NOTE-1506: Sánchez-Cañizares — Decision-making through irruption theory

Contribution

It gives Irruption Theory, which the author says “lacks a formal, ‘axiomatic’ grounding” (p. 2), a mathematical stand-in: the HKB potential with time-dependent parameters a(t), b(t). It sets three regimes in sequence and computes means and standard deviations over initial conditions. It claims, for the first time, that HKB fits decision-making results, specifically “neural variability quenching through time-changing control parameters” (p. 18). It also revises Froese’s pairing of irruption and absorption, as IAT*.

Key insight

A decision is modelled as a change in the number of attractors: a landscape with two wells opens (one per option), then closes to one. Variability across trials rises while two wells are open and falls once one remains. The mind does not push the state. It sets the landscape: “Through ‘irruption,’ the mind formally sets up the relevant attractor landscape in the brain. Through the ‘absorption’ of physiological variability (reduction of the number of attractors), conscious experience is associated with its distinctive brain signature” (p. 7). “Mind and brain drive the dynamics, but none completely controls it” (p. 7).

Assumptions

• The model (eqs. 1, 2, p. 8): V(φ) = −a cos φ − b cos 2φ, φ̇ = −∂V/∂φ = −a sin φ − 2b sin 2φ, plus a noise term √q ξ(t), with φ the relative phase of two oscillators, here two neuronal assemblies firing at nearly the same frequency (p. 13).
• The schedule (eq. 3, p. 8): a = b = 0 for 0 ≤ t < 1; a = b = 1 for 1 ≤ t < 2; a = 1, b = 0 for 2 ≤ t ≤ 15; q = 1 throughout (Fig. 2, 3 captions).
• The statistics (eqs. 4–7, p. 11) are over the initial condition φ(0), taken uniform on (−π, π]. The text says “the differences that justify a statistical treatment in a deterministic equation stem from the different initial conditions” (p. 10), although a noise term is present.
• The noise is described as ξ(t) ∈ [−1, 1], with zero mean and delta-correlated (pp. 8–9). Bounded values and delta correlation do not go together for continuous-time white noise; the paper does not say which it simulated. Footnote 10 (p. 9) uses “noise” more broadly, for uncontrolled and “possibly biased” influences that “can modify parameters a and b”; the white-noise term cannot.
• IAT is a modelling assumption*, “not intended as a broader conceptual revision of IAT” but restricted to “binary decision-making tasks” (pp. 2, 20).

Key results

• The bifurcation (p. 8). With b = 1, there are two minima, at 0 and π, for |a| < 4, and one otherwise. Checked: V″ = a cos φ + 4b cos 2φ is positive at π iff a < 4b and at 0 iff a > −4b. This is the anti-phase stability boundary b/a = 1/4 of THEORY-1227.
• The simulated trend (pp. 10–12, Figs. 2, 3). In 0 ≤ t < 1, cos φ has mean 0. In 1 ≤ t < 2 the standard deviations of φ and cos φ rise (“a bump”) as paths split between the wells. For t ≥ 2 “the average values of φ(t) and cos φ(t) tend to 0 and 1, respectively, and their standard deviations smoothly decrease to 0 after their bump” (p. 12). “The influence of white noise is less relevant than the influence of changes in parameters a and b.”
• Re-simulation (this reading). 20,000 paths, Euler–Maruyama with dt = 10⁻³, φ(0) uniform, eq. (3), Gaussian white noise √q dW.
  • q = 0: the SD of cos φ goes 0.71 (t = 0.99), 0.93 (t = 1.5), 0.98 (t = 1.99), 0.82 (t = 5), 0.12 (t = 10), 0.01 (t = 15), and the mean of cos φ reaches 1.00. The paper’s description holds.
  • q = 1, as stated: the SD of cos φ goes 0.71, 0.87, 0.89, 0.48 (t = 5), then stays at 0.40 (t = 10, 15), with the mean of cos φ at 0.70. These are the stationary values of p(φ) ∝ exp(2a cos φ / q) with a = 1 (mean 0.698, SD 0.405, computed). So under the stated noise the rise and fall is reproduced, but the deviation does not go to 0. The paper’s Fig. 3 must have used a weaker effective noise than q = 1 Gaussian white noise, perhaps the bounded ξ, or else its “decrease to 0” is wrong for q = 1.
• Sergent et al. 2021 (pp. 13–15; Fig. 4). 20 participants classified a vowel in noise at several SNRs. At the critical SNR near −9 dB, the ERP’s SD at Cz peaks around 400–500 ms. The author reads the trend off their Fig. 1F, since “we do not possess a graph of the standard deviation over time for fixed SNR” (p. 14). He proposes a/b as a proxy for SNR. Two mismatches are stated: the ERP returns to baseline faster than the model (fn. 17), and at low SNR the experimental SD is smaller than at moderate to high SNR, “which our HKB model does not reproduce” (p. 15).
• Wolff et al. 2019 and 2021 (pp. 15–16; Fig. 5). 34 participants made moral trade-off decisions. Trial-to-trial variability peaks near 100 ms and declines after 200 ms; “the quantitative prediction makes little sense at this level, the qualitative agreement … is remarkable” (p. 16). The model cannot address pre-stimulus effects, because σ_cos φ is constant before t = 1.
• Hua et al. 2025 (pp. 16–18; Fig. 6). On-task deliberate thought raised pre-stimulus variability and post-stimulus inter-trial phase coherence and shortened response times. A direct comparison “is not possible” (p. 17). The author re-maps the model so that the prepared participant starts at a = b = 1.
• The reversal of Froese 2024 (fn. 12, p. 10). Froese hypothesized that irruption coincides with decreases in shared variance and absorption with increases. “Here, however, we claim the opposite”, citing critical fluctuations and slowing before transitions.

Claims

id	claim	strength	support
C1	With b = 1, the HKB potential has two attractors (φ = 0, π) for |a| < 4 and one otherwise	strong	analytic (p. 8); checked here
C2	Switching a, b from zero to bistable to monostable makes the across-initial-condition SD of cos φ rise then fall	strong	simulation (Fig. 3); reproduced here
C3	That SD falls to 0 after the switch to one attractor, with q = 1	weak	Fig. 3 and p. 12; contradicted by re-simulation with Gaussian white noise, where it settles near 0.40
C4	The model qualitatively matches the rise and fall of ERP trial-to-trial variability in Sergent et al. 2021 and Wolff et al. 2019, 2021	weak	visual comparison with reproduced figures; the author: “only qualitative” (p. 18); two stated mismatches (p. 15, fn. 17)
C5	In IAT*, irruption is the onset of a bistable landscape and absorption the passage to a single attractor, both correlated in mind and brain	weak	proposed as a modelling assumption (pp. 6, 20)
C6	The mind’s causal role is to select the coarse-graining, setting the control parameters, without disturbing physical efficient causation	weak	asserted (pp. 3, 19); defended by the parameters’ lack of a microscopic derivation (p. 19)

Concepts

• IAT (Irruption-Absorption Theory) — Froese 2024: a mind-to-matter difference shows in matter as an “unintelligible increase in measurable differences” (irruption), matter-to-mind as a decrease (absorption), in anticorrelated pairs (pp. 2, 5).
• IAT* — the author’s modification: irruption in mind and brain go together, as do absorption in both; either can be cause or effect (p. 6, fn. 5).
• Control parameters — a, b (or k = b/a, Kelso’s original control parameter, fn. 8), here made functions of time and attributed ultimately to the mind.
• Neural variability quenching — the drop in trial-to-trial variability after stimulus onset, which the paper aims to explain (p. 18).

Connections

• Haken, Kelso and Bunz 1985 (LIT-588, NOTE-1424, THEORY-1227) supply the potential; Schöner, Haken and Kelso 1986 its stochastic version (fn. 11).
• Sánchez-Cañizares 2024 (LIT-1132, Unavailable) is the “proof of concept” this extends; the 2026 paper’s novelty is the time-dependent parameters (p. 8). It quotes its Fig. 6 as a “cycle of relaxation, fixation of possibilities, and decision-making” (p. 10).
• Froese 2023 (LIT-1950, NOTE-1521) and Froese 2024 (Entropy 26:288, not held).
• Hölken 2024 (LIT-1957) is cited once, for the criticism that IT does not explain “how underdetermination arises in material systems and how the scaling-up of irruptions causally affects normative behavior” (fn. 6, p. 6). This 2026 paper does not answer it. It places the mind at the parameters instead of in microscopic noise, so it needs no scaling-up.
• Juarrero 2023, Context Changes Everything (open access, not held), is cited for “constraints’ causality”, which IAT “seems to initially reject” and which the author says “need not be the case” (fn. 7, p. 6). Kelso 1995 is cited for order parameters.
• Kane is not cited, and neither is chaos.

Bearing on the record

• LIT-1132. Its status note expected this paper “may cover the same ground”. It does not reproduce the 2024 paper. It extends it, with time-dependent parameters and a comparison with experiments, and it restates the 2024 result only in a sentence (p. 19): the HKB model “can instantiate some of IT’s predictions”. LIT-1132 stays unread.
• What it concedes. The registration says A22 “summarizes what it concedes”; A22’s text is not in the record, so this reading lists the concessions in the paper’s words, against which A22 can be checked:
  • IAT* is a restricted modelling assumption, not a revision of IAT (pp. 2, 20).
  • “IAT* cannot be tested with empirical results without further assumptions” (p. 7).
  • “The HKB model, as a scientific model, does not require the existence of IAT* to work” (p. 7).
  • The agreement with experiment is “only qualitative” (p. 18), with two stated mismatches.
  • Changes in a and b “are likely consequences of external parameters, like SNR, but they are also the effect of internal brain dynamics” (p. 15).
  • Making IAT* testable and not ad hoc needs “much more” (p. 20).
• CLAIM-533. This is the irruption-theory reading A21 put first “for the chaos line”, but it has no chaos in it. Its dynamics is a one-dimensional gradient flow, φ̇ = −∂V/∂φ, which cannot be chaotic, and its openness is bistability plus noise. On the claim’s terms the agent has no input channel into φ at all. Its only channel is to the parameters a(t), b(t), that is, to the landscape. That is a different control architecture from the one the claim discusses: it controls which outcomes exist, not which one the state reaches. The paper does not ask how the choice between the two wells is made in the bistable interval. In its own simulation that is settled by the initial condition and noise (pp. 10–11), and the mind’s contribution is to remove one well (b → 0). So the paper locates control exactly where CLAIM-533 says it must be, in a channel to the slow structure, but without any account of whose input sets the parameters.
• CLAIM-478. The model’s control parameters are the slow structure that constrains the fast phase, the claim’s scale separation, and the paper attributes setting them to the mind. Its fn. 12 appeals to critical fluctuations and slowing near transitions, which is CLAIM-478’s second half.
• QUESTION-071. If what the mind does is set a(t), b(t), and those are also set by SNR and internal dynamics (p. 15), nothing in the measured dynamics tells a mind-set parameter change from one set by stimulus or by another party. The paper treats that as the black box’s virtue (p. 20).
• No instruction for machine-learning practice.

Limitations

• Qualitative throughout. No parameters are fitted, and the comparisons are by eye against figures reproduced from other papers.
• The noise is underspecified, and with the stated q = 1 the paper’s description of Fig. 3 does not hold (above).
• The statistics are over initial conditions, which the author distinguishes from noise (fn. 13), but trial-to-trial variability in the experiments has no such clear counterpart.
• The mental attribution is not tested and is said to be untestable on its own: it rests on the absence of a microscopic derivation of a and b (p. 19), which is an argument from ignorance unless such a derivation is impossible.
• Schöner et al. 1986 and the four experimental papers were not checked here.

Open questions

• Which noise produced Fig. 3, and does the variability-quenching match survive with noise strong enough to give realistic stationary variability?
• Can a(t), b(t) be estimated from data trial by trial, and does any variable other than stimulus and task predict them? That is the test the mental attribution would need (THEORY-1302).