Threshold-linear rate source and polyglot fidelity evidence

This page records the scientific scope, algebraic equation, five-runtime parity, failure atomicity, and controlled benchmark used to promote ThresholdLinearRateNeuron to the polyglot-complete catalogue.

Scientific scope

Gerstner, Kistler, Naud, and Paninski (2014), doi:10.1017/CBO9781107447615, define the piecewise-linear gain (F(h)=[h]_+=\max(0,h)) in Section 18.2, Eq. 18.23. The maintained SC-NeuroCore transfer exposes an explicit translation and scale:

[ r = g\max(0,I-\theta). ]

The citation supports the underlying population-rate gain function. It does not turn this algebraic transfer into a spiking neuron, a fitted single-cell model, or a temporal population ODE.

Algebraic contract

Every non-empty evaluation overwrites the cached output with the transfer for the current input. The previous r is validated before execution but does not enter the right-hand side. An empty batch preserves it.

All runtimes enforce finite r, theta, gain, and current, with non-negative r and gain. They compute the complete candidate before visible mutation and reject non-finite output. Reset clears r while preserving theta and gain.

Executable parity matrix

Runtime	Executed surface	Enrolled result
Python	public scalar and atomic batch	reference
Rust engine	PyO3 modular batch, zero crate-root delta	bit-exact
Rust safety	independently compiled module	bit-exact; 6/6 tests pass
Julia	simulate_trace through JuliaCall	bit-exact
Go	service plus generated C-shared ABI	bit-exact
Mojo	exported shared-library C ABI	bit-exact

The configured parity case is r=0.25, theta=1.5, gain=2, and I=3. Each of the 200,000 output values is exactly 3.0; the canonical little-endian float64 trace SHA-256 in every runtime is cdb90f105692311ba359cfbf0574faa23586215e1a253ddcad29276b9bf69402.

The focused cohort also executes below-threshold, threshold-equality, and above-threshold branches, empty batches, explicit-unavailable backends, overflow rejection, corrupted runtime state, and invalid Go/Mojo contracts whose caller buffers must remain unchanged.

Generated fixed-point co-simulation

The paired TOML and JSON schemas reproduce the configured hand transfer exactly. The production equation compiler lowers the same schema to Q16.16 Verilog with theta=1.5 and gain=2.0. Icarus Verilog co-simulation drives 193 representable inputs from -4.0 through 8.0 in 1/16 increments and establishes cycle-exact public r_out words across the below-threshold, equality, and above-threshold branches. spike_out remains zero throughout.

This is an H1 generated-RTL result for the declared algebraic transfer. It does not recast positive rates as binary events.

Controlled benchmark

benchmarks/bench_model_threshold_linear_rate.py measures the complete 200,000-value trace through each public dispatcher five times after warm-up. It fails if any runtime is absent, the standalone Rust-safety binary fails, or any trace value or final cached output differs from Python.

The run was pinned to logical CPU 4, but the CPU was not exclusively isolated and host load was high. The artifact records every raw sample, exact source and native-binary hashes, tool versions, affinity, and load. These are local regression timings, not production, cross-host, hardware, or universal-ranking claims.

Backend	Median call	Median ns/evaluation	Trace mismatches
Python	1.621 ms	8.107	0
Mojo	3.388 ms	16.938	0
Rust	3.892 ms	19.458	0
Go	9.824 ms	49.122	0
Julia	12.425 ms	62.123	0

Python uses a vectorised fill for this constant-input algebraic workload and was the shortest raw call. The compiled-backend dispatcher policy keeps Python as its always-available floor; among native lanes the measured order is Mojo, Rust, Go, then Julia.

The dedicated threshold-linear-rate-five-backend-local-regression evidence gate adds no failure. The aggregate repository report still records 66 older source-hash mismatches, including prior gates whose shared bridge/engine sources changed; those inherited refreshes are not presented as Model35 failures.

Boundaries

• Positive output is continuous rate, not a binary event or spike count.
• r is a cached observable, not integrated state or memory.
• The book equation supports the rectified gain; the explicit threshold and gain are the declared translated/scaled form.
• The benchmark is local non-exclusive regression evidence.
• The generated Q16.16 claim is cycle-exact co-simulation only. No formal equivalence, synthesis, timing, device, or PPA result is claimed.

Reproduction

go test ./services -run ThresholdLinearRate -count=1
rustc --edition 2021 --test \
  src/sc_neurocore/accel/rust/safety/threshold_linear_rate.rs \
  -o /tmp/threshold_linear_rate_tests
/tmp/threshold_linear_rate_tests

PYTHONPATH=bridge:src:. .venv/bin/python -m pytest -q \
  tests/test_model_threshold_linear_rate_dynamics.py \
  tests/test_model_threshold_linear_rate_batch.py \
  tests/test_model_threshold_linear_rate_descriptor_schema.py \
  tests/test_model_threshold_linear_rate_validation.py \
  tests/test_cosim_threshold_linear_rate.py \
  tests/test_threshold_linear_rate_backend_loading.py \
  tests/test_threshold_linear_rate_backends.py \
  tests/test_bench_threshold_linear_rate.py

taskset -c 4 env PYTHONPATH=bridge:src:. .venv/bin/python \
  benchmarks/bench_model_threshold_linear_rate.py \
  --json benchmarks/results/local_python_2026-07-14_threshold_linear_rate.json