# Research release

## Scope

Positive Transverse Response in Quadratic Population Networks studies a centered quadratic population network in dimension five with signed trainable output weights and a specified compact initialization. The paper proves a nonnegative transported transverse response and a residual barrier, with independently checked interval obligations. The linearized weak residual converges to zero without crossing it. Finite mixing is controlled on each compact interval before the limiting exposure.

## Reproducible evidence

The barrier evidence covers eight signed atoms, 140 angular cells and 400 time steps. A separate 40-digit interval checker verifies initialization, continuous trajectory inclusion, endpoint remainders, derivatives, angular weights, aggregation and positivity. A 50-digit checker verifies the freeze comparison, including J8 > 25 and the integrated-capacity threshold 8141/10000 > 4/5. Exact symbolic checks verify the response identities and perturbation coefficients.

Ordinary numerical experiments compare both probe pairs against the linearized response, sweep the mixing parameter, and vary time step, quadrature and arithmetic precision. The scientific figures are generated from the accompanying machine-readable data. These experiments are not proofs.

The GS capacity perturbation is first order in mixing; the block capacity is even and has a quadratic remainder. Coupling has a separately justified time-weighted remainder. The public theorem statements keep their initialization, model and time-horizon assumptions explicit.

## Package

The package contains the manuscript source and PDF, computational evidence, independent checkers, regression tests, experimental outputs and plotting code. RELEASE_MANIFEST.json binds the files and checker reports by hash. See [reproduction](REPRODUCING.md), [verification scope](VERIFICATION.md), [claim map](CLAIM_MAP.json), and [related work](NOVELTY.md).

Uniform finite-mixing behavior through infinite physical time remains open. Finite-width and finite-sample results, arbitrary initialization and general neural architectures are outside scope. This is computer-assisted mathematics with documented trusted software, not proof-assistant verification.