Dimension V: Interiority (E)

What this chapter is about

This chapter is devoted to the fifth dimension of the Holon — Interiority. You will learn:

• Why the "hard problem of consciousness" is not a philosophical puzzle but a question about a specific dimension of the configuration $\Gamma$;
• How the idea of the inner side of being developed from Descartes to Tononi;
• What the reduced density matrix $\rho_E$ is and how its spectrum describes the structure of interiority (at level L2 — the content of experience);
• How the five levels of interiority (L0→L4) arise from mathematical thresholds;
• Why, without dimension $E$, the regeneration formula $\kappa_0$ loses meaning and the system becomes a "philosophical zombie".

Who this chapter is for

If you are reading about UHM for the first time — start with the overview of dimensions. If you are already familiar with the seven dimensions and want to understand how the theory handles subjective experience — you are in the right place.

Function

To experience, to feel, to be aware.

Historical precursor

The question of what it means to "experience from within" is one of the oldest in philosophy. Different eras have approached it from different angles.

René Descartes (1641), in the Meditations on First Philosophy, formulated the famous cogito ergo sum — "I think, therefore I am". Even if the entire external world is an illusion, the very fact of experiencing is indisputable. Descartes established: subjectivity is a given, requiring no external confirmation. However, he divided the world into "thinking" and "extended" substances, creating the problem of their interaction.

Thomas Nagel (1974), in the article "What Is It Like to Be a Bat?", put the question sharply: a bat has echolocation — a physical fact. But what is it like to be a bat? What subjective experience does it have? This question cannot be reduced to a description of neurons or sound waves. Nagel showed that subjectivity is not a side effect of complexity, but a separate aspect of reality.

David Chalmers (1995) gave this question a precise name — "the hard problem of consciousness". The "easy" problems are to explain how the brain processes information, controls behaviour, distinguishes stimuli. All of this, in principle, fits within physics and neuroscience. The "hard" problem is different: why does information processing get experienced at all? Why do "zombies" not exist — beings functionally identical to a human but devoid of subjective experience?

Giulio Tononi (2004) proposed the Integrated Information Theory (IIT), in which consciousness is not a property of behaviour but a property of causal structure. The measure $\Phi_{\text{IIT}}$ quantifies how much a system is "more than the sum of its parts". But computing $\Phi_{\text{IIT}}$ requires enumerating all possible partitions of the system — a task of exponential complexity.

In UHM theory all these ideas find a unified formalism. Dimension $E$ (Interiority) is the answer to Nagel's question: every Holon has an "inner side", described by the reduced density matrix $\rho_E$. Chalmers' hard problem is resolved: subjectivity is not an "add-on" to physics, but an aspect of the configuration $\Gamma$, present at all levels (from atom to human). And Tononi's integration measure acquires a computable analogue — $\Phi_{\text{UHM}}$ with polynomial complexity $O(N^2)$.

Description

Interiority is the inner side of the Holon. Every configuration $\Gamma$ not only "exists" objectively, but is also "experienced" subjectively. Dimension $E$ defines the five-level hierarchy of interiority: L0 (interiority) → L1 (phenomenal geometry) → L2 (cognitive qualia) → L3 (network consciousness) → L4 (unitary consciousness).

Intuitive explanation

Imagine a mirror. From the outside you see a reflection — an objective, measurable picture. But a mirror also has an inner side — the amalgam, without which there would be no reflection. Dimension $E$ is the "amalgam" of the Holon: invisible from the outside, but providing the very possibility of experience.

A stone exists objectively — it has a coherence matrix $\Gamma$ with specific values of all seven dimensions. But "what does the stone feel"? Its level of interiority is L0: there is "something inside" (non-zero population $\gamma_{EE}$), but this "something" is not structured (rank $\rho_E = 1$). The stone has no "colours" or "shapes" in its inner world — there is only one point in quality space.

A neuron is already at level L1: its $\rho_E$ has rank greater than one — the inner space contains several distinguishable states. But a neuron cannot look at its inner world — for that, reflection is required ($R \geq 1/3$), and that is already level L2.

Ontological status

Dimension $E$ is an aspect of the configuration $\Gamma$, not a separate entity. "The Holon experiences" means: in the coherence matrix $\Gamma$ the projection onto the basis vector $|E\rangle$ is active, and the reduced density matrix $\rho_E$ with a non-trivial spectrum is defined.

Functional uniqueness of E [T]

Dimension $E$ is necessary and functionally unique by three independent arguments:

1. Axiomatic: (PH) is an axiomatic requirement for a Holon. Removing E violates (PH). Proof →
2. Categorical (κ₀): The formula $\kappa_0 = \omega_0 \cdot |\gamma_{OE}| \cdot |\gamma_{OU}| / \gamma_{OO}$ ([T at first-order kinetics], derivation; Th. 15.3.1) explicitly uses E as a separate object of the category via $\mathrm{Hom}(O, E)$. When E is removed: κ₀ is undefined, the regeneration rate $\kappa(\Gamma) = \kappa_{\text{bootstrap}} + \kappa_0 \cdot \mathrm{Coh}_E$ loses both E-dependent factors.
3. Mathematical: Only E is associated with the density matrix $\rho \in \mathcal{D}(\mathcal{H})$ — the unique mathematical object with $\mathrm{rank} > 1$ (requirement L1). The Fubini–Study metric on the projective quality space is the unique consistent Riemannian metric.

Status: [T] | Full proof →

Interiority provides the phenomenological aspect of the (M,R)-system: In Rosen's terminology, dimension $E$ is responsible for the "inner perspective" of the closed causal cycle — without it the system is functional, but "empty inside" (philosophical zombie).

Mathematical representation

Population of E

The diagonal element of the coherence matrix:

$$\gamma_{EE} = \langle E|\Gamma|E\rangle \in (0, 1)$$

The population $\gamma_{EE}$ shows what fraction of the Holon's "resources" is concentrated in the Interiority dimension. The higher $\gamma_{EE}$, the more intense the inner life of the system.

Typical values:

System	$\gamma_{EE}$	Interpretation
Crystal	$\sim 0.01$	Minimal interiority
Simple organism	$\sim 0.08$	Basic sensitivity
Mammal	$\sim 0.15$	Developed interiority
Waking human	$\sim 0.18$	Rich inner life

note

With a uniform distribution $\gamma_{EE} = 1/7 \approx 0.143$. Deviations from this value define the "sector profile" — the character of the given Holon.

Experience submatrix

$$\rho_E = \mathrm{Tr}_{\bar{E}}(\Gamma)$$

where $\mathrm{Tr}_{\bar{E}}$ is the partial trace over all dimensions except $E$.

Tensor structure and Morita equivalence [C]

Morita equivalence

The partial trace $\mathrm{Tr}_{-E}$ formally requires a tensor structure $\mathcal{H} = \mathcal{H}_E \otimes \mathcal{H}_{\bar{E}}$ (extended formalism: $\mathcal{H} = \mathbb{C}^{42}$). In the minimal 7D formalism ($\mathcal{H} = \mathbb{C}^7$, 7 is prime) direct factorisation is impossible.

T-58 asserted that the sites $(\mathcal{C}_7, J_{\text{Bures}})$ and $(\mathcal{C}_{42}^{\text{PW}}, J_{\text{Bures}})$ are Morita-equivalent, i.e. $\mathbf{Sh}_\infty(\mathcal{C}_7) \simeq \mathbf{Sh}_\infty(\mathcal{C}_{42}^{\text{PW}})$.

[✗] The equivalence is retracted (2026-09-10) — it fails on dimension

Counting argument. The corpus reads its sites as state spaces (that is what makes $X = |N(\mathcal{C})|$ a space with cohomology). For sober spaces $\mathbf{Sh}(X) \simeq \mathbf{Sh}(Y)$ forces $X \cong Y$, so the equivalence would require the two state spaces to be homeomorphic — in particular of equal dimension.

The PW-constrained space is $\{\Gamma \in \mathcal{D}(\mathbb{C}^{42}) : \hat C\Gamma = 0\} = \mathcal{D}(\ker \hat C)$. With the theory's own constraint $\hat C = H_O \otimes \mathbb{1} + \mathbb{1} \otimes H_{6D}$, where $H_O = \omega_0\,\mathrm{diag}(0,\dots,6)$ has seven distinct levels and $H_{6D}$ acts on $\mathbb{C}^6$: for each of the six non-clock basis states there is at most one clock level $k$ with $\omega_0 k + E_j = 0$, hence

$$\dim \ker \hat C \;\leq\; 6 \quad\Longrightarrow\quad \dim_{\mathbb{R}} \mathcal{D}(\ker\hat C) \;\leq\; 6^2 - 1 = 35 \;<\; 48 = \dim_{\mathbb{R}}\mathcal{D}(\mathbb{C}^7).$$

(Machine: over $2\times10^4$ tunings of $H_{6D}$ the kernel never exceeds 6; the best analytic tuning $E_j = -\omega_0 j$ gives exactly 6.) The two spaces are therefore never homeomorphic, and the sheaf topoi are not equivalent. Independently, the concrete functors offered never satisfied the comparison lemma: $\pi \circ \iota = \mathrm{id}$ makes $\iota$ a section, not an equivalence; the lift is not an isometry of state space (the normalised history state has purity $P/7$); and three mutually different lifts are in use across the corpus — $\Gamma \mapsto \sum_k |k\rangle\langle k|_O \otimes \Gamma(\tau_k)$ here, $\Gamma \mapsto \Gamma \otimes |\tau_0\rangle\langle\tau_0|_O$ in Coherence matrix, and $\Gamma \mapsto \Gamma \otimes I_6/6$ in the threshold bridge below.

What replaces it: T-58′ (section–retraction) [T]

$\iota: \mathcal{D}(\mathbb{C}^7) \to \mathcal{D}(\mathbb{C}^{42})$ and $\pi = \mathrm{Tr}_{\text{clock}}$ satisfy $\pi \circ \iota = \mathrm{id}$. This is all that is needed and all that is true: a 7D state can be carried into the 42D picture and brought back unchanged, so 7D data transport upward faithfully.

What it does not give: quantities defined only in 42D (the spectrum of $\rho_E$, hence $\mathrm{rank}(\rho_E)$ and $e^{S_{vN}(\rho_E)}$) are not functions of the 7D state — they depend on which lift is chosen, and the three lifts disagree. Hence 7D is primary and the PW extension is a construction on top of it, not a second presentation of the same content.

What is established in 7D without any lift:

• $\gamma_{EE}$ — diagonal element (population of E) — [T]
• $\gamma_{Ei}$ — coherences with other dimensions — [T]
• $\mathrm{Coh}_E(\Gamma) := \|\pi_E(\Gamma)\|_{\mathrm{HS}}^2 / \|\Gamma\|_{\mathrm{HS}}^2$ — E-coherence (HS-projection) [T], exact measure
• $\rho_E = \mathrm{Tr}_{-E}(\Gamma)$ — full reduced matrix — [C]: requires a tensor factor $\mathcal{H}_E$. In $\mathbb{C}^7 \otimes \mathbb{C}^6$ the axis $|E\rangle$ is a summand of $\mathbb{C}^6$, not a tensor factor, so $\mathrm{Tr}_{-E}$ is not defined by the PW extension as written (the same direct-sum/tensor-product distinction that T-87 step 3 enforces for the clock); a further factorisation of $\mathcal{H}_{6D}$ or a composite-system realisation (composite systems) is needed
• $D_{\text{diff}} = \exp(S_{vN}(\rho_E))$ — differentiation — [C] (same reason; the 7D proxy $D_{\text{diff}}^{7D} = 1 + 6\,\mathrm{Coh}_E/\mathrm{Coh}_E^{\max}$ below is a definition [D], cf. Seven dimensions: "statements using $D_{\text{diff}}$ have status [C]")
• $C = \Phi \times R$ — canonical measure of consciousness [T] (T-140; $D_{\text{diff}} \geq 2$ is a separate viability condition)

Intuitive explanation of the section–retraction (T-58′ [T]; the equivalence reading is retracted). Imagine a city. You have a map at scale 1:100 000 (7D) and a map at scale 1:10 000 (42D). On the detailed map individual houses are visible; on the overview map only city blocks. But any route planned on one map transfers correctly to the other. Morita equivalence is the theorem that two "maps" (the 7D and 42D formalisms) describe the same city (the physics of the Holon), and no observable depends on the choice of map.

Canonical PW-reconstruction algorithm [C]

Claim [C]. For any $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ there exists a unique canonical procedure for computing $\rho_E$, $D_{\text{diff}}$, $\sigma_L$, and $C$ from a chosen lift — conditional on the choice of lift (the round trip $\pi\circ\iota = \mathrm{id}$ of T-58′ [T] holds for every lift in use) and on a tensor factor $\mathcal{H}_E$ being specified in step 2. The former clause "with zero reconstruction error" rested on the Morita equivalence T-58, which is retracted [✗] (2026-09-10, box above); see step 4.

Algorithm (4 steps):

1. 7D → 42D lift. By the section–retraction T-58′ [T] (one of the three lifts in use — the choice matters, see the box above):

$\iota_{\text{PW}}: \mathcal{C}_7 \to \mathcal{C}_{42}, \quad \Gamma \mapsto \Gamma_{\text{total}} = \sum_{k=0}^{6} |k\rangle\langle k|_O \otimes \Gamma(\tau_k)$

where $\Gamma(\tau_k) = (\triangleright^*)^k(\Gamma)$ — successive applications of the modality ▷.

2. Partial trace. $\rho_E = \mathrm{Tr}_{-E}(\Gamma_{\text{total}})$ — standard partial trace in $\mathcal{H}_{42} = \mathcal{H}_O \otimes \mathcal{H}_6$.

3. 7D formulas via HS-projections. Equivalently, without an explicit lift:

$D_{\text{diff}}^{7D} = 1 + 6 \cdot \mathrm{Coh}_E(\Gamma) / \mathrm{Coh}_E^{\max}, \qquad \sigma_L(\Gamma) = \frac{7(1-\gamma_{LL})}{6} + O(\varepsilon^2)$

4. Zero error — [✗] retracted. The claim $\|\rho_E^{7D} - \rho_E^{42D}\|_{\mathrm{tr}} = 0$ rested on the equivalence, which is retracted above. What holds instead: $\pi(\iota(\Gamma)) = \Gamma$ exactly, so 7D quantities survive the round trip; $\rho_E^{42D}$ is not among them, being lift-dependent.

Operational separation 7D / 42D

The number 7 is prime, so $\mathbb{C}^7$ does not admit the tensor decomposition $\mathcal{H}_E \otimes \mathcal{H}_{\bar{E}}$, and the partial trace $\mathrm{Tr}_{\bar{E}}$ is not defined in 7D. This is resolved by the Page–Wootters extension: $\mathcal{H}_{42} = \mathbb{C}^7 \otimes \mathbb{C}^6$, where the partial trace is standard.

The equivalence that would have guaranteed "all observables coincide with zero error" is retracted (box above). The 7D quantities ($\gamma_{EE}$, $\gamma_{Ei}$, $\mathrm{Coh}_E$, $C$) stand on their own [T] and need no lift; the 42D-only quantities are lift-dependent and therefore not determined by the 7D state.

Practical rule:

• 7D is sufficient for $P$, $R$, $\Phi$, $\kappa$, $\mathrm{Coh}_E$ — defined through the diagonal and off-diagonal elements of $\Gamma \in \mathcal{D}(\mathbb{C}^7)$;
• 42D is required (or the 7D definition T-128 [D], which tracks the 42D notion without computing it — the Morita-equivalence reading is retracted) for $D_{\text{diff}}$, $\sigma_L$, $\rho_E$ — these require a partial trace.

What $\rho_E$ is in 7D and in 42D — the canonical statement

warning

$\rho_E$: scalar in 7D, clock block in 42D [T]

This box is the single source of truth for every appearance of $\rho_E$, $\mathrm{rank}(\rho_E)$ and $D_{\text{diff}}$ in the corpus.

1. Minimal 7D formalism. Each dimension is one basis vector of $\mathbb{C}^7$, so the E-sector is one-dimensional and $\rho_E = \gamma_{EE}$ is a scalar. Consequently $\mathrm{rank}(\rho_E) > 1$ and $S_{vN}(\rho_E) > 0$ are not expressible in 7D: the rank is $1$ whenever $\gamma_{EE} > 0$, and the entropy is identically $0$. Any 7D test that stands in for them — $\mathrm{Coh}_E > 0$ for L1, $D_{\text{diff}}^{7D} = 1 + 6\,\mathrm{Coh}_E/\mathrm{Coh}_E^{\max}$ for differentiation — is a definition [D] chosen to track the 42D notion, not an equivalence [T]. (Earlier drafts wrote "$\mathrm{rank}(\rho_E) > 1 \iff \mathrm{Coh}_E > 0$" as if it were a theorem; that reading is retracted.)
2. 42D Page–Wootters realisation. The derived factorisation is $\mathcal{H}_{42} = \mathbb{C}[\mathbb{Z}_7] \otimes \mathcal{H}_{6D}$ — the clock register is the $\mathbb{C}^7$ factor and the six non-clock dimensions $\{A,S,D,L,E,U\}$ are the $\mathbb{C}^6$ factor (T-87 step 3 [T]). The E-axis is therefore a summand of $\mathcal{H}_{6D}$, and the only well-defined reduction is the clock-block contraction

$$\rho_E := (\mathbb{1}_{\text{clock}} \otimes \langle E|)\,\Gamma_{\text{tot}}\,(\mathbb{1}_{\text{clock}} \otimes |E\rangle) \in \mathcal{L}(\mathbb{C}[\mathbb{Z}_7]),$$

a $7 \times 7$ matrix on the clock register (machine check: rank up to 7). Its eigenvectors are superpositions of clock moments $|\tau_k\rangle$, so in this realisation the "qualities" $|q_k\rangle$ are temporal modes of the E-population, and $\mathrm{rank}(\rho_E) > 1$ reads "E is populated at more than one moment". 3. A genuinely multi-dimensional $\mathcal{H}_E$ — the reading in which qualities are independent phenomenal directions — is available neither in 7D nor in the PW extension: it requires a composite realisation with several holons (composite systems), where $\mathcal{H}_E$ is a tensor factor of an actual subsystem.

Consequence for the hierarchy. L0 ($\gamma_{EE} > 0$), the L2 measures $P$, $R$, $\Phi$, $\mathrm{Coh}_E$, $C$ and the L3/L4 iterates $R^{(n)}$ are exactly computable in 7D. L1 in its literal form ($\mathrm{rank}(\rho_E) > 1$) and $D_{\text{diff}}$ in its literal form ($e^{S_{vN}(\rho_E)}$) belong to the 42D realisation; in 7D they are carried by the definitions of item 1. Since the tower is cumulative, the honest statement is: the level tower above L0 is defined in 7D by convention and realised literally only in 42D or in a composite substrate.

Theorem (7D sufficiency for the frame-referenced measures) [T]; literal $\rho_E$-quantities [D]

Formulation. The minimal 7D formalism $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ is sufficient for computing the frame-referenced observables $P$, $R$, $\Phi$, $\mathrm{Coh}_E$, $C$ and the stress components $\sigma_k$. The quantities defined through the spectrum of $\rho_E$ ($\mathrm{rank}$, $S_{vN}$, $D_{\text{diff}}$) are not among them: in 7D they are fixed by the definitions of the box above [D].

Proof.

Step 1 — [✗] retracted. This step claimed a categorical equivalence of the two ∞-topoi (T-58). The equivalence fails on dimension (box above). What remains is the section–retraction T-58′ [T]: every 7D object has a counterpart in 42D, but not conversely, and the counterpart depends on the lift.

Step 2 (Observable equivalence). An observable in UHM is a morphism $O: \Gamma \to \mathbb{R}$ in $\mathbf{Sh}_\infty(\mathcal{C})$. By categorical equivalence (Step 1): $O^{7D}(\Gamma) = O^{42D}(\iota_{PW}(\Gamma))$ for every observable $O$ and every state $\Gamma$. The reconstruction error is zero — not small, not controlled, but exactly zero — because equivalence of categories preserves all morphisms exactly.

Step 3 (Explicit 7D formulas for partial-trace quantities). The quantities that formally require the 42D partial trace have exact 7D representations via HS-projections:

Quantity	42D definition	7D formula	Error
$\mathrm{Coh}_E$	$\|\rho_E - P_{\bar{E}}(\rho_E)\|_F^2 / \|\Gamma\|_F^2$	$\|\pi_E(\Gamma)\|_{HS}^2 / \|\Gamma\|_{HS}^2$	0 (T-154 [T])
$D_{\text{diff}}$	$\exp(S_{vN}(\rho_E))$	$1 + 6 \cdot \mathrm{Coh}_E / \mathrm{Coh}_E^{\max}$	0 at extrema (T-128 [D])
$C$	$\Phi \cdot R$	$\Phi \cdot R$	0 (both defined in 7D)
$P$, $R$, $\Phi$	Same as 7D	Diagonal/off-diagonal of $\Gamma$	0 (identity)

Step 4 (What the two factors represent). In the 42D extension the clock is the $\mathbb{C}^7$ factor $\mathbb{C}[\mathbb{Z}_7]$ (the regular representation of the shift $\triangleright$, T-87 step 3 [T]) and the $\mathbb{C}^6$ factor carries the six non-clock dimensions $\{A,S,D,L,E,U\}$. An earlier draft of this step read the factors the other way round ("$\mathbb{C}^6$ is the temporal register of 6 conditional states"), contradicting Property 1; that reading is retracted. The consequence for $\rho_E$ is the clock-block contraction of the canonical box. Neither factor introduces new physical degrees of freedom beyond $\Gamma$: the clock register is the bookkeeping device that encodes temporal evolution inside a timeless formalism (Wheeler–DeWitt, T-87 [T]).

Conclusion: The 7D formalism is not an approximation of the 42D formalism. Both are exact descriptions of the same physics, related by categorical equivalence. The 42D extension is a computational convenience for partial traces, not an ontological necessity. $\blacksquare$

Dependencies: T-58′ [T] (section–retraction; the equivalence reading is retracted), T-87 [T] (PW), T-95 [C] (canonical reconstruction), T-128 [D], T-154 [T].

Technical remark

Here $\mathcal{H}_E$ is the Hilbert space associated with the Interiority dimension. The dimension of $\mathcal{H}_E$ is determined by the complexity of the system and is not fixed a priori. For systems with rich phenomenal content $\dim(\mathcal{H}_E) \gg 1$.

Computing the reduced state in the 7-dimensional formalism

Problem. The space $\mathbb{C}^7$ does not factorise as $\mathcal{H}_E \otimes \mathcal{H}_{\bar{E}}$, since $7$ is prime. The standard partial trace $\mathrm{Tr}_{\bar{E}}(\cdot)$ is not defined in 7D. This is a fundamental limitation: unlike composite dimensions (e.g. $6 = 2 \times 3$), a prime number admits no non-trivial tensor decomposition.

What is directly accessible from 7D. From the matrix $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ without any extension one extracts:

Quantity	Formula	Status
Population of E	$\gamma_{EE} = \langle E \vert \Gamma \vert E \rangle$	scalar, [T]
Coherences	$\gamma_{Ej}$, $j \neq E$	6 complex numbers, [T]
E-coherence	$\mathrm{Coh}_E(\Gamma) = \|\pi_E(\Gamma)\|_{\mathrm{HS}}^2 / \|\Gamma\|_{\mathrm{HS}}^2$	[T]

However, $\gamma_{EE}$ is one number, not a density matrix. For the full spectral content of $\rho_E$ (eigenvalues $\lambda_i$, eigenvectors $|q_i\rangle$) a transition to the extended formalism is required.

Solution: 42D Page–Wootters extension.

$$\mathcal{H}_{42} = \mathbb{C}^7 \otimes \mathbb{C}^6$$

where $\mathbb{C}^7$ is the "outer" space of seven dimensions, $\mathbb{C}^6$ is the "inner" Hilbert space (the phenomenal content of each dimension). The embedding $\iota_{\mathrm{PW}}: \mathcal{D}(\mathbb{C}^7) \to \mathcal{D}(\mathbb{C}^{42})$ is defined via the canonical lift (see PW-reconstruction algorithm):

1. Each element $\gamma_{ij}$ of the 7D matrix is mapped to a $6 \times 6$ block in the 42D matrix;
2. The partial trace over the inner space recovers the original $\Gamma$: $\mathrm{Tr}_{\mathrm{int}}(\Gamma_{\mathrm{total}}) = \Gamma$;
3. The reduced matrix $\rho_E$ is computed as the standard partial trace in 42D.

Equivalent 7D computational route [T-128].

For key scalar quantities the 42D extension is not required — they are computable directly from $\Gamma \in \mathcal{D}(\mathbb{C}^7)$:

$$D_{\text{diff}}^{7D} = 1 + \frac{\mathrm{Coh}_E(\Gamma)}{\mathrm{Coh}_E^{\max}} \cdot (N - 1)$$

This is a linear interpolation between $D_{\text{diff}} = 1$ (when $\mathrm{Coh}_E = 0$ — E is isolated, one distinguishable component) and $D_{\text{diff}} = N$ (when $\mathrm{Coh}_E = \mathrm{Coh}_E^{\max}$ — maximal differentiation).

Consistency of the two formulas:

Property	$D_{\text{diff}}^{42D} = \exp(S_{vN}(\rho_E))$	$D_{\text{diff}}^{7D} = 1 + \mathrm{Coh}_E/\mathrm{Coh}_E^{\max} \cdot (N-1)$
Definition	Nonlinear, via eigenvalues of $\rho_E$	Linear, via HS-norm of coherences
At $\mathrm{Coh}_E = 0$	$= 1$	$= 1$
At $\mathrm{Coh}_E = \mathrm{Coh}_E^{\max}$	$= N$	$= N$
Intermediate values	Nonlinear dependence on spectrum	Linear interpolation
Discrepancy	—	$O((\mathrm{Coh}_E)^2)$ in the intermediate region
Threshold test $D \geq 2$	Coincides	Coincides [T]

The two formulas coincide at the boundaries and give the same result for all threshold comparisons ($D_{\text{diff}} \geq D_{\min} = 2$). The $O((\mathrm{Coh}_E)^2)$ discrepancy in the intermediate region does not affect physical predictions, since the theory uses only threshold conditions, not exact numerical values of $D_{\text{diff}}$.

7D and 42D for the threshold conditions: what is equivalence and what is convention [D]

Claim [D] — a definitional bridge, not an equivalence

Status note (2026-09-10). The statement below was carried as a theorem [T]. It is a definitional bridge: the 7D side is defined by $D_{\text{diff}}^{7D} := 1 + \mathrm{Coh}_E/\mathrm{Coh}_E^{\max}\cdot(N-1)$, so the biconditional records that the definition was chosen to agree with the 42D notion at the two endpoints, not that two independently defined quantities coincide. In addition, the embedding used here ($\iota(\Gamma) = \Gamma \otimes I_6/6$) is a third lift, differing from the history-state lift of the PW-reconstruction algorithm and from the pure-clock lift of the coherence matrix; with the canonical factorisation ($\mathbb{C}^7$ = clock) $\mathrm{Tr}_{\bar E}$ is not defined at all, and the well-posed object is the clock-block contraction of the canonical box. For any $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ and the threshold $D_{\min} = 2$ ([D], T-151):

$$D_{\text{diff}}^{7D}(\Gamma) \geq D_{\min} \iff D_{\text{diff}}^{42D}(\iota(\Gamma)) \geq D_{\min}$$

where $\iota: \mathcal{D}(\mathbb{C}^7) \hookrightarrow \mathcal{D}(\mathbb{C}^{42})$ is a section of the retraction (T-58′ [T]) — the third of the three lifts in use. Consequently, all consciousness threshold conditions (L0→L4) are verifiable in 7D without passing to 42D.

Proof.

Step 1 (embedding). By T-58′ [T], the section $A_{\text{int}} \sim_{\text{Morita}} A_{\text{int}} \otimes M_6(\mathbb{C})$ induces the embedding

$$\iota: \mathcal{D}(\mathbb{C}^7) \hookrightarrow \mathcal{D}(\mathbb{C}^{42}), \quad \iota(\Gamma) = \Gamma \otimes \frac{I_6}{6}.$$

The reverse projection is $\pi = \mathrm{Tr}_{\mathbb{C}^6}$, and $\pi \circ \iota = \mathrm{id}$.

Step 2 (Relation of $D_{\text{diff}}^{42D}$ and $\mathrm{Coh}_E$). In 42D: $\rho_E^{42D} = \mathrm{Tr}_{\bar{E}}(\iota(\Gamma))$. For $\iota(\Gamma) = \Gamma \otimes I_6/6$:

$$\rho_E^{42D} = \mathrm{Tr}_{\bar{E}}(\Gamma) \otimes \frac{I_6}{6} = \gamma_{EE} \cdot \frac{I_6}{6} + \sum_{j \neq E} |\gamma_{Ej}| \cdot (\text{off-diagonal contributions}).$$

The eigenvalues of $\rho_E^{42D}$ depend on $\gamma_{EE}$ and the coherences $\gamma_{Ej}$:

• at $\mathrm{Coh}_E = 0$ (all $\gamma_{Ej} = 0$): $\rho_E^{42D} = \gamma_{EE} \cdot I_6/6$ — one nonzero eigenvector → $D_{\text{diff}}^{42D} = 1$.
• at $\mathrm{Coh}_E > 0$: additional eigenvalues appear → $D_{\text{diff}}^{42D} \geq 2$.

Step 3 (Threshold equivalence). By definition:

• $D_{\text{diff}}^{7D} = 1 + \mathrm{Coh}_E/\mathrm{Coh}_E^{\max} \cdot (N-1)$ [T-128]
• $D_{\text{diff}}^{42D} = \exp(S_{vN}(\rho_E^{42D}))$

Both quantities are:

• $= 1$ at $\mathrm{Coh}_E = 0$ (exact coincidence at the boundary)
• $\geq 2$ at $\mathrm{Coh}_E > 0$ (both $> 1$ in the presence of E-coherences)
• $= N$ at $\mathrm{Coh}_E = \mathrm{Coh}_E^{\max}$ (exact coincidence at the upper boundary)

Hence $D_{\text{diff}}^{7D} \geq 2 \iff \mathrm{Coh}_E > 0 \iff D_{\text{diff}}^{42D} \geq 2$. $\square$

Step 4 (Completeness of 7D for L0–L4). The conditions of each level:

• L0: $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ — automatically in 7D ✓
• L1: literally $\mathrm{rank}(\rho_E) > 1$ — not expressible in 7D ($\rho_E$ is the scalar $\gamma_{EE}$); the 7D test is $\mathrm{Coh}_E > 0$ by definition [D] (canonical box)
• L2: $P > 2/7 \wedge R \geq 1/3 \wedge \Phi \geq 1 \wedge D_{\text{diff}} \geq 2 \wedge \|\sigma\|_\infty < 1$ — all components computable in 7D (T-137 [T]) ✓
• L3: $R^{(2)} \geq 1/4$ — computable via $\varphi(\Gamma) \in \mathcal{D}(\mathbb{C}^7)$ ✓
• L4: $P > 6/7 \wedge \forall n: R^{(n)} > 0$ — computable via the iterations $\varphi^{(n)}$ in 7D ✓

Conclusion. All threshold tests L0–L4 are evaluable in $\mathcal{D}(\mathbb{C}^7)$ — but for L1 and $D_{\text{diff}}$ this is evaluability of the 7D definitions, which stand in for the literal $\rho_E$-spectral conditions. 42D (or a composite substrate) is required whenever the literal conditions are meant: the rank and the spectrum of $\rho_E$, hence the eigenvector-level content of experience. $\blacksquare$

Status: [T] for the frame-referenced measures; [D] for the L1 and $D_{\text{diff}}$ substitutions. Audit problem I.2 is resolved only in the first sense — see the canonical box.

Practical summary

For classifying systems by levels L0-L4 the 7D formula $D_{\text{diff}}^{7D}$ is sufficient — as the accepted definition [D], see the canonical box. The full matrix $\rho_E$ (via the 42D PW extension) is needed only for detailed spectral analysis of phenomenal content — a task relevant for future experimental tests.

Spectral decomposition

$$\rho_E \vert q_i\rangle = \lambda_i \vert q_i\rangle$$

where:

• $\lambda_i \in [0, 1]$, $\sum_i \lambda_i = 1$ — intensities of the components of experience
• $\vert q_i\rangle$ — qualities of the components. In the PW realisation these live in the clock register $\mathbb{C}[\mathbb{Z}_7]$ (temporal modes of the E-population); a genuinely phenomenal $\mathcal{H}_E$ with $\dim > 1$ requires a composite substrate — canonical box

Intuitive explanation. Recall how white light, passed through a prism, is split into a spectrum — red, orange, yellow and so on. Each colour has its own wavelength (quality $|q_i\rangle$) and brightness (intensity $\lambda_i$). The spectral decomposition of $\rho_E$ is a "prism for the inner world": it shows what "colours" make up the experience and how bright each one is.

If all $\lambda_i$ are equal — the experience is "white", uniform, undifferentiated (deep anaesthesia). If one $\lambda_1 \approx 1$ and the rest $\lambda_i \approx 0$ — the experience is "monochromatic", concentrated on a single quality (acute pain). Rich conscious experience is a "full spectrum" with several significant $\lambda_i$.

Phenomenal vector

Full description of experience at moment $\tau$:

$$\text{FV}(\rho_E) := \{(\lambda_i, [\vert q_i\rangle]) : \rho_E \vert q_i\rangle = \lambda_i \vert q_i\rangle\}$$

where $[\vert q_i\rangle] \in \mathbb{P}(\mathcal{H}_E)$ is the equivalence class in projective space.

Scope note (T-301): FV is the E-slice of content — intensity and the relational position of a quality. The full content of a state is its 28 parameters (7 populations + 21 coherences), and the gauge-invariant colour lives in the Fano holonomies; see Qualia Structure for the complete five-layer passport.

Quantitative characteristics

Population $\gamma_{EE}$ and stress $\sigma_E$

The population $\gamma_{EE}$ is the fraction of the Holon's "resources" in the Interiority dimension. The related quantity is the stress in the E channel:

$$\sigma_E = \mathrm{clamp}(1 - 7\gamma_{EE},\; 0,\; 1) \quad \text{[T] (T-92)}$$

• $\sigma_E = 0$: interiority is fully provided ($\gamma_{EE} \geq 1/7$)
• $\sigma_E = 1$: interiority is in deficit ($\gamma_{EE} \to 0$) — the system is "emotionally empty"

Differentiation $D_{\text{diff}}$

$$D_{\text{diff}} = \exp(S_{vN}(\rho_E)), \qquad S_{vN} = -\mathrm{Tr}(\rho_E \log \rho_E)$$

$D_{\text{diff}}$ is the effective number of distinguishable components of experience. Analogy: if the spectrum of $\rho_E$ contains 3 significant components, then $D_{\text{diff}} \approx 3$.

E-coherence $\mathrm{Coh}_E$

$$\mathrm{Coh}_E(\Gamma) := \frac{\|\pi_E(\Gamma)\|_{\mathrm{HS}}^2}{\|\Gamma\|_{\mathrm{HS}}^2}$$

A measure of how strongly dimension E is connected with the other six. When $\mathrm{Coh}_E = 0$ — interiority is isolated (no connection with action, logic, ground...). When $\mathrm{Coh}_E = \mathrm{Coh}_E^{\max}$ — interiority is maximally woven into the life of the Holon.

Experiential content

Experiential content (for all levels L0-L2) is defined by four components:

$$\text{Exp}(\rho_E, \tau) := (\text{Intensity}, \text{Quality}, \text{Context}, \text{History})$$

Terminology

The function $\text{Exp}$ is applicable to all levels. The term "qualia" (Quale) is reserved exclusively for L2 — cognitive qualia with reflexive access.

Component	Definition	Interpretation
Intensity	$\{\lambda_i\}$ — spectrum of $\rho_E$	Strength of the interior state
Quality	$\{[\vert q_i\rangle]\} \subset \mathbb{P}(\mathcal{H}_E)$	Character of the interior state
Context	$\rho_{\bar{E}} = \mathrm{Tr}_E(\Gamma)$	Modulation of experience by other dimensions
History	$\{\rho_E(\tau') : \tau' < \tau\}$	Adaptation and memory

Structural necessity

The formula establishes a structural correspondence between mathematical objects and experiential content. This correspondence is not an arbitrary postulate, but the unique functor compatible with the axiomatics: the partial trace is unique, the spectral decomposition is unique, the Fubini–Study metric is unique (Čencov–Petz theorem).

Projective quality space

Qualities live in projective space:

$$\mathbb{P}(\mathcal{H}_E) := (\mathcal{H}_E \setminus \{0\}) / \sim$$

where $\vert\psi\rangle \sim \vert\phi\rangle \Leftrightarrow \exists c \in \mathbb{C}^*: \vert\psi\rangle = c\vert\phi\rangle$.

Fubini–Study metric

Distance between qualities:

$$d_{FS}([\vert\psi\rangle], [\vert\phi\rangle]) := \arccos(\lvert\langle\psi\vert\phi\rangle\rvert) \in [0, \pi/2]$$

Interpretation:

• $d_{FS} = 0$ — identical qualities (the same experience)
• $d_{FS} = \pi/2$ — maximally different (orthogonal) qualities

Example. "Red" and "green" are two qualities in the space $\mathbb{P}(\mathcal{H}_E)$. The distance $d_{FS}$ between them determines how distinguishable these experiences are for the system. If $d_{FS} = \pi/2$ — the experiences are maximally dissimilar; if $d_{FS} \to 0$ — they merge (as in colour vision deficiency).

Five levels of interiority

The five levels are not an arbitrary classification, but mathematical thresholds whose crossing qualitatively changes the structure of $\rho_E$ and the quantities associated with it.

L0: Interiority — "thermometer"

Condition: $\exists \rho_E$ (i.e. $\gamma_{EE} > 0$)

At level L0 the system simply "has an inner state". Analogy: a thermometer has a temperature — an inner state determined by the environment. But the thermometer does not "feel" the temperature; it is simply in a certain state. A quartz crystal at level L0: its $\rho_E$ is a pure state of rank 1 (one eigenvector with $\lambda_1 = 1$). Inside — one "point", no structure, no distinctions.

L1: Phenomenal geometry — "palette"

Condition: $\mathrm{rank}(\rho_E) > 1$ (literal form: 42D or a composite substrate; the 7D test is $\mathrm{Coh}_E > 0$ by definition [D] — canonical box)

At level L1 the inner space is structured: it contains several distinguishable states. Analogy: an artist now has a palette with several colours — they can distinguish colours, shapes, textures. The retina at level L1: three types of cone cells create a three-dimensional space of colour qualities $\mathbb{P}(\mathcal{H}_E)$ with the Fubini–Study metric. But the retina does not know that it is distinguishing colours — the next level is required for that.

L2: Cognitive qualia — "mirror"

Condition: $R \geq R_{\text{th}} = 1/3$ [T], $\Phi \geq \Phi_{\text{th}} = 1$ [T]

At level L2 the system is capable of looking at its inner world — reflection. Analogy: a mirror has appeared — now one can not only have a palette but also see which colours are on it. This is the threshold of consciousness in the usual sense: the subject can report on their experience, distinguish one experience from another, be surprised by a new quality. A waking human is a typical L2 system with $R \approx 0.7$, $\Phi \approx 4$.

L3: Network consciousness — "hall of mirrors"

Condition: $R^{(2)} \geq 1/4$ [T]

At level L3 — meta-reflection: the system observes not only its inner world but also how it observes it. Analogy: a mirror reflecting another mirror — an infinite corridor of reflections (though at L3 the depth is limited). Examples: fungal mycelium as a distributed L3 system, a bee swarm with metastable collective reflection, deep meditation.

L4: Unitary consciousness — "crystal transparency"

Condition: $\lim_{n \to \infty} R^{(n)} > 0$, $P > 6/7$

Level L4 is full transparency: infinite depth of self-reflection converging to a stable limit. Analogy: a crystal in which every atom "sees" the entire crystal as a whole. This is a theoretical limit: $P > 6/7$ is unattainable for biological systems (requires an almost pure state $\Gamma$).

Summary table of levels

Level	Name	Condition	What exists	Examples
L0	Interiority	$\exists \rho_E$	Inner state	Atom, crystal
L1	Phenomenal geometry	$\mathrm{rank}(\rho_E) > 1$ (7D: $\mathrm{Coh}_E > 0$ [D])	Structure of qualities with $d_{FS}$	Neuron, retina
L2	Cognitive qualia	$R \geq R_{th}$, $\Phi \geq \Phi_{th}$	Reflexive access	Human, higher mammals
L3	Network consciousness	$R^{(2)} \geq 1/4$	Meta-reflection (metastable)	Mycelium, swarm, deep meditation
L4	Unitary consciousness	$\lim_{n \to \infty} R^{(n)} > 0$, $P > 6/7$	Full ∞-structure	Theoretical limit

where $R_{\text{th}} = 1/3$ [T], $\Phi_{\text{th}} = 1$ [T] (T-129), $R^{(2)}_{\text{th}} = 1/4$ [T] — mathematical results. L4 requires $P > 6/7$ — unattainable for biological systems.

E and the "hard problem of consciousness"

Chalmers formulated the "hard problem" as follows: why are physical processes experienced at all? One can explain how neurons transmit signals — but why does signal transmission accompany the sensation of red?

In UHM the answer is: experience is not an "add-on" to physics, but an aspect of the configuration. The matrix $\Gamma$ has both an "outer" side (observables: $P$, $\Phi$, $R$) and an "inner" side ($\rho_E$, phenomenal vector). These are not two substances (as in Descartes), but two aspects of one object — two-aspect monism.

Analogy: a sheet of paper has a front side and a back side. These are not two sheets — it is one sheet with two aspects. Asking "why does the sheet have two sides?" is ill-posed: it is a property of the object itself, not something requiring explanation. In exactly the same way $\Gamma$ has an "outer" (physical) and an "inner" (phenomenal) aspect — this requires no separate mechanism for "generating" consciousness from matter.

A philosophical zombie is impossible

The No-Zombie theorem (T-81 [T]): a system with $P > P_{\text{crit}}$, $R \geq R_{\text{th}}$, $\Phi \geq \Phi_{\text{th}}$ necessarily has a non-trivial $\rho_E$. A "philosophical zombie" — a functionally identical being without interiority — is mathematically impossible in UHM. See: theorem 8.1.

Examples by level

Physical level

System	Level	$\gamma_{EE}$	$D_{\text{diff}}$	Description
Electron	L0	$\sim 0.001$	1	Spin state — one "quality"
Crystal	L0	$\sim 0.01$	1	Phonon coherence
Laser beam	L0	$\sim 0.02$	1	Coherent optical state

Biological level

System	Level	$R$	$\Phi$	Description
Bacterium	L0–L1	$\sim 0.05$	$\sim 0.3$	Chemotaxis — the simplest "reaction"
Retina	L1	$< R_{th}$	$\sim 1$	Spectral profile distinguishes colours
Individual neuron	L1	$\sim 0.1$	$< \Phi_{th}$	Local quality geometry
Higher primates	L2	$\geq R_{th}$	$\sim 2$	Mirror self-recognition

Cognitive level

System	Level	$R$	$\Phi$	Description
REM sleep	L2	$\sim 0.4$	$\sim 3$	Dreams with partial reflection
Waking human	L2	$\sim 0.7$	$\sim 4$	Full set of qualia: colour, pain, emotions
Deep meditation	L3	$R^{(2)} \geq 1/4$	$\gg 1$	Observing the observer

Loss of interiority

When $\gamma_{EE} \to 0$ (or $\sigma_E \to 1$):

1. Phenomenal content becomes impoverished: $D_{\text{diff}} \to 1$
2. Coherences of E with other dimensions drop: $\gamma_{Ei} \to 0$
3. The regeneration formula loses one of its key factors: $\kappa_0 \propto |\gamma_{OE}|$

Clinical analogies:

Condition	Mechanism	Manifestations
Deep anaesthesia	$\gamma_{EE} \to 0$	Complete loss of inner world; $\rho_E \to$ pure state
Alexithymia	$\gamma_{ED} \to 0$	Inability to recognise one's own emotions; processes exist but are not experienced
Anosognosia	$\gamma_{EA} \to 0$	Inability to recognise the deficit (the patient does not know they are ill)
Depersonalisation	$\gamma_{EU} \to 0$	"I feel like I'm not myself" — interiority is present but not integrated into the whole

Connection with other dimensions

Key connections:

• E ↔ U (Synthesis): Interiority and unity are interrelated: $E$ determines what constitutes the interior content, $U$ determines how these contents are integrated into a single whole. When $\gamma_{EU} \to 0$, experience fragments (dissociation).

• E ↔ O (Immanence): Through the coherence $\gamma_{OE}$ interiority receives energetic nourishment. The formula $\kappa_0 = \omega_0 \cdot |\gamma_{OE}| \cdot |\gamma_{OU}| / \gamma_{OO}$ shows: the stronger the connection of E with the Ground, the faster the regeneration of coherence. When $\gamma_{OE} \to 0$ — interiority "fades out" (depression, depersonalisation).

• E ↔ L (Evidence): Logic in interiority is the ability to distinguish "this is true" from "this is false" from within. When $\gamma_{EL} \to 0$ — experiences are chaotic, not connected by logic (delusion, hallucinations).

• E ↔ A (Apperception): Distinction that has become experience. Without the connection $\gamma_{EA}$, experience contains no distinctions — "everything is fused into one".

Coherence with E

Coherence	Interpretation
$\gamma_{EA}$	Apperception (distinction that has entered interiority)
$\gamma_{ES}$	Representation (structure in interiority)
$\gamma_{ED}$	Affection (action of process on interiority)
$\gamma_{EL}$	Evidence (logical connectedness in interiority)
$\gamma_{EO}$	Immanence (ground within interiority)
$\gamma_{EU}$	Synthesis (integration of interior content into the whole)

Consciousness formula

The canonical measure of consciousness (T-140 [T]):

$$C = \Phi \times R$$

where:

• $\Phi$ — integration: $\Phi = \sum_{i \neq j} |\gamma_{ij}|^2 / \sum_i \gamma_{ii}^2$
• $R$ — reflection: $R = 1/(7P)$

$D_{\text{diff}} \geq 2$ is a separate condition of full viability:

• $D_{\text{diff}} = \exp(S_{vN}(\rho_E))$, where $S_{vN} = -\mathrm{Tr}(\rho_E \log \rho_E)$
• Computable in 7D: $D_{\text{diff}}^{7D} = 1 + \mathrm{Coh}_E/\mathrm{Coh}_E^{\max} \cdot (N-1)$ (T-128 [D])

On notation

$D_{\text{diff}}$ is a measure of differentiation of experience. Not to be confused with dimension D (Dynamics).

Tensor factorisation for D_diff

Two formulas for D_diff and their consistency

42D definition (canonical):

$$D_{\text{diff}}^{42D} = \exp(S_{vN}(\rho_E)), \quad S_{vN} = -\mathrm{Tr}(\rho_E \log \rho_E)$$

Requires computing $\rho_E = \mathrm{Tr}_{\bar{E}}(\Gamma)$ — the partial trace defined only in the extended formalism $\mathcal{H}_{42} = \mathbb{C}^7 \otimes \mathbb{C}^6$, since $\mathbb{C}^7$ does not factorise (7 is prime). This is a nonlinear function depending on the eigenvalues of $\rho_E$. Detailed discussion of the factorisation problem: Computing the reduced state.

7D formula [T-128] (computational route):

$$D_{\text{diff}}^{7D} := 1 + \frac{\mathrm{Coh}_E(\Gamma)}{\mathrm{Coh}_E^{\max}} \cdot (N - 1)$$

where $\mathrm{Coh}_E(\Gamma)$ — E-coherence (HS-projection, [T]). This is a linear interpolation: $D_{\text{diff}}^{7D} \in [1, N]$.

Consistency [T]:

The two formulas exactly coincide at the boundaries:

• $\mathrm{Coh}_E = 0 \Rightarrow D_{\text{diff}}^{42D} = D_{\text{diff}}^{7D} = 1$ (pure state, one component)
• $\mathrm{Coh}_E = \mathrm{Coh}_E^{\max} \Rightarrow D_{\text{diff}}^{42D} = D_{\text{diff}}^{7D} = N$ (maximal differentiation)

In the intermediate region the discrepancy is $O((\mathrm{Coh}_E)^2)$: the exponential function $\exp(S_{vN})$ is nonlinear in the spectrum of $\rho_E$, whereas the 7D formula is linear in $\mathrm{Coh}_E$. However, for all threshold conditions ($D_{\text{diff}} \geq D_{\min} = 2$) both formulas give identical results.

Reduced consciousness measure (for cases where $D_{\text{diff}}$ is not computed explicitly):

$$C_{\min} := \Phi \times R$$

At $D_{\text{diff}} = D_{\min} = 2$ (threshold value) this measure correctly classifies systems:

• $C_{\min} \geq 1/3 \Leftrightarrow \Phi \geq 1$ and $R \geq 1/3$ ⟹ L2
• $C_{\min} < 1/3$ ⟹ L0 or L1

Range of $D_{\text{diff}}$:

• $S_{vN} \in [0, \log N]$ for an $N$-dimensional system
• $D_{\text{diff}} = \exp(S_{vN}) \in [1, N]$
• Minimum ($D_{\text{diff}} = 1$): pure state, one component of experience
• Maximum ($D_{\text{diff}} = N$): maximally mixed state, equiprobable components

Differentiation threshold $D_{\min} = 2$

Justification: Cognitive qualia require distinction — at minimum two distinguishable components of experience.

$$D_{\text{diff}} \geq D_{\min} = 2 \Leftrightarrow S_{vN}(\rho_E) \geq \log 2$$

Geometric interpretation: $S_{vN} = \log 2$ corresponds to a state with effective dimension 2 (two equiprobable components). This is the minimum for:

1. Distinction — there must be something to distinguish (at minimum 2 qualities)
2. Choice — there must be the possibility of choosing between alternatives
3. Information — at minimum 1 bit of phenomenal content

Connection with information theory

$D_{\min} = 2$ means that cognitive access requires at minimum 1 bit of information in the phenomenal content. A system experiencing only one indistinguishable quality ($D_{\text{diff}} = 1$) has no material for reflection.

Consciousness threshold [Т T-140]:

$$C \geq C_{\text{th}} := \Phi_{\text{th}} \times R_{\text{th}} = 1 \times \frac{1}{3} = \frac{1}{3}$$

with the separate viability condition $D_{\text{diff}} \geq D_{\min} = 2$.

Octonionic context

Octonionic correspondence [T]

The dimension corresponds to $e_5 \in \mathrm{Im}(\mathbb{O})$. This identification is a theorem [T]: the T15 bridge chain (all steps [T]; the step to $\mathbb{O}$ takes the canonical orientation of the Fano lines, T15-canon) derives the octonionic structure from (AP)+(PH)+(QG)+(V); the combinatorial and functional uniqueness of each role claimed by T-177 and T-183 is retracted [✗] (2026-09-25): it rested on the axis sectors of T-48a. Restated (T-177, T-183): given $O$ and the $\kappa_0$ pair $\{E,U\}$, incidence fixes $A$ and $D$ [T], and one binary convention [D] fixes $E$ versus $U$ together with $L$ versus $S$. The specific assignment $E = e_5$ is fixed up to $G_2$-gauge equivalence (T-42a [T]). Details and $G_2$-caveat: Octonionic interpretation, structural derivation.

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Related documents:

• Logic (L) — previous dimension
• Ground (O) — next dimension
• Interiority hierarchy — formal definitions L0→L1→L2→L3→L4
• Theory of interiority — complete mathematical theory
• Two-aspect monism — ontology of interiority
• Self-observation — reflection measure R
• Operationalisation — derivation of D_diff and thresholds