Waveform-Dependent ROS Dynamics in a Dose-Matched Toy Model: Closed Form, Ablation, and a Retraction

We study a four-variable toy model of photobiomodulation bioenergetics (active cytochrome-c-oxidase, ATP, reactive oxygen species, and an entropy index) under four dose-matched optical drives: continuous wave (CW), 10 Hz and 40 Hz square waves, and a φ-harmonic frequency ladder. Two results are established by direct integration, reproduced on two machines: (1) Biphasic fluence response (CW). Final ROS rises with peak flux, peaks at Φpeak = 75 (4.931011 µM), and falls at 100 (4.456755 µM), while ATP rises monotonically (0.839864 → 69.949778 µM). The biphasic shape is a property of the imposed production term (§4.1), not an emergent discovery. (2) ATP is waveform-insensitive; ROS is waveform-sensitive. At matched average dose, final ATP spans only 0.004063 µM across all four drives, while final ROS spans 4.873872 (CW) to 0.935305 (φ-ladder). For the φ-ladder we decompose the ROS reduction by ablation: of the total drop from CW (3.938568 µM), 96.2% is waveform amplitude distribution (the ladder with the coherence factor removed yields 1.083459 µM) and 3.8% is the model's hand-inserted coherence factor (which brings it to 0.935305 µM). No φ-specific effect is claimed: an integer-harmonic ladder behaves the same way (1.542436 µM), and crest factor is shown to be neither necessary nor sufficient — the explanatory variable is the cycle mean ⟨G⟩ of the biphasic production term, for which we prove a closed form for square waves, ROS*(p) = 1/5 + (24/5)·exp(19/25 − p·exp(−6/25)), verified symbolically and numerically on two machines. An earlier claim associated with this framework — that φ-harmonic entrainment reduces ROS "by over 500%" — is retracted (§4.3). Version 4.1 — Corrected Merge Edition (2026-10-28). This version merges the verified analysis of the v4.0 draft (closed form, ablation, retraction) onto the two-machine-verified result tables of v3.3 and supersedes both. v4.0 must not be circulated. Nothing in this document is a biological or therapeutic claim; the model is an exploratory toy system with asserted, unsourced parameters; every result describes the behavior of the equations, and only that.