What Counts as One Loop

The previous post argued that Miller, Brincat, and Roy's wave theory makes a sufficiency claim RSP has to reject: organized, globally integrated wave dynamics is not by itself consciousness, for the same reason integration and broadcasting aren't. That disagreement is real and it stands. But it occupies about one page out of twenty-two pages of argument, and treating it as the whole story does the rest of the paper a disservice. The mechanism underneath — spatial computing, the specific proposal about how alpha/beta rhythms route information across cortex — is the most detailed candidate implementation RSP's formal appendixes have ever been offered. It deserves its own accounting.

This post is that accounting. It gets more specific than the reply did, and it ends somewhere I didn't expect: at a free parameter in RSP's own mathematics that a wave substrate could plausibly fix.

The Clean Mapping, and an Honest Caveat

Spatial computing theory says that alpha/beta rhythms (~13–30 Hz), generated in higher frontoparietal cortex, function as temporary inhibitory "stencils" laid across the cortical surface. They suppress gamma (~30–80 Hz) and its associated spiking in targeted locations. Feedforward sensory content, carried by gamma, is expressed wherever the stencil leaves gaps. Where information shows up in spiking is therefore determined by the interaction between spatially localized bottom-up drive and broadly patterned top-down suppression.

Read that next to Appendix A and the correspondence is immediate. Hierarchical predictive coding says ascending prediction errors are gain-modulated by precision, so expected reliability determines which errors propagate upward and revise beliefs. Appendix A specifies that operation formally and says nothing whatsoever about what physically applies the gain. Stencils are a gain-setting mechanism on ascending signals. Same operation, one level of description lower.

The honest caveat: this is not an RSP-specific discovery, and presenting it as one would be misleading. Bastos et al. (2012), which Miller's paper cites in exactly this context, already mapped alpha/beta to descending feedback and gamma to ascending prediction error within canonical predictive-coding microcircuits. The correspondence belongs to the predictive processing literature generally. RSP inherits it by being a predictive-coding architecture, not by having anticipated anything. What RSP contributes comes later in this post; it isn't here.

Where It Lands in the Level Structure

The more interesting question is which of RSP's levels the mechanism actually reaches. Here the paper's own empirical results are unusually informative.

Miller and colleagues cite recent work (Chen et al., 2026) confirming several spatial computing predictions in primate prefrontal cortex. The alpha/beta patterns they found encoded, in the paper's description, learned and internally generated contextual information: order within a sequence, the mode of eventual behavioral report, and learned category structure. Those patterns inversely mirrored the spatial distribution of sensory information in spiking, and predicted behavioral errors through mismatches in contextual representation.

Every one of those variables is Level 3a. Sequence order, report mode, learned categories — these are conceptual and contextual structures being applied downward to organize sensory content. The confirmed empirical content of spatial computing is a mechanism for 3a issuing gating commands to Level 2 and Level 1, with the frontoparietal origin of the stencils matching where RSP would expect conceptual control to live.

What the mechanism does not touch, anywhere, is 3b. Nothing in a stencil requires that it model the system generating it. A stencil encoding "this is the recall condition, not the match condition" is a context label, not a self-model. This restates the reply post's argument with the mechanism in hand, and the mechanistic version is sharper: spatial computing is a strong implementation candidate for precisely the parts of RSP that are not doing the phenomenal work.

An Audit of the Timescales

Here the comparison turns quantitative, and I have to correct something.

RSP is often summarized with Level 1 running at roughly 100 ms. That figure comes from early drafts and has already been walked back in the published papers. Part II's revision, prompted by review, states the position plainly: Level 1 is sub-second to early-second rather than the ~100 ms ERP figure of earlier drafts. The fastest brainstem updates may approach 100 ms, but conscious-affect stabilization is slower. Cardiac-evoked potentials run 200–500 ms, and cardiac and respiratory cycles span 1–4 seconds. Appendix D, working from control theory rather than evoked potentials, independently lands nearby: Level 1 gets a time constant of roughly 0.5–2 seconds, Level 2 roughly 5–30 seconds, Level 3 roughly 60–3600 seconds.

Now line those up against the wave bands:

Every oscillatory band runs faster than RSP's fastest level, and faster than even the most aggressive reading of Level 1. Taking Appendix D's control-theoretic numbers, RSP's entire bandwidth range spans roughly 0.0003 to 2 Hz. Miller's entire oscillatory range spans 1 to 80 Hz. The two barely touch at the edges.

A caution about that comparison, since the units invite a mistake I nearly made. Appendix D's bandwidth figures are control bandwidths — the maximum disturbance frequency a loop can track — not oscillation frequencies. A loop with 1 Hz bandwidth does not oscillate at 1 Hz. So the numerical overlap between Level 1's 0.5–2 Hz and the delta band's 1–4 Hz is a coincidence of units, not a physical identity, and it would be an overclaim to say Level 1 "is" delta.

What the audit does establish is structural. Waves cannot be directly implementing Level 2 or Level 3 dynamics, because they cycle between ten and a thousand times too fast. Whatever the relationship is, it is not identity. The two frameworks stack; they do not compete for the same slot.

Carrier and Content

The resolution I would propose — and this is my construction, not a claim either paper makes — is that the oscillation is a carrier and the represented state is the stencil configuration, which persists across many cycles. What changes slowly is the spatial pattern of suppression, not the rhythm carrying it. A radio station holds a stable identity while its carrier oscillates millions of times a second; nothing about the carrier's rate constrains how long the station stays on the air.

Miller's own treatment supports this reading. The traveling-wave section describes transitions between coding subspaces — sensory encoding shifting into working-memory storage — as discrete state changes, events that happen to a configuration rather than updates occurring once per cycle. The configuration is the thing with a lifetime.

If that is right, it converts a conceptual fit into a measurable one. RSP's level timescales become predictions about stencil dwell time: a stencil implementing a Level 2 schema should hold its configuration on the order of 5–30 seconds; one implementing Level 1 gating, on the order of 0.5–2 seconds. Those are far enough apart to distinguish, and Chen et al.'s recordings are exactly the kind of data that would contain the answer. Whether the dwell-time distributions cluster in the right places is, as far as I can tell, an unasked question.

The Free Parameter

This is the part I find genuinely interesting, and reaching it required correcting my own first reading.

Appendix D formalizes the strange loop as a fixed-point problem. The recursive self-model updates through a mapping that blends the current self-model with the average predicted self-as-seen-by-others, and the appendix proves this mapping is a contraction when social reflection is itself contractive, since averaging across multiple perspectives dampens extremes. Convergence is geometric in recursion depth. For plausible contraction coefficients the model converges in roughly seven iterations, which the appendix notes sits comfortably against an observed maximum recursion depth of three to five.

My first instinct was that this created a tension with an oscillatory substrate: contraction mappings settle to fixed points, and oscillators by definition do not settle. That instinct was wrong, and it is worth naming as wrong. The contraction does not run on continuous time. It runs on a discrete iteration index — recursion depth, not seconds. There is no conflict between a state converging over iterations and a carrier oscillating throughout.

Correcting the error exposes something better. If the contraction runs on an iteration counter, the formalism contains a question it never answers: what physically constitutes one iteration? Appendix D is substrate-neutral on this. The index is abstract. Nothing in the mathematics says which event in the brain corresponds to advancing it by one.

Spatial computing offers a candidate. If one iteration of the self-model update is one stencil reconfiguration — one traveling-wave sweep transitioning the population into a new subspace — then the abstract depth index acquires a physical clock. Recursion depth becomes countable in wave events rather than posited. And the claim that recursive self-modeling bottoms out at three to five iterations becomes a claim about how many reconfigurations occur inside a Level 3 window, which is the sort of thing an electrode can see.

The confidence here needs stating carefully. Neither paper proposes this. Miller and colleagues never mention recursion depth; RSP never mentions traveling waves. The identification is mine, it is speculative, and it could fail in at least two ways: stencil reconfigurations might prove far too frequent to match a depth of three to five over Level 3 timescales, or self-model iterations might not correspond to any single wave event at all. Both are testable objections, which is the main argument for taking the proposal seriously enough to check.

What Goes Back the Other Way

The exchange is not one-directional, and the reciprocal contribution is more specific than the reply post had room to make it.

Spatial computing has no principled account of stencil persistence. The theory explains how a stencil routes information, and why the same population can be reused across contexts by swapping the pattern. It does not explain why one configuration holds for half a second and another for thirty. Within that framework, dwell time is whatever it turns out to be.

Cascade control does have an account. Appendix D's timescale separation is not a description of observed rates; it is a stability requirement. The inner loop must be fast relative to the outer loop or the cascade goes unstable, and the appendix puts a number on it: each level roughly five times slower than the one inside it, so that every controller sees its inner loop as effectively instantaneous. Loop gains and stability margins set settling times. If stencils are the physical form of the gating, their persistence is not free — it is constrained by what keeps the cascade stable.

That is a real trade. Spatial computing supplies the physical event; cascade control supplies the reason the event lasts as long as it does.

Scope, Honestly

Three things in this post are grounded in the sources: the stencil mechanism and its frequency bands, Chen et al.'s confirmed variables and their 3a character, and RSP's timescale and contraction formalisms as the appendixes actually state them. Two things are mine, and should be read as proposals rather than findings: the carrier-versus-configuration reading of how waves could carry slow states, and the identification of one self-model iteration with one stencil reconfiguration.

One correction belongs on the record rather than in a quiet edit. The ~100 ms figure for Level 1 circulates in summaries of RSP and is no longer the position. Part II hedged it to sub-second-to-early-second under review, and Appendix D's control-theoretic estimate of 0.5–2 seconds agrees. Anyone comparing RSP's timescales against oscillatory data should use the corrected figures. The comparison in this post is stronger with them, not weaker: the gap between wave rates and control rates is wider than the old number implied, which is precisely why the carrier reading is needed at all.