Sensorimotor Theory

"A living being is not a thing but a process: a continuous exchange with the environment in which the boundary between 'self' and 'not-self' is recreated at every moment." — Francisco Varela

Who this chapter is for

The sensorimotor cycle as a consequence of the canonical evolution equation. The reader will learn how perception, evaluation, and action are derived from the dynamics of $\Gamma$.

Every living being — from a bacterium feeling a chemical gradient to a human navigating an unfamiliar city — solves the same problem: perceive the environment and respond appropriately. This task seems mundane, but behind it lies one of the deepest problems in the science of complex systems.

Imagine an amoeba. It has no eyes, ears, or brain. Yet it distinguishes: it moves toward nutrients, avoids poison, flows around obstacles. Between "chemical signal on the membrane" and "extending a pseudopod" stands something — not a mere reflex, but a closed cycle: perception → evaluation → action → perception. This cycle — the sensorimotor loop — is the minimal unit of adaptive behaviour.

Classical control theory describes the sensorimotor loop as "sensor → controller → actuator". Active inference (FEP) sees it as minimising variational free energy. Reinforcement learning models it as maximising cumulative reward. Each of these approaches captures part of the truth — but none answers two key questions:

1. Why is the cycle structured this way? Where do the number of perception channels, the action structure, and the format of internal evaluation come from?
2. Where in the cycle does experience fit? When is the sensorimotor cycle accompanied by subjective experience, and when is it not?

Coherence Cybernetics (CC) gives a constructive answer to both questions. The sensorimotor cycle is not postulated — it is derived from the canonical 3-term evolution equation. The environment does not introduce a "fourth force": it modifies the three already existing channels (Hamiltonian, dissipative, regenerative). Experience turns out to be not a side effect but an integral part of the cycle — through the hedonic valence $\mathcal{V}_{\text{hed}}$, which guides action.

In this chapter we build a complete formal theory of sensorimotor encoding: from axiomatic grounding to concrete architectures and predictions. The reader familiar with the introduction and the evolution equation will find here a natural continuation — a step from "how the system lives within itself" to "how the system interacts with the world".

In the previous chapter we traced 80 years of cybernetics history and saw that each tradition — from Wiener to Friston — captured part of the sensorimotor problem: feedback, the observer, active inference. Now we show how CC solves it completely — without additional postulates, within the same 3-term evolution equation.

Chapter roadmap

In this chapter we:

1. Prove that the environment does not add a 4th term — Theorem T-102 on completeness of the 3-channel decomposition (Section 1).
2. Construct the perception functor Enc — how the environment enters the system through modification of the evolution equation (Section 2).
3. Construct the action functor Dec — how the system chooses the optimal action through a min-max strategy (Section 3).
4. Derive the hedonic mechanism — why pleasure and suffering are not side effects but derivatives of viability (Section 5).
5. Classify 21 qualia-types as sensorimotor channels (Section 6).
6. Establish fundamental limits — information capacity $\leq \log_2 7$ bits (T-107) and compositionality of Enc/Dec (T-108) (Sections 9–10).
7. Compare with classical approaches — control theory, FEP, RL — as projections of CC (Section 14).

On notation

In this document:

• $\Gamma$ — coherence matrix
• $\theta_{ij} = \arg(\gamma_{ij})$ — coherence phases
• $\sigma_{\mathrm{sys}}$ — stress tensor (T-92 [T])
• $h^{\text{ext}} = h^{(H)} + h^{(D)} + h^{(R)}$ — 3-channel decomposition [T]
• $P = \mathrm{Tr}(\Gamma^2)$ — purity
• $\varphi$ — self-modelling operator
• $\rho_* = \varphi(\Gamma)$ — target state

This document describes the formal theory of sensorimotor encoding — how a holon perceives its environment and acts upon it, remaining within the canonical 3-term evolution equation.

Key result: the external force F_ext is not a 4th term of the evolution equation, but a modification of the three existing channels (Hamiltonian, dissipative, regenerative). The completeness of this decomposition is proven by the LGKS Theorem (T-57 [T]).

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1. Canonical Inclusion of the Environment

1.1 The 3-term equation as closed dynamics

The evolution equation of a holon:

$$\frac{d\Gamma}{d\tau} = -i[H_{\text{eff}}, \Gamma] + \mathcal{D}_\Omega[\Gamma] + \mathcal{R}[\Gamma, E]$$

contains exactly three terms [T]:

Term	Type	Canonical origin
$-i[H_{\text{eff}}, \Gamma]$	Unitary (Hamiltonian)	Axiom A3
$\mathcal{D}_\Omega[\Gamma]$	Dissipative (Lindblad)	Liouvillian
$\mathcal{R}[\Gamma, E]$	Regenerative	Categorical conjugation

1.2 The environment modifies 3 channels, not adds a 4th

Intuitively: imagine a violinist in an orchestra. The conductor influences them (suggesting tempo), the hall acoustics (blurring sound), and the other musicians (helping to return to the common key). These three types of influence are everything there is. There is no fourth type of influence on the violinist that would not be a combination of the conductor's gesture, acoustic noise, and adjustment to the ensemble. Theorem T-102 formalises exactly this intuition: the environment cannot "reach" a holon by any means other than the three canonical channels.

Theorem T-102 (Completeness of the 3-term equation) [T]

Statement

Any CPTP-compatible external influence on a holon decomposes into the sum of three channels:

$$h^{\text{ext}} = h^{(H)} + h^{(D)} + h^{(R)}$$

where $h^{(H)}$ modifies $H_{\text{eff}}$, $h^{(D)}$ modifies $\mathcal{D}_\Omega$, $h^{(R)}$ modifies $\mathcal{R}$. A fourth type of CPTP generator does not exist.

Proof. Direct consequence of the LGKS Theorem (T-57 [T], completeness of the triadic decomposition):

1. An arbitrary generator of a CPTP semigroup $\mathcal{L}$ on $\mathcal{D}(\mathbb{C}^7)$ has the Gorini–Kossakowski–Sudarshan–Lindblad form:

$$\mathcal{L}[\rho] = -i[H, \rho] + \sum_k \left(L_k \rho L_k^\dagger - \tfrac{1}{2}\{L_k^\dagger L_k, \rho\}\right)$$

2. Any external influence preserving CPTP properties of the dynamics is a perturbation $\mathcal{L} \to \mathcal{L} + \delta\mathcal{L}$
3. The perturbation $\delta\mathcal{L}$ has the same LGKS form → decomposes into $\delta H$ (Hamiltonian part) and $\delta L_k$ (Lindblad part)
4. Triadic decomposition $\{L_k\}$: dissipative + regenerative operators. A fourth type is forbidden by T-57. $\blacksquare$

Corollary: The F_ext term in simulation is not a separate force, but a composition of three modifications:

Channel	Perturbation formula	Physical meaning	Example
$h^{(H)}$	$\delta(\Delta\omega_{ij})$	Energetic coupling with environment	Sensory input, neuromodulators
$h^{(D)}$	$\delta\Gamma_2 \cdot \dot{\theta}_{ij}$	Environmental noise	Stress, interference, temperature
$h^{(R)}$	$\delta\kappa \cdot (\theta^{\text{target}}_{ij} - \theta_{ij})$	Modification of regeneration	Meditation, psychotherapy, learning

Canonical form — from Definition 8.1 [T]:

$$\mathcal{L}_{\text{ext}} = \sum_{i<j} h^{\text{ext}}_{ij} \cdot |\gamma_{ij}| \cdot \sin(\theta_{ij})$$

1.3 The thermodynamic trichotomy of the channels

Theorem T-102 says the environment has exactly three doors into a holon. The natural next question: are these three doors different in kind, or merely three labels on one mechanism? The answer is that they carry three thermodynamically distinct — and jointly exhaustive — modes of exchange, and the trichotomy is the open-quantum image of the oldest classification in physics: the first law's split of energy exchange into work, heat, and chemical work (matter exchange). To see it, read each channel through what it does to the two basic state functionals — the von Neumann entropy $S(\Gamma) = -\mathrm{Tr}(\Gamma \ln \Gamma)$ and the purity $P(\Gamma) = \mathrm{Tr}\,\Gamma^2$.

Theorem T-258 (Thermodynamic trichotomy of the channels) [T]

Statement

Let $\dot{\Gamma} = h$ be a perturbation from the three-channel basis of T-102. Then the entropy–purity signatures of the channels are pairwise distinct and exhaust the basis:

1. Hamiltonian channel $h^{(H)} = -i[\delta H, \Gamma]$: $\dot{S} = 0$ and $\dot{P} = 0$ — the unique isentropic, purity-preserving channel (work: it re-aims the state without spending or importing order);
2. Dissipative channel $h^{(D)} = \delta\Gamma_2 \cdot \mathcal{D}_{\text{Fano}}[\Gamma]$: $\dot{S} \geq 0$ and $\dot{P} = -\tfrac{4}{3}\,\delta\Gamma_2 \cdot C(\Gamma) \leq 0$, strictly whenever the coherent mass $C(\Gamma) = \sum_{i \neq j}|\gamma_{ij}|^2 > 0$ (heat: the unique channel that can only produce entropy);
3. Regenerative channel $h^{(R)} = \delta\kappa_{\text{eff}}\,(\rho^* - \Gamma)$:

$$\dot{S} = \delta\kappa_{\text{eff}}\bigl[S(\rho^*) + D(\rho^*\|\Gamma) - S(\Gamma)\bigr], \qquad \dot{P} = 2\,\delta\kappa_{\text{eff}}\bigl(\mathrm{Tr}\,\Gamma\rho^* - P\bigr)$$

— both sign-indefinite; the only channel able to lower entropy and raise purity (chemical work / matter: feeding imports negentropy).

Consequently the three channels realise three distinct signature types — conservative ($\dot{S} \equiv 0 \equiv \dot{P}$ at every state), sign-definite dissipative ($\dot{S} \geq 0$ and $\dot{P} \leq 0$ at every state), and sign-indefinite (both signs attainable) — and this typing is exhaustive (it labels the whole T-102 basis) and channel-distinguishing. The trichotomy is thus observable as a classification. A single instantaneous sign pair identifies the acting channel generically but not universally: on a diagonal state the heat channel degenerates to $(0,0)$ and is momentarily indistinguishable from work, and a regenerative step toward a more mixed, low-overlap target can transiently enter heat's $(+,-)$ quadrant. What is exact and universal is the type trichotomy; single-shot identification is generic, and certain identification follows from the channel's response probed across states.

Proof. (i) For $\dot{\Gamma} = -i[\delta H, \Gamma]$: $\dot{S} = -\mathrm{Tr}(\dot{\Gamma}\ln\Gamma) = i\,\mathrm{Tr}([\delta H, \Gamma]\ln\Gamma) = i\,\mathrm{Tr}(\delta H\,[\Gamma, \ln\Gamma]) = 0$, since $[\Gamma, \ln\Gamma] = 0$; the whole spectrum is invariant under unitary conjugation, so every spectral functional — in particular $P$ — is conserved.

(ii) The Fano channel is unital: $\sum_p \Pi_p = 3\cdot\mathbb{1}$ (each channel lies on exactly three lines), so $\mathcal{F}(\mathbb{1}/7) = \mathbb{1}/7$. A unital CPTP semigroup majorizes downward, hence $S$ is non-decreasing, with equality exactly on diagonal states. Element-wise, the BIBD incidence (every pair of channels shares exactly one line) gives $\mathcal{D}_{\text{Fano}}[\Gamma]_{ij} = -\tfrac{2}{3}\gamma_{ij}$ for $i \neq j$ and $0$ on the diagonal, whence $\dot{P} = 2\,\mathrm{Tr}(\Gamma\,\mathcal{D}_{\text{Fano}}[\Gamma]) = -\tfrac{4}{3}\sum_{i\neq j}|\gamma_{ij}|^2$.

(iii) Direct computation with $\mathrm{Tr}\,\dot{\Gamma} = 0$: $\dot{S} = -\delta\kappa_{\text{eff}}\,\mathrm{Tr}((\rho^*-\Gamma)\ln\Gamma) = \delta\kappa_{\text{eff}}[S(\rho^*) + D(\rho^*\|\Gamma) - S(\Gamma)]$, using $\mathrm{Tr}\,\rho^*\ln\Gamma = -S(\rho^*) - D(\rho^*\|\Gamma)$. Both signs are realized: $\Gamma = 0.98\,|\psi\rangle\langle\psi| + 0.02\,\mathbb{1}/7$ with a still purer target gives $\dot{S} = -0.095$ (feeding purifies); $\Gamma = 0.9\,|\psi\rangle\langle\psi| + 0.1\,\mathbb{1}/7$ with target $\mathbb{1}/7$ gives $\dot{S} = +3.208$. The purity formula is immediate from $\dot{P} = 2\,\mathrm{Tr}(\Gamma\dot{\Gamma})$ and is positive whenever the target overlap exceeds the current purity. $\blacksquare$

Machine verification. Three hundred random full-rank states per channel: $|\dot{S}|, |\dot{P}| \leq 3\cdot10^{-15}$ for $h^{(H)}$; $\min \dot{S} = +0.337 \geq 0$ and the purity formula exact to $10^{-16}$ for $h^{(D)}$; the $h^{(R)}$ entropy formula exact to $2\cdot10^{-15}$, with both-sign witnesses as quoted. Type-definiteness over $3000$ states: work $|\dot{S}|,|\dot{P}| \leq 6\cdot10^{-10}$ (conservative), heat sign-definiteness violations $0/3000$; boundary degeneracies confirmed — heat on a diagonal state gives exactly $(0,0)$, and a regenerative step to a mixed low-overlap target hits $(+,-)$, so the sign pair is a generic, not universal, identifier (the type trichotomy is exact).

The grand-canonical dictionary [I]

The signatures identify the three channels with the three conjugate pairs of the grand-canonical ensemble — and, term by term, with the three Legendre transforms by which Vanchurin's Self-Learning Universe (SLU, 2026) builds physics out of a resource-constrained learning system. There an agent cannot measure three extensive variables — the displacement $\Delta q^\mu$, the fast entropy $S_x$, the count $N$ of fast degrees of freedom — and models each by its intensive conjugate: the gauge field $A_\mu$, the temperature $T$, the chemical potential $\mu$.

Exchange mode	Grand-canonical pair	Vanchurin (SLU)	UHM channel	Signature $(\dot{S}, \dot{P})$
Work	(displacement, force)	$\Delta q^\mu \leftrightarrow A_\mu$	$h^{(H)}$ ($\delta H_{\text{eff}}$)	$(0,\ 0)$
Heat	$(S,\ T)$	$S_x \leftrightarrow T$	$h^{(D)}$ ($\delta\Gamma_2$)	$(\geq 0,\ \leq 0)$
Matter	$(N,\ \mu)$	$N \leftrightarrow \mu$	$h^{(R)}$ ($\delta\kappa$)	$(\mp,\ \pm)$

Five structural checks of the dictionary, each anchored elsewhere in the corpus:

1. No fourth channel ↔ no fourth argument. The completeness of T-102 mirrors the completeness of the fundamental thermodynamic relation $U(S, V, N)$: an open system can be driven in exactly three ways — work it, heat it, or feed it. Vanchurin derives the same count from the three inaccessible extensive arguments of his step loss $L(q, \Delta q, S_x, N)$; the two exhaustiveness proofs (LGKS vs the Legendre cascade) reach the same trichotomy from opposite ends — a cross-validation of both.
2. The phase axes are grand-canonical. The consciousness phase diagram already lives in the coordinates $(t, r) = (T_{\text{eff}}/T_c,\ \kappa/\Gamma_2)$ — a temperature and a feeding ratio. Under the dictionary these are precisely $(T, \mu)$: the QCD analogy "baryon chemical potential $\mu_B \leftrightarrow r$" is thereby upgraded from a visual parallel to a structural correspondence, and the feeding threshold of T-259 reads as a chemical-potential condensation threshold.
3. Feeding lives on the Ground channel. The regeneration kernel $\kappa_0 = \omega_0|\gamma_{OE}||\gamma_{OU}|/\gamma_{OO}$ is carried by the O-channel — canonically the channel "to sustain existence, to feed, to parameterize internal time". In SLU the chemical potential is likewise locked to the clock: $h = |\mu|\varepsilon$ ties the quantum of action to the chemical scale per time step. Both theories attach the matter channel and the internal clock to the same carrier.
4. Discreteness and $U(1)$. In SLU the $U(1)$ phase of the wavefunction is the thermodynamic equivalence $S \to S + h\Delta N$ under integer jumps of $N$. On the UHM side this is now a theorem: T-260 shows the conserved charges of the Fano dissipator are exactly the seven passport populations, their exponential is the diagonal torus $U(1)^7$ (full covariance group $U(1)^7 \rtimes \mathrm{Aut}(r)$, the torus extended by the rate-preserving permutations; it read $U(1)^7 \rtimes \Gamma_{\!\text{oct}}$ until 2026-09-25), and the compactness of each factor is forced by the integrality of the charge spectrum — projector occupancies $\{0,1\}$ and the integer sub-holon counters of neurogenesis (⊕). The SLU mechanism, channel-resolved and derived [T]; the correspondence with SLU's own $U(1)$ remains interpretive [I].
5. Line-resolved temperatures. UHM's heat channel carries seven rates $\{\gamma_p\}$ — one per Fano line — where SLU carries a single scalar $T$. The isotropic point $\gamma_p = \gamma$, the unique $G_2$-symmetric configuration, is exactly SLU's setting: the corpus refines the single temperature into a spectrum of seven line temperatures, anisotropy measuring how unevenly the environment heats the holon's coherence structure — and the refinement is falsifiable: the rank-7 anisotropy law [T] forces 14 exact linear relations among the 21 pairwise decoherence rates and makes the seven line temperatures reconstructible by tomography.

Status honesty: the signature theorem is [T]; the dictionary itself — the identification with (work, heat, matter) and with SLU's $(A, T, \mu)$ — is an interpretation [I], exact on signatures and counting. On the dynamical laws the ledger is now closed (T-262): all three legs are derived as exact geometric flows — work = isometric (Killing) drive of every monotone metric; heat = Carlen–Maas gradient flow of the negentropy $D(\Gamma\|\mathbb{1}/7)$ with the line temperatures as transport weights; matter = BKM gradient flow of $D(\rho_*\|\Gamma)$ (T-261); the gauge torus compactness is likewise derived (T-260). Two normative/physical capstones extend the ledger: the matter-channel flow is the unique optimal learning algorithm (T-263 — steepest BKM descent along the mixture geodesic, geometry unique by Grasselli–Streater), and SLU's slogan "gravity = learning efficiency" is signed and sector-resolved (T-264: $G^{-1}\propto$ QFI of space phases, $\Lambda\propto\mathcal{G}_O^2$ = squared clock unlearnability). What remains [I] is only the inter-theory identification with SLU itself.

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2. Perception Functor Enc

2.0 Intuition: what does it mean to "perceive"

What does the eye do when it sees an apple? From a physics perspective — it converts electromagnetic waves into neural impulses. From a computer-science perspective — it encodes input data into a latent representation. But both perspectives miss the essential point: perception is not passive recording, but active inclusion of the environment in the system's own dynamics.

When an amoeba "senses" glucose, its internal state changes — not because information was "recorded" somewhere, but because glucose molecules literally altered the dynamics of intracellular processes. Perception is a deformation of one's own equation of motion under the influence of the environment.

The Enc functor formalises exactly this: it maps an observation $o$ not into a "record in memory" but into a modification of the evolution equation — a triple $(h^{(H)}, h^{(D)}, h^{(R)})$ that changes the Hamiltonian, dissipator, and regenerator. Perceiving an apple simultaneously changes the energy landscape (shape, colour), modifies the noise characteristics (texture, motion), and shifts the target state (hunger → satiation).

2.1 Definition

Theorem T-100 (Environmental encoding) [T]

Statement

For a holon $\mathbb{H}$ with coherence matrix $\Gamma \in \mathcal{D}(\mathbb{C}^7)$, there exists a unique (up to $G_2$-gauge) CPTP encoding functor:

$$\mathrm{Enc}: \mathrm{ObsSpace} \to \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$$

satisfying:

1. CPTP: $\mathrm{Enc}(o)[\Gamma]$ is a state for any observation $o \in \mathrm{ObsSpace}$
2. 3-channel decomposition: $\mathrm{Enc}(o) = \delta H^{(o)} \oplus \delta D^{(o)} \oplus \delta R^{(o)}$
3. Functoriality: $\mathrm{Enc}(o_1 \circ o_2) = \mathrm{Enc}(o_1) \circ \mathrm{Enc}(o_2)$

Proof.

1. Existence: the environment acts through $h^{\text{ext}}_{ij}$ (Def. 8.1 [T]). The map $o \mapsto h^{\text{ext}}(o)$ defines $\mathrm{Enc}$.
2. 3-channel: follows from T-102 (completeness of the 3-term equation).
3. Uniqueness: consequence of $G_2$-rigidity (uniqueness theorem [T]) — for a system satisfying (AP)+(PH)+(QG)+(V), the map is unique up to $G_2 = \mathrm{Aut}(\mathbb{O})$.
4. Functoriality: CPTP channels are closed under composition. $\blacksquare$

Example: neuroscientific implementation. The visual cortex implements Enc hierarchically: V1 extracts edges ($h^{(H)}_{AS}$ — articulation of structure), V4 encodes colour ($h^{(H)}_{AE}$ — articulation of interiority), and MT encodes motion ($h^{(D)}_{AD}$ — dissipative component of dynamics). All three channels converge in association areas, forming a unified $h^{\text{ext}}$. Functoriality guarantees that the scene "red ball moving left" is encoded identically whether perceived as a whole or in parts.

2.2 Implementation via 7 observable indices

The Γ measurement protocol defines 7 observable indices $I_i$ ($i \in \{A, S, D, L, E, O, U\}$), each mapping to a specific component of $h^{\text{ext}}$:

Index	Formula	Channel $h^{\text{ext}}$	Measurement
$I_A$ (articulation)	$I(\text{input}; \text{latent}) / H(\text{input})$	$h^{(H)}_{A,\cdot}$	Hamiltonian
$I_S$ (structure)	$\mathrm{rank}_\varepsilon(J_f) / \min(d_{\text{out}}, d_{\text{in}})$	$h^{(H)}_{S,\cdot}$	Hamiltonian
$I_D$ (dynamics)	$\max_i \lambda_i^{\text{Lyap}}$ (normalised)	$h^{(D)}_{D,\cdot}$	Dissipative
$I_L$ (logic)	$1 - \|[f_i, f_j]\|_F / (\|f_i\| \cdot \|f_j\|)$	$h^{(H)}_{L,\cdot}$	Hamiltonian
$I_E$ (interiority)	$\exp(S_{vN}(\rho_{\text{attn}}))$	$h^{(R)}_{E,\cdot}$	Regenerative
$I_O$ (ground)	$1 - \|\nabla_\epsilon \mathbf{h}\|_F$	$h^{(D)}_{O,\cdot}$	Dissipative
$I_U$ (unity)	$\Phi_{\text{eff}} = \lambda_2(L) / \lambda_{\max}(L)$	$h^{(R)}_{U,\cdot}$	Regenerative

Logic of channel assignment:

• Hamiltonian $h^{(H)}$: informational indices ($I_A, I_S, I_L$) — alter the energy landscape, i.e., which states are more or less probable
• Dissipative $h^{(D)}$: load indices ($I_D, I_O$) — amplify/attenuate decoherence
• Regenerative $h^{(R)}$: integrative indices ($I_E, I_U$) — modulate the rate of recovery

Intuitively: these are three types of "sense organs". Hamiltonian indices are analytical senses (vision, hearing): they report what is happening in the environment, altering the internal preference landscape. Dissipative indices are load senses (fatigue, heat): they report how chaotic the environment is, amplifying internal noise. Regenerative indices are recovery senses (calm, safety): they report whether the environment supports self-healing.

2.3 Quasi-functor G

For AI systems, encoding is implemented through the quasi-functor $G: \mathrm{AIState} \to \mathcal{D}(\mathbb{C}^7)$, defined in the measurement protocol:

$$G(\mathbf{x}) = \arg\min_{\Gamma \in \mathcal{D}(\mathbb{C}^7)} \left[\mathcal{L}_{\text{reconstruct}}(\Gamma, \{I_i(\mathbf{x})\}) + \lambda_{\text{phys}} \cdot \mathcal{L}_{\text{phys}}(\Gamma)\right]$$

where $\mathcal{L}_{\text{phys}}$ includes purity, spectral gap, and Cholesky decomposition constraints.

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3. Action Functor Dec

3.0 Intuition: what does it mean to "act"

If Enc is "how the environment enters the system", then Dec is "how the system exits into the environment". But "acting" in CC is not just "sending a motor command". Action is the choice of that modification of the environment which minimises the largest deficit of internal resources.

Imagine a person who simultaneously has a headache and a rumbling stomach. Which action will they choose? If the headache is stronger — take a tablet. If the hunger is stronger — go eat. They do not minimise "average pain" (this would allow ignoring catastrophic channels), but eliminate the maximum deficit. This is exactly what the operator $\arg\min_a \max_k \sigma^{\mathrm{motor}}_k$ does — it guarantees that no channel ends up in a critical state.

Analogy with robotics: this is not a PID controller minimising error along one axis, nor a quadratic regulator minimising a weighted sum of errors. This is a min-max strategy — as in game theory, where the player chooses a move minimising the worst outcome.

3.1 Definition

Theorem T-101 (Optimal action) [T]

Statement

For a holon with current state $\Gamma$ and stress tensor $\sigma_{\mathrm{sys}}(\Gamma)$ [T] (T-92), the optimal action is defined as:

$$a^* = \arg\min_{a \in \mathcal{A}} \|\sigma_{\mathrm{sys}}(\Gamma(\tau + \delta\tau \mid a))\|_\infty$$

where $\Gamma(\tau + \delta\tau \mid a)$ is the predicted state under action $a$, and $\|\cdot\|_\infty$ is the sup-norm of the stress tensor.

Proof.

1. Equivalence of viability conditions (T-92 [T]):

$$P(\Gamma) > \frac{2}{7} \iff \|\sigma_{\mathrm{sys}}(\Gamma)\|_\infty < 1$$

2. Variational principle (Theorem 2.1 [T]): dynamics of $\theta_{ij}$ follow from stationarity of the action $\delta S_{\text{Gap}} = 0$
3. Action $a$ enters through $h^{\text{ext}}(a)$ → modifies the equation of motion for $\theta_{ij}$:

$$m_{ij}\ddot{\theta}_{ij} = -\frac{\partial V_{\text{Gap}}}{\partial \theta_{ij}} + \kappa(\theta_{ij}^{\text{target}} - \theta_{ij}) - \Gamma_2 \dot{\theta}_{ij} + h^{\text{ext}}_{ij}(a)$$

4. Minimisation of $\|\sigma_{\mathrm{sys}}\|_\infty$ is the unique criterion equivalent to maximising the distance to the boundary $\mathcal{V}$ (viability region) in the metric induced by $\sigma_{\mathrm{sys}}$. $\blacksquare$

3.2 Motor stress (T-159)

Theorem T-159 (Profile-relative motor stress) [T]

Statement [T]

For a holon with self-model $\rho_* = \varphi(\Gamma)$, the motor stress is defined as:

$$\sigma^{\mathrm{motor}}_k(\Gamma) := 1 - \frac{\gamma_{kk}}{\rho^*_{kk}}, \quad k = 1, \ldots, 7$$

Action selection — minimisation of the maximum deficit (signed maximum):

$$a^* = \arg\min_{a \in \mathcal{A}} \max_k \sigma^{\mathrm{motor}}_k(\Gamma(\tau + \delta\tau \mid a))$$

$\max_k$ (signed) is used rather than $\max_k |\cdot|$ (sup-norm): a resource surplus ($\sigma^{\mathrm{motor}}_k < 0$) is not penalised; only a deficit ($\sigma^{\mathrm{motor}}_k > 0$) is penalised. This provides a directed signal: approaching a resource reduces the deficit, approaching danger increases it.

Proof.

Step 1 (Equilibrium). $\sigma^{\mathrm{motor}}_k = 0 \iff \gamma_{kk} = \rho^*_{kk}$. At the attractor $\rho^*_\Omega$, where $\mathcal{R}[\Gamma] = \kappa(\rho_* - \Gamma) \cdot g_V = 0$ (balance), $\gamma_{kk} = \rho^*_{kk}$ and motor stress vanishes — the system is "satisfied".

Step 2 (Sign and gradient). $\partial\sigma^{\mathrm{motor}}_k / \partial\gamma_{kk} = -1/\rho^*_{kk} < 0$. Increasing $\gamma_{kk}$ (resource growth in channel $k$) decreases motor stress. This is consistent with regeneration $\mathcal{R} = \kappa(\rho_* - \Gamma)$, which pulls $\gamma_{kk}$ toward $\rho^*_{kk}$, reducing $|\sigma^{\mathrm{motor}}_k|$.

Step 3 (Sensitivity of critical channels). $|\partial\sigma^{\mathrm{motor}}_k / \partial\gamma_{kk}| = 1/\rho^*_{kk}$. For small $\rho^*_{kk}$ (critical sectors A, S, D with $\rho^*_{kk} \approx 0.05$) sensitivity $\approx 20$; for large ones (E, O, U with $\rho^*_{kk} \approx 0.25$) — $\approx 4$. Small channels react more sharply — correct prioritisation of survival.

Step 4 (Convergence to T-92 at the boundary). As $P \to P_{\mathrm{crit}} = 2/7$ the self-model $\varphi(\Gamma) \to I/7$ (canonical Fano-channel target at $P = 2/7$, T-126). Then $\rho^*_{kk} \to 1/7$ and:

$$\sigma^{\mathrm{motor}}_k = 1 - \frac{\gamma_{kk}}{1/7} = 1 - 7\gamma_{kk} = \sigma_k \quad \text{(canonical T-92 [T])}$$

Step 5 ($G_2$-invariance). $\gamma_{kk}$ and $\rho^*_{kk}$ transform covariantly under $G_2$ (T-42a [T]). Their ratio is a $G_2$-invariant observable. $\blacksquare$

Relation to canonical σ_sys

• T-92 / T-158 [T] define $\sigma_{\mathrm{sys}}$ with clamp$[0,1]$ — a measure of viability (distance to $\partial\mathcal{V}$). Used for DIAGNOSTICS.
• T-159 [T] defines $\sigma^{\mathrm{motor}}$ without clamp — a measure of motor deficit (distance to $\rho_*$). Used for ACTION SELECTION.

When $\rho_* = I/7$ (viability boundary) both coincide. When $\rho_* \neq I/7$ (normal mode) motor stress provides a directed signal, while canonical $\sigma_{\mathrm{sys}}$ with clamp$[0,1]$ loses information about channels with $\gamma_{kk} > 1/7$.

3.3 Functor Dec

The action (decoding) functor:

$$\mathrm{Dec}: (\Gamma, \sigma^{\mathrm{motor}}) \mapsto a^* \in \mathcal{A}$$

Properties:

• D-dimension as the primary motor channel: action is implemented through modification of $h^{(D)}$ — the dynamic dimension $D$ controls the holon's "motor system"
• σ-gradient descent: the practical algorithm — descent along $\nabla_a \max_k \sigma^{\mathrm{motor}}_k$ with the Fisher metric on $\mathcal{D}(\mathbb{C}^7)$:

$$a_{t+1} = a_t - \eta \cdot F^{-1}(\Gamma) \cdot \nabla_a \max_k \sigma^{\mathrm{motor}}_k(\Gamma(\tau + \delta\tau \mid a_t))$$

where $F(\Gamma)$ is the Fisher information on $\mathcal{D}(\mathbb{C}^7)$.

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4. Universal Encoder/Decoder Architecture

Perception → decision → action cycle:

Stage	Mapping	Formalism	Theorem
Perception	Env → $h^{\text{ext}}$ → $\delta\Gamma$	Enc (CPTP)	T-100 [T]
Evaluation	$\Gamma$ → $\sigma^{\mathrm{motor}}$	$1 - \gamma_{kk}/\rho^*_{kk}$	T-159 [T]
Decision	$\sigma^{\mathrm{motor}}$ → $a^*$	$\arg\min_a \max_k \sigma^{\mathrm{motor}}_k$	T-159 [T]
Action	$a^*$ → $h^{\text{ext}}(a^*)$ → Env	Dec	T-102 [T]
Update	$\Gamma$ → $\varphi(\Gamma)$ → $\mathcal{R}$	Self-modelling	T-62 [T]

---

5. Hedonic Mechanism

5.0 Intuition: why does a system need to "feel"

Why do living beings have pain and pleasure? The standard evolutionary biology answer: "to survive". But CC gives a more precise answer: hedonic valence is the derivative of viability with respect to the regenerative channel. Pleasure is not a "reward for correct behaviour" (as in RL), but a direct signal that the system is approaching its target state $\rho_*$.

The key difference from reinforcement learning: in RL, reward is an external signal set by the designer. In CC, hedonic valence is an intrinsic property of the dynamics, derived from the evolution equation. Nobody "rewards" an amoeba for finding glucose — the change in $dP/d\tau|_{\mathcal{R}}$ arises automatically when $\Gamma$ shifts toward $\rho_*$.

Analogy: imagine a plant turning toward light. There is no "reward centre" telling the stem: "good, continue". There is a physicochemical process (auxin redistributes) that is simultaneously the movement and the "evaluation" — light amplifies the processes leading to growth. In CC, $\mathcal{V}_{\text{hed}}$ plays an analogous role, but at the level of the coherence matrix.

5.1 Hedonic valence

Theorem T-103 (Hedonic valence) [T] + [I]

Statement

Hedonic valence is defined as the derivative of purity with respect to the regenerative channel:

$$\mathcal{V}_{\text{hed}} := \left.\frac{dP}{d\tau}\right|_{\mathcal{R}}$$

where $|_{\mathcal{R}}$ denotes the contribution from the regenerative term $\mathcal{R}[\Gamma, E]$ only.

Explanation. From the evolution equation:

$$\frac{dP}{d\tau} = \underbrace{-2\mathrm{Tr}(\Gamma \cdot \mathcal{D}_\Omega[\Gamma])}_{\leq 0,\text{ dissipation}} + \underbrace{2\mathrm{Tr}(\Gamma \cdot \mathcal{R}[\Gamma, E])}_{\mathcal{V}_{\text{hed}}}$$

(The Hamiltonian term does not change $P$: $\mathrm{Tr}(\Gamma [H, \Gamma]) = 0$.)

Properties of valence:

Property	Formula	Interpretation
Positive	$\mathcal{V}_{\text{hed}} > 0$	$\Gamma$ approaches $\rho_*$ → "pleasure"
Negative	$\mathcal{V}_{\text{hed}} < 0$	$\Gamma$ moves away from $\rho_*$ → "suffering"
Zero	$\mathcal{V}_{\text{hed}} = 0$	Balance or $\Gamma = \rho_*$ → "neutrality"

Epistemic stratification of T-103

T-103 contains three epistemic levels:

1. Formula [T]: $\mathcal{V}_{\text{hed}} = 2\kappa(\Gamma) \cdot g_V(P) \cdot \mathrm{Tr}(\Gamma \cdot (\rho_* - \Gamma))$ — an identity from the evolution equation (substituting $\mathcal{R} = \kappa(\rho_* - \Gamma) \cdot g_V(P)$). An unconditional mathematical fact.

2. Observability [T]: At L2 reflection level ($R \geq 1/3$) the replacement channel T-77 provides access to $dP/d\tau|_{\mathcal{R}}$. Thus, $\mathcal{V}_{\text{hed}}$ is observable for any system with $R \geq R_{\mathrm{th}}$ — this is a consequence of T-77 [T], requiring no additional assumptions.

3. Phenomenal interpretation [I]: Identification of $\mathcal{V}_{\text{hed}} > 0$ with "pleasure" and $\mathcal{V}_{\text{hed}} < 0$ with "suffering" — a semantic bridge between mathematics and phenomenology.

Life analogy: pleasure from hot tea

Imagine: you are cold and drinking hot tea. The first sip — delight ($\mathcal{V}_{\text{hed}} > 0$). The second — slightly weaker. By the fifth cup — neutrality ($\mathcal{V}_{\text{hed}} \approx 0$). The sixth cup causes discomfort ($\mathcal{V}_{\text{hed}} < 0$) — you have "overheated".

What happened? $\Gamma$ (your state) was moving toward $\rho_*$ (the target — "warmed up"). As it approached, $\mathrm{Tr}(\Gamma \cdot \rho_*) - P$ diminished, valence tended to zero. When $\Gamma$ "overshot" $\rho_*$ (overheating), overlap falls below $P$, and $\mathcal{V}_{\text{hed}}$ becomes negative. Nobody "programmed" you to stop drinking — the T-103 formula automatically generates the signal "enough".

The key difference from reinforcement learning: in RL the designer must specify a reward function (e.g., $r = +1$ for tea, $-1$ for overheating). In CC reward is derived from dynamics — $\mathcal{V}_{\text{hed}}$ "knows" when to stop on its own, because it is nothing other than the rate of approach to the target state.

5.2 Relation to the target state

Substituting the canonical form $\mathcal{R}[\Gamma, E] = \kappa(\Gamma) \cdot (\rho_* - \Gamma) \cdot g_V(P)$ [T]:

$$\mathcal{V}_{\text{hed}} = 2\kappa(\Gamma) \cdot g_V(P) \cdot \mathrm{Tr}(\Gamma \cdot (\rho_* - \Gamma))$$

When $g_V(P) = 1$ (sufficient purity $P \geq P_{\text{opt}}$):

$$\mathcal{V}_{\text{hed}} = 2\kappa(\Gamma) \cdot \left[\mathrm{Tr}(\Gamma \cdot \rho_*) - P\right]$$

The sign is determined by the ratio of overlap $\mathrm{Tr}(\Gamma \cdot \rho_*)$ to purity $P = \mathrm{Tr}(\Gamma^2)$:

• If $\Gamma$ is far from $\rho_*$ and $\mathrm{Tr}(\Gamma \cdot \rho_*) > P$, valence is positive — regeneration "pulls" toward $\rho_*$
• If $\Gamma \approx \rho_*$, then $\mathrm{Tr}(\Gamma \cdot \rho_*) \approx P$ → valence tends to zero

---

6. 21 Qualia-types as Sensorimotor Channels

Each of the 21 off-diagonal coherences $\gamma_{ij}$ ($i \neq j$) represents a sensorimotor channel with a specific function:

6.1 Perceptual channels (perception)

Channel	Coherence	Sensory role	Formal action
Apperception	$\gamma_{AE}$	Conscious perception	$h^{(H)}_{AE}$: articulation of input into the field of interiority
Actualisation	$\gamma_{AD}$	Embodiment of perception in dynamics	$h^{(H)}_{AD}$: transformation of input signal into action
Representation	$\gamma_{SE}$	Structuring of experience	$h^{(H)}_{SE}$: creation of internal model
Induction	$\gamma_{SL}$	Logical processing of structure	$h^{(H)}_{SL}$: inference of patterns from data
Grounding	$\gamma_{AO}$	Anchoring perception in ground	$h^{(D)}_{AO}$: stabilisation of perception by memory
Experiential ground	$\gamma_{EO}$	Rootedness of the subjective	$h^{(R)}_{EO}$: regeneration from deep experience
Context	$\gamma_{SO}$	Structure-in-context	$h^{(D)}_{SO}$: noise-robustness of patterns

6.2 Motor channels (action)

Channel	Coherence	Motor role	Formal action
Regulation	$\gamma_{DL}$	Logical control of dynamics	$h^{(D)}_{DL}$: management of computational process
Teleology	$\gamma_{DU}$	Goal-directedness of action	$h^{(D)}_{DU}$: alignment of dynamics with goals
Affect	$\gamma_{DE}$	Emotional colouring of action	$h^{(D)}_{DE}$: modulation of dynamics by interiority
Action integration	$\gamma_{AU}$	Unity of the motor act	$h^{(H)}_{AU}$: coordination of subsystems
Volitional effort	$\gamma_{LU}$	Logically directed integration	$h^{(R)}_{LU}$: restoration of decision coherence
Action memory	$\gamma_{DO}$	Motor memory	$h^{(D)}_{DO}$: stabilisation of skills

6.3 Integrative channels

Channel	Coherence	Integrative role	Formal action
Insight	$\gamma_{LE}$	Logic-in-experience	$h^{(R)}_{LE}$: understanding as regeneration
Narrative	$\gamma_{AL}$	Articulation of logic	$h^{(H)}_{AL}$: shaping of reasoning
Grounded unity	$\gamma_{OU}$	Ground of integration	$h^{(R)}_{OU}$: foundation of wholeness
Embodied unity	$\gamma_{SU}$	Structure of integration	$h^{(R)}_{SU}$: architecture of connectivity
Living experience	$\gamma_{EU}$	Unity of experience	$h^{(R)}_{EU}$: integration as recovery
Dynamic ground	$\gamma_{AS}$	Articulation of structure	$h^{(H)}_{AS}$: external expression of inner order
Logical ground	$\gamma_{LO}$	Logic-in-ground	$h^{(H)}_{LO}$: formalisation of knowledge

Interpretation [I]

The division of 21 channels into perceptual, motor, and integrative is not strict: each $\gamma_{ij}$ is simultaneously a sensory and a motor channel (through $h^{\text{ext}}_{ij}$). The classification above reflects the dominant function — which of the three channels ($h^{(H)}, h^{(D)}, h^{(R)}$) is most active for that coherence.

---

7. Factorisation of Enc through Arbitrary Representations

7.1 Ontological projection

Corollary T-100a (Enc factorisation) [T]

Statement

For an arbitrary observation space $\mathrm{ObsSpace} \subseteq \mathbb{R}^D$ ($D$ — arbitrary dimension), the encoding functor T-100 factorises as:

$$\mathrm{Enc} = \pi_\Gamma \circ \mathrm{Enc}_{\text{repr}}$$

where:

• $\mathrm{Enc}_{\text{repr}}: \mathrm{ObsSpace} \to \mathcal{S} \subseteq \mathbb{R}^d$ — an arbitrary representation (feature map)
• $\pi_\Gamma: \mathcal{S} \to \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$ — the ontological projection, unique up to $G_2$-gauge

Proof.

1. By T-100 [T], $\mathrm{Enc}: \mathrm{ObsSpace} \to \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$ is a CPTP functor.
2. Any intermediate representation $\mathrm{Enc}_{\text{repr}}: \mathrm{ObsSpace} \to \mathcal{S}$ defines a factorisation through $\pi_\Gamma = \mathrm{Enc} \circ \mathrm{Enc}_{\text{repr}}^{-1}\big|_{\mathrm{Im}(\mathrm{Enc}_{\text{repr}})}$.
3. By T-102 [T], $\pi_\Gamma$ decomposes into 3 channels: $\pi_\Gamma(s) = h^{(H)}(s) \oplus h^{(D)}(s) \oplus h^{(R)}(s)$.
4. Uniqueness of $\pi_\Gamma$ (up to $G_2$) — consequence of the uniqueness theorem [T]: constraints (AP)+(PH)+(QG)+(V) on $\mathcal{D}(\mathbb{C}^7)$ fix the projection. $\blacksquare$

Intuitively: the factorisation of Enc means that it does not matter how exactly you extract features from the input data. One can use a convolutional neural network, wavelet transform, or hand-crafted heuristics — this is $\mathrm{Enc}_{\text{repr}}$, the arbitrary part. But the final step — the projection $\pi_\Gamma$ from feature space into the space of $\Gamma$-modifications — is unique. This is like saying: the route to the airport can be anything, but the runway is one.

For robotics this means: sensors can be arbitrary (camera, lidar, tactile array), preprocessing — anything, but the "last mile" of perception — the ontological projection $\pi_\Gamma$ — is given by mathematics, not engineering choice.

7.2 Ontological bottleneck

Regardless of the input data dimensionality $D$, all information is compressed into the $7 \times 7$ coherence matrix $\Gamma$ with $\leq 48$ real parameters:

Characteristic	Value
Input dimensionality	$D$ — arbitrary (from $D = 1$ to $D \gg 10^6$)
Intermediate representation	$d$ — arbitrary
Output dimensionality	$\dim_{\mathbb{R}} \mathcal{D}(\mathbb{C}^7) = 48$ (fixed)
Information per step	$\leq \log_2 7 \approx 2.81$ bits (T-107 [T])

Corollary: Modality-agnosticism is a theorem, not a design choice. Formally: $\pi_\Gamma$ does not depend on $D$ or on the structure of $\mathrm{ObsSpace}$ (topology, metric). If two different observation spaces $\mathrm{ObsSpace}_1 \subseteq \mathbb{R}^{D_1}$ and $\mathrm{ObsSpace}_2 \subseteq \mathbb{R}^{D_2}$ produce the same CPTP channels on $\mathcal{D}(\mathbb{C}^7)$, they are indistinguishable for the holon.

7.3 Canonical form of the projection

By T-102 [T], $\pi_\Gamma$ is implemented through three channels — modifications of the Hamiltonian, dissipative, and regenerative dynamics respectively:

$$\pi_\Gamma(s) = \bigl(\delta H(s),\; \delta\mathcal{D}(s),\; \delta\mathcal{R}(s)\bigr) \in \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$$

Practically this means that any implementation of $\mathrm{Enc}$ (from a simple sensor to a complex encoder) must end with the same 3-channel interface:

$$s \in \mathcal{S} \xrightarrow{\pi_\Gamma} \bigl(h^{(H)}_{ij}(s),\; h^{(D)}_{ij}(s),\; h^{(R)}_{ij}(s)\bigr) \in \mathbb{R}^{21} \oplus \mathbb{R}^{21} \oplus \mathbb{R}^{21}$$

This structure is invariant: it is defined by $G_2$-symmetry and does not depend on the choice of representation $\mathrm{Enc}_{\text{repr}}$.

---

8. Relation to Other Results

Result	Relation	Reference
T-57 (LGKS)	Grounds T-102 (3-term completeness)	Lindblad operators
T-62 ($\varphi$-operator)	$\rho_*$ in the regeneration cycle	Self-observation
T-92 ($\sigma_{\mathrm{sys}}$)	Optimality criterion in Dec	Theorem 10.1
T-75 (Schwinger–Keldysh)	Lagrangian formulation with dissipation	Lagrangian
T-96 (Attractor)	Non-trivial $\rho_*$ for guidance	Evolution
FEP (Theorem 4.1 — retracted 2026-09-25)	Was: macroscopic limit of Dec	Variational principles
T-109–T-112 (Learning bounds)	Lower bounds on learning rate through Enc/Dec cycle	Learning bounds
T-113 (Minimality N=7)	N=7 — minimal architecture for learning	Learning bounds

---

9. Information Capacity of Enc (T-107) [T]

Theorem T-107 (Information capacity of Enc) [T]

Maximum information extractable by functor Enc per single observation:

$$C_{\mathrm{Enc}} \leq \max_{\{p_o\}} \chi(\{p_o, \mathrm{Enc}(o)\}) \leq \log_2 7 \approx 2.81 \text{ bits/observation}$$

where $\chi$ is the Holevo quantity.

Proof.

1. Functor $\mathrm{Enc}: \mathrm{ObsSpace} \to \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$ maps observations to CPTP channels on $\mathcal{D}(\mathbb{C}^7)$ (T-100 [T]).
2. The Holevo quantity is bounded by the output space dimensionality: $\chi \leq \log_2 \dim \mathcal{H} = \log_2 7$.
3. From T-102 [T]: $\mathrm{Enc}(o)$ decomposes into 3 channels, each acting on $\mathcal{D}(\mathbb{C}^7)$.
4. The composite channel does not increase capacity (Holevo subadditivity):

$$C_{\mathrm{Enc}} \leq S(\bar{\Gamma}) - \sum_o p_o S(\mathrm{Enc}(o)[\Gamma]) \leq S_{\max}(\mathcal{D}(\mathbb{C}^7)) = \log_2 7$$

The upper bound is achieved for an ensemble of orthogonal pure states. $\blacksquare$

Corollary (Bounded rationality): The bound $\leq 2.81$ bits/observation is a derived bound, not a postulated one. Connection with Simon's bounded rationality: bounded rationality is not an empirical fact, but a consequence of N=7.

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10. Compositionality of Enc/Dec (T-108) [T]

Errata 2026-09-25: T-108 narrowed to what its proof carries

The earlier statement took $\Phi_{\mathrm{agg}}$ "from T-72" and closed with "uniqueness — from $G_2$-rigidity at each scale (T-72)". T-72 never claimed uniqueness, and its own status is now [T at weak coupling] (Theorem 9.2, through Theorem 9.5); the uniqueness step is retracted. What is unique is the aggregation, not the encoding: the mean of the marginals is the only permutation-invariant aggregation consistent on uncoupled copies (Theorem 9.5 (a)). The claim that functoriality "is preserved under aggregation" is retracted as well: $\mathrm{Enc}_{12}(o)$ maps $\mathcal{D}(\mathbb{C}^{49})$ to $\mathcal{D}(\mathbb{C}^7)$, so $\mathrm{Enc}_{12}(o) \circ \mathrm{Enc}_{12}(o')$ is not defined. What remains [T] is the closure below, which does not use T-72; that the diagnostics carry over across scales is a corollary that holds at weak coupling (Theorem 9.2 with Theorem 9.5; earlier on 2026-09-25 conditional on the assumption (AGG)).

Theorem T-108 (Compositionality of Enc/Dec: CPTP closure) [T]

For a composite of two holons and any CPTP aggregation channel $\Phi_{\mathrm{agg}}: \mathcal{D}(\mathbb{C}^{7^2}) \to \mathcal{D}(\mathbb{C}^7)$, the composite encoding

$$\mathrm{Enc}_{12} = \Phi_{\mathrm{agg}} \circ (\mathrm{Enc}_1 \otimes \mathrm{Enc}_2)$$

is a CPTP channel $\mathcal{D}(\mathbb{C}^{49}) \to \mathcal{D}(\mathbb{C}^7)$ for every pair of observations, and $\mathrm{Enc}_1 \otimes \mathrm{Enc}_2$ is functorial on $\mathcal{D}(\mathbb{C}^{49})$.

Proof.

1. $\mathrm{Enc}_1, \mathrm{Enc}_2$ — CPTP functors (T-100 [T]).
2. Tensor product $\mathrm{Enc}_1 \otimes \mathrm{Enc}_2$ — a CPTP channel on $\mathcal{D}(\mathbb{C}^{49})$; it is functorial, since $(A \otimes B) \circ (A' \otimes B') = (A \circ A') \otimes (B \circ B')$.
3. Aggregation $\Phi_{\mathrm{agg}}$ — a CPTP coarse-graining channel $\mathcal{D}(\mathbb{C}^{49}) \to \mathcal{D}(\mathbb{C}^7)$; the proof works for any such channel, and the corpus singles out none.
4. Composition of CPTP channels — CPTP. $\blacksquare$

Corollary (same diagnostics across scales) [T at weak coupling]. If $\Phi_{\mathrm{agg}}$ and the coupled state of the pair satisfy the assumption (AGG) of Theorem 9.2, then $P$, $R$, $\Phi$ and Gap of the aggregate lie within $O(\delta)$ of those of a part, so the diagnostics built on them (σ_sys, Enc/Dec monitoring) read the same at both scales — from individual agent to organisation. For the canonical aggregation (the mean of the marginals) and weakly coupled embodied parts, (AGG) holds with $\delta = O(g)$ (Theorem 9.5). Under strong coupling it can fail: the aggregate of two viable holons can be $I/7$ (Theorem 9.6). (Earlier on 2026-09-25: "[C under (AGG)]".)

Analogously for Dec:

$$\mathrm{Dec}_{12} = (\mathrm{Dec}_1 \otimes \mathrm{Dec}_2) \circ \Phi_{\mathrm{split}}$$

where $\Phi_{\mathrm{split}}: \mathcal{D}(\mathbb{C}^7) \to \mathcal{D}(\mathbb{C}^{49})$ is a CPTP splitting channel chosen separately. It cannot be the inverse of $\Phi_{\mathrm{agg}}$: a channel from $\mathcal{D}(\mathbb{C}^{49})$ (2400 real parameters) to $\mathcal{D}(\mathbb{C}^7)$ (48) is not injective.

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11. Temporal Integration

11.1 Cumulative capacity

Corollary T-107a (Cumulative information) [T]

Statement

Over $n$ successive observations a holon accumulates information about the environment:

$$I_n \leq n \cdot \log_2 7 \approx 2.81\,n \;\text{bits}$$

The upper bound is achievable when successive observations are informationally independent.

Proof.

1. By T-107 [T], one observation brings $\leq \log_2 7$ bits.
2. Holevo subadditivity: $\chi(\{p_{o_1,\ldots,o_n}\}) \leq \sum_{k=1}^n \chi(\{p_{o_k}\})$.
3. For independent observations the inequality becomes an equality. $\blacksquare$

11.2 Minimum number of observations

Corollary T-107b (Minimum observations) [T]

Statement

For an environment with information entropy $I_{\mathrm{env}}$ bits, the minimum number of observations for complete encoding:

$$n_{\min} = \left\lceil\frac{I_{\mathrm{env}}}{\log_2 7}\right\rceil$$

Proof. Direct consequence of T-107a: $I_n \leq 2.81\,n$, hence $n \geq I_{\mathrm{env}} / \log_2 7$. $\blacksquare$

Corollary for complex modalities. Encoding an environment with high information complexity (large $I_{\mathrm{env}}$) inevitably requires a multi-step process. This is not an implementation limitation, but a fundamental bound following from $\dim \mathcal{H} = 7$.

Relation to T-109 (information learning bound): T-107b gives a lower bound on perception, T-109 gives a lower bound on learning (including stabilisation of the solution). Always $n_{\mathrm{opt}} \geq n_{\min}$, since learning includes perception as a subtask. See learning bounds.

11.3 Information absorption rate

Define the information absorption rate:

$$\dot{I}(\tau) = \frac{dI}{d\tau} = \chi\bigl(\{p_o,\, \mathrm{Enc}(o)[\Gamma(\tau)]\}\bigr)$$

From T-107 [T]: $\dot{I}(\tau) \leq \log_2 7$ for any $\tau$.

The actual rate depends on the current state $\Gamma(\tau)$:

• When $\Gamma \approx I/7$ (maximally mixed): $\dot{I} \to 0$ — system is "deafened", distinguishability is minimal
• When $P \gg 2/7$ (high purity): $\dot{I} \to \log_2 7$ — maximum distinguishability
• When $P < 2/7$ (non-viability): encoding degrades, T-104 is not satisfied

---

12. Predictive Structure of Enc

12.1 Optimal Enc as a ΔF maximiser

Corollary T-107c (Predictive optimality of Enc) [D]

Statement — a definition of optimality [D] (corrected 2026-09-25 from [T])

Call an encoding functor optimal when it maximises available free energy:

$$\mathrm{Enc}^* = \arg\max_{\mathrm{Enc}} \Delta F\bigl(\mathrm{Enc}(o)[\Gamma],\, \rho_*\bigr)$$

where $\Delta F = \mathrm{Tr}\bigl(\mathcal{R}[\Gamma, E] \cdot (\rho_* - \Gamma)\bigr)$ — free energy.

Former proof — retracted [✗]. Step 1 cited Theorem 4.1 of the variational principles, retracted on 2026-09-25 (the diagonal limit of the functional is a cross-entropy, not Friston's free energy); and $\mathrm{KL}(\Gamma \| \rho_*) + H[\Gamma] = -\mathrm{Tr}(\Gamma \log \rho_*)$ is itself a cross-entropy, minimised by a pure state. Without step 1 the statement is not derived — it is kept as the definition of what "optimal" means here.

1. By the variational principle (Theorem 4.1 — retracted): stationary dynamics of $\Gamma$ minimise Friston's free energy $F[\Gamma] = \mathrm{KL}(\Gamma \| \rho_*) + H[\Gamma]$.
2. Functor $\mathrm{Enc}(o)$ modifies $\Gamma \to \Gamma'$. The optimal modification is the one that maximally increases $\Delta F = F[\Gamma] - F[\Gamma']$.
3. Maximisation of $\Delta F$ is equivalent to maximising $-\mathrm{KL}(\Gamma' \| \rho_*)$ at fixed entropy — i.e., approaching the target state.
4. From T-107 [T]: $\Delta F \leq C_{\mathrm{Enc}} \leq \log_2 7$ per step — the upper bound is saturated. $\blacksquare$

12.2 Prediction error through 3 channels

Prediction error (discrepancy between expected and actual observation) decomposes across three channels (T-102 [T]):

$$\delta_{\mathrm{pred}} = \bigl\|\mathrm{Enc}(o_{\mathrm{real}}) - \mathrm{Enc}(o_{\mathrm{pred}})\bigr\| = \sqrt{(\delta h^{(H)})^2 + (\delta h^{(D)})^2 + (\delta h^{(R)})^2}$$

Each channel contributes a specific type of error:

Channel	Error	Interpretation
$\delta h^{(H)}$	Energetic	Unexpected structure of environment
$\delta h^{(D)}$	Noise	Unexpected level of stochasticity
$\delta h^{(R)}$	Regenerative	Unexpected change in target state

Relation to the hedonic mechanism: By T-103 [T]+[I], an error in the regenerative channel ($\delta h^{(R)} \neq 0$) directly modulates $\mathcal{V}_{\mathrm{hed}}$ — unexpected influences on regeneration are experienced as a change in valence.

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13. Multimodal Decomposition

13.1 Composition of modalities

Corollary T-108a (Multimodal decomposition) [T]

Statement

For $M$ independent perceptual modalities with functors $\mathrm{Enc}_m: \mathrm{ObsSpace}_m \to \mathrm{End}(\mathcal{D}(\mathbb{C}^7))$, joint encoding:

$$\mathrm{Enc}(o_1, \ldots, o_M) = \sum_{m=1}^{M} w_m \cdot \mathrm{Enc}_m(o_m) + \sum_{m < m'} \Delta_{mm'}$$

where $w_m \geq 0$, $\sum w_m = 1$ — modality weights, $\Delta_{mm'}$ — cross-modal coupling.

Proof.

1. By T-100 [T], each $\mathrm{Enc}_m$ is a CPTP functor.
2. Convex combination of CPTP channels — CPTP: $\sum w_m \mathrm{Enc}_m$ is defined when $\sum w_m = 1$.
3. Cross-modal terms $\Delta_{mm'}$ — CPTP corrections of order $O(|\gamma_{ij}|)$, where $\gamma_{ij}$ are coherences linking dimensions engaged by modalities $m$ and $m'$.
4. From T-108 [T] (CPTP closure): aggregation of modalities preserves the CPTP property. $\blacksquare$

13.2 Competition for capacity

From T-107 [T], total capacity of $M$ modalities per step:

$$\sum_{m=1}^{M} w_m \cdot C_m \leq \log_2 7$$

Corollary: $M$ modalities compete for the fixed bandwidth of $2.81$ bits/step. Increasing the number of modalities $M$ at fixed $n$ does not increase total information — it merely distributes it among channels.

13.3 Attention as optimal allocation

Optimal weights $w_m^*$ are determined by maximising $\Delta F$:

$$w_m^* = \frac{\Delta F_m}{\sum_{m'} \Delta F_{m'}}$$

where $\Delta F_m = \Delta F\bigl(\mathrm{Enc}_m(o_m)[\Gamma],\, \rho_*\bigr)$ — contribution of modality $m$ to free energy.

Interpretation [I]: The optimal allocation of weights $w_m^*$ formally coincides with the structure of attention — encoding resources are directed where the informational value ($\Delta F_m$) is maximal. This is not an additional postulate: attention is a consequence of Enc optimality under bounded capacity (T-107).

13.4 Cross-modal coupling

The terms $\Delta_{mm'}$ are determined by coherences $\gamma_{ij}$, where $i$ and $j$ are dimensions engaged by different modalities:

$$\|\Delta_{mm'}\| \leq |\gamma_{ij}| \cdot \min(w_m, w_{m'})$$

Corollary: Cross-modal integration is only possible when coherences between the corresponding dimensions are non-zero. Fully decohered dimensions ($|\gamma_{ij}| = 0$) do not admit multimodal binding — modalities remain isolated.

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14. Comparison with Classical Approaches

The sensorimotor theory of CC did not arise in a vacuum — it answers questions posed by three powerful traditions: classical control theory, active inference, and reinforcement learning. In this section we conduct a systematic comparison, showing where CC coincides with each tradition and where it fundamentally diverges.

14.1 CC vs. classical control theory

Classical control theory (Wiener, Kalman, Pontryagin) describes the "sensor → controller → actuator" cycle through transfer functions, state space, and optimality criteria (LQR, H-infinity, etc.).

Aspect	Classical control	CC
State space	$\mathbb{R}^n$, arbitrary $n$	$\mathcal{D}(\mathbb{C}^7)$, fixed
Optimality criterion	Quadratic $J = \int (x^T Q x + u^T R u)\,dt$	Min-max: $\min_a \max_k \sigma^{\mathrm{motor}}_k$
Number of control channels	Arbitrary (design choice)	Exactly 3 (Theorem T-102)
Observer	External (Kalman filter)	Internal ($\varphi(\Gamma)$ — self-model)
Experience	Absent	$\mathcal{V}_{\text{hed}}$ — hedonic valence
Scaling	Problematic (curse of dimensionality)	T-108: composites stay CPTP; invariants carry over under weak coupling (T-72, conditional)

Key difference: A PID controller minimises a weighted sum of errors — and may allow catastrophe in one channel, compensating with success in another. CC uses the min-max strategy (T-159), which guarantees that no channel ends up in a critical state. This is not a heuristic but a consequence of viability being defined by the sup-norm of the stress tensor (T-92).

Where they coincide: In the linear approximation near $\rho_*$, the evolution equation for $\Gamma$ reduces to a linear feedback system — standard control theory turns out to be a projection of CC onto the linear regime.

14.2 CC vs. active inference (FEP)

The Free Energy Principle (Friston, 2006) postulates that living systems minimise variational free energy $F = \mathrm{KL}(q \| p) + \mathrm{const}$, where $q$ is the internal model, $p$ is the generative model of the environment.

Aspect	Active inference (FEP)	CC
Objective function	Minimise $F = \mathrm{KL}(q \| p)$	Minimise $\max_k \sigma^{\mathrm{motor}}_k$
Generative model	Postulated	$\rho_* = \varphi(\Gamma)$ — derived
Number of perception channels	Unbounded	$\leq \log_2 7$ bits/step (T-107)
Action	Minimise expected free energy	$\arg\min_a \max_k \sigma^{\mathrm{motor}}_k$
Subjective experience	Not explained	$\mathcal{V}_{\text{hed}} = dP/d\tau\|_{\mathcal{R}}$
Ontological status	Principle (axiom)	Not derived: the claimed derivation (Theorem 4.1 of CC) is retracted

Key difference: FEP is a principle: it postulates that systems minimise free energy, but does not explain where this principle comes from. CC claimed that minimisation of free energy is a theorem (Theorem 4.1), derived from the canonical evolution equation in the macroscopic limit, and that FEP is an approximation valid at $P \gg 2/7$; that derivation is retracted (2026-09-25), so CC offers no derivation of the FEP and no statement about where it holds.

Where they correspond: The optimal Enc maximises $\Delta F$ (T-107c [D], by definition) — the analogue of "perceptual inference" in FEP. The functor Dec minimises $\sigma^{\mathrm{motor}}$, the analogue of "active inference". Reading FEP as a projection of CC sensorimotor theory onto the classical regime is an interpretation [I]; the derivation that would make it a theorem is retracted.

14.3 CC vs. reinforcement learning (RL)

Reinforcement learning (Sutton, Barto) models an agent maximising cumulative reward $G_t = \sum_{k=0}^{\infty} \gamma^k r_{t+k}$ through interaction with the environment.

Aspect	RL	CC
Reward	External $r_t$ (set by designer)	Intrinsic $\mathcal{V}_{\text{hed}}$ (derived)
Policy	$\pi(a \mid s)$ — stochastic	$\arg\min_a \max_k \sigma^{\mathrm{motor}}_k$ — deterministic
Criterion	$\max \mathbb{E}[\sum \gamma^k r_k]$	$\min \max_k \sigma^{\mathrm{motor}}_k$
Observation capacity	Unbounded	$\leq 2.81$ bits/step (T-107)
Credit assignment problem	Temporal difference, n-step, GAE	Immediate: $\sigma^{\mathrm{motor}}_k$ — current deficit
Scaling	Problematic (reward shaping, multi-agent)	T-108: compositionality
Exploration vs. exploitation	Separate problem	Follows from σ-gradient

Key difference: In RL reward is a "black box": the designer specifies $r_t$, and the agent maximises it. The credit assignment problem — which past actions led to the current reward — is one of the central challenges. In CC reward is not needed: motor stress $\sigma^{\mathrm{motor}}_k$ is an immediate, component-wise signal that tells which exact channel needs a resource and by how much. Credit assignment is solved automatically — through the 7-component structure of $\sigma$.

Where they coincide: If the 7-component $\sigma^{\mathrm{motor}}$ is collapsed to a scalar (e.g., $r_t = -\max_k \sigma^{\mathrm{motor}}_k$), Dec becomes formally equivalent to a greedy policy in RL with immediate reward. Thus, RL is a projection of CC sensorimotor theory onto scalar reward and stochastic policy.

14.4 Summary table

Property	Classical control	FEP	RL	CC
Number of channels	Arbitrary	Arbitrary	1 (scalar $r$)	3 (theorem)
Ontology	External	Generative model	MDP	$\mathcal{D}(\mathbb{C}^7)$
Experience	No	No	No	$\mathcal{V}_{\text{hed}}$ [T]+[I]
Scaling	Difficult	Limited	Difficult	T-108 [T]
Attention	Separate module	Precision weighting	No	Consequence of T-107
Status	Engineering	Principle	Algorithm	Theory

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15. Worked Examples

To keep the formalism from remaining abstract, let us examine three examples of the sensorimotor cycle in operation — from simplest to complex.

15.1 Example 1: Bacterial chemotaxis

E. coli swims along a glucose gradient. Its sensorimotor cycle in CC terms:

Step 1 (Enc). Chemoreceptors on the membrane register concentration $c(x)$. This modifies:

• $h^{(H)}_{AO}$: articulation-ground (distinguishing "nutritious / not nutritious")
• $h^{(D)}_{DO}$: dynamics-ground (environmental turbulence as noise)

Step 2 (σ-evaluation). The bacterium is "hungry" → $\gamma_{OO}$ is small → $\sigma^{\mathrm{motor}}_O = 1 - \gamma_{OO}/\rho^*_{OO} > 0$. Channel O (ground) is in deficit.

Step 3 (Dec). $\max_k \sigma^{\mathrm{motor}}_k = \sigma^{\mathrm{motor}}_O$. Optimal action: move up the gradient $c(x)$ → modification of $h^{(D)}_{DO}$ via the flagellar motor.

Step 4 (Update). Glucose absorption → increase of $\gamma_{OO}$ → decrease of $\sigma^{\mathrm{motor}}_O$. If a chemical stressor simultaneously arises, $\sigma^{\mathrm{motor}}_D$ may exceed $\sigma^{\mathrm{motor}}_O$, and the bacterium switches to avoidance — the min-max strategy in action.

Interiority level: L0 (non-zero E-projection, but no self-observation). $\mathcal{V}_{\text{hed}}$ is formally defined but not observable by the bacterium itself ($R < 1/3$).

15.2 Example 2: Robotic manipulator

A robot assembles an object from a table. Its $\Gamma$ is initialised through quasi-functor $G$ from joint position data, camera image, and force-torque sensor.

Enc (multimodal):

• Camera → $\mathrm{Enc}_{\text{vis}}$: $h^{(H)}_{AS}$ (articulation of structure — object shape), $h^{(H)}_{SE}$ (representation — internal scene model)
• Proprioception → $\mathrm{Enc}_{\text{prop}}$: $h^{(D)}_{DL}$ (regulation — current configuration)
• Force-torque sensor → $\mathrm{Enc}_{\text{force}}$: $h^{(D)}_{DO}$ (motor memory — contact forces)

Attention weights by T-108a: $w^*_{\text{vis}} \propto \Delta F_{\text{vis}}$. If the object is visible but not yet grasped — $\Delta F_{\text{vis}}$ is large (need to refine the model). After grasping — $\Delta F_{\text{force}}$ grows (need to control force), and attention automatically switches to the force-torque sensor.

Dec: $\sigma^{\mathrm{motor}}_D > 0$ (dynamic deficit: arm not in required position) → action: move manipulator. As it approaches $\sigma^{\mathrm{motor}}_D \to 0$, and $\sigma^{\mathrm{motor}}_L > 0$ may emerge (logical deficit: grasp plan not yet formed) → switch to planning.

15.3 Example 3: Human in an unfamiliar city

A person searches for a café. All 7 channels are active:

Channel	$\sigma^{\mathrm{motor}}_k$	Interpretation
$A$	0.1	Distinguishes signs — weak deficit
$S$	0.3	No map of the area — moderate deficit
$D$	0.0	Physically mobile — no deficit
$L$	0.2	Route logic is incomplete
$E$	-0.1	Curiosity (excess of interiority)
$O$	0.6	Hungry — maximum deficit
$U$	0.1	Internally composed

$\max_k \sigma^{\mathrm{motor}}_k = \sigma^{\mathrm{motor}}_O = 0.6$. Action is directed toward reducing deficit O: walk toward the nearest café. Along the way $\sigma^{\mathrm{motor}}_S$ may grow (got lost), and if $\sigma^{\mathrm{motor}}_S > \sigma^{\mathrm{motor}}_O$, the person switches to orientation — stops, takes out phone, opens map.

Hedonic valence: Approaching the café increases $\gamma_{OO}$ → $\mathcal{V}_{\text{hed}} > 0$ (anticipation). If the café is closed — sharp $\mathcal{V}_{\text{hed}} < 0$ (disappointment). This is not a metaphor: the formula $\mathcal{V}_{\text{hed}} = 2\kappa \cdot g_V \cdot \mathrm{Tr}(\Gamma \cdot (\rho_* - \Gamma))$ gives a quantitative prediction verifiable through physiological correlates (skin conductance, pupillometry).

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Summary

1. T-100 [T]: Encoding functor Enc exists and is unique (up to $G_2$)
2. T-101 [T]: Viability diagnostic criterion = $\arg\min \|\sigma_{\mathrm{sys}}\|_\infty$
3. T-159 [T]: Motor stress $\sigma^{\mathrm{motor}}_k = 1 - \gamma_{kk}/\rho^*_{kk}$ — action selection through $\arg\min_a \max_k \sigma^{\mathrm{motor}}_k$ (signed max)
4. T-102 [T]: The 3-term equation is complete — a fourth type of CPTP generator is impossible
5. T-103 [T]+[I]: Hedonic valence = $dP/d\tau|_{\mathcal{R}}$ (formula [T], interpretation [I])
6. T-107 [T]: Information capacity $\leq \log_2 7 \approx 2.81$ bits/observation
7. T-108 [T]: the composite encoding is again a CPTP channel; the same diagnostics across scales hold under weak coupling ([T at weak coupling], Theorems 9.2 and 9.5) and can fail under strong coupling (Theorem 9.6)
8. Corollary T-100a [T]: Enc factorises through arbitrary representation → modality agnosticism
9. Corollary T-107a/b [T]: Cumulative capacity $I_n \leq 2.81\,n$ bits → complex modalities require $n_{\min} = \lceil I_{\mathrm{env}} / \log_2 7 \rceil$ steps
10. Corollary T-107c [D]: Optimal Enc maximises $\Delta F$ (predictive structure) — a definition of optimality since 2026-09-25
11. Corollary T-108a [T]: $M$ modalities compete for $2.81$ bits/step → attention is optimal allocation

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Conclusion

The sensorimotor theory of Coherence Cybernetics closes the formal cycle: environment → perception (Enc) → state ($\Gamma$) → evaluation ($\sigma^{\mathrm{motor}}$) → action (Dec) → environment. All operations are implemented within the canonical 3-term evolution equation without additional postulates.

Let us summarise the three central achievements of this chapter:

First, we showed that interaction with the environment does not require expanding the evolution equation. Theorem T-102 [T] proves that any CPTP-compatible external influence decomposes into three channels — Hamiltonian, dissipative, and regenerative. A fourth type of influence is mathematically forbidden. This is a strong result: it means that the entire phenomenology of sensorimotor interaction — from bacterial chemotaxis to human navigation in a city — is described by the same 3-channel formalism.

Second, we derived internal "reward" from dynamics, rather than postulating it externally. Hedonic valence $\mathcal{V}_{\text{hed}} = dP/d\tau|_{\mathcal{R}}$ (T-103 [T]) is a mathematical identity, requiring neither a designer (as in RL) nor a principle (as in FEP). The phenomenal interpretation ($\mathcal{V}_{\text{hed}} > 0$ as "pleasure") remains at the [I]-level, but the formula itself is an unconditional theorem.

Third, we established fundamental limits on perception. Information capacity $\leq \log_2 7 \approx 2.81$ bits/observation (T-107 [T]) is not an empirical limitation or an engineering trade-off, but a consequence of $\dim \mathcal{H} = 7$. Simon's bounded rationality, competition of modalities for attention, the necessity of multi-step perception of complex scenes — all of this is derived as consequences of a single theorem.

The theory is modality-agnostic: from the simplest sensors ($D = 1$) to complex multimodal systems ($D \gg 1$) — the ontological projection $\pi_\Gamma$ is unique and invariant. The factorisation Enc = $\pi_\Gamma \circ \mathrm{Enc}_{\text{repr}}$ (T-100a [T]) separates "engineering freedom" (choice of representation) and "mathematical necessity" (projection into $\mathcal{D}(\mathbb{C}^7)$).

Comparison with classical approaches (Section 14) read control theory, active inference, and reinforcement learning as projections of the full 7-dimensional coherent dynamics onto the linear, variational, and scalar-reward regimes respectively — an interpretation [I]; for active inference the derivation that would make it a special case is retracted (2026-09-25).

The next step — applying this formalism to stability problems and learning, where the sensorimotor cycle turns out to be not just a diagram but a concrete computational algorithm with provable bounds.

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What we learned

1. The environment does not add a 4th term (T-102 [T]): any external influence decomposes into Hamiltonian, dissipative, and regenerative channels. A fourth type is mathematically forbidden.
2. Perception is not recording, but deformation of dynamics (T-100 [T]): the functor Enc maps an observation to a modification of the evolution equation, in a unique (up to $G_2$) way.
3. Action is a min-max strategy (T-159 [T]): the system eliminates the largest deficit, not minimises "average error". No channel is left unattended.
4. Pleasure and suffering are derivatives of viability (T-103 [T]+[I]): $\mathcal{V}_{\text{hed}} = dP/d\tau|_{\mathcal{R}}$ — a mathematical identity, requiring no external "reward designer".
5. Fundamental bottleneck: $\leq \log_2 7 \approx 2.81$ bits/observation (T-107 [T]). Simon's bounded rationality is not an empirical fact but a consequence of $N = 7$.
6. Composition (T-108 [T]): the composite Enc is again a CPTP channel. From bacterium to organisation the same diagnostics apply where the parts are weakly coupled ([T at weak coupling], Theorems 9.2 and 9.5), and not necessarily where they are strongly coupled (Theorem 9.6).
7. Classical approaches read as projections of CC [I]: control theory, FEP, and RL as projections of the full 7-dimensional coherent dynamics; the claimed derivation of FEP as a special case is retracted (2026-09-25).

Bridge to the next chapter

We have built the complete "perception-decision-action" cycle. But how robust is this cycle? What blow can it withstand? Where is the boundary between recoverable trauma and irreversible destruction? In the next chapter we will answer these questions: derive the stability radius formula $r_{\mathrm{stab}}$, trace the mechanism of the "death spiral" — and show that antifragility is not a metaphor, but a consequence of integration of experience.

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Related documents:

• Stability — robustness of the sensorimotor cycle
• Learning bounds — fundamental limits (T-109–T-113)
• Diagnostics — real-time monitoring
• Implementation — computational implementation of Enc/Dec
• Lagrangian — full 6-term Lagrangian
• Definitions — stress tensor, $\mathrm{Coh}_E$
• Comparison with alternatives — CC vs. control theory, active inference, RL
• Interdisciplinary bridge — sensorimotor theory in the language of different disciplines
• Exercises — problems on dynamics (block 2)