unit price used Planning quantity, 8-edge board Extended
ALD1106PBL matched N-MOS array; two twin edges/package $9.29 4 $37.16
AD633ANZ exact four-quadrant contrast multiplier $21.48 0, 1, or 8 $0 / $21.48 / $171.84
TLV274IPWR quad rail-to-rail op amp $1.43 6–8 $8.58–$11.44
LM13700 dual OTA; output nudge/current sources $1.70 1–2 $1.70–$3.40
LM339-class comparator sign/deadband channels about $0.67 2–4 about $1.34–$2.68
CD4066-class switch optional static freeze/routing about $0.73–$0.97 2–4 about $1.46–$3.88
Film capacitors weight storage about $0.63 at 1 µF 8–12 about $5–$8
Diodes, resistors, trims rectifiers, limits, biasing lot $15–$40
PCB/protoboard, headers, test points physical implementation lot $35–$120
Power rails, protection, spare parts laboratory overhead lot $30–$90

Resulting envelopes:

Variant Estimated build cost Recommendation
One-edge exact/sign calibration tile $70–$130 Build first
8-edge sign-only $140–$250 Cheapest useful network
8-edge sign + one exact reference channel $170–$300 Recommended MVP
8-edge all-exact AD633 $300–$500 Only after reference channel works
32-edge sign-update network $450–$900 Replication-class nonlinear demo
Reciprocal two-token attention add-on +$100–$250 Phase 2 only

Aggressive transistor substitution

A CD4007UBE costs about $0.89 and contains a CMOS dual complementary pair plus inverter. It can be explored on the one-edge tile, but it is not the main-board recommendation: matching, body connections, and device operating region become the dominant uncertainty. Saving roughly $30 of matched-transistor cost on an eight-edge board is not worth sacrificing the experiment’s interpretability.


6. Delete list for the original plan

For the academic MVP, delete these entirely:

Keep or replace as follows:

Original function MVP replacement
MVM/crossbar patchable reciprocal nonlinear resistor network
state integrators natural RC/KCL state dynamics
sequential free/nudged phases two continuously operating physical replicas
DAC nudge OTA current injection
digital phase memory simultaneous physical state comparison
outer-product update/programming local capacitor charge/discharge
FPGA control manual boundary selection and learn/freeze
ADC telemetry buffered scope/DMM observation outside loop

7. Experimental matrix

The cleanest paper is not merely “it learned.” It should isolate the choices that make the clockless implementation possible.

Factor 1 — nudge type

Factor 2 — local update

Factor 3 — device quality

Factor 4 — nudge magnitude

Factor 5 — operating conditions

Proposed acceptance criteria, to be declared before network training:

  1. no periodic control signal in the learning loop;
  2. no processor, ADC, DAC, or stored free-phase sample used for an update;
  3. exact reference channel has a stable zero-contrast null;
  4. sign channel agrees with the exact channel on update direction in at least 95% of the specified operating grid;
  5. frozen weights remain within the chosen tolerance over the measurement interval;
  6. loss decreases consistently across multiple resets, with all failures reported;
  7. EP-current and CL-voltage modes are labeled separately.

8. Six-week execution plan

Week 1 — SPICE and one-edge schematic

Week 2 — one-edge breadboard

Week 3 — one-edge characterization

Week 4 — first four-edge card

Week 5 — eight-edge network

Week 6 — robustness and paper figures


9. Claims this MVP can and cannot support

It can support

It cannot support

A passive reciprocal network avoids the adjoint/transposed-Jacobian problem by construction, but it also excludes normal non-reciprocal attention. The transformer connection must therefore use tied, energy-based attention or be deferred to an active-adjoint phase.


10. Why this is the better academic MVP

It tests one scientific claim at a time. Every expensive item in the original plan exists to preserve a nearly complete transformer block. That is appropriate after the physical learning primitive is established, not before it.

The proposed sequence produces publishable intermediate results even if the final nonlinear task fails:

The main recommendation is therefore:

Build the $70–$130 one-edge tile, then the $170–$300 eight-edge hybrid board. Do not purchase a CIM evaluation module, FPGA, DAC/ADC bank, or softmax hardware for the MVP.


11. Primary references used

  1. Sam Dillavou et al., “Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial,” arXiv:2311.00537v2 / PNAS (2024). Demonstrated a fully analogue, clockless nonlinear twin-resistor network; 32 twin edges; local capacitive learning; XOR and nonlinear regression. Its implemented rule is Coupled Learning.
  2. Sam Dillavou et al., “Understanding and Embracing Imperfection in Physical Learning Networks,” arXiv:2505.22887v2 (2026 revision). Characterizes bias, limit cycles, and system-agnostic suppression of analogue learning imperfections.
  3. Joshua A. McGinnis, Xinbo Li, and Yoichiro Mori, “Coercivity and Local Convergence of Physical Learning in Linear Circuits,” arXiv:2606.15443 (2026). Distinguishes EP’s force/current nudge from CL’s voltage constraint and analyzes their gradient-flow structure.
  4. Seung-Hyun Oh et al., “Memristor Crossbar Circuits Implementing Equilibrium Propagation for On-Device Learning,” Micromachines 14(7):1367 (2023). Proposes simultaneous free/nudged networks and a sign-only fixed-step update to remove analogue multipliers; results are simulation-oriented for memristor programming.
  5. I. M. Elfadel and J. L. Wyatt, “The ‘Softmax’ Nonlinearity: Derivation Using Statistical Mechanics and Useful Properties as a Multiterminal Analog Circuit Element,” NeurIPS 1993. Establishes reciprocal/passive circuit properties of an entropic softmax element.
  6. “How to Train Your Resistive Network: Generalized Equilibrium Propagation and Analytical Learning,” arXiv:2602.03546 (2026). Gives a single-network projector-based route for linear resistive networks; useful as a future replica-reduction direction, not the first nonlinear build.