The Self-Modelling Operator φ

This chapter describes how a system builds a model of itself — one of the central questions in consciousness science, philosophy, and cybernetics alike. What does it mean to "know oneself"? How can a system composed of parts encompass itself as a whole — including the very mechanism by which it does so?

The operator $\varphi$ is the mathematical answer to this question. It takes the current state of the Holon (the coherence matrix $\Gamma$) and returns a model of that state — an approximate reflection constructed by the system itself. When the reflection coincides with the original ($\varphi(\Gamma^*) = \Gamma^*$), the system achieves self-consistency — its self-model is exact.

DRY: Master definition of φ

This is the canonical definition of the self-modelling operator $\varphi$ in the theory section. The full formalisation, proofs of the equivalence of the three definitions, and the fixed-point theorem are in Formalisation of the operator φ.

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Historical Precursors

The problem of self-reference is one of the deepest in intellectual history.

Douglas Hofstadter in the book Gödel, Escher, Bach (1979) described strange loops — structures that, ascending through levels of a hierarchy, unexpectedly return to the starting point. The Gödel number encodes statements about numbers through numbers. Escher's hands draw each other. Bach's canon climbs through keys and returns to the original. Hofstadter suggested that precisely such self-referential loops underlie consciousness.

Robert Rosen (1991) in Life Itself formalised the idea of closure under efficient causation: a living system is a system that is its own model. His (M,R)-systems anticipated the autopoietic axiom of UHM.

Karl Friston (2006–) in the framework of the Free Energy Principle showed that living systems minimise free energy, which is equivalent to building a predictive model of the environment (and of themselves). The variational definition of φ in UHM is a formal analogue of Friston's principle, but derived from axioms rather than postulated.

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Intuitive Explanation: A Mirror for the Holon

Imagine that the Holon is a creature living in a room without external mirrors. The only way to "see" itself is to build an internal model: to picture how it looks, based on what it feels.

The operator $\varphi$ is that "mirror". The Holon looks into it ($\varphi(\Gamma)$) and sees an approximate reflection of itself. But the mirror is imperfect:

• It may be cloudy — losing detail (the base decohering form $\varphi_{\text{base}}$, which erases all connections between dimensions)
• It may be defocused — seeing not individual pixels but groups of 3 (the Fano form $\varphi_{\text{coh}}$, which preserves connections but weakens them)

The fixed point $\Gamma^*$ is the state in which the reflection coincides with the original. A Holon in state $\Gamma^*$ sees itself exactly as it is. This is the state of complete self-consistency.

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Bootstrap: How the Apparent Circularity Is Resolved

At first glance, the definition of φ appears to be a vicious circle: φ defines the self-model $\Gamma^*$, while $\Gamma^*$ enters the definition of φ. But this is not a vicious circle — it is a bootstrap, a self-consistent construction.

Analogy: a recursive picture. Imagine a painter painting a picture that depicts a painter painting a picture that depicts... The definition seems infinitely recursive. But if one finds that picture in which the depicted picture coincides with the picture itself — the recursion closes. That is the fixed point.

Mathematically, the circularity is resolved rigorously:

Bootstrap nature of the definition of φ

The operator φ defines the "self-model" of the system, i.e. φ(Γ) ≈ Γ — the system models itself. This appears to be a circular definition. The circularity is resolved via the fixed-point theorem: the operator φ is defined independently (as the left adjoint to the inclusion of subobjects), and for the canonical $\varphi_{\mathrm{coh}}$ the fixed point Γ* with φ(Γ*) = Γ* exists and is unique — it is $I/7$ — because $\|\varphi_{\mathrm{coh}}(\Gamma) - I/7\|_F \leq k\|\Gamma - I/7\|_F$ with $k \leq 6/7$. Corrected 2026-09-25: the box said "by Banach's theorem (φ is a contractive mapping with parameter k < 1)"; $k$ multiplies the deviation from $I/7$ and is not a Lipschitz constant — $\varphi_{\mathrm{coh}}$ stretches Frobenius distances by up to $9/8$, on diagonal states with $P = 4/7$, and by $54/49$ at pure states (evolution, split-step method; "up to $54/49$" until 2026-09-28; which maps do contract: three maps), and a self-model that keeps an isolated holon alive has several fixed points. A detailed account of the resolution of circularity is in Formalisation of the operator φ: resolution of circularity.

Which φ contracts: three maps, and the fed loop

Three maps called φ in the corpus behave differently in the Frobenius norm. Write $x = \Gamma - I/7$, $t = 7\lVert x\rVert_F^2$, so that $P = (1+t)/7$ and $R = 1/(7P) = 1/(1+t)$ (T-126).

Map	Deviation from $I/7$	Lipschitz constant	Fixed point
canonical $\varphi_{\mathrm{coh}} = k\,\mathcal{P}_\alpha(\Gamma) + (1-k)\,I/7$, $k = 1 - R$	$k\,\mathcal{P}_\alpha x$, factor $\leq 6/7$	$9/8$ for every $\alpha$: radial derivative $t(t+3)/(1+t)^2$, largest at $t = 3$, $P = 4/7$; $54/49$ at pure states	$I/7$
dissipative replacement form (T-62, T-249), $R\,\Gamma + (1-R)\,I/7$	$R\,x$	$1$: radial derivative $(1-t)/(1+t)^2 \in [-1/8, 1]$	$I/7$
replacement toward the previous map, $(1-k^2)\,\Gamma + k^2\,I/7$	$\bigl(1 - (1-R)^2\bigr)\,x$	$1$: radial derivative $1 - t^2(t+5)/(1+t)^3 \in [-17/108, 1]$	$I/7$

The third map is the one on which the closed form $R_\varphi = 1 - (1-R)^4\,\lVert\Gamma - I/7\rVert_F^2/\lVert\Gamma\rVert_F^2$ of the self-observation table holds exactly; the second gives $R_\varphi = 1 - (1-R)^3$, and $\varphi_{\mathrm{coh}}$ gives neither. It is also the "canonical φ of UHM" of Foundations of Mathematics, Part XVIII, ch. 9 (Theorems 9.5–9.8, "radial contraction", "global Banach of the loop"). The two corpora therefore do not contradict each other: "not a contraction, $54/49$" concerns $\varphi_{\mathrm{coh}}$, "non-expanding" concerns the replacement forms. For the fed loop $L_H(\Gamma) = U\,[(1-\mu)\,\varphi(\Gamma) + \mu\,\Theta]\,U^\dagger$ with $U = e^{-iHt}$ the feed cancels in differences and $U$ is an isometry, so $\mathrm{Lip}(L_H) = (1-\mu)\,\mathrm{Lip}(\varphi)$: at most $1 - \mu$ for the replacement forms, a Banach contraction for every $\mu > 0$ and every $H$; at most $\tfrac98(1-\mu)$ for $\varphi_{\mathrm{coh}}$, a contraction for $\mu > 1/9$ — at $\mu = 0.05$ two diagonal states near $P = 4/7$ move apart by the factor $1.069$. All three maps have the single fixed point $I/7$, and none of them keeps an isolated holon alive (dead isolation; the living self-models are $\varphi_s$ and $\varphi_J$, below). Numbers: test_three_self_model_maps_lipschitz_constants_and_the_fed_loop.

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Definition

The self-modelling operator $\varphi: \mathcal{D}(\mathcal{H}) \to \mathcal{D}(\mathcal{H})$ is defined in three equivalent ways:

#	Definition	Formula
1	Categorical	$\varphi \dashv i: \text{Sub}(\Gamma) \hookrightarrow \mathbf{Sh}_\infty(\mathcal{C})$
2	Dynamical	$\varphi(\Gamma) = \Pi_0[\Gamma]$ — projector onto the multiplicity-1 zero mode of the linearised full generator $\mathcal{L}_\Omega'\|_{\rho^*_\Omega}$
3	Idempotent	$\varphi \circ \varphi = \varphi$, $\exists \Gamma^*: \varphi(\Gamma^*) = \Gamma^*$

Convention for definition 2 (self-referential-ρ* fix, T-96)

"$\lim_{\tau\to\infty}e^{\tau\mathcal{L}_\Omega}$" must not be read as the constant map onto the dissipative attractor $I/7$: the full generator $\mathcal{L}_\Omega=\mathcal{L}_0+\mathcal{R}$ is nonlinear and non-primitive, with a nontrivial fixed point $\rho^*_\Omega\neq I/7$. Definition 2 is the $\Gamma$-dependent projector onto the zero mode of the linearised generator; only the linear part $\mathcal{L}_0$ is primitive (with unique stationary state $I/7$) — a property of the dynamics, not of $\varphi$.

The Three Definitions in Plain Language

Each of the three definitions answers the same question — "how does the system build a model of itself?" — but from a different point of view.

Definition 1 (Categorical): "Best approximation from below". Imagine you have a complex object (the Holon) and a collection of simpler objects (subobjects of the classifier). The categorical φ is the way to find the best approximation of the complex object through the simpler ones. "Left adjoint to inclusion" is the mathematical way of saying "optimal projection onto a subset". Analogy: you describe your appearance to a friend over the phone. From an infinite number of details you select the most important (height, hair colour, build). That is the "best approximation" — $\varphi$ of your full appearance.

Definition 2 (Dynamical): "The self-model the dynamics singles out". Linearise the full (regenerative) dynamics about its nontrivial fixed point $\rho^*_\Omega$ and project onto the surviving zero mode. Unlike a plain funnel — where every ball ends at the same lowest point (that would be the dissipative $I/7$, giving a trivial constant $\varphi$) — the regenerative term $\mathcal{R}$ keeps $\varphi(\Gamma)$ dependent on $\Gamma$: different inputs yield different self-models. The fixed point of $\varphi$ is $\rho^*_\Omega\neq I/7$.

Definition 3 (Idempotent): "A double reflection adds nothing new". If you look in a mirror twice, you see the same thing as the first time. $\varphi \circ \varphi = \varphi$ means that the model of the model coincides with the model. Analogy: photograph a photograph — you get (approximately) the same photograph.

Theorem: Equivalence of the definitions of φ

The three definitions specify the same operator $\varphi$. Proof → | Status: [T]

Base Form φ_base (Decohering Self-Observation)

For a Holon with $\mathcal{H} = \mathbb{C}^7$, the base (decohering) form is:

$$\varphi_{\text{base}}(\Gamma) = \sum_{k=1}^{7} \Pi_k \, \Gamma \, \Pi_k = \mathrm{diag}(\Gamma)$$

where $\Pi_k = |e_k\rangle\langle e_k|$ are projectors onto the basis dimensions.

Φ_base is insufficient as the canonical form

This form destroys all coherences ($\gamma_{ij} \to 0$ for $i \neq j$), which is incompatible with viability at uniform weights. The canonical form for living systems is the generalised operator $\varphi_{\text{coh}}$ with Fano structure (see below). The canonical form in Formalisation of the operator φ uses $\varphi_{\text{UHM}} = k \cdot \mathcal{P}_{\text{pred}} + (1-k) \cdot I/7$, which when $\mathcal{P}_{\text{pred}} = \mathcal{P}_{\text{base}}$ coincides with $\varphi_{\text{base}}$ (with anchor $I/7$). Generalisation to $\mathcal{P}_{\text{pred}} = \mathcal{P}_\alpha$ (convex combination of $\mathcal{P}_{\text{base}}$ and $\mathcal{P}_{\text{Fano}}$) gives $\varphi_{\text{coh}}$.

Свойства

1. CPTP channel: $\varphi$ is a completely positive, trace-preserving map
2. Idempotence (of ideal φ): $\varphi \circ \varphi = \varphi$ — for the idempotent definition (Definition 3). The canonical form $\varphi_{\text{coh}}$ with compression parameter $k = 1 - R < 1$ [T] is not idempotent; it contracts toward $I/7$, $\|\varphi_{\text{coh}}(\Gamma) - I/7\|_F \leq k\|\Gamma - I/7\|_F$, but is not a contraction of the state space (Lipschitz constant $9/8$, attained at $P = 4/7$ on the ray to a pure state, $54/49$ at the pure state; "contractive mapping" until 2026-09-25); the idempotent projection is the limit $\lim_{n\to\infty} \varphi_{\text{coh}}^n$, the constant map onto $I/7$
3. Purity monotonicity: $P(\varphi_{\text{base}}(\Gamma)) \leq P(\Gamma)$ for the base form (decoherence decreases purity); $P(\varphi_{\text{coh}}(\Gamma))$ depends on the parameter $\alpha$ — at $\alpha < 1$ the Fano component partially preserves coherences. The fixed point of the canonical $\varphi_{\mathrm{coh}}$ is $I/7$, with $P = 1/7$ (the value $2/7$ printed here earlier is retracted, see below)
4. Fixed point: $\exists! \, \Gamma^*_{\mathrm{coh}}: \varphi_{\mathrm{coh}}(\Gamma^*_{\mathrm{coh}}) = \Gamma^*_{\mathrm{coh}}$, namely $\Gamma^*_{\mathrm{coh}} = I/7$

Theorem: Fixed point of φ_coh (corrected 2026-09-25)

The canonical $\varphi_{\mathrm{coh}}$ (anchor $I/7$, $k = 1 - R < 1$) has exactly one fixed point, $\Gamma^*_{\mathrm{coh}} = I/7$, with $P = 1/7$.

Proof. A fixed point has $\gamma_{ij} = \tfrac{k(1-\alpha)}{3}\gamma_{ij}$ for $i \neq j$, and $k(1-\alpha)/3 < 1$, so $\gamma_{ij} = 0$; on the diagonal $\gamma_{ii} = k\gamma_{ii} + (1-k)/7$, and $1 - k = R > 0$ gives $\gamma_{ii} = 1/7$. $\blacksquare$ This agrees with Corollary 2.1 of Formalisation of φ (uniform anchor, fixed point $I/7$). Status: [T]. The earlier statement "$P(\Gamma^*_{\mathrm{coh}}) = P_{\text{crit}} = 2/7$" is retracted [✗]: 200 iterations of $\varphi_{\mathrm{coh}}$ from a random pure state end at $P = 1/7$ to $10^{-12}$ (test_unital_self_model_keeps_an_isolated_holon_dead).

Distinction between fixed points

For the canonical $\varphi_{\mathrm{coh}}$ the fixed point of the self-model and the attractor of the dissipator coincide: both are $I/7$. Corrected 2026-09-25: the box said that $\Gamma^*_{\mathrm{coh}}$, with $P = 2/7$, differs from $\rho^*_{\mathrm{diss}} = I/7$. They differ for a self-model with a non-unital anchor — for the self-registering $\varphi_s$ below every basis state is a fixed point. The canonical definition of the reflexion measure R uses $\rho^*_{\mathrm{diss}} = I/7$: $R = 1/(7P)$. Details: stratification of definitions.

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Necessity of generalised φ for living systems

Canonical φ_base is insufficient

The canonical $\varphi_{\text{base}}$ (decohering self-observation, projection onto the diagonal) destroys all coherences: $[\varphi_{\text{base}}(\Gamma)]_{ij} = 0$ for $i \neq j$. This is incompatible with viability: when $\gamma_{ii} \approx 1/7$ we get $P \approx 1/7 < P_{\text{crit}} = 2/7$. To achieve $P > P_{\text{crit}}$ without coherences, a pathological localisation of one dimension is required.

Theorem: Necessity of a coherence-preserving φ

A living self-model must preserve coherences: $\exists\, (i,j): [\varphi(\Gamma)]_{ij} \neq 0$. A generalised $\varphi_{\text{coh}}$ is required. Proof → | Status: [T]

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Canonical construction of φ_coh from the Fano structure

Why the Fano channel is needed: defocused vision

Before turning to formulas, let us understand why the Fano structure is needed.

Imagine that the Holon's mirror can operate in two modes:

• Pixel mode ($\varphi_{\text{base}}$): the mirror sees each "pixel" (dimension) separately, but completely loses the connections between pixels. As if you cut a photograph into 7 squares and shuffled them — you know the content of each square, but not how they are connected.
• Defocused mode ($\mathcal{P}_{\text{Fano}}$): the mirror sees not individual pixels, but groups of 3 (Fano lines). This is like defocused vision — you lose fine details, but preserve connections between dimensions. Each group of three dimensions is observed as a whole.

Why groups of 3? Because the Fano plane PG(2,2) is the unique structure on 7 points where every pair of points lies on exactly one line of 3 points. This is the maximally democratic observation: no pair of dimensions is privileged.

Key result: the pixel mirror kills the system (with uniform weights, purity drops below the threshold $P_{\text{crit}} = 2/7$). The defocused mirror preserves life, because it preserves the connections (coherences) between dimensions. Living self-observation must be partially defocused.

Mathematics of channel mixing

Why the convex combination $\mathcal{P}_\alpha = \alpha\,\mathcal{P}_{\text{base}} + (1-\alpha)\,\mathcal{P}_{\text{Fano}}$ works:

1. $\mathcal{P}_{\text{base}}$ alone: destroys all coherences → with uniform weights $P \approx 1/7 < P_{\text{crit}}$ → the system dies
2. $\mathcal{P}_{\text{Fano}}$ alone: coherences are scaled by $1/3$, phases are preserved → $P$ remains above the threshold
3. Convex combination: $\mathcal{P}_\alpha$ is a CPTP channel (a convex combination of CPTP channels is CPTP)
4. At $\alpha = 0$: pure Fano, maximum coherence preservation, but less accurate predictive model
5. At $\alpha = 1$: pure atomic, ideal predictive accuracy, but the system dies
6. No proven principle fixes $\alpha$. The variational principle was said to find an optimum $\alpha^* \in (0,1)$ balancing accuracy and survivability; that claim is retracted (2026-09-25; see the α* section below) — the functional it used is minimised at $\alpha = 0$

Two types of classifier atoms

DRY: Master definition

Complete definitions of atomic and Fano Lindblad operators are in Lindblad Operators. Below are the key formulas needed for the construction of φ_coh.

The classifier Ω contains not only atomic subobjects $S_k = |k\rangle\langle k|$, but also composite ones. The Fano plane $PG(2,2)$ defines 7 linear subobjects — projections onto 3-dimensional subspaces:

$$\Pi_p = \sum_{i \in \mathrm{line}_p} |i\rangle\langle i|, \quad p = 1, \ldots, 7$$

Theorem: Completeness of Fano atoms

Each dimension lies on exactly 3 Fano lines. Therefore: $\sum_{p=1}^{7} \Pi_p = 3I$. Proof → | Status: [T]

Fano predictive channel $\mathcal{P}_{\text{Fano}}$

For each Fano line $p = (i,j,k)$ a Lindblad operator is defined:

$$L_p^{\text{Fano}} := \frac{1}{\sqrt{3}}\,\Pi_p = \frac{1}{\sqrt{3}}(|i\rangle\langle i| + |j\rangle\langle j| + |k\rangle\langle k|)$$

The Fano predictive channel:

$$\mathcal{P}_{\text{Fano}}(\Gamma) := \sum_{p=1}^{7} L_p^{\text{Fano}}\,\Gamma\,(L_p^{\text{Fano}})^\dagger = \frac{1}{3}\sum_{p=1}^{7} \Pi_p\,\Gamma\,\Pi_p$$

CPTP verification

$\sum (L_p^{\text{Fano}})^\dagger L_p^{\text{Fano}} = I$ — full proof in Lindblad Operators.

Theorem: The Fano channel preserves coherences

Theorem: Preservation of coherences by the Fano channel

For an arbitrary coherence matrix $\Gamma$:

(a) Diagonal elements are preserved exactly: $[\mathcal{P}_{\text{Fano}}(\Gamma)]_{ii} = \gamma_{ii}$

(b) Coherences are preserved with coefficient $1/3$: $[\mathcal{P}_{\text{Fano}}(\Gamma)]_{ij} = \frac{1}{3}\gamma_{ij}$ for $i \neq j$

(c) Phases of coherences are preserved exactly: $\arg([\mathcal{P}_{\text{Fano}}(\Gamma)]_{ij}) = \arg(\gamma_{ij})$

Key difference from $\varphi_{\text{base}}$: the Fano channel scales coherence amplitudes without phase distortion, whereas $\varphi_{\text{base}}$ destroys them entirely. Proof → | Status: [T]

Canonical form of φ_coh

Theorem: Canonical form of φ_coh

Canonical coherence-preserving self-modelling:

$$\varphi_{\text{coh}}(\Gamma) = k \cdot \left[\alpha \cdot \mathcal{P}_{\text{base}}(\Gamma) + (1 - \alpha) \cdot \mathcal{P}_{\text{Fano}}(\Gamma)\right] + (1 - k) \cdot \Gamma_{\text{anchor}}$$

where:

• $\mathcal{P}_{\text{base}}(\Gamma) = \sum_m P_m\,\Gamma\,P_m = \mathrm{diag}(\Gamma)$ — atomic channel (from φ formalisation)
• $\mathcal{P}_{\text{Fano}}(\Gamma) = \frac{1}{3}\sum_p \Pi_p\,\Gamma\,\Pi_p$ — Fano channel
• $\alpha \in [0, 1]$ — decoherence depth parameter (balance between atomic and Fano observation)
• $k = 1 - R$ — compression parameter determined by the reflexion measure $R = 1 - \|\Gamma - \rho^*\|_F^2/\|\Gamma\|_F^2$ [T]. Not a free parameter
• $\Gamma_{\text{anchor}} = \rho^*_{\mathrm{diss}} = I/7$ — anchor state, coinciding with the attractor of the dissipative part $\mathcal{L}_0$. This choice is dictated by the primitivity of $\mathcal{L}_0$ [T-39a]: the unique stationary state of the linear dynamics is the maximally mixed $I/7$. Under full compression ($k \to 1$, $R \to 0$) the self-model tends to $I/7$ — the state of complete absence of information about itself.

$\mathcal{P}_\alpha = \alpha\,\mathcal{P}_{\text{base}} + (1-\alpha)\,\mathcal{P}_{\text{Fano}}$ — a convex combination of CPTP channels, hence CPTP. Proof → | Status: [T]

Target coherences of φ_coh

Theorem: Target coherences of φ_coh

(a) Magnitude of the target coherence (with diagonal anchor): $|\gamma_{ij}^{\text{target}}| = \frac{k(1-\alpha)}{3} \cdot |\gamma_{ij}|$

(b) Target phase is preserved: $\theta_{ij}^{\text{target}} = \theta_{ij}$

(c) Target Gap is preserved: $\mathrm{Gap}^{\text{target}}(i,j) = \mathrm{Gap}(i,j)$

The canonical $\varphi_{\text{coh}}$ does not seek to change the Gap — it reproduces the Gap with a reduced amplitude, scaling coherences without phase distortion. Proof → | Status: [T]

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Explicit coefficients $c_{mn}$

General form of the coherence-preserving channel from the definition of $\varphi_{\text{coh}}$:

$$\mathcal{P}_{\text{coh}}(\Gamma) = \sum_{m,n} c_{mn}\,|m\rangle\langle n|\,\Gamma\,|n\rangle\langle m|$$

tip

Theorem: Explicit coefficients $c_{mn}$

The coefficients of the canonical $\varphi_{\text{coh}}$, given the Fano weight $\alpha$:

$$c_{mn} = \begin{cases} k & m = n \text{ (the atomic and the Fano channel both keep the diagonal)} \\ (1-\alpha) k / 3 & m \neq n \end{cases}$$

and the anchor adds $(1-k)\,[\Gamma_{\text{anchor}}]_{mn}$. Every pair $(m,n)$ lies on exactly one Fano line, so a third case "$0$ for $(m,n)$ not on a common Fano line" is empty. Corrected 2026-09-25: the box printed $c_{mm} = \alpha^* k$ and that empty third case; the diagonal coefficient is $\alpha k + (1-\alpha)k = k$ (as in $G_2$-structure, Theorem 10.5), and $\alpha^*$ is retracted.

The coefficients are determined through:

• The Fano structure $PG(2,2)$ (algebraic geometry)
• The Fano weight $\alpha$ — a free parameter; its variational value $\alpha^* \approx 1 - 2/(7P)$ is retracted (see below)
• The compression parameter $k$ (from φ formalisation)

Proof → | Status: [T]

Kraus operators (7 + 7 + 49; corrected 2026-09-25)

Atomic operators (7): $K_m^{(\text{atom})} = \sqrt{\alpha k} \cdot |m\rangle\langle m|$. Fano operators (7): $K_p^{(\text{Fano})} = \sqrt{(1-\alpha) k / 3} \cdot \Pi_p$. Anchor operators (49), with $\Gamma_{\text{anchor}} = \sum_i \lambda_i |\psi_i\rangle\langle\psi_i|$: $K_{ij}^{(\text{anch})} = \sqrt{(1-k)\lambda_i} \cdot |\psi_i\rangle\langle j|$. Verification: $\sum_m (K_m^{(\text{atom})})^\dagger K_m^{(\text{atom})} = \alpha k \cdot I$; $\sum_p (K_p^{(\text{Fano})})^\dagger K_p^{(\text{Fano})} = \tfrac{(1-\alpha)k}{3} \cdot 3I$ (every point lies on three lines); $\sum_{i,j} (K_{ij}^{(\text{anch})})^\dagger K_{ij}^{(\text{anch})} = (1-k) \cdot I$; total $I$. The 63 operators reproduce $\varphi_{\text{coh}}$ to $3 \times 10^{-16}$ on 50 random states. The former set — $K_m^{(\text{atom})} = \sqrt{\alpha^* k/7}\,|m\rangle\langle m|$, one anchor $K_0 = \sqrt{(1-k)/7}\,I$ — was not trace-preserving: $\sum_m |m\rangle\langle m| = I$, not $7I$, and $K_0^\dagger K_0 = \tfrac{1-k}{7}\,I$; for $\alpha = 0.4$, $k = 0.8$ it misses $I$ by $1.18$ in Frobenius norm, and a single multiple of $I$ cannot implement the replacement $\Gamma \mapsto (1-k)\,\Gamma_{\text{anchor}}$. Same correction as $G_2$-structure, Theorem 10.5.

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Variational definition of α*

Theorem: Variational definition of α* — retracted 2026-09-25 [✗]

The optimal parameter $\alpha^*$ is determined by the variational principle:

$$\alpha^* = \arg\min_{\alpha \in [0,1]} \mathcal{F}[\mathcal{P}_\alpha; \Gamma] = \arg\min_{\alpha} \left[S_{\text{spec}}(\mathcal{P}_\alpha(\Gamma)) + D_{KL}(\mathcal{P}_\alpha(\Gamma) \| \Gamma)\right]$$

Approximate formula for a system with purity $P > P_{\text{crit}}$:

$$\alpha^* \approx 1 - \frac{P_{\text{crit}}}{P} = 1 - \frac{2}{7P}$$

Purity $P$	$\alpha^*$	Interpretation
$P = 1$ (pure state)	$\approx 0.71$	Substantial Fano contribution
$P = 0.5$	$\approx 0.43$	Balance of atomic and Fano
$P \to P_{\text{crit}}$	$\to 0$	Almost purely Fano (minimal coherence destruction)

Proof → | Status: retracted [✗]

Why retracted. $\mathcal{F}[\psi;\Gamma] = -\mathrm{Tr}(\psi(\Gamma)\log\Gamma)$ is linear in $\psi(\Gamma)$, and $\mathcal{P}_\alpha(\Gamma) = \Delta(\Gamma) + \tfrac{1-\alpha}{3}(\Gamma - \Delta(\Gamma))$, with $\Delta$ the diagonal part, is affine in $\alpha$. Hence $\mathcal{F}(\alpha) = \mathcal{F}(0) + \tfrac{\alpha}{3}\left[D_{KL}(\Gamma\|\Delta\Gamma) + D_{KL}(\Delta\Gamma\|\Gamma)\right]$ for full-rank $\Gamma$: the minimum over $[0,1]$ is at $\alpha = 0$ (pure Fano) whenever $\Gamma$ has coherences, and there is no interior optimum $1 - 2/(7P)$. Checked on 400 random states with $P$ from $0.19$ to $0.91$: minimum at $\alpha = 0$ in all 400. The weight $\alpha$ of the canonical $\varphi_{\text{coh}}$ is therefore a free parameter; no principle in the corpus fixes it.

Physical meaning of the balance

At $\alpha = 1$ (purely atomic channel) — maximum predictive accuracy, but complete destruction of coherences. At $\alpha = 0$ (purely Fano) — coherences preserved with coefficient $1/3$, but a less accurate predictive model. The optimum $\alpha^* \in (0,1)$ is a balance between predictive accuracy and structure preservation. [✗] The functional of the box above has no such interior optimum; the trade-off is real, but it is not resolved by that functional.

Sketch of the derivation of α*

The functional $\mathcal{F}[\alpha] = S_{\text{spec}}(\mathcal{P}_\alpha(\Gamma)) + D_{KL}(\mathcal{P}_\alpha(\Gamma) \| \Gamma)$.

The channel $\mathcal{P}_\alpha$ acts as follows: the diagonal is preserved, coherences $\gamma_{ij} \mapsto \frac{(1-\alpha)}{3}\gamma_{ij}$. Therefore the purity of the self-model: $P_\alpha \approx P_{\text{diag}} + \left(\frac{1-\alpha}{3}\right)^2 P_{\text{coh}}$.

The spectral entropy $S_{\text{spec}}$ increases as $\alpha$ decreases (weakening coherences → mixing). The Kullback–Leibler divergence $D_{KL}$ increases as $\alpha$ increases (greater deviation from $\Gamma$). The stationarity condition $\partial\mathcal{F}/\partial\alpha = 0$ for typical $\Gamma$ with purity $P$ gives:

$\alpha^* \approx 1 - \frac{P_{\text{crit}}}{P} = 1 - \frac{2}{7P}$

The formula is approximate — the exact solution requires numerical optimisation for arbitrary $\Gamma$.

[✗] Retracted sketch (2026-09-25): $S_{\text{spec}} + D_{KL}$ is not the sum of a falling and a rising term here — it is affine in $\alpha$ with slope $\tfrac13[D_{KL}(\Gamma\|\Delta\Gamma) + D_{KL}(\Delta\Gamma\|\Gamma)] \geq 0$, so $\partial\mathcal{F}/\partial\alpha = 0$ has no solution unless $\Gamma$ is diagonal.

Numerical example

Let $\Gamma$ have purity $P = 0.4$ (a viable system). We compute (step 1 uses the retracted formula for $\alpha^*$, so the numbers below illustrate one choice of the free weight $\alpha$, not a derived value):

1. Parameter $\alpha^*$: $\alpha^* \approx 1 - 2/(7 \times 0.4) = 1 - 0.714 = 0.286$
2. Reflexion measure: $R = 1/(7P) = 1/2.8 \approx 0.357$
3. Compression parameter: $k = 1 - R = 0.643$
4. Target coherence: $|\gamma_{ij}^{\text{target}}| = \frac{k(1-\alpha^*)}{3}|\gamma_{ij}| = \frac{0.643 \times 0.714}{3}|\gamma_{ij}| \approx 0.153\,|\gamma_{ij}|$

The self-model retains ~15% of each coherence amplitude — a "defocused" but not destroyed reflection. Under iteration the purity of the self-model converges to $1/7$, not to $P_{\text{crit}} = 2/7$ (corrected 2026-09-25; the text said that the viability threshold acts as an attractor of self-modelling): the canonical $\varphi_{\mathrm{coh}}$ is unital, and its only fixed point is $I/7$.

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Why an isolated holon needs a non-unital self-model: the self-registering form φ_s

The canonical $\varphi_{\mathrm{coh}}$ preserves coherences, and that is necessary for life (Fano channel, Theorem 9.1); it is not sufficient. Its anchor $I/7$ makes it unital, and a unital self-model cannot raise purity: an isolated holon regenerating toward $\varphi_{\mathrm{coh}}(\Gamma)$ dies whatever $\kappa$ is (dead isolation [T]).

Theorem (Symmetric linear self-models are unital) [T]

A linear CPTP self-model covariant under $G_2$, or under the frame group $\Gamma_{\mathrm{oct}}$, is unital. So is $\varphi_{\mathrm{coh}}$ for every $\alpha$ and $k$.

Proof. $\Phi(I)$ commutes with an irreducible representation, so it is a multiple of $I$ (Schur), equal to $I$ by trace preservation; $G_2$ and $\Gamma_{\mathrm{oct}}$ act irreducibly on $\mathbb{C}^7$ (evolution, dead isolation, item 3). $\blacksquare$

So an anchor that keeps a holon alive either breaks the symmetry from outside — the environmental anchor of an embodied holon (T-148) — or depends on the state itself, which lets a covariant law have non-symmetric fixed points.

Lemma (Intrinsic anchors are spectral) [T]

Let $\sigma$ send states to states with $\sigma(U\Gamma U^\dagger) = U\sigma(\Gamma)U^\dagger$ for every unitary $U$ — the anchor refers to nothing outside the holon. If $\Gamma = \sum_i \lambda_i |i\rangle\langle i|$ has distinct eigenvalues, then $\sigma(\Gamma) = \sum_i s_i |i\rangle\langle i|$ is diagonal in the eigenbasis of $\Gamma$, and $\mathrm{Tr}(\Gamma\sigma(\Gamma)) = \sum_i \lambda_i s_i$. The constant anchor $s_i = 1/7$ gives overlap $1/7 < P$ for every $\Gamma \neq I/7$; the choice $s_i = \lambda_i$ ($\sigma(\Gamma) = \Gamma$) makes the regeneration target at $\Gamma$ equal to the image of $\Gamma$ under the unital $k\mathcal{P}_\alpha + R\,\mathrm{id}$; the choice $s_i = \lambda_i^2/\sum_j \lambda_j^2$ gives $\mathrm{Tr}\,\Gamma^3/\mathrm{Tr}\,\Gamma^2 \geq P$, with equality only for a flat spectrum.

Proof. The unitaries $U = \sum_i e^{i\theta_i}|i\rangle\langle i|$ fix $\Gamma$, so $U\sigma(\Gamma)U^\dagger = \sigma(\Gamma)$ for all phases $\theta_i$, which forces $\sigma(\Gamma)$ to be diagonal in $\{|i\rangle\}$. The overlap is then $\sum_i\lambda_i s_i$. For $s_i \propto \lambda_i^2$ the inequality $\sum_i\lambda_i^3 \geq (\sum_i\lambda_i^2)^2$ is Chebyshev's sum inequality with the weights $\lambda_i$. $\blacksquare$

The weights $s \propto \lambda^q$ order the intrinsic anchors: $q = 0$ is the canonical $I/7$, $q = 1$ is the state itself, and both are dead by the theorem of dead isolation; $q = 2$ is the lowest degree that sharpens. At $q \to \infty$ the anchor becomes the projector onto the top eigenvector of $\Gamma$ — the minimiser, over CPTP channels, of the cross-entropy $-\mathrm{Tr}(\psi(\Gamma)\log\Gamma)$ of the retracted variational principle — which is discontinuous where the top eigenvalue is degenerate.

Definition [D] (self-registering self-model).

$$\varphi_s(\Gamma) = k\,\mathcal{P}_\alpha(\Gamma) + R\,\frac{\Gamma^2}{\mathrm{Tr}\,\Gamma^2}, \qquad R = \frac{1}{7P},\quad k = 1 - R .$$

The anchor $\Gamma^2/\mathrm{Tr}\,\Gamma^2 = \sqrt{\Gamma}\,\Gamma\,\sqrt{\Gamma}/\mathrm{Tr}(\Gamma^2)$ is the Lüders update of $\Gamma$ on the effect $\Gamma$: the state the holon is left in after registering its own state, the one effect it has without an outside reference. It is smooth on all states ($\mathrm{Tr}\,\Gamma^2 \geq 1/7$) and differs from $\varphi_{\mathrm{coh}}$ only in the anchor. With it an isolated holon has at least seven self-sustaining attractors with $P > 2/7$ (evolution [T]); at $H = 0$ they are the basis states, where $\varphi_s(e_m) = e_m$ — exact self-knowledge. That the self-model of a physical holon is $\varphi_s$ rather than $\varphi_{\mathrm{coh}}$ is not derived from the axioms [Pr]; that it must be non-unital for an isolated holon to live is [T].

What the axioms fix about the anchor, and the collineation anchor φ_J

The attractors of $\varphi_s$ lie above the conscious window, and they are localised. The reason is not the choice $q = 2$: a self-model covariant under the diagonal unitaries — every intrinsic anchor, every anchor built from the Fano projectors — holds no hyperbolic attractor in $\mathcal{V}_{\mathrm{full}}$ near $H = 0$ (phase-reference obstruction [T]). Such a self-model can reach the window by purity — the Fano-line registration of the same theorem holds $P = 1/3$ for every $\kappa$ — but not $\Phi \geq 1$.

Theorem (Symmetric anchors) [T]

Let a self-model have the replacement form $\varphi(\Gamma) = k\,\mathcal{P}_\alpha(\Gamma) + R\,\rho_a$ with an anchor $\rho_a$ independent of $\Gamma$.

1. If $\varphi$ is covariant under $G_2$, under the signed frame group $\Gamma_{\mathrm{oct}}$, or under the full group of monomial unitaries that permute the Fano lines (permutations with arbitrary phases), then $\rho_a = I/7$ and $\varphi$ is unital.
2. If $\varphi$ is covariant under the 168 collineations of the Fano plane acting as permutations of the basis, then $\rho_a = (1 - t)\,I/7 + t\,uu^\dagger$ with $u = (1, \dots, 1)/\sqrt7$ and $t \in [-1/6, 1]$; it is unital only for $t = 0$ and pure only for $t = 1$.

Proof. $\mathcal{P}_\alpha$ is covariant under each of these groups (they map Fano lines to lines, and $\mathcal{P}_{\mathrm{base}}$ commutes with monomial unitaries), so covariance of $\varphi$ is $M\rho_aM^\dagger = \rho_a$ for every $M$ of the group, i.e. $\rho_a$ lies in the commutant. For $G_2$ and $\Gamma_{\mathrm{oct}}$ the commutant is $\mathbb{C}I$ (dead isolation, item 3); for the monomial group the diagonal phases force $\rho_a$ diagonal and the 2-transitive permutations force its diagonal constant. The 168 collineations act 2-transitively on the points, so the permutation representation has two orbits on pairs of indices — equal and distinct — and its commutant is spanned by $I$ and $J$ ($J/7 = uu^\dagger$). The eigenvalues of $(1 - t)I/7 + t\,uu^\dagger$ are $(1 + 6t)/7$ and $(1 - t)/7$, which gives $t \in [-1/6, 1]$. $\blacksquare$

What the adjunction leaves open. The self-modelling adjunction $\varphi \dashv i$ and the terminal object make $\varphi$ a CPTP left adjoint; they do not fix its anchor — every anchor above yields a CPTP channel of the same form, and so does every intrinsic anchor, including the Lüders update $\Gamma^2/\mathrm{Tr}\,\Gamma^2$ of $\varphi_s$. Four further routes from the categorical side leave the anchor open or kill the holon [T]. The state the terminal object supplies — the dual of the unique discarding map — is $I/7$, dead. Lawvere's fixed-point theorem, like Brouwer's, gives a fixed point of $\varphi$ but says nothing about the anchor. A reflexive anchor, equal to the fixed point of the self-model it defines ($\rho_a = k\mathcal{P}_\alpha(\rho_a) + R\rho_a$), satisfies $\mathcal{P}_\alpha(\rho_a) = \rho_a$, so it is diagonal: phase-covariant, hence outside $\mathcal{V}_{\mathrm{full}}$ near $H = 0$ by the obstruction. And keeping as much of the frame group as possible — a maximal subgroup of $\Gamma_{\mathrm{oct}}$ — also fixes only diagonal anchors: $\Gamma_{\mathrm{oct}} = 2^3{\cdot}\mathrm{GL}(3,2)$ is a non-split extension (Gaschütz: its Sylow 2-subgroup of order 64 has no complement to the sign group $N$), $N$ is an irreducible $\mathrm{GL}(3,2)$-module, so every maximal subgroup contains $N$, and the commutant of $N$ is the diagonal.

What does fix the anchor is the frame group read at the level where it is compatible with life.

Theorem T-334 (The collineation anchor, derived up to gauge) [T]

Let the self-model have the replacement form $\varphi(\Gamma) = k\,\mathcal{P}_\alpha(\Gamma) + R\,\rho_a$ with a $\Gamma$-independent anchor, and let the isolated holon evolve by the gated dynamics with the Fano dissipator (evolution); $c = (1 - \alpha)/3$.

1. Frame covariance and life. $\varphi$ is $\Gamma_{\mathrm{oct}}$-covariant, or $\mathcal{P}_\alpha \circ \varphi$ is ($\alpha \lt 1$), only for $\rho_a = I/7$, which is dead. The atomic reading $\mathcal{P}_{\mathrm{base}} \circ \varphi$ is $\Gamma_{\mathrm{oct}}$-covariant if and only if $\mathrm{diag}\,\rho_a = I/7$; the stationary diagonal at $H = 0$ then is $I/7$ as well, so $\sigma_k = 0$ on every axis.
2. Viability depends on one number. If $\mathrm{diag}\,\rho_a = I/7$, a hyperbolic sink in $\mathcal{V}_{\mathrm{full}}$ at $H = 0$ exists exactly for $\kappa > \kappa_c(s)$, where $s = P(\rho_a) - 1/7$ is the coherent purity of the anchor; $\kappa_c(s)$ is that of the family $(1-t)I/7 + t\,uu^\dagger$ with $t = \sqrt{7s/6}$, strictly decreasing in $s$, and finite exactly for $s > (2 - c)^2/7$.
3. Most viable = most informative. Hence among anchors that privilege no axis the one whose living range of $\kappa$ contains every other one's is a pure state with uniform diagonal, $\rho_a = D\,uu^\dagger D^\dagger$ with $D$ a diagonal unitary, $u = (1, \dots, 1)/\sqrt7$. Maximal viability and maximal information (purity) select the same anchor.
4. Gauge. Diagonal unitaries commute with $\mathcal{D}_\Omega$ and $\mathcal{P}_\alpha$ and leave $P$, $R$, $g_V$ invariant, so conjugation by $D$ carries the dynamics with anchor $uu^\dagger$ and Hamiltonian $D^\dagger H D$ onto that with anchor $D\,uu^\dagger D^\dagger$ and $H$. At $H = 0$ the anchors $D\,uu^\dagger D^\dagger$ give conjugate flows; $\varphi_J$ is unique up to this gauge of the $H$-free dynamics (and the weight $\alpha$).
5. Symmetry. $uu^\dagger$ is fixed by all $5040$ permutation matrices, and the $H$-free dynamics is covariant under all of them (every pair of axes lies on exactly one Fano line, so $\mathcal{D}_\Omega$ and $\mathcal{P}_\alpha$ treat all pairs alike — Fano channel, Theorem 11.1). Of the 168 collineations acting as permutations only 21 (the group $7{:}3$) lie in $\Gamma_{\mathrm{oct}}$; the whole $\Gamma_{\mathrm{oct}}$ moves $uu^\dagger$ over its 64 sign rephasings $D\,uu^\dagger D$, $D = \mathrm{diag}(\pm1)$. The states whose $\Gamma_{\mathrm{oct}}$-orbit stays inside their gauge orbit are exactly $D\bigl((1-t)I/7 + t\,uu^\dagger\bigr)D^\dagger$, $t \in [-1/6, 1]$.
6. One clause. The following are equivalent: (Eq-V) below; $\rho_a = D\,uu^\dagger D^\dagger$ for a diagonal unitary $D$; the anchor has the largest integration of any state, $\Phi(\rho_a) = P_{\mathrm{coh}}/P_{\mathrm{diag}} = 6$; it is maximally coherent in the frame, $C_{\mathrm{rel}}(\rho_a) = S(\mathrm{diag}\,\rho_a) - S(\rho_a) = \log 7$; its coherent purity is $s = 6/7$. Neither half of (Eq-V) suffices alone: (Eq) leaves every anchor with uniform diagonal, and viability alone, over all constant anchors, does not pick $\varphi_J$ — anchors with non-uniform diagonal come arbitrarily close to the $H = 0$ rate floor of T-336, $1.25$–$1.27$ times below $\kappa_c(\alpha)$, and reach it only in the limit where their attractor tends to $\Phi = 1$. (Eq) is the terminal object read on the axes: $\mathrm{diag}\,\rho_a = I/7$ says that the atomic reading $\mathcal{P}_{\mathrm{base}}(\rho_a)$ of the anchor is the state the terminal object supplies.

Proof. (1) Covariance of $\varphi$ puts $\rho_a$ in the commutant $\mathbb{C}I$ (theorem above). $\mathcal{P}_\alpha \circ \varphi = k\,\mathcal{P}_\alpha^2 + R\,\mathcal{P}_\alpha(\rho_a)$ is covariant iff $\mathcal{P}_\alpha(\rho_a)$ is $\Gamma_{\mathrm{oct}}$-invariant, i.e. $\mathcal{P}_\alpha(\rho_a) = I/7$; $\mathcal{P}_\alpha$ keeps the diagonal and multiplies coherences by $c \neq 0$, so $\rho_a = I/7$. $\mathcal{P}_{\mathrm{base}} \circ \varphi = k\,\mathcal{P}_{\mathrm{base}} + R\,\mathrm{diag}\,\rho_a$: signs act trivially on diagonals and the permutation part is transitive on the axes, so covariance is $\mathrm{diag}\,\rho_a = I/7$. The diagonal of the stationarity equation is $\kappa g_V R\,(\mathrm{diag}\,\rho_a - \mathrm{diag}\,\Gamma) = 0$ (T-335). (2) By T-335 the stationary states with $P > 2/7$ are $(1 - \eta)I/7 + \eta\rho_a$ with $P = 1/7 + \eta^2 s$, and every coherence obeys the same scalar equation; with $\xi = \eta t$ it is the equation $\kappa Q_t(\xi) = 2/3$ of the family, $Q_t(\xi) = (6\xi^2 - 1)\bigl[(t - c\xi)/(\xi(1 + 6\xi^2)) - (1 - c)\bigr]$. $Q_t$ grows strictly with $t$ at each $\xi > 1/\sqrt6$, so $\max Q_t$ grows and $\kappa_c = 2/(3\max Q_t)$ falls; the threshold $t > (2 - c)/\sqrt6$ is the one of the living attractor theorem, i.e. $s > (2-c)^2/7$. The upper edge $P \le 3/7$ and $\Phi = 7P - 1 \ge 1$ hold as there. (3) $s \le 6/7$ with equality iff $\rho_a$ is pure; a pure state $\psi\psi^\dagger$ with $|\psi_k|^2 = 1/7$ is $D\,uu^\dagger D^\dagger$. (4) Direct. (5) $u$ is fixed by every permutation matrix. The rest is a finite check (test_frame_covariance_modulo_gauge_fixes_the_collineation_anchor) plus the following argument. A Hermitian matrix with all off-diagonal moduli nonzero is fixed up to diagonal gauge by its diagonal, the moduli $|\rho_{ij}|$ and the triangle fluxes $\arg(\rho_{ij}\rho_{jk}\rho_{ki})$, and signs in $\Gamma_{\mathrm{oct}}$ do not change these. $\mathrm{GL}(3,2)$ is 2-transitive on points, so the diagonal and the moduli are constant; it is transitive on lines and on ordered non-collinear triples, and contains elements reversing the orientation of a triangle of each kind, so each kind carries one flux $f_L$ or $f_N$ in $\{0, \pi\}$. Four points containing a line bound a tetrahedron with one collinear and three non-collinear faces, so $f_L = f_N$ (enumeration of all $2^{15}$ gauge classes of signings of $K_7$ gives exactly the two patterns). Flux $0$ is $t > 0$, flux $\pi$ is $t \lt 0$; if some modulus vanishes all do, $t = 0$. (6) (Eq-V) $\Leftrightarrow$ $\rho_a = D\,uu^\dagger D^\dagger$ is items 1–3: uniform diagonal is (Eq), and among such anchors $\kappa_c(s)$ decreases strictly in $s \leq 6/7$, with equality only for a pure anchor. $\Phi = s/P_{\mathrm{diag}}$ with $s = P - P_{\mathrm{diag}} \leq 1 - P_{\mathrm{diag}}$ and $P_{\mathrm{diag}} \geq 1/7$, so $\Phi \leq 6$, with equality iff $P = 1$ and $P_{\mathrm{diag}} = 1/7$; $S(\mathrm{diag}\,\rho) \leq \log 7$ with equality iff the diagonal is uniform, and $S(\rho) \geq 0$ with equality iff $\rho$ is pure (T. Baumgratz, M. Cramer, M. B. Plenio, "Quantifying coherence", Phys. Rev. Lett. 113, 140401 (2014)). Anchors with uniform diagonal need $\kappa > \kappa_c(\alpha)$ by items 2–3, so those approaching the floor have non-uniform diagonal; the ratios are $16.63/13.11$, $29.25/23.21$, $59.34/47.35$ at $\alpha = 0, 1/2, 1$ (test_one_clause_principle_for_the_anchor_is_maximal_integration). $\blacksquare$

The remaining principle. What the axioms, the frame decision and the theorems above leave is one principle [Pr], weaker than the two it replaces:

• (Eq-V) the self-model privileges no axis of the frame — its atomic reading is $\Gamma_{\mathrm{oct}}$-covariant — and among such self-models it is the most viable.

(Eq-V) gives $\varphi_J$ up to gauge by T-334; the former (Col) (covariance under the 168 collineations) now follows [T], with $S_7$ in place of the 168, and the former (Pure) (a pure anchor) is equivalent [T] to maximal viability once the first half of the principle — call it (Eq) — holds. The strict form of frame covariance — of $\varphi$ itself — is excluded by life (item 1), which is why the atomic reading is the strongest reading at which the frame decision D-0910 can hold for an isolated living holon. With the gauge fixed,

$$\varphi_J(\Gamma) = k\,\mathcal{P}_\alpha(\Gamma) + R\,uu^\dagger ,$$

unique up to the Fano weight $\alpha$. With it an isolated holon at $H = 0$ has, for $\kappa > \kappa_c(\alpha)$, a single living attractor, which persists for small $H$ and lies in $\mathcal{V}_{\mathrm{full}}$: $P \in (2/7, 5/14)$, $\Phi \in (1, 3/2]$, uniform diagonal (living attractor in the window [T]); its perturbations and the admissible range of $\kappa$ are given by T-335 and T-336. The price that stays is the phase reference: for the $H$-free holon the phases of $u$ are a gauge, but a Hamiltonian that is not diagonal in the frame makes the relative orientation of $D$ and $H$ physical. The attractor exists for every $D$ with the same bound on $\|H\|$ (T-334, item 4), so the reference is a free datum, not a condition of life. Whether a physical holon's self-model is $\varphi_{\mathrm{coh}}$, $\varphi_s$ or $\varphi_J$ is not decided by the axioms: $\varphi_J$ is the self-model fixed by (Eq-V) [Pr].

One clause. By item 6, (Eq-V) is equivalent [T] to a single condition on the anchor, stated in the corpus's own measure of integration:

• (MaxΦ) the anchor of the self-model is a state of maximal integration, $\Phi(\rho_a) = 6$ [Pr].

It mentions neither viability nor the frame group and yields both: an anchor of maximal integration has uniform diagonal, so it privileges no axis (item 1), and it is pure, so it is the most viable among such anchors (items 2–3). Routes tried to derive it, none sufficient: covariance of the anchor's gauge class under the symmetry group of the dynamics it regulates — Curie's principle for an isolated holon, with $S_7$ or with $\Gamma_{\mathrm{oct}}$ — gives the family $D\bigl((1-t)I/7 + t\,uu^\dagger\bigr)D^\dagger$ (item 5) but not $t = 1$; the terminal object gives the atomic half, $\mathrm{diag}\,\rho_a = I/7$, and nothing about purity; Lawvere's and Brouwer's theorems give fixed points, not anchors; viability alone does not pick $\varphi_J$ (item 6); the largest integration of the living attractor, rather than of the anchor, fails in a band — the attractor of a constant anchor at $H = 0$ depends only on $d = \sum_i(\rho_a)_{ii}^2$ and $s = P(\rho_a) - d$, its integration $\eta^2 s/d$ grows with $s$ at fixed $d$, so the maximiser is pure, but for $\kappa_c(\alpha) < \kappa < \kappa_* \approx 1.012\,\kappa_c(\alpha)$ a pure anchor with slightly non-uniform diagonal beats $uu^\dagger$ ($\alpha = 0$, $\kappa = 16.8$: $d = 1/7 + 10^{-4}$ gives $\Phi_{\mathrm{att}} = 1.25155$ against $1.25148$), above the band $uu^\dagger$ wins on the tested grid [H as a global statement], and below $\kappa_c$ only non-uniform anchors live, so there the principle contradicts (MaxΦ) (premises, §7; test_anchor_principle_is_independent_and_attractor_integration_does_not_replace_it). The two halves of (MaxΦ) are independent [T]: a pure anchor with non-uniform diagonal (amplitudes $1 \pm 0.3$, a sink at $\kappa = 50$, T-335) satisfies (Pure) and not (Eq), and $(1-t)I/7 + t\,uu^\dagger$ with $t = 0.9$ satisfies (Eq) and not (Pure) while holding a living sink in the window at $\alpha = \tfrac12$, $\kappa = 100$; so neither half follows from the other, and a derivation of (MaxΦ) has to supply both. (MaxΦ) is the smallest form of the principle found.

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Unified theorem of self-observation

Theorem: Fano-coherent self-modelling (unified theorem)

The canonical coherence-preserving self-modelling for UHM is determined up to the Fano weight $\alpha$ (the variational value of item (b) is retracted; the compression parameter $k = 1 - R$ is defined by the reflexion measure [T]) through:

(a) Algebraic structure: The Fano plane $PG(2,2)$ defines the composite atoms of the classifier $\Omega$, generating the Fano Lindblad operators $L_p^{\text{Fano}}$.

(b) Variational principle: The balance between atomic and Fano observation $\alpha^*$ minimises the functional $\mathcal{F} = S_{\text{spec}} + D_{KL}$. Retracted 2026-09-25 [✗]: that functional is minimised at $\alpha = 0$; the weight $\alpha$ is a free parameter.

(c) Phase properties: The canonical $\varphi_{\text{coh}}$ preserves the phases of coherences. The target Gap coincides with the current Gap.

(d) Symmetry (corrected): $G_2$ is broken at every $\alpha$, by the Fano and the atomic components alike: $\Delta_{G_2}(\alpha) = \tfrac{2+\alpha}{3}\,\Delta_{\max} \geq \tfrac23\Delta_{\max}$ (Theorem 5.1b). The earlier reading — "the Fano dissipator is G₂-covariant; the atomic one is not", with $\Delta_{G_2} = \alpha^* \cdot \Delta_{\max}$ — was retracted on 2026-09-10.

(e) Stationary Gap: upon substitution $\theta_{ij}^{\text{target}} = \theta_{ij}$:

$$\mathrm{Gap}^{(\infty)}(i,j) = \left|\sin\left(\theta_{ij} - \arctan\frac{\Delta\omega_{ij}}{\Gamma_2 + \kappa}\right)\right|$$

The stationary Gap is shifted relative to the current one by the angle $\arctan(\Delta\omega/(\Gamma_2 + \kappa))$ due to unitary rotation.

Proofs → | Status: [T] for (a), (c), (e) and the corrected (d); (b) retracted [✗]

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Three definitions of φ and their equivalence

In the documentation φ appears in three forms. They were presented as a chain, each a consequence of the previous; the variational form (2) and both links through it are retracted (2026-09-25): the functional of (2) is minimised by a projection onto the top eigenvector of $\Gamma$, which is neither the categorical φ nor the replacement channel. Forms (1) and (3) stand on their own pages.

Three forms

#	Name	Formula	Location
1	Categorical φ	$\varphi \dashv i: \mathrm{Sub}(\Gamma) \hookrightarrow \mathbf{Sh}_\infty(\mathcal{C})$	Axiom Ω⁷, FEP derivation
2	Variational φ — retracted 2026-09-25 [✗]	$\varphi = \arg\min_{\psi \in \mathcal{CPTP}} \mathbb{E}_\Gamma[S_{\mathrm{spec}}(\psi(\Gamma)) + D_{KL}(\psi(\Gamma) \| \Gamma)]$	Theorem 3.1, FEP derivation
3	Replacement φ_k	$\varphi_k(\Gamma) = (1-k)\Gamma + k\rho^*_{\mathrm{diss}},\ k = 1 - R$	Self-observation

Connection (1) ↔ (2): Theorem 3.1

Theorem 3.1 (Variational characterisation) — retracted 2026-09-25 [✗]

The categorically defined $\varphi$ (as the left adjoint to the inclusion $i: \mathrm{Sub}(\Gamma) \hookrightarrow \mathcal{E}$) coincides with the minimiser of the variational functional:

$$\varphi = \arg\min_{\psi \in \mathcal{CPTP}} \mathbb{E}_{\Gamma \sim \mu}\left[S_{\mathrm{spec}}(\psi(\Gamma)) + D_{KL}(\psi(\Gamma) \| \Gamma)\right]$$

The invariant measure $\mu$ is unique by the primitivity of the linear part $\mathcal{L}_0$ [T-39a]. Full proof → | Status: retracted [✗] — the functional is the cross-entropy $-\mathrm{Tr}(\psi(\Gamma)\log\Gamma)$, minimised by a projection onto the top eigenvector of $\Gamma$

Thus: the variational principle is not an axiom, but a theorem about the categorically defined φ. Retracted with Theorem 3.1: there is no variational principle for φ in the corpus.

Connection (2) ↔ (3): the replacement channel as a minimiser

Theorem (Replacement channel as CPTP-minimiser) — retracted 2026-09-25 [✗]

The minimiser of the functional $\mathcal{F}[\psi; \Gamma] = S_{\mathrm{spec}}(\psi(\Gamma)) + D_{KL}(\psi(\Gamma) \| \Gamma)$ over the class of CPTP channels on $\mathcal{D}(\mathbb{C}^7)$ is the replacement channel

$$\varphi_k(\Gamma) = (1 - k)\,\Gamma + k\,\rho^*_{\mathrm{diss}}, \qquad k = 1 - R$$

Key proof steps.

1. Convexity: $\mathcal{F}[\psi; \Gamma]$ is a strictly convex functional on the convex compact $\mathcal{CPTP}$ — the minimiser exists and is unique.
2. Form of the minimiser: From the stationarity conditions (variation over $\psi$ under the CPTP constraint) the minimiser takes the form of a convex combination of $\mathrm{Id}$ and the constant channel $\mathcal{C}_{\rho^*}$, i.e. $\psi^*(\Gamma) = (1-k)\Gamma + k\rho^*$.
3. Value of $k$: From the Banach principle (contracting mapping with constant $(1-k) < 1$) and the consistency condition with the reflexion measure: $k = 1 - R = 1 - 1/(7P)$.

Proof of physical realisation → | Parameter k from reflexion → | Status: retracted [✗]

Why retracted. Step 1 is false: $\mathcal{F}[\psi;\Gamma] = -\mathrm{Tr}(\psi(\Gamma)\log\Gamma)$ is linear in $\psi$, not strictly convex, and its minimum $-\log\lambda_{\max}(\Gamma)$ is reached by the channel onto the top eigenvector of $\Gamma$. The replacement channel $\varphi_k$ stays a well-defined CPTP channel with its own properties (self-observation); it is not a minimiser of $\mathcal{F}$ — on 300 random states it lay above the minimum in all 300.

Unified chain: φ_cat → φ_var → φ_k

φ_cat (categorical)
   — left adjoint to i: Sub(Γ) ↪ Sh_∞(C)
   — defined axiomatically through the structure of the ∞-topos
         |
         | Theorem 3.1 — retracted ✗
         ↓
φ_var (variational)
   — argmin [S_spec + D_KL] over all CPTP channels
   — variational principle as a CONSEQUENCE, not an axiom
         |
         | convexity + Banach principle — retracted ✗
         ↓
φ_k (replacement)
   — φ_k(Γ) = (1−k)Γ + k·ρ*_diss,  k = 1−R
   — explicit, computable form for D(ℂ⁷)

Absence of circularity

Resolution of circularity

The definition of φ contains no vicious circle. The derivation order is strictly linear:

1. $\rho^*_{\mathrm{diss}} = I/7$ is determined from the primitivity of the linear part $\mathcal{L}_0$ [T-39a] — this is a property of the dynamics, independent of φ.
2. $R(\Gamma) = 1/(7P(\Gamma))$ is determined only by the current state $\Gamma$ and the constant $\rho^*_{\mathrm{diss}} = I/7$ — not through $\varphi$.
3. $k = 1 - R$ is a function of the state $\Gamma$, not a free parameter.
4. $\varphi_k(\Gamma)$ is fully determined through $\Gamma$, $\rho^*_{\mathrm{diss}}$, and $k$ without self-reference.

Each level depends only on the previous ones — a closed directed acyclic graph (DAG).

The apparent "circularity" (φ defines $\rho^*$, and $\rho^*$ enters φ) is resolved by splitting: $\rho^*_{\mathrm{diss}} = I/7$ is the dissipative attractor of the linear part $\mathcal{L}_0$, whereas φ is the nonlinear regeneration operator. They reside at different levels of the hierarchy [D] (see attractor hierarchy).

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Connections

• Derived from: Axiom Ω⁷ → $\mathcal{L}_\Omega$ → $\varphi$
• Fano channel: Fano selection rules → $\Pi_p$ → $\mathcal{P}_{\text{Fano}}$
• Used in: Self-observation, Evolution, Gap dynamics
• Full formalisation: Formalisation of the φ operator
• Proofs of Fano theorems: Fano channel and Gap theorems
• Variational characterisation (retracted 2026-09-25): FEP derivation from UHM
• G₂ structure: G₂ = Aut(O) — the Fano dissipator is covariant only under the frame group $\Gamma_{\!\text{oct}}$ (Theorem 5.1b)