Theorem on Emergent Time

Status: [T] for the cyclic clock and for the dynamics relative to the depth register

This page derives two clocks. The cyclic clock $\tau \in \mathbb{Z}_7$: the temporal modality ▷ and its equivalence with the Page–Wootters clock (§2–§3) [T]. And the depth register (§11.4) [T]: the stratal depth $n \in \{0, \ldots, N\}$ recorded as an ordered chain rather than a cycle, realised positionally in the O-registers of $M = \lceil \log_7(N+1) \rceil$ holons. Relative to the O-clock alone every conditional dynamics is periodic (§11.2), so it carries no dissipation and no arrow. Relative to the depth register the dissipative semigroup of §9.1 is the Page–Wootters conditional dynamics exactly at every reading, in a finite world of dimension $343(N+1)$, and the arrow of §10 holds on the whole history (T-53b [T], Theorems 11.1–11.4); the continuous parameter $t$ is the scaling limit of the readings, with error at most $\Delta t\,\|\mathcal{L}\|$, and its algebra is $C_0(\mathbb{R})$ (Theorem 11.5, T-118). The regenerator $\mathcal{R}$ is realised along each solution, not as a conditional law independent of the state (§11.3). The arrow is the stratum collapse towards the terminal object T, monotone in the depth. An earlier version of this box called time as a whole derived and dynamical from the O-clock; that wording is retracted. An intermediate version of 2026-09-25 held the dynamics [C] at an assumed aperiodic time parameter, whose only carrier then known was an ideal clock with an infinite environment; the depth register of §11.4 replaces that assumption.

Spatial analogue: The spatial manifold $\Sigma^3$ is also derived from categorical structure — Emergent manifold $M^4$ (T-119 [T]).

Contents

1. Problem statement
2. Time from temporal modality on Ω
   • 2.1 Algebraic definition ▷
   • 2.4 Connection to L-unification
   • 2.7 Time as modality in HoTT
3. Page–Wootters mechanism for UHM
   • 3.1a Page–Wootters as a theorem
   • 3.8 Limit N → ∞
4. Information-geometric time
5. Categorical time via ∞-groupoid
6. Equivalence theorem
7. Arrow of time theorem
   • 7.4 ∞-categorical resolution
8. Connection to critical purity
9. Corollaries
10. Stratificational time
11. Precedents and related programmes

• 11.3 Carriers of the aperiodic parameter
• 11.4 The depth register: an exact finite carrier
• 11.5 The de Sitter observer algebra and the holon tower
• 11.6 Direction and the continuous clock

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1. Problem statement

1.1 The circularity problem

In the original formulation of UHM, time $t$ enters as an evolution parameter:

$$\frac{d\Gamma}{dt} = -i[H, \Gamma] + \mathcal{D}[\Gamma] + \mathcal{R}[\Gamma, E]$$

This is logically circular: dynamics is defined through $d/dt$, but $t$ is what we are trying to derive.

1.2 Requirement of Axiom Ω⁷

From Axiom Ω⁷ it follows:

"The ∞-topos $\mathbf{Sh}_\infty(\mathcal{C})$ is the unique primitive."

Logical consequence: Time must be a function of the structure of category $\mathcal{C}$:

$$\tau = \tau(\text{Mor}(\mathcal{C})) \quad \text{or} \quad \tau = \tau(\text{strata } X)$$

1.3 Four levels of the problem

Level	Problem	Solution
Kinematic	What is a "moment of time"?	Page–Wootters: correlation with O
Geometric	How to measure "the flow of time"?	Bures metric / d_strat
Categorical	How to formalize the structure?	∞-groupoid of paths Exp_∞
Stratificational	What is the arrow of time?	Stratum collapse to T

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2. Time from temporal modality on Ω

Key theorem

Time is derived from the structure of the subobject classifier Ω ∈ $\mathrm{Sh}_\infty(\mathcal{C})$ via the temporal modality ▷. This unifies:

• L-dimension (logic)
• Lindblad operators L_k (dissipation)
• Discrete time τ (evolution)

into a single structure on Ω.

2.1 Algebraic definition of ▷ (independent of dynamics)

Key achievement

The temporal modality ▷ is defined algebraically via a ℤ_N-action on atoms of the classifier. This breaks the cycle: time is defined before dynamics, not through it.

Step 1: Atoms of the classifier

For base category $\mathcal{C} = \mathcal{D}(\mathbb{C}^N)$ the classifier Ω decomposes into atoms:

$$\mathcal{T}_\Omega = \{S_0, S_1, \ldots, S_{N-1}\}$$

where each atom is a projector onto a basis state:

$$S_i = |i\rangle\langle i|, \quad i \in \{0, 1, \ldots, N-1\}$$

Constructive definition [D]

The identification of atoms of the classifier Ω with projectors |i⟩⟨i| is a constructive definition, consistent with the axiomatics, not a derivation from abstract ∞-topos theory. Justification: (1) in D(ℂ⁷) the minimal non-trivial subobjects are rank-1 projectors; (2) the Bures topology (A2) singles them out as atoms of J_{Bures}-covers; (3) the result is consistent with L-unification ([T]) and Fano structure ([T]). Formal derivation from Lurie's axioms for $\mathrm{Sh}_\infty(\mathcal{C})$ is [P] (open program).

Step 2: ℤ_N-action on atoms

On the set of atoms, the cyclic shift is defined:

$$\triangleright: \mathcal{T}_\Omega \to \mathcal{T}_\Omega, \quad \triangleright(S_i) := S_{(i+1) \mod N}$$

Step 3: Extension to Ω

A permutation of the atoms of a finite Boolean algebra induces a unique Boolean automorphism, so ▷ extends canonically to the decidable fragment $\mathrm{Dec}(\Omega) \cong 2^7$ generated by the atoms:

$$\triangleright: \mathrm{Dec}(\Omega) \to \mathrm{Dec}(\Omega), \quad \triangleright\Big(\bigvee_{i \in I} S_i\Big) := \bigvee_{i \in I} S_{(i+1) \mod N}, \qquad I \subseteq \{0, \ldots, N-1\},$$

and further to the 0-truncation $\tau_{\leq 0}(\Omega)$ (Heyting algebra) and to the full ∞-groupoid $\Omega$ as the induced automorphism. ($\Omega$ is a Heyting algebra, not a vector space: no linear combinations $\sum_i \alpha_i S_i$ are formed — an earlier draft wrote the extension in that form.)

Choices involved [D]

Two definitional inputs enter here, and both are named as such: (1) the identification of the atoms with the basis projectors $|i\rangle\langle i|$ (box above); (2) the cyclic order of the atoms used by ▷. On seven labelled atoms there are $6! = 720$ free transitive $\mathbb{Z}_7$-actions ($120$ up to the choice of generator); compatibility with the Fano structure restricts ▷ to the Singer cycles — the elements of order 7 of $\mathrm{Aut}(PG(2,2)) \cong \mathrm{PSL}(2,7)$, which form 8 subgroups of order 7 (this group is the image on the axes of the frame group $\Gamma_{\!\text{oct}} \subset G_2$, not a subgroup of it; the text read "$\subset G_2$" until 2026-09-25). In the cyclic labelling used for the generation structure (Fano lines $\{k, k+1, k+3\}$, fermion generations) the shift $S_i \mapsto S_{i+1}$ is such a cycle. "Unique up to the choice of generator" therefore holds after the cyclic labelling is fixed, not before.

Properties of algebraic ▷:

1. Monotonicity: $p \leq q \Rightarrow \triangleright p \leq \triangleright q$
2. Cyclicity: $\triangleright^N = \text{Id}$ on $\mathrm{Dec}(\Omega)$ (exact equality at the 0-truncated level; on the full ∞-groupoid $\Omega$ it is a natural isomorphism $\triangleright^N \simeq \text{Id}$, Theorem 2.7.1)
3. Compatibility with logic: $\triangleright(p \land q) = \triangleright p \land \triangleright q$

Physical interpretation

For a predicate $\chi: \Gamma \to \Omega$, the value $\triangleright\chi$ means "χ is true at the next moment of time". The definition of time precedes dynamics.

2.2 Generation of discrete time

Theorem (Time from iteration of ▷)

Discrete time $\tau \in \mathbb{Z}_N$ arises as the iterated application of modality ▷:

$$\tau_n := \underbrace{\triangleright \circ \cdots \circ \triangleright}_{n \text{ times}}(now) = \triangleright^n(now)$$

where $now \in \Omega$ is the predicate "now" (current moment).

For N = 7 (UHM):

$$\tau_n = \triangleright^n(now), \quad n \in \{0, 1, 2, 3, 4, 5, 6\}$$

Cyclic structure:

$$\triangleright^7(now) = now \quad (\text{mod } \mathbb{Z}_7)$$

which corresponds to the $S^1$ topology of time for finite-dimensional systems.

2.3 Consistency with Page–Wootters

Theorem (Equivalence of constructions)

Two definitions of discrete time are equivalent:

(a) Page–Wootters (§3):

$$|\tau_n\rangle_O = \frac{1}{\sqrt{7}} \sum_{k=0}^{6} e^{-2\pi i k n / 7} |E_k\rangle_O$$

(b) Temporal modality:

$$\tau_n = \triangleright^n(now)$$

Equivalence is established by the isomorphism:

$$\mathcal{H}_O \cong \Gamma(\Omega, \mathcal{O}_\Omega)$$

(global sections of the structure sheaf on Ω).

Proof.

We construct an explicit $\mathbb{Z}_7$-equivariant isomorphism between:

• Page–Wootters (PW) picture: $\mathcal{H}_O \cong \mathbb{C}^7$ with clock basis $\{|\tau_n\rangle\}_{n=0}^{6}$;
• Modal picture: $\mathbb{Z}_7$-orbit of the predicate $now$ under the temporal modality $\triangleright$.

Step 1 (Unitarity of the shift operator $V_O$).

The clock shift operator is defined on the clock basis:

$$V_O |\tau_n\rangle := |\tau_{n+1 \bmod 7}\rangle, \quad n \in \mathbb{Z}_7.$$

In the energy basis $\{|E_k\rangle\}_{k=0}^6$, the operator $V_O$ is diagonal: $V_O |E_k\rangle = \omega^k |E_k\rangle$, where $\omega = e^{2\pi i/7}$ is a primitive 7th root of unity.

Verification. Apply to $|\tau_n\rangle = \frac{1}{\sqrt{7}}\sum_k e^{-2\pi i k n/7} |E_k\rangle$:

$$V_O |\tau_n\rangle = \frac{1}{\sqrt{7}} \sum_k e^{-2\pi i k n/7} \omega^k |E_k\rangle = \frac{1}{\sqrt{7}} \sum_k e^{-2\pi i k n/7} e^{2\pi i k/7} |E_k\rangle$$ $$= \frac{1}{\sqrt{7}} \sum_k e^{-2\pi i k (n-1)/7} |E_k\rangle = |\tau_{n-1}\rangle.$$

(The sign depends on the phase convention of DFT.) With the convention $|\tau_n\rangle = \frac{1}{\sqrt{7}}\sum_k e^{2\pi i k n/7}|E_k\rangle$ we get $V_O|\tau_n\rangle = |\tau_{n+1}\rangle$.

Unitarity $V_O^\dagger V_O = V_O V_O^\dagger = I_7$ follows from the fact that $V_O$ in the energy basis is a diagonal unitary matrix with $|V_O^{(k,k)}| = |\omega^k| = 1$.

Cyclicity $V_O^7 = I_7$: $V_O^7 |E_k\rangle = \omega^{7k} |E_k\rangle = |E_k\rangle$ (since $\omega^7 = 1$). $\square$

Step 2 ($\mathbb{Z}_7$-representation structure on $\mathcal{H}_O$).

The operator $V_O$ defines a unitary representation of the group $\mathbb{Z}_7$ on $\mathcal{H}_O$:

$$\rho_{PW}: \mathbb{Z}_7 \to U(\mathcal{H}_O), \quad \rho_{PW}(k) := V_O^k.$$

Decomposition into irreducibles. By the Peter-Weyl theorem, $\rho_{PW}$ decomposes into 7 one-dimensional representations: $\mathcal{H}_O = \bigoplus_{k=0}^6 \mathbb{C}|E_k\rangle$, where $V_O$ acts on $|E_k\rangle$ by multiplication by $\omega^k$. This is the regular representation of $\mathbb{Z}_7$. $\square$

Step 3 (Modal representation structure on $\Omega$).

In the $\infty$-topos $\mathbf{Sh}_\infty(\mathcal{C})$, the subobject classifier $\Omega$ has a temporal modality $\triangleright: \Omega \to \Omega$ — an endomorphism satisfying:

(M1) $\triangleright$ is an automorphism of $\Omega$ (invertible);

(M2) $\triangleright^7 = \mathrm{id}_\Omega$ (cyclicity of time $\mathbb{Z}_7$, follows from the clock register of A5 (T-87, steps 1–3) and finite-dimensionality of $\mathcal{D}(\mathbb{C}^7)$);

(M3) For the predicate $now \in \mathrm{Hom}(*, \Omega)$, the orbit $\{\triangleright^n(now)\}_{n=0}^{6}$ contains 7 distinct elements.

Verification of (M3). If $\triangleright^m(now) = now$ for some $0 < m < 7$, then the order of $\triangleright$ would divide $m$. But the order of $\triangleright$ is 7 (prime by (M2)), hence $m$ is a multiple of 7, which is impossible for $0 < m < 7$. Contradiction. $\square$

The orbit $\{\triangleright^n(now)\}_{n=0}^{6}$ is the regular representation of $\mathbb{Z}_7$ in the space of predicates $\mathrm{Hom}(*, \Omega)$, since $\mathbb{Z}_7$ acts transitively and freely.

Step 4 (Construction of the equivariant isomorphism).

Define the linear map:

$$\Psi: \mathcal{H}_O \to \mathrm{span}_\mathbb{C}\{\triangleright^n(now) : n \in \mathbb{Z}_7\}$$

on the clock basis:

$$\Psi(|\tau_n\rangle) := \triangleright^n(now), \quad n \in \mathbb{Z}_7,$$

and extend linearly to $\mathcal{H}_O$.

$\mathbb{Z}_7$-equivariance. For any $k \in \mathbb{Z}_7$:

$$\Psi(V_O^k |\tau_n\rangle) = \Psi(|\tau_{n+k}\rangle) = \triangleright^{n+k}(now) = \triangleright^k(\triangleright^n(now)) = \triangleright^k(\Psi(|\tau_n\rangle)).$$

Hence $\Psi \circ V_O = \triangleright \circ \Psi$. $\square$

Bijectivity. $\Psi$ maps the orthonormal basis $\{|\tau_n\rangle\}_{n=0}^{6}$ to the family $\{\triangleright^n(now)\}_{n=0}^{6}$, which by (M3) contains 7 distinct elements. Since both spaces are 7-dimensional (as complex vector spaces with $\mathbb{Z}_7$-action), $\Psi$ is a bijection. $\square$

Unitarity. We induce an inner product on the right-hand side by requiring $\{\triangleright^n(now)\}_{n=0}^{6}$ to be an orthonormal basis. Then $\Psi$ is a unitary operator (preserves the inner product by construction). $\square$

Step 5 (Correspondence with structure sheaves).

The isomorphism $\Psi$ extends to an isomorphism:

$$\mathcal{H}_O \cong \Gamma(\Omega, \mathcal{O}_\Omega),$$

where $\mathcal{O}_\Omega$ is the structure sheaf on $\Omega$ whose sections are "functions on the time axis" $\mathbb{Z}_7$. The global sections are $\mathbb{C}$-valued functions on $\mathbb{Z}_7$, i.e. $\mathbb{C}^7$ as a $\mathbb{Z}_7$-module.

The isomorphism $\Psi$ is a special case of a general fact: any two free transitive actions of a finite group $G$ on sets of size $|G|$ are isomorphic as $G$-sets, hence their permutation representations are both isomorphic to the regular representation $\mathbb{C}[G]$. For abelian $G$ the regular representation decomposes as the direct sum of all $|G|$ one-dimensional characters, each once (Peter–Weyl for finite groups) — exactly as computed in Step 2. What is identified here are the two regular representations of $\mathbb{Z}_7$, not irreducibles: the irreducible representations of $\mathbb{Z}_7$ are one-dimensional. (An earlier draft stated "every irreducible representation of a finite abelian group is isomorphic to the regular one"; that sentence was false and is retracted.)

Conclusion. The map $\Psi: \mathcal{H}_O \cong \Gamma(\Omega, \mathcal{O}_\Omega)$ is a $\mathbb{Z}_7$-equivariant unitary isomorphism mapping:

• $|\tau_n\rangle_O$ (Page-Wootters) $\leftrightarrow$ $\triangleright^n(now)$ (temporal modality);
• $V_O$ (shift operator) $\leftrightarrow$ $\triangleright$ (modal operator);
• Energy basis $\{|E_k\rangle\}$ $\leftrightarrow$ characters $\{\chi_k: \mathbb{Z}_7 \to \mathbb{C}^*\}$ of the group $\mathbb{Z}_7$.

The two pictures of time are mathematically identical. $\blacksquare$

Status: [T]. The equivalence theorem for Page-Wootters and temporal modality is proven with full rigor.

Results used:

• Peter-Weyl theorem for finite abelian groups (regular representation of $\mathbb{Z}_n$);
• Discrete Fourier transform (standard convention);
• the clock register of A5, $\mathcal{H}_O \cong \mathbb{C}[\mathbb{Z}_7]$ (T-87, steps 1–3; the Page–Wootters constraint of step 4 is not used here).

Consistency check:

• Dependencies: the clock register of T-87, representation theory of $\mathbb{Z}_7$ — standard;
• No circularities: proof uses only the structure of $\mathbb{C}^7$ + unitary $\mathbb{Z}_7$-action;
• Consistent with the case $M=1$ of composite clocks (§3.8), where $\mathbb{Z}_7$-cyclicity is immediate.

2.4 Connection to L-unification

Central theorem: Dynamics as predicate evolution

The evolution of system Γ(τ) is equivalent to the evolution of logical predicates χ ∈ L under the action of ▷.

Definition (Dual Liouvillian):

For a predicate $\chi \in L = \Omega \cap \Gamma$, its evolution is defined by the dual logical Liouvillian:

$$\frac{d\chi}{d\tau} = \mathcal{L}_\Omega^*[\chi]$$

where $\mathcal{L}_\Omega^*$ is the adjoint operator to the logical Liouvillian:

$$\langle \mathcal{L}_\Omega^*[\chi], \Gamma \rangle = \langle \chi, \mathcal{L}_\Omega[\Gamma] \rangle$$

Explicit form of the dual Liouvillian:

$$\mathcal{L}_\Omega^*[\chi] = i[H_{eff}, \chi] + \sum_k \gamma_k \left( L_k^\dagger \chi L_k - \frac{1}{2}\{L_k^\dagger L_k, \chi\} \right)$$

Interpretation:

Picture	Evolution	QM analogue
Schrödinger	$\frac{d\Gamma}{d\tau} = \mathcal{L}_\Omega[\Gamma]$	States evolve
Heisenberg	$\frac{d\chi}{d\tau} = \mathcal{L}_\Omega^*[\chi]$	Predicates evolve

2.5 Temporal modal operators

In the ∞-topos $\mathrm{Sh}_\infty(\mathcal{C})$, standard temporal operators are defined:

Definition (Temporal logic):

$$\Diamond \phi := \exists \tau' > \tau_{now}. \phi(\tau') \quad \text{(sometime in the future)}$$ $$\Box \phi := \forall \tau' > \tau_{now}. \phi(\tau') \quad \text{(always in the future)}$$

Connection to ▷:

$$\Diamond \phi = \bigvee_{n=0}^{N-1} \triangleright^n(\phi)$$ $$\Box \phi = \bigwedge_{n=0}^{N-1} \triangleright^n(\phi)$$

2.6 Diagram: unification via Ω

Related sections

• Internal logic of Ω — definition of the classifier and L-unification
• Logical Liouvillian — direct picture of evolution
• Dimension L — logical dimension of the Holon

2.7 Time as modality in HoTT

Internal language of the ∞-topos

HoTT (Homotopy Type Theory) is the internal language of ∞-toposes. In this language, time is defined as a modality on types, not as an external parameter.

Definition (Temporal modality in HoTT):

In homotopy type theory, the temporal modality is an operation on types:

$$\triangleright: \mathcal{U} \to \mathcal{U}$$

where $\mathcal{U}$ is the universe of types.

Key advantage of the HoTT formulation:

Aspect	Traditional approach	HoTT approach
Time	External parameter t ∈ ℝ	Modality ▷ on types
Moment	Value t₀	Application of ▷^n to a type
Evolution	dΓ/dt = ...	Morphism Γ → ▷(Γ)
Dependency	Dynamics defines time	Time defines dynamics

Theorem 2.7.1 (Time from modal structure):

Let $\mathfrak{T} = (\mathbf{Sh}_\infty(\mathcal{C}), J_{Bures}, \omega_0)$ be the unique primitive of UHM. Then:

1. Temporal modality ▷: Ob(Sh_∞) → Ob(Sh_∞) — endofunctor
2. Cyclicity: $\triangleright^N \simeq \text{Id}$ (natural isomorphism)
3. Minimality: $\triangleright^k \not\simeq \text{Id}$ for 0 < k < N

Corollaries:

• $\tau \in \mathbb{Z}_N$ arises as the set of isomorphism classes of $\triangleright^k$
• Dynamics is defined by morphisms $\Gamma \to \triangleright(\Gamma)$
• Page–Wootters is formally Axiom 5; its clock register is constructed from T-53 (T-87, steps 1–3), while its constraint is an assumption (T-87, step 4, [C]; see §3.1a)

Proof:

(a) The orbit of the ▷-action on Ω defines N points: $\{\Omega, \triangleright(\Omega), \ldots, \triangleright^{N-1}(\Omega)\}$

(b) The quotient $\Omega / \triangleright$ is isomorphic to a point (contractibility of the ∞-topos)

(c) The clock space $\mathcal{H}_O := \text{span}\{|\tau_k\rangle : k \in \mathbb{Z}_N\}$ is derived as the basis of eigenstates of the time generator $T$, where $\triangleright = e^{2\pi i T / N}$

(d) The tensor decomposition $\mathcal{H} = \mathcal{H}_O \otimes \mathcal{H}_{rest}$ is induced by the factorization $\Omega = \Omega_O \times \Omega_{rest}$

∎

Connection to HoTT

Temporal modalities in homotopy type theory are a standard tool for formalizing time in the internal language of ∞-toposes.

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3. Page–Wootters mechanism for UHM

Status: half constructed, half assumed

The Page–Wootters mechanism is formally Axiom 5. Its tensor structure $\mathcal{H}_O \otimes \mathcal{H}_{rest}$ is built from the finite spectral triple of T-53: the Wedderburn decomposition of the algebra $A_{\text{int}}$ isolates the clock summand (the KO-dimension-6 claim of T-53 is retracted and not needed), and the clock register is the regular representation $\mathbb{C}[\mathbb{Z}_7]$ of the shift ▷ (T-87, steps 1–3). Its constraint $\hat{C}\,\Gamma_{total} = 0$ is not derived: stationarity of a mixed global state gives only $[\hat{C}, \Gamma_{total}] = 0$, and the constraint is the further assumption $\mathrm{supp}\,\Gamma_{total} \subseteq \ker \hat{C}$ (T-87, step 4, [C]; §3.1a). An earlier version of this box called A5 derivable from A1–A4; that claim is retracted, because its constraint half is assumed.

See honest axiomatics and derivation of A5 from spectral triple.

3.1 The idea of the mechanism (standard formulation)

In quantum gravity, the following construction is used:

Full system: $\mathcal{H}_{total} = \mathcal{H}_C \otimes \mathcal{H}_S$

• $\mathcal{H}_C$ — clock subsystem
• $\mathcal{H}_S$ — the rest of the system

Wheeler–DeWitt condition: $\hat{H}_{total} |\Psi\rangle = 0$

Time arises as correlation between the clock and the system.

3.1a Page–Wootters: constructed clock, assumed constraint

The clock register is constructed from T-53; the constraint is assumed

The tensor decomposition $\mathcal{H} = \mathcal{H}_O \otimes \mathcal{H}_{rest}$ is formally Axiom 5 in honest axiomatics. Its clock factor is constructed from spectral triple T-53 (spacetime): the Wedderburn decomposition of the algebra $A_{\text{int}} = \mathbb{C} \oplus M_3(\mathbb{C}) \oplus M_3(\mathbb{C})$ isolates the clock summand (the KO-dimension-6 claim of T-53 is retracted and not needed), and the register $\mathcal{H}_O \cong \mathbb{C}[\mathbb{Z}_7]$ is the regular representation of ▷ (T-87, steps 1–3).

The constraint is a separate assumption. Stationarity of a mixed global state means $[\hat{C}, \Gamma_{total}] = 0$, that is, $\Gamma_{total}$ is block-diagonal over the eigenspaces of $\hat{C}$. The constraint $\hat{C}\,\Gamma_{total} = 0$ says more: all the weight of $\Gamma_{total}$ lies in $\ker \hat{C}$. A state spread over two eigenvalues $c_a \neq c_b$ of $\hat{C}$, $\Gamma_{total} = \tfrac12(|a\rangle\langle a| + |b\rangle\langle b|)$, commutes with $\hat{C}$, yet $(\hat{C} - c)\,\Gamma_{total} \neq 0$ for every shift $c$. For a pure state the two conditions agree once the energy is shifted to zero, which is the case Page and Wootters treat (§11.1). Hence A5 follows from A1–A4 only together with the assumption $\mathrm{supp}\,\Gamma_{total} \subseteq \ker \hat{C}$, which is the constraint itself: step 4 of T-87 is [C]. The earlier wording of this box — "the constraint $\hat{C}\Gamma = 0$ follows from stationarity. Thus A5 is a consequence of A1–A4" — is retracted. Details: derivation of A5 from spectral triple.

Axiom 5 (Page–Wootters):

Let ▷: $\mathrm{Sh}_\infty(\mathcal{C})$ → $\mathrm{Sh}_\infty(\mathcal{C})$ be the temporal modality. It is postulated:

1. Clock space: $\mathcal{H}_O := \text{span}\{|\tau_k\rangle : \triangleright^k(|0\rangle) = \zeta^k |\tau_k\rangle\}$

2. Remainder: $\mathcal{H}_{rest} := \mathcal{H} / \mathcal{H}_O$

3. Tensor structure: $\mathcal{H} \cong \mathcal{H}_O \otimes \mathcal{H}_{rest}$ (postulated isomorphism)

4. Constraint: $\hat{C} = H_O \otimes \mathbb{1} + \mathbb{1} \otimes H_{rest} + H_{int}$, where $H_O = \omega_0 \cdot T$ (generator of ▷)

5. Conditional states: $\Gamma(\tau) = \text{Tr}_O[(|\tau\rangle\langle\tau| \otimes \mathbb{1}) \cdot \Gamma_{total}] / p(\tau)$

Theorem (Consistency of Page–Wootters with ▷):

If Axiom 5 holds, then the conditional states evolve according to: $\Gamma(\tau_{n+1}) = \triangleright^*(\Gamma(\tau_n)) + O(H_{int})$

This is consistency, not a derivation.

Proof:

(a) Operator $T := (1/2\pi i) \log(\triangleright)$ is defined on Spec(Ω) and has eigenvalues $\{0, 1, \ldots, N-1\}$

(b) The eigensubspaces of T form a direct sum: $\mathcal{H} = \bigoplus_k \mathcal{H}_k$

(c) Dimension O is defined as $\dim(\mathcal{H}_O) = N$ (orbit of ▷-action). By construction, $\mathcal{H}_O$ is the clock space

(d) Invariance under a global time shift gives the commutator condition $[T \otimes \mathbb{1} + \mathbb{1} \otimes T', \Gamma_{total}] = 0;$ the constraint $\hat{C} \cdot \Gamma = 0$ follows only if, in addition, $\Gamma_{total}$ is supported in the kernel of the generator (box above). An earlier wording derived the constraint from the invariance alone; that step is retracted.

(e) The conditional state formula is the standard consequence of the tensor structure

∎

3.2 Adaptation for UHM

In the 7D structure of UHM, the natural candidate for the role of a clock is dimension O (Foundation).

Justification:

• O — connection to the quantum vacuum
• O participates in regeneration: $\kappa_0 = \|\mathrm{Nat}(\mathcal{D}_\Omega, \mathcal{R})\|$ (see categorical derivation of κ₀)
• Physically: O is the "source" feeding the dynamics

3.3 Formal construction

Step 1: Decomposition of Γ

$$\Gamma_{total} \in \mathcal{L}(\mathcal{H}_O \otimes \mathcal{H}_{6D})$$

where $\mathcal{H}_{6D} = \text{span}\{|A\rangle, |S\rangle, |D\rangle, |L\rangle, |E\rangle, |U\rangle\}$.

Step 2: Page–Wootters constraint

$$\hat{C} \cdot \Gamma_{total} = 0$$

where the constraint operator:

$$\hat{C} = H_O \otimes \mathbb{1}_{6D} + \mathbb{1}_O \otimes H_{6D} + H_{int}$$

Step 3: Conditional state

Definition 3.1 (Internal time)

Internal time $\tau$ is defined via conditional states:

$$\Gamma(\tau) := \frac{\text{Tr}_O\left[ (|\tau\rangle\langle \tau|_O \otimes \mathbb{1}_{6D}) \cdot \Gamma_{total} \right]}{p(\tau)}$$

where:

• $|\tau\rangle_O$ — basis of eigenstates of clock O
• $p(\tau) = \text{Tr}\left[ (|\tau\rangle\langle \tau|_O \otimes \mathbb{1}_{6D}) \cdot \Gamma_{total} \right]$ — normalization

3.4 Page–Wootters theorem

Theorem 3.1 (Emergent dynamics)

Let $\Gamma_{total}$ satisfy the constraint $\hat{C} \cdot \Gamma_{total} = 0$. Then the conditional states $\Gamma(\tau)$ evolve according to:

$$\frac{d\Gamma(\tau)}{d\tau} = -i[H_{eff}, \Gamma(\tau)] + \text{corrections}$$

where $H_{eff}$ is the effective Hamiltonian arising from $H_{int}$.

Corollary: Time $\tau$ is not an external parameter, but a parametrization of correlations within the global state $\Gamma_{total}$.

The "corrections" are not small in general. For $H_{int} = 0$ the conditional states are related by a unitary step, $\Gamma(\tau_{n+1}) = e^{-iH_{6D}\delta\tau}\,\Gamma(\tau_n)\,e^{iH_{6D}\delta\tau}$. When the constraint couples clock and system, A. R. H. Smith and M. Ahmadi proved for clocks with continuous spectrum that the conditional state obeys a time-nonlocal Schrödinger equation, in which the system Hamiltonian is replaced by an integral operator ("Quantizing time: interacting clocks and systems", Quantum 3, 160 (2019), arXiv:1712.00081); the local generator $H_{eff}(\tau) = H_{6D} + \langle\tau|H_{int}|\tau\rangle_O$ of §3.6 is at best the leading term in $H_{int}$, and the corpus has not shown that the seven-level clock escapes the nonlocal form.

Retracted: "exact time" (T-186(b))

An earlier version of this box said that the Cohesive Closure Theorem eliminates the $O(H_{\text{int}})$ correction, the conditional states being "exact sections of the flat projection $\flat(\Gamma_{\text{total}})$" evolved by the counit of $\Pi \dashv \flat$. That claim is retracted: with interaction the exact conditional dynamics is time-nonlocal (paragraph above), relative to a clock of period seven ticks it is periodic in $\tau$ (§11.2), and a counit of an adjunction carries no information about $H_{int}$ that could remove either. T-186(b) is withdrawn [✗].

3.5 Clock basis for 7D

For $\dim(\mathcal{H}_O) = 7$:

$$|\tau_n\rangle = \frac{1}{\sqrt{7}} \sum_{k=0}^6 e^{-2\pi i k n / 7} |E_k\rangle, \quad n = 0, 1, \ldots, 6$$

where $|E_k\rangle_O$ are eigenstates of $H_O$.

3.6 Explicit constructions for UHM

Complete formulas for the 7D UHM system are defined in the respective master documents:

Construction	Formula	Master definition
Clock Hamiltonian	$H_O = \omega_0 \sum_{k=0}^{6} k \vert k\rangle\langle k\vert_O$	dimension-o#гамильтониан-часов-h_o
Shift operator	$V_O = \sum_{k=0}^{5} \vert k+1\rangle\langle k\vert + \vert 0\rangle\langle 6\vert$	dimension-o#оператор-сдвига-v_o
C*-algebra of clocks	$\mathcal{A}_O = C^*(H_O, V_O) \cong M_7(\mathbb{C})$	dimension-o#c-алгебра-часов-a_o
Interaction Hamiltonian	$H_{int} = \lambda_E(a_O^\dagger \otimes \vert E\rangle\langle E\vert + h.c.) + \ldots$	axiom-omega#гамильтониан-взаимодействия
Full constraint	$\hat{C} = H_O \otimes \mathbb{1}_{6D} + \mathbb{1}_O \otimes H_{6D} + H_{int}$	axiom-omega#свойство-2
Effective Hamiltonian	$H_{eff}(\tau) = H_{6D} + \langle\tau\vert H_{int}\vert\tau\rangle_O$	evolution#вывод-h_eff

The last row is exact only for $H_{int} = 0$; with interaction it is the leading term of a time-nonlocal law (§3.4).

3.7 Discreteness of time for finite systems

Fundamental discreteness

For $N = 7$ time is fundamentally discrete, not continuous.

Practical significance

Question: If τ ∈ ℤ₇ is discrete, why does the evolution equation use dΓ/dτ (a derivative)?

Answer:

1. Minimal formalism (N=7): τ is discrete, equations are difference equations (Δτ instead of dτ)
2. Macroscopic limit (N → ∞): τ approaches a continuum, equations are differential
3. Practice: The differential form is a convenient approximation when Δτ ≪ the characteristic timescales of the system

For implementations: Use the discrete form: Γ(τ+1) = Γ(τ) + Δτ·(...) with step Δτ = 2π/(7ω₀).

Common misconception: "7 ticks of the universe"

$\dim(\mathcal{H}_O) = 7$ is the dimensionality of the clock Hilbert space, not the cardinality of the set of moments. The distinction:

• Clock basis: 7 orthogonal states $|\tau_n\rangle_O$ — basis of $\mathcal{H}_O$, analogous to 7 divisions on a clock face
• Moments of time: $\tau \in \mathbb{Z}_7$ — a cyclic group. The system passes through cycles $\tau_0 \to \tau_1 \to \cdots \to \tau_6 \to \tau_0 \to \cdots$ indefinitely, like clock hands with 7 divisions
• Chronon: $\delta\tau = 2\pi/(7\omega_0)$ — the minimal quantum of subjective time, determined by the characteristic frequency $\omega_0$ of the system, not by the number 7

For composite systems the clock Hilbert space grows as $7^M$, but the summed clock of $M$ holons has only $6M+1$ distinguishable readings and keeps the period $2\pi/\omega_0$ (composite clocks below). An earlier sentence here, "$N_{\text{eff}} = \dim(\mathcal{H}_O^{\text{composite}}) \gg 7$ gives quasi-continuity of macroscopic time", is retracted: the number of readings is not the dimension of the space.

Theorem (Discreteness of time): For a finite-dimensional system with $\dim(\mathcal{H}_O) = N$, the internal time takes values from the cyclic group:

$$\tau \in \mathbb{Z}_N = \{0, 1, 2, \ldots, N-1\}$$

For UHM with $N = 7$:

$$\tau \in \mathbb{Z}_7 = \{0, 1, 2, 3, 4, 5, 6\}$$

Corollaries:

Property	Discrete time ($N = 7$)	Continuous limit ($N \to \infty$)
Set of times	$\mathbb{Z}_7$ (7 moments)	$S^1$ or $\mathbb{R}$
Topology	Discrete, cyclic	Continual
Chronon (minimal quantum)	$\delta\tau = 2\pi/(7\omega_0)$	$\delta\tau \to 0$
Fundamental group	$\pi_1 \cong \mathbb{Z}_7$	$\pi_1 \cong \mathbb{Z}$
Evolution equation	Difference	Differential

Interpretation:

1. Quantization of the present: There exists a minimal "quantum" of subjective time — chronon
2. Cyclic time: Time locally has the structure of $\mathbb{Z}_7$, not $\mathbb{R}$
3. Emergent continuity: Continual time is the macroscopic approximation for $N \gg 1$

3.8 Limit N → ∞ and connection to physics

Clarification: Algebraic, not topological limit

As $N \to \infty$, the discrete time $\tau \in \mathbb{Z}_N$ transitions to continuous time algebraically, not topologically.

Topological error: $\lim_{N \to \infty} \mathbb{Z}_N \neq U(1)$ topologically!

• Projective limit $\hat{\mathbb{Z}} = \varprojlim_N \mathbb{Z}_N$ — totally disconnected space
• $U(1) \cong S^1$ — connected space
• They are topologically distinct

Correct formulation of the limit:

Definition (Scaled limit): $t := \lim_{N \to \infty} \tau_n \cdot \delta\tau(N) = \lim_{N \to \infty} \tau_n \cdot \frac{2\pi}{N \cdot \omega_0}$

This is a scaled limit, not a topological one.

Theorem on algebraic limit

Theorem (Algebraic limit ℂ[ℤ_N] → C(S¹))

As $N \to \infty$, the group algebra $\mathbb{C}[\mathbb{Z}_N]$ converges to the algebra of continuous functions on the circle:

$$\lim_{N \to \infty} \mathbb{C}[\mathbb{Z}_N] \cong C(S^1)$$

as C*-algebras (algebraically, not topologically).

Proof:

(a) Structure of the group algebra:

$$\mathbb{C}[\mathbb{Z}_N] = \text{span}\{e_k : k = 0, 1, \ldots, N-1\}, \quad e_k \cdot e_l = e_{(k+l) \mod N}$$

(b) Fourier transform:

Isomorphism $\mathcal{F}: \mathbb{C}[\mathbb{Z}_N] \to \mathbb{C}^N$:

$$\mathcal{F}(e_k) = \left(\zeta^{0 \cdot k}, \zeta^{1 \cdot k}, \ldots, \zeta^{(N-1) \cdot k}\right), \quad \zeta = e^{2\pi i/N}$$

(c) Limiting transition:

As $N \to \infty$, the spectrum $\text{Spec}(\mathbb{C}[\mathbb{Z}_N]) = \mathbb{Z}_N$ becomes dense in $S^1$:

$$\left\{e^{2\pi i k/N} : k = 0, \ldots, N-1\right\} \xrightarrow{N \to \infty} S^1$$

(d) C-isomorphism:*

By the Gelfand–Naimark theorem:

$$\mathbb{C}[\mathbb{Z}_N] \cong C(\text{Spec}(\mathbb{C}[\mathbb{Z}_N])) \xrightarrow{N \to \infty} C(S^1)$$

∎

Chronon as a function of N:

$$\delta\tau(N) = \frac{2\pi}{N \cdot \omega_0}$$

N	$\delta\tau$	Interpretation
7	$\approx 0.9/\omega_0$	UHM chronon (minimal quantum of subjective time)
100	$\approx 0.063/\omega_0$	Mesoscopic limit
$\infty$	0	Classical limit (continuous time)

Correspondence theorem (classical limit)

Theorem (Classical limit of averages)

For any observable $A$:

$$\lim_{N \to \infty} \langle A(\tau_n) \rangle_N = \langle A(t) \rangle_{\text{classical}}$$

where $t = \tau_n \cdot \delta\tau(N)$.

Proof:

Average over discrete time:

$$\langle A(\tau_n) \rangle_N = \mathrm{Tr}\left[A \cdot \Gamma(\tau_n)\right]$$

As $N \to \infty$ with $\tau_n / N \to t/T$ (where $T = 2\pi/\omega_0$):

$$\lim_{N \to \infty} \langle A(\tau_n) \rangle_N = \mathrm{Tr}\left[A \cdot \Gamma(t)\right] = \langle A(t) \rangle_{\text{classical}}$$

∎

Corollary for UHM:

Classical continuous time on a circle is the macroscopic approximation of discrete internal time: the readings become dense in $S^1$, whose circumference $2\pi/\omega_0$ does not change. A line $\mathbb{R}$ is not obtained this way (see the end of this subsection).

Theorem (Continuous limit — algebraic):

In the limit $N \to \infty$ at fixed $\omega_0$:

1. $\delta\tau = 2\pi/(N\omega_0) \to 0$ (chronon vanishes)
2. $\mathbb{Z}_N \cdot \delta\tau$ fills the circle of circumference $2\pi/\omega_0$; the period does not grow
3. Algebraic convergence: $\mathbb{C}[\mathbb{Z}_N] \to C(S^1)$ (group algebras, not groups!)

(An earlier version held the product $N\omega_0$ fixed while also claiming $\delta\tau \to 0$; with $N\omega_0$ fixed the chronon $\delta\tau = 2\pi/(N\omega_0)$ stays fixed and only the period $2\pi/\omega_0$ grows. That combination is retracted.)

Key clarification: The transition is algebraic (group algebras $\mathbb{C}[\mathbb{Z}_N] \to C(S^1)$), not topological ($\mathbb{Z}_N \not\to U(1)$).

Theorem on composite clocks and continuous limit

warning

Retracted: $N_{\text{eff}} = 7^M$ readings for $M$ holons

An earlier version stated as a theorem [T] that a system of $M$ holons has an effective clock with $N_{\text{eff}} = 7^M$ readings and chronon $\delta\tau_{\text{eff}} = 2\pi/(N_{\text{eff}}\,\omega_{\text{eff}})$. This is false for the generator its own proof uses. With $T_{comp} = \sum_{m=1}^{M} \mathbb{1}^{\otimes(m-1)} \otimes T^{(m)} \otimes \mathbb{1}^{\otimes(M-m)}$ and each $T^{(m)}$ of spectrum $\{0, 1, \ldots, 6\}$, the spectrum of $T_{comp}$ is the set of integers $\{0, 1, \ldots, 6M\}$: $6M+1$ values with large multiplicities. Hence $e^{-2\pi i\,T_{comp}} = \mathbb{1}$, and the composite clock has the same period $2\pi/\omega_0$ as one holon. The orbit $e^{-i\omega_0 t\,T_{comp}}|\psi\rangle$ of any state lies in a subspace of dimension at most $6M+1$ (one direction per distinct eigenvalue), so it contains at most $6M+1$ mutually orthogonal, that is perfectly distinguishable, readings. The dimension $7^M$ of $\bigotimes_m \mathcal{H}_O^{(m)}$ is not the number of readings; $7^M$ readings would require clock frequencies in the ratio $1 : 7 : 7^2 : \cdots$ (a positional clock), which a composite of identical holons does not have. For $M = 1, 2, 3, 4$ the summed clock has $7, 13, 19, 25$ distinct eigenvalues, against $7, 49, 343, 2401$ claimed (regression check in website/scripts/check_core_numbers.py).

What holds instead (elementary; checked numerically for $M \leq 4$). For $M$ holons with identical clocks $H_O^{(m)} = \omega_0 T^{(m)}$ the summed clock has $6M+1$ distinguishable readings, period $2\pi/\omega_0$ and finest orthogonal resolution

$$\delta\tau_M = \frac{2\pi}{(6M+1)\,\omega_0},$$

which shrinks like $1/M$, not like $7^{-M}$. As $M \to \infty$ the readings become dense in a circle of fixed circumference — the algebraic limit $\mathbb{C}[\mathbb{Z}_N] \to C(S^1)$ above — but the circle does not unroll into a line: composite O-clocks do not supply an aperiodic time (§11.2).

Where $7^M$ readings do exist. The same $M$ O-registers carry $7^M$ perfectly distinguishable, linearly ordered readings when they are used as the digits of one number, $n = \sum_{m=1}^{M} \tau_m 7^{m-1}$, and the step $n-1 \to n$ is the odometer carry rather than the flow of a summed generator; the constraint is then of Feynman–Kitaev type, not $H_O \otimes 1 + 1 \otimes H_{6D}$. This positional depth register (§11.4) has no period: its readings form the chain $0 < 1 < \cdots < 7^M - 1$. The retraction above stands for the summed clock; the $7^M$ readings belong to the positional register only.

Theorem (Convergence of discrete dynamics to continuous) [T]

Let $\mathcal{L}_\Omega$ be the logical Liouvillian with $\|\mathcal{L}_\Omega\| \leq \Lambda$. Then the discrete evolution $T_{\delta\tau} = e^{\delta\tau \cdot \mathcal{L}_\Omega}$ converges to the continuous Lindblad equation:

$$\left\| \frac{\Gamma(\tau + \delta\tau) - \Gamma(\tau)}{\delta\tau} - \mathcal{L}_\Omega[\Gamma(\tau)] \right\| \leq \frac{\Lambda^2 \cdot \delta\tau}{2}$$

For $M$ holons with the resolution $\delta\tau_M = 2\pi/((6M+1)\omega_0)$ of the summed clock the per-step error is of order $(6M+1)^{-2}$: small polynomially, not exponentially. (An earlier version used $\delta\tau_{\text{eff}} \sim 7^{-M}/\omega_0$ and an error $\sim 7^{-2M}$; both are retracted with the composite-clock statement above.)

Proof: Standard estimate via Taylor formula for the exponential: $e^{h\mathcal{L}} = \mathbb{1} + h\mathcal{L} + O(h^2 \|\mathcal{L}\|^2)$.

Substituting $h = \delta\tau_M = 2\pi/((6M+1) \omega_0)$:

$$\left\| T_{\delta\tau}[\Gamma] - \Gamma - \delta\tau \cdot \mathcal{L}_\Omega[\Gamma] \right\| \leq \frac{(2\pi)^2 \Lambda^2}{2 \cdot (6M+1)^{2} \cdot \omega_0^2}$$

As $M \to \infty$ this tends to zero like $M^{-2}$. $\quad\blacksquare$

Physical interpretation:

System	M	Readings $6M+1$	$\delta\tau_M$	Continuity
Single holon	1	7	$\approx 0.9/\omega_0$	Discrete
Neuron ($\sim 10^4$ molecules)	$\sim 10^4$	$\approx 6 \cdot 10^4$	$\approx 1.0 \cdot 10^{-4}/\omega_0$	Quasi-continuous, on a circle
Macroscopic system	$\gg 1$	$6M+1$	$\to 0$	Continuous circle $S^1$ of circumference $2\pi/\omega_0$, not $\mathbb{R}$

(The earlier table gave $N_{\text{eff}} = 7^{10^4}$ and $\delta\tau \sim 10^{-8450}/\omega_0$ for a neuron and "Continuous ($\mathbb{R}$)" for a macroscopic system; these entries are retracted.)

Connection to the chronon:

Scale	Chronon	Time
Subjective (N = 7)	$\delta\tau \sim 1/\omega_0$	Discrete, $\mathbb{Z}_7$
Neural (N ~ 10⁸)	$\delta\tau \sim 10^{-8}/\omega_0$	Quasi-continuous
Physical (N → ∞)	$\delta\tau \to 0$	Continuous circle for the summed O-clock; $\mathbb{R}$ from the depth register (§11.4)

Corollary for interpretation:

Physical (Newtonian) time $t \in \mathbb{R}$ is not the limit of the O-clock readings as $N \to \infty$: at fixed $\omega_0$ that limit is a circle of circumference $2\pi/\omega_0$. The line $\mathbb{R}$ is the scaling limit of the readings of the depth register (§11.4, Theorem 11.5), which is where the dissipative dynamics runs (§9.1). (An earlier sentence here called $t \in \mathbb{R}$ the limit of the O-clock's internal time; it is retracted.) For the Holon with N = 7 time is fundamentally discrete, which is consistent with:

• Discreteness of states of consciousness
• Finite information capacity
• Topology of ∞-groupoid $\mathbf{Exp}_\infty$

Connection to categorical structure

Discreteness of time leads to a discrete ∞-groupoid $\mathbf{Exp}^{disc}_\infty$ instead of a continuous one. See Categorical formalism.

---

4. Information-geometric time

4.1 Bures metric

The space of density matrices $\mathcal{D}(\mathcal{H})$ has a natural Riemannian structure.

Definition 4.1 (Bures metric)

$$ds_B^2(\Gamma, \Gamma + d\Gamma) = \frac{1}{2} \text{Tr}\left[ d\Gamma \cdot L_\Gamma(d\Gamma) \right]$$

where $L_\Gamma$ is the solution of the Lyapunov equation:

$$\Gamma \cdot L_\Gamma(X) + L_\Gamma(X) \cdot \Gamma = X$$

Explicit formula for the distance (Bures angle):

$$d_B(\Gamma_1, \Gamma_2) = \arccos\left( \sqrt{\mathrm{Fid}(\Gamma_1, \Gamma_2)} \right)$$

where $\mathrm{Fid}(\Gamma_1, \Gamma_2) = \left(\mathrm{Tr}\sqrt{\sqrt{\Gamma_1} \Gamma_2 \sqrt{\Gamma_1}}\right)^2$ — fidelity.

4.2 Geometric time

Definition 4.2 (Information time)

Between two configurations $\Gamma_1$ and $\Gamma_2$, the information time:

$$\tau(\Gamma_1, \Gamma_2) := \inf_{\gamma} \int_0^1 \sqrt{g_{\mu\nu}^B \dot{\gamma}^\mu \dot{\gamma}^\nu} \, ds$$

where the infimum is taken over all paths $\gamma: [0,1] \to \mathcal{D}(\mathcal{H})$ connecting $\Gamma_1$ and $\Gamma_2$.

4.3 Flow of time

Theorem 4.1 (Speed of time flow)

Let $\{\Gamma(\sigma)\}_{\sigma \in [0,1]}$ be a continuous family of states. The speed of flow of internal time:

$$\frac{dt_{int}}{d\sigma} = \left\| \frac{d\Gamma}{d\sigma} \right\|_B$$

Interpretation: "The flow of time" is the rate of change of Γ in the Bures metric. Time "flows faster" when Γ changes more.

4.4 Correspondence with dynamics

Theorem 4.2 (Connection to Hamiltonian)

For unitary evolution $\Gamma(t) = U(t) \Gamma_0 U^\dagger(t)$ with $U(t) = e^{-iHt}$:

$$\frac{dt_{int}}{dt} = \sqrt{\text{Tr}([H, \Gamma] \cdot L_\Gamma([H, \Gamma]))}$$

For $\Gamma$ close to a pure state $|\psi\rangle\langle\psi|$:

$$\frac{dt_{int}}{dt} \approx 2 \Delta H, \quad \Delta H = \sqrt{\langle H^2 \rangle - \langle H \rangle^2}$$

Corollary: The time-energy uncertainty relation:

$$\Delta t_{int} \cdot \Delta H \geq \frac{1}{2}$$

is derived from the geometry of the state space, not postulated.

---

5. Categorical time via ∞-groupoid

5.1 ∞-groupoid of experiential paths

Definition 5.1 (∞-category Exp_∞)

∞-category $\mathbf{Exp}_\infty$ is defined as:

0-cells (objects):

$$\text{Ob}(\mathbf{Exp}_\infty) = \mathcal{E} = \Delta^{N-1} \times_{\text{Spec}} \mathbb{P}(\mathcal{H}_E)^N \times \mathcal{C}$$

(History Hist is not included — it is derived as the structure of the ∞-groupoid)

1-morphisms:

$$\text{Mor}_1(\mathcal{Q}_1, \mathcal{Q}_2) = \{\gamma: [0,1] \to \mathcal{E} \mid \gamma(0) = \mathcal{Q}_1, \gamma(1) = \mathcal{Q}_2\}$$

2-morphisms:

$$\text{Mor}_2(\gamma_1, \gamma_2) = \text{homotopies between } \gamma_1 \text{ and } \gamma_2$$

n-morphisms:

$$\text{Mor}_n = n\text{-parameter families of paths}$$

5.2 Time as a 1-morphism

Definition 5.2 (Categorical time)

Time is a 1-morphism in $\mathbf{Exp}_\infty$:

$$\tau: \mathcal{Q}_1 \to \mathcal{Q}_2$$

Direction of time — choice of orientation on 1-morphisms.

Equivalent moments of time — 2-isomorphic 1-morphisms.

5.3 Theorem on internal time

Theorem 5.1 (Time as a path)

In the ∞-groupoid $\mathbf{Exp}_\infty$:

1. History — automatically arises as the loop space:

   $$\text{Hist}(\mathcal{Q}) := \Omega_\mathcal{Q}(\mathbf{Exp}_\infty) = \{\gamma: S^1 \to \mathcal{E} \mid \gamma(0) = \gamma(1) = \mathcal{Q}\}$$

2. Temporal structure — homotopy type:

   $$\pi_1(\mathbf{Exp}_\infty, \mathcal{Q}) = \text{"cyclic time" at point } \mathcal{Q}$$

3. Arrow of time — orientation σ on 1-morphisms.

5.4 ∞-topos of sheaves

Definition 5.3 (∞-topos Sh_∞(Exp))

∞-topos $\mathbf{Sh}_\infty(\mathbf{Exp})$ — category of ∞-sheaves on $\mathbf{Exp}_\infty$:

1. ∞-topology: Cover = family of paths covering a neighborhood
2. ∞-sheaf: Functor $F: \mathbf{Exp}_\infty^{op} \to \mathbf{Spaces}$, satisfying the descent condition

Theorem 5.2 (Existence of ∞-topos)

$\mathbf{Sh}_\infty(\mathbf{Exp})$ is an ∞-topos and has:

1. Internal logic: Homotopy type theory (HoTT)
2. Internal time: Modality of type "in the future", "in the past"
3. Subobject classifier: ∞-groupoid of truth values

Corollary: The logic of experiential content is temporal modal logic, derivable from the internal structure of the ∞-topos.

---

6. Equivalence theorem

6.1 Three aspects of emergent time

Aspect	Mechanism	Time as...
Relational	Page–Wootters	Correlation between O and the remaining dimensions
Geometric	Bures metric	Distance in state space
Categorical	∞-groupoid	1-morphism in $\mathbf{Exp}_\infty$

6.2 Main theorem

Theorem 6.1 (Emergence of time in UHM)

Let $\Gamma_{total}$ be the global coherence matrix satisfying:

1. Axiom Ω⁷ (∞-topos as primitive)
2. Axiom (AP+PH+QG+V) (autopoiesis, phenomenology, quantum foundation, viability)
3. Constraint $\hat{C} \cdot \Gamma_{total} = 0$ (Page–Wootters)

Then:

(a) Kinematic time:

$$\tau := \text{parameter of conditional states } \Gamma(\tau) = \text{Tr}_O[|\tau\rangle\langle\tau| \cdot \Gamma_{total}] / p(\tau)$$

is equivalent to

(b) Geometric time:

$$t_{int} := \int d_B(\Gamma(\sigma), \Gamma(\sigma + d\sigma))$$

in the limit of small intervals.

(c) Categorical time:

$$\tau \in \text{Mor}_1(\mathcal{Q}_1, \mathcal{Q}_2) \subset \mathbf{Exp}_\infty$$

with natural orientation σ.

Proof.

Step 1 (PW ↔ Bures): PW clock parameter and Bures metric

Lemma 6.1. For the PW flow of conditional states $\Gamma(\tau)$ the parameter $\tau$ is connected to the Bures metric:

$$d\tau \propto d_B(\Gamma(\tau), \Gamma(\tau + d\tau)).$$

Proof. The conditional state $\Gamma(\tau) = \mathrm{Tr}_O[|\tau\rangle\langle\tau|\cdot\Gamma_{\text{total}}]/p(\tau)$ evolves under the shift $\tau \to \tau + d\tau$ via the action of $V_O$ on the clock register. Infinitesimal shift operator: $V_O = e^{-i H_O d\tau}$. Hence:

$$d\Gamma = -i[H_O^{\text{eff}}, \Gamma] d\tau + O(d\tau^2),$$

where $H_O^{\text{eff}}$ is the effective Hamiltonian of the conditional state. The Bures metric:

$$d_B^2(\Gamma, \Gamma + d\Gamma) = \tfrac{1}{2} \mathrm{Tr}[d\Gamma \cdot L_\Gamma(d\Gamma)] = \tfrac{1}{2}\|[H_O^{\text{eff}}, \Gamma]\|^2_{L_\Gamma} d\tau^2,$$

where $L_\Gamma$ is the symmetric logarithmic derivative. For regular $\Gamma$ the norm $\|[H_O^{\text{eff}}, \Gamma]\|_{L_\Gamma}$ is finite and positive, hence:

$$d\tau = d_B / \|[H_O^{\text{eff}}, \Gamma]\|_{L_\Gamma}. \quad \square$$

Step 2 (Bures ↔ Categorical): Geodesics as 1-morphisms

Lemma 6.2. The geodesics of the Bures metric on $\mathcal{D}(\mathbb{C}^7)$ correspond to minimal 1-morphisms in $\mathbf{Exp}_\infty$.

Proof. By definition of $\mathbf{Exp}_\infty$ (categorical formalism §10), 1-morphisms $\gamma: \mathcal{Q}_1 \to \mathcal{Q}_2$ are continuous paths $\gamma: [0,1] \to \mathcal{E}$. The space $\mathcal{E}$ is equipped with the Bures metric via the functor $F: \mathbf{DensityMat} \to \mathbf{Exp}$ (§5 categorical-formalism [T]).

The minimal length in $\mathbf{Exp}_\infty$ is a geodesic of the Bures metric:

$$\gamma_{\min} = \arg\min_\gamma \int_0^1 \|\dot\gamma(s)\|_B \, ds.$$

By the Petz-Uhlmann theorem (Uhlmann 1992): the Bures metric geodesics on $\mathcal{D}(\mathcal{H})$ have an explicit parametrization via pure purifications $|\psi(s)\rangle \in \mathcal{H} \otimes \mathcal{H}'$. $\square$

Step 3 (PW ↔ Stratificational): retracted

warning

Retracted: Lemma 6.3 (the stratificational index is a $\mathbb{Z}_7$-set)

An earlier version asserted that the stratificational parameter is a free transitive $\mathbb{Z}_7$-set canonically isomorphic to the Page–Wootters tick, because "the operators $\pi_\tau$ are cyclically closed: $\pi_6 \circ \ldots \circ \pi_0 = \mathrm{id}$" and $V_O \leftrightarrow \pi$. This contradicts the coarsening it describes. A coarsening loses information — it is not an equivalence, $\ker \pi_n \neq 0$ (T-53c) — so no composite of coarsenings is the identity: $\pi^7 = \mathrm{id}$ would make $\pi$ invertible with inverse $\pi^6$. The stratificational index is the depth $n \in \mathbb{N}$ of §10.3, which grows along the flow; it is not a $\mathbb{Z}_7$-torsor, and its only relation to the tick is the surjection $n \mapsto \tau = n \bmod 7$, which is not a bijection. Lemma 6.3 and the stratificational leg of the equivalence are withdrawn [✗].

Step 4 (What the equivalences give)

Combining Lemmas 6.1 and 6.2:

$$\text{PW} \xrightarrow{\text{Lemma 6.1}} \text{Bures} \xrightarrow{\text{Lemma 6.2}} \text{Categorical (}\mathbf{Exp}_\infty\text{)}$$

The Page–Wootters, information-geometric and categorical constructions are matched through the common parameter $\tau$; the stratificational construction is not a fourth copy of $\tau \in \mathbb{Z}_7$ but carries the depth $n$ with $\tau = n \bmod 7$.

Conclusion

Three constructions of emergent time (PW, Bures, Categorical) describe one cyclic structure $\tau \in \mathbb{Z}_7$; the fourth (stratificational) is a monotone index over it, not isomorphic to it. $\blacksquare$

Status: [T] for the PW ↔ modal isomorphism of §2.3 and for Lemmas 6.1–6.2 as correspondences of label sets. An earlier status line said "Lemmas 6.1, 6.2, 6.3 are explicitly established"; Lemma 6.3 is retracted (box above), and T-53a is narrowed accordingly.

Results used:

• Page-Wootters equivalence §2.3 [T] ($\mathbb{Z}_7$-equivariant isomorphism $\mathcal{H}_O \simeq \Gamma(\Omega, \mathcal{O}_\Omega)$);
• Petz-Uhlmann theorem on geodesics of the Bures metric (Uhlmann 1992);
• Chentsov-Petz framework: Bures = Petz-minimal (extremal) metric within the monotone family (Petz 1996 — the quantum family is not a singleton; Bures is selected by extremality);
• Categorical formalism §5, §10 [T] (functor $F: \mathbf{DensityMat} \to \mathbf{Exp}$).

Consistency check:

• No circularities; the retracted Lemma 6.3 had relied on a cyclic evolution over $\mathbb{Z}_7$, which contradicts the irreversibility of §10;
• The PW, Bures and categorical constructions describe the same structure $\mathbb{Z}_7$ — cyclicity of the UHM clock; the arrow is carried by the depth $n$ (§10.3);
• Consistent with Page-Wootters equivalence §2.3 [T] and with T-53d [T].

---

7. Arrow of time theorem

The circularity problem: what is resolved and what is not

In early versions of UHM there was a circularity problem: the CPTP structure already encoded temporal asymmetry. The ∞-categorical structure of §7.4 reorganises it:

1. The arrow of time is the stratum collapse to the terminal object T, monotone in the parameter $t$ of the dissipative semigroup (§10.4), not in the Page–Wootters tick
2. That the CPTP property is a consequence of the orientation towards T, rather than a postulate, is an open hypothesis [H] (box in §7.1)
3. Free will arises from the flat (zero-mode) directions $\dim\ker(\mathcal H_\Gamma)$ of the free energy (not the multiplicity of paths in the contractible Map(Γ, T))

An earlier version of this box declared the problem "RESOLVED" with item 2 as a result; that claim is retracted. See §7.4 ∞-categorical resolution.

7.1 Categorical formulation

Theorem 7.1 (Arrow of time for unital channels) [T]

For any path γ: [0,1] → $\mathcal{D}(\mathcal{H})$ in state space:

$$\sigma(\gamma) \cdot \Delta S_{vN}(\gamma) \geq 0$$

where:

• $\sigma(\gamma) = +1$, if the path is induced by a unital CPTP channel, $\Phi(\mathbb{1}) = \mathbb{1}$
• $\sigma(\gamma) = -1$, if the path requires inverting such a channel
• $\Delta S_{vN}(\gamma) = S_{vN}(\Gamma(1)) - S_{vN}(\Gamma(0))$

For an arbitrary CPTP channel $\Phi$ the monotone quantity is the relative entropy to a fixed point: if $\Phi(\sigma) = \sigma$, then $D(\Phi(\Gamma)\,\|\,\sigma) \leq D(\Gamma\,\|\,\sigma)$.

Proof:

Relative entropy does not increase under any CPTP map: $D(\Phi(\Gamma)\,\|\,\Phi(\sigma)) \leq D(\Gamma\,\|\,\sigma)$ (Lindblad 1975; Uhlmann 1977). If $\Phi$ is unital, then $\Phi(\mathbb{1}/d) = \mathbb{1}/d$, and with $\sigma = \mathbb{1}/d$ the inequality reads $\log d - S_{vN}(\Phi(\Gamma)) \leq \log d - S_{vN}(\Gamma)$, that is,

$$\Phi \text{ — unital CPTP} \Rightarrow S_{vN}(\Phi(\Gamma)) \geq S_{vN}(\Gamma).$$

The dissipator of UHM has Hermitian Lindblad operators (the pointer projectors) and is therefore unital, so the linear part $\mathcal{L}_0$ of the evolution raises $S_{vN}$ (§10.4, part 1).

Retracted: "CPTP channels do not decrease von Neumann entropy"

An earlier version of this theorem stated $\Phi$ CPTP $\Rightarrow S_{vN}(\Phi(\Gamma)) \geq S_{vN}(\Gamma)$ for every channel, "from strong subadditivity and contractivity". This is false for non-unital channels: the reset channel $X \mapsto \mathrm{Tr}(X)\,|0\rangle\langle 0|$ is CPTP and maps $\mathbb{1}/7$, with $S_{vN} = \log 7 \approx 1.95$, to a pure state, with $S_{vN} = 0$. The same page already relies on the correct version — regeneration lowers entropy locally (§7.3), and §10.4 uses the monotonicity only for the unital part. The statement is retracted and replaced by the theorem above.

Status clarification

The CPTP property of evolution channels in this section is used, not derived. The full derivation of CPTP from ∞-categorical structure (orientation towards terminal T → entropy monotonicity → CPTP) is [H] (open hypothesis). Standard status: CPTP is postulated at the physics level (Lindblad, 1976) and is consistent with the axiomatics A1–A5.

∎

7.2 Physical interpretation

Corollary: Paths induced by unital channels do not decrease entropy; decreasing it along such a path would require inverting a unital channel. Non-unital channels are physical and can lower entropy — the reset channel above, and regeneration toward a purer $\rho_*$ (§7.3). (An earlier corollary said that every physically realizable path increases entropy; it is retracted with Theorem 7.1's old form.)

7.3 Connection to regeneration

Theorem 7.2 (Local arrow of time)

Regeneration $\mathcal{R}[\Gamma, E]$ locally decreases entropy, but only when:

$$\Delta S_{vN}^{local} < 0 \Rightarrow \Delta F_{env \to sys} > 0$$

Total entropy (system + energy source) grows:

$$\Delta S_{vN}^{total} = \Delta S_{vN}^{sys} + \Delta S_{vN}^{source} \geq 0$$

Corollary: The gate $g_V(P)$ in the regenerative term (refining $\Theta(\Delta F)$ from Landauer) is not a postulate, but a consequence of the CPTP structure, thermodynamics and V-preservation.

7.4 ∞-categorical resolution

The circularity problem is fully resolved in the ∞-categorical formulation of UHM.

Reformulation in ∞-category

In the ∞-category $\mathcal{C}_\infty$ the terminal object T is defined by the condition:

$$\text{Map}_{\mathcal{C}_\infty}(\Gamma, T) \simeq *$$

Key distinction:

• In a 1-category: Hom(Γ, T) = {f} — a unique morphism
• In an ∞-category: Map(Γ, T) ≃ * — a set of morphisms, all equivalent

Theorem 7.3 (Arrow of time as structure of ∞-category)

The arrow of time is described by the following structure:

1. Terminal object T exists and is unique (attractor)
2. All morphisms are oriented towards T — this defines the direction
3. CPTP structure as a consequence [H]: channels that increase "distance" to T are excluded (open hypothesis, §7.1)

Formally:

$$\sigma(\gamma) = +1 \Leftrightarrow \gamma \text{ decreases } d_{strat}(\Gamma, T)$$

Proof:

1. Stratification X = ⊔S_α with terminal stratum S_0 = {T}

2. Stratum collapse along the stratal depth $n \in \mathbb{N}$ (§10.3 — not the cyclic tick $\tau \in \mathbb{Z}_7$) defines a canonical direction:

   $$\dim(X_n) \geq \dim(X_{n+1}) \to \dim(\{T\}) = 0$$

3. Morphisms violating this order do not exist in the ∞-category (no inverse morphisms in stratification)

4. [H] That the CPTP property follows from this order is the open hypothesis of §7.1. An earlier step 4 read "channels increasing entropy are the only realizable morphisms in the category with terminal object T"; it is retracted — non-unital channels are realizable and can lower entropy (§7.1).

Steps 1–3 describe the order; the direction itself is that of the semigroup parameter $t$ of §10.4.

∎

Free will in a deterministic structure

Theorem 7.4 (Multiplicity of paths)

Although the goal (T) is unique, there is a multiplicity of equivalent paths:

$$|\text{Mor}_1(\Gamma, T)| \text{ can be arbitrarily large}$$

provided all paths are connected by 2-morphisms (homotopies).

Physical interpretation:

Aspect	1-category (determinism)	∞-category (UHM)
Goal	Unique (T)	Unique (T)
Path	Unique (f)	Set of equivalent
Choice	Absent	Choice of path
Freedom	Illusion	Freedom = choice of homotopy class

Free will is not the choice of goal (the goal $T$ is inevitable), but the latitude among flat directions of the free energy:

$$\mathrm{Freedom}(\Gamma) := \dim\ker(\mathcal{H}_\Gamma) + 1$$

(Not $\pi_0(\mathrm{Map}(\Gamma, T))$: the mapping space into the terminal object is contractible, so $\pi_0=1$ — see Consequences §Free will.)

where π₀ is the set of connected components of the path space.

Connection to categorical formalism

For detailed exposition of the ∞-categorical structure see Categorical formalism.

---

8. Connection to critical purity

8.1 Temporal interpretation of P_crit

Theorem 8.1 (Connection of P_crit to time)

Critical purity $P_{crit} = 2/7$ is connected to the minimal speed of time flow:

$$P > P_{crit} \Leftrightarrow \frac{d\tau}{d\sigma} > \frac{d\tau}{d\sigma}\bigg|_{min}$$

where $\frac{d\tau}{d\sigma}\big|_{min}$ is the minimal speed, below which the system "falls out" of temporal dynamics.

Proof.

Definition 8.1 (Emergent time velocity). For $\Gamma \in \mathcal{D}(\mathbb{C}^7)$ define:

$$v_\tau(\Gamma) := \|[H_O, \Gamma]\|_F,$$

where $H_O$ is the O-sector Hamiltonian (generator of Page-Wootters time evolution), $\|\cdot\|_F$ is the Frobenius norm.

Physical meaning: $v_\tau$ is the rate of state change under the O-sector time operator. It is a $\mathbb{Z}_7$-invariant measure of "time flow".

Step 1 (Upper bound via purity).

Lemma 8.1. For any $\Gamma \in \mathcal{D}(\mathbb{C}^7)$:

$$v_\tau(\Gamma) \leq 2 \|H_O\|_{\text{op}} \cdot \sqrt{P(\Gamma) - \tfrac{1}{7}}.$$

Proof. We use the commutator inequality for Hermitian operators (see Bhatia, Matrix Analysis 1997, §IX.1):

$$\|[A, B]\|_F \leq 2 \|A\|_{\text{op}} \cdot \|B - \lambda I\|_F \quad \text{for any } \lambda \in \mathbb{R}.$$

Apply to $A = H_O$, $B = \Gamma$, $\lambda = \frac{\mathrm{Tr}(\Gamma)}{N} = \frac{1}{7}$ (for $N=7$):

$$\|[H_O, \Gamma]\|_F \leq 2 \|H_O\|_{\text{op}} \cdot \left\| \Gamma - \tfrac{1}{7} I_7 \right\|_F.$$

Compute $\|\Gamma - \tfrac{1}{7} I_7\|_F^2$:

$$\|\Gamma - \tfrac{1}{7} I_7\|_F^2 = \mathrm{Tr}\left( \Gamma^2 - \tfrac{2}{7}\Gamma + \tfrac{1}{49} I_7 \right) = P(\Gamma) - \tfrac{2}{7} + \tfrac{1}{7} = P(\Gamma) - \tfrac{1}{7}.$$

(Using $\mathrm{Tr}(\Gamma) = 1$ and $\mathrm{Tr}(I_7) = 7$.) Hence:

$$v_\tau(\Gamma) = \|[H_O, \Gamma]\|_F \leq 2 \|H_O\|_{\text{op}} \cdot \sqrt{P(\Gamma) - \tfrac{1}{7}}. \quad \square$$

Step 2 (Vanishing at maximal mixture).

Corollary 8.1. $v_\tau(I_7/7) = 0$.

Proof. At $\Gamma = I_7/7$: $\|\Gamma - \tfrac{1}{7}I_7\|_F = 0$, hence by Lemma 8.1: $v_\tau \leq 0$. Since $v_\tau \geq 0$ (Frobenius norm), $v_\tau(I_7/7) = 0$.

Direct verification: $[H_O, I_7/7] = H_O - H_O = 0$, hence $v_\tau = 0$. $\square$

Step 3 (Behaviour as $P \to 1/7$).

As $P(\Gamma) \to 1/7$ we have $\Gamma \to I_7/7$, and by Lemma 8.1:

$$v_\tau(\Gamma) \to 0 \quad \text{as } P(\Gamma) \to 1/7.$$

Rate of decay: $v_\tau(\Gamma) = O(\sqrt{P(\Gamma) - 1/7})$. $\square$

Step 4 (Connection to viability threshold $P_{\text{crit}} = 2/7$).

Remark (threshold distinction). The threshold $P_{\text{crit}} = 2/7$ is the viability threshold (by T-39 [T]), not the time-freezing threshold. Direct connection:

• $P = 1/7$: critical point $I_7/7$, $v_\tau = 0$ (time freezes);
• $P = 2/7$: viability threshold, $\|\Gamma - I_7/7\|_F = \sqrt{1/7}$ (minimum distance from $I/7$ for viable states);
• $P > 2/7$: viable region, $\|\Gamma - I_7/7\|_F > \sqrt{1/7}$ strictly.

Step 5 (Minimum $v_\tau$ on the viable set).

For $\Gamma \in \mathcal{V} = \{P(\Gamma) > 2/7\}$ the upper bound on $v_\tau$ is bounded away from zero:

$$v_\tau(\Gamma) \leq 2\|H_O\|_{\text{op}} \cdot \sqrt{P(\Gamma) - \tfrac{1}{7}} \leq 2\|H_O\|_{\text{op}} \cdot \sqrt{1 - \tfrac{1}{7}} = 2\|H_O\|_{\text{op}} \cdot \sqrt{\tfrac{6}{7}}.$$

Remark. A lower bound $v_\tau(\Gamma) \geq v_\tau^{\min} > 0$ is not guaranteed by the condition $P > 2/7$ alone: a state could be diagonal in the O-energy basis, in which case $[H_O, \Gamma] = 0$, $v_\tau = 0$, even though $P > 2/7$. For a strict lower bound an additional off-diagonality condition in the O-basis is needed.

Step 6 (Autonomous UHM dynamics).

Under autonomous UHM dynamics $\dot\Gamma = \mathcal{L}_\Omega[\Gamma]$ with regeneration $\mathcal{R}$ [T-62 [T]]:

• The attractor $\rho^* = \varphi(\Gamma_0)$ does not coincide with $I_7/7$ (by T-96 [T], $\rho^* \neq I/7$ for nontrivial initial $\Gamma_0$);
• $\rho^*$ has nontrivial O-coherences: $[\rho^*, H_O] \neq 0$ in general;
• Consequently $v_\tau(\rho^*) > 0$ for typical attractor.

Hence in the dynamical stationary regime UHM systems have $v_\tau > 0$ (time continues to flow). $\square$

Step 7 (Dynamical refinement — connection to T-53d [T]).

Steps 1–6 give a kinematic statement (upper bound on $v_\tau$ via $P$). The dynamical statement — about behaviour at the UHM attractor — constitutes a separate theorem T-53d [T]:

$$v_{\text{int}}(\rho^*) \propto (P(\rho^*) - P_{\text{crit}})^{1/2}, \quad P_{\text{crit}} = 2/7.$$

Consistency of kinematics and dynamics. From Step 5:

$$v_\tau^2 = \|[H_O, \Gamma]\|_F^2 = 2\omega_0^2 \sum_{i \neq O} |\gamma_{Oi}|^2$$

(with $H_O = \omega_0 |O\rangle\langle O|$ in the dimension basis $\{O, A, S, D, L, E, U\}$). Hence $v_\tau^2 = \tfrac{1}{2} v_{\text{int}}^2$ — both measures differ by a fixed factor.

Distinction between statements:

Level	Estimate	Condition	Status
Kinematics (Steps 1-6)	$v_\tau \leq 2\|H_O\|\sqrt{P - 1/7}$ (upper)	Any $\Gamma \in \mathcal{D}(\mathbb{C}^7)$	[T]
Dynamics (T-53d)	$v_\tau \propto (P - 2/7)^{1/2}$ (exact asymptotic)	$\Gamma$ at UHM attractor	[T]

Conclusion. Both statements are correct and complement each other:

• Kinematically: $v_\tau = 0$ is possible only for states with $\gamma_{Oi} = 0$ for all $i \neq O$ (diagonal in O-basis). A special case is $\Gamma = I/7$ with $P = 1/7$.
• Dynamically: at the UHM attractor $\rho^*$ such diagonal states are reached only in the limit $P \to P_{\text{crit}} = 2/7$, and the time speed scales as $(P - 2/7)^{1/2}$ (critical slowing down, Landau theory).

The original statement of Theorem 8.1 $(P > P_{\text{crit}} \Leftrightarrow v_\tau > v_\tau^{\min})$ follows from the combination of the kinematic bound and the dynamical scaling law T-53d. $\blacksquare$

Status: [T]. Theorem 8.1 is fully proven: kinematic upper bound + dynamical scaling (T-53d [T]).

Results used:

• Commutator inequality (Bhatia, Matrix Analysis, 1997, §IX.1);
• T-39 [T] ($P_{\text{crit}} = 2/7$);
• T-53d [T] (critical slowing down of time at UHM attractor);
• T-62 [T] (φ as CPTP channel);
• T-96 [T] ($\rho^* \neq I/7$ for nontrivial systems).

Consistency check:

• Dependencies: T-39, T-53d, T-62, T-96 — all [T], no circularities;
• Consistent with T-53d (core/operators/emergent-time.md): $v_\tau^2 = \tfrac{1}{2} v_{\text{int}}^2$;
• Consistent with statements in dimension-d.md, viability.md, temporal-consciousness.md about time freezing as $P \to P_{\text{crit}} = 2/7$ (this is the dynamical result at UHM attractor);
• Consistent with the evolution equation (§2.4) and the attraction theorem (T-39a [T]).

8.2 Interpretation

Viability ($P > 2/7$) means that the Holon continues to exist in time.

At $P \leq 2/7$ the system loses coherence and "spreads" over the state space — for it, time ceases to be well-defined.

---

9. Corollaries

9.1 Modification of the evolution equation

Old form (with external t):

$$\frac{d\Gamma}{dt} = -i[H, \Gamma] + \mathcal{D}[\Gamma] + \mathcal{R}[\Gamma, E]$$

Retracted: the full equation in the O-clock's own time

An earlier version presented the equation below, written in the Page–Wootters tick $\tau$, as a consequence of the structure of $\Gamma_{total}$ and not a postulate. It is not a consequence. The tick ranges over $\mathbb{Z}_7$, so any evolution in it satisfies $\Gamma(\tau + 7) = \Gamma(\tau)$. Along the full equation the free energy is a Lyapunov functional, never increasing and stationary only at the attractor (Theorem 10.1, T-261), and a non-increasing function on a cycle is constant; relative to the O-clock the conditional states would therefore all coincide with a stationary state. Dynamics relative to a periodic clock is necessarily periodic (L. Chataignier, P. A. Höhn, M. P. E. Lock, F. M. Mele, New J. Phys. 28, 034504 (2026); §11.2). The Page–Wootters construction itself gives, for $H_{int} = 0$, a unitary step between ticks and no dissipator (§3.4). The equation holds in an aperiodic parameter $t$ — the parameter of the Lindblad semigroup — and the physical carrier of $t$ is not the O-clock. The candidate carriers are compared in §11.3; the carrier is the depth register of §11.4, relative to which the conditional states obey the equation exactly at every reading (T-53b, Theorems 11.1 and 11.3). (An intermediate version of this box, of the same day, left T-53b conditional on an assumed aperiodic parameter; the assumption is discharged there.)

Form in an aperiodic parameter [T] (relative to the depth register, §11.4):

$$\frac{d\Gamma(t)}{dt} = -i[H_{eff}, \Gamma(t)] + \mathcal{D}[\Gamma(t)] + \mathcal{R}[\Gamma(t), E]$$

where:

• $t \in \mathbb{R}_{\geq 0}$ — the aperiodic semigroup parameter; at finite resolution $t_n = n\,\Delta t$ is the reading $n$ of the depth register (§11.4), and the Page–Wootters tick is its lowest digit $\tau = n \bmod 7$ (§10.3)
• $H_{eff}$ — effective Hamiltonian from constraint $\hat{C}$, exact for $H_{int} = 0$ and the leading term otherwise (§3.4)
• The dissipator and the regenerator are the axiomatic dynamics in $t$ (evolution); what is derived is that a finite timeless world reproduces them as conditional states relative to the depth register — the linear part as a state-independent law (Theorem 11.1), the full flow along each solution (Theorem 11.3)

9.2 Extended role of dimension O

Dimension O now has a dual role:

1. Energy source: Provides $\Delta F > 0$ for regeneration
2. Internal clock: Parametrizes internal time via the Page–Wootters mechanism

9.3 Extended categorical structure

                    G                           F
DensityMat_C  ──────────► DensityMat  ────────────► Exp
    │                         │                      │
    │ constraint               │ CPTP                 │ induced
    ▼                         ▼                      ▼
DensityMat_C  ──────────► DensityMat  ────────────► Exp

                                        ↓ embed

                              Exp_∞ (∞-groupoid)
                                        ↓ sheafify

                              Sh_∞(Exp) (∞-topos)

where:

• DensityMat_C — category with Page–Wootters constraint
• G — functor "conditional states"
• Exp_∞ — ∞-groupoid of paths
• Sh_∞(Exp) — ∞-topos of sheaves

9.4 Experimental predictions

Prediction	Formula	Theor. status	Exp. status
Time slowdown at decoherence	$\frac{d\tau_{int}}{dt_{ext}} \propto (P - P_{crit})^{1/2}$	[T] Corollary of T.8.1	Requires verification
Discreteness of internal time	$\tau \in \{\tau_1, \ldots, \tau_7\}$	[T] Corollary of §3.7	Requires verification
Temporal entanglement	$\Gamma_{12,total} \neq \Gamma_{1} \otimes \Gamma_{2}$ even when $\Gamma_{12}(\tau) = \Gamma_1(\tau) \otimes \Gamma_2(\tau)$	[T] Corollary of P-W	Requires verification

On statuses

• Theor. status [T]: Prediction is mathematically derived from the UHM formalism
• Exp. status: Prediction requires experimental verification

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10. Stratificational time

10.1 Base space as nerve of category

From Axiom Ω⁷ the base space is defined as:

$$X := |N(\mathcal{C})|$$

where $N(\mathcal{C})$ is the nerve of the category of Holons.

10.2 Stratification of X

Space X is stratified:

$$X = \bigsqcup_{\alpha \in A} S_\alpha$$

where:

• $S_0 = \{T\}$ — terminal object (attractor Γ*)
• $S_1$ — edges (morphisms to T)
• $S_n$ — n-simplices

10.3 Temporal stratification: two indices, one arrow

Two different indices are attached to a holon and must not be confused:

• the cyclic Page–Wootters tick $\tau \in \mathbb{Z}_7$ (§2–§3): a kinematic label of the clock register, periodic by construction ($\triangleright^7 = \mathrm{Id}$). No function of $\tau$ alone can be strictly monotone — a monotone function on a cycle is constant;
• the stratal (thermodynamic) depth $n \in \mathbb{N}$: the number of coarsening steps $\pi: \mathcal{C}_n \to \mathcal{C}_{n-1}$ applied along the dissipative flow — the cumulative count of elapsed ticks ($\tau = n \bmod 7$), not reduced modulo 7. It measures how far the state has descended toward $T$.

We stratify $X$ by depth: $X = \bigsqcup_{n \in \mathbb{N}} X_n$, where $X_n$ is the stratum reached after $n$ coarsenings. Since a coarsening never raises the dimension of a stratum, $\dim(X_n) \geq \dim(X_{n+1})$ holds by construction; the content of the arrow is that the dynamics realises the coarsenings and never their inverses — the Lyapunov statement below.

10.4 Arrow of time theorem (stratificational)

Theorem 10.1 (Arrow of time) [T]

Along the axiomatic evolution $\mathcal{L}_\Omega$ there is a functional that never increases and is stationary only at the attractor:

1. Dissipative part $\mathcal{L}_0 = -i[H_{\text{eff}},\cdot] + \mathcal{D}_\Omega$ — a unital CPTP semigroup with fixed point $I/7$: the relative entropy $D(\Gamma(t)\,\|\,I/7) = \log 7 - S_{vN}(\Gamma(t))$ is non-increasing (data-processing inequality; Lindblad 1975, Petz–Ruskai monotonicity), hence $S_{vN}$ is non-decreasing and the purity $P$ non-increasing, and $D \to 0$ by primitivity (T-39a).
2. Full flow with regeneration: the free energy $F(\Gamma)$ is a Lyapunov functional with an exact dissipation identity (H-theorem, T-261 [T]), decreasing toward the attractor $\rho_*$ (T-96).

Consequently the stratal depth $n(t)$ is non-decreasing in $t$: $\dim(X_{n(t)}) \geq \dim(X_{n(t')})$ for $t \leq t'$, with equality only at stationarity. The arrow is the direction of increasing depth, not the cyclic tick: the Page–Wootters label $\tau \in \mathbb{Z}_7$ returns to itself after seven ticks, the depth $n$ does not. Here $t$ is the parameter of the Lindblad semigroup. The O-clock does not supply it (§11.2); the depth register does: relative to its readings the conditional states follow the semigroup exactly, and $D(\cdot\,\|\,I/7)$ and the purity decrease along the whole recorded history (§11.4, Theorem 11.1(c)).

Proof. (1) is the monotonicity of relative entropy to the fixed point of a CPTP semigroup, applied to the unital $\mathcal{L}_0$ whose unique stationary state is $I/7$ (primitivity, T-39a): with $\sigma = I/7$ fixed, $D(\mathcal{E}_t\Gamma\,\|\,I/7) \leq D(\Gamma\,\|\,I/7)$, and $D(\Gamma\,\|\,I/7) = \log 7 - S_{vN}(\Gamma)$. (2) is the H-theorem of T-261. Since the coarsenings $\pi$ are the CPTP steps of this flow, the depth cannot decrease along it, and $\dim$ is non-increasing along coarsenings by definition of the stratification. $\blacksquare$

Interpretation:

Arrow of time = progressive collapse of higher strata towards the terminal object T, measured by the depth $n$; the cyclic tick $\tau = n \bmod 7$ is what the O-clock can read, the depth $n$ is what grows. The depth is a winding number of the O-clock, and a winding number is not an invariant observable relative to a periodic clock (§11.2), so the O-clock does not measure the arrow; the depth register of §11.4, whose higher digits are exactly these winding numbers, does. (An earlier draft indexed the collapse by $\tau \in \mathbb{Z}_7$ itself; on a cycle the inequality $\dim(X_\tau) \geq \dim(X_{\tau+1})$ forces all $\dim(X_\tau)$ to be equal, so that formulation carried no arrow.)

10.5 Connection to thermodynamics

Stratificational time	Thermodynamics
dim(X_n) decreases with the depth $n$	Entropy of the unital part grows (§7.1)
X_n → {T}	System → equilibrium
Stratum collapse	Structural dissipation

10.6 Stratified metric

Definition (Metric d_strat):

$$d_{strat}(\omega_1, \omega_2) = \inf_\gamma \int_\gamma ds_\alpha$$

where:

• γ — path through strata
• ds_α — Connes metric on stratum S_α

Theorem 10.2: d_strat is consistent with the Bures metric:

$$d_{strat}(\Gamma_1, \Gamma_2) \asymp d_B(\Gamma_1, \Gamma_2)$$

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11. Precedents and related programmes

The Page–Wootters mechanism on which §3 rests is more than forty years old, and it has a literature that this page did not cite: the classic objections to it, answers to those objections, experiments, and results about the kind of clock UHM uses — a finite, periodic one. This section says what each work established, how it stands and on whose judgment, which UHM construction it bears on, and where UHM differs; §11.2 then checks the clock $\tau \in \mathbb{Z}_7$ and the arrow of §10 against these results. Every mapping between UHM and these works is an interpretation [I] unless a theorem is named.

Terms used below. A constraint $\hat{C}|\Psi\rangle = 0$ (an equation of Wheeler–DeWitt type) says that the global state is annihilated by the total Hamiltonian, so nothing in it changes with an external time. The conditional state is the state of the rest of the system given a clock reading (Definition 3.1). An ideal clock has a Hamiltonian whose spectrum is the whole real line and a reading that runs monotonically; a periodic clock returns to its initial state after a fixed period. A relational observable is a gauge-invariant quantity of the form "the value of $A$ when the clock reads $t$".

11.1 The mechanism, its objections and the answers

• Page and Wootters (1983); Wootters (1984). D. N. Page, W. K. Wootters, "Evolution without evolution: dynamics described by stationary observables", Phys. Rev. D 27, 2885–2892 (1983); W. K. Wootters, "'Time' replaced by quantum correlations", Int. J. Theor. Phys. 23, 701–711 (1984). The universe is in a stationary state; one subsystem serves as a clock, and the state of the rest conditioned on a clock reading evolves by the Schrödinger equation. Wootters argued that coordinate time is unobservable while clock time is observable, so every statement about evolution can be replaced by a statement about correlations between a clock and another system, all stored in one timeless state. Standing: the starting point of the relational-time literature below; C. Marletto and V. Vedral (2017) call the model elegant but write that it "has never been developed further, because it was criticised for generating severe ambiguities". For UHM: §3.3 adopts the mechanism as it stands; UHM's additions are the choice of clock — the seven-dimensional O-register — and the claim that the split $\mathcal{H}_O \otimes \mathcal{H}_{6D}$ is derived (T-87). One step of that claim needs an extra assumption. Page and Wootters take the global state to be a stationary state, that is, an eigenstate of the total Hamiltonian, and the constraint form $\hat{C}\Psi = 0$ follows after the energy is shifted to zero. For a mixed global state, stationarity gives only $[\hat{C}, \Gamma_{\text{total}}] = 0$, whereas the constraint $\hat{C}\,\Gamma_{\text{total}} = 0$ of §3.3 requires the state to lie in the zero eigenspace of $\hat{C}$; §3.1a used to state that the constraint "follows from stationarity" without this assumption; it now names the assumption, and step 4 of T-87 is [C].
• Kuchař (1992). K. V. Kuchař, "Time and interpretations of quantum gravity", in Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, eds. G. Kunstatter, D. Vincent, J. Williams, World Scientific 1992; reprinted in Int. J. Mod. Phys. D 20, Suppl. 1, 3–86 (2011). The classic review of the problem of time. Its objections to the conditional-probability reading of Page–Wootters, as paraphrased by Höhn, Smith and Lock (2021, §VIII C): (1) for a relativistic particle it gives a wrong localisation probability; (2) conditioning on a clock reading uses operators that do not commute with the constraint, so it appears to violate the constraint; (3) it gives wrong two-time propagators — after one conditioning the clock is "stuck" and time does not flow.
• Unruh and Wald (1989). W. G. Unruh, R. M. Wald, "Time and the interpretation of canonical quantum gravity", Phys. Rev. D 40, 2598–2614 (1989). They proved that in ordinary Schrödinger quantum mechanics, for a system whose Hamiltonian is bounded below, no dynamical variable can correlate monotonically with the Schrödinger time parameter: a clock made of such a system is never ideal. Standing: an accepted theorem; Höhn, Smith and Lock (2021, §III) present it as the refinement of Pauli's observation that no self-adjoint time operator is canonically conjugate to a bounded Hamiltonian. Its consequence for UHM is taken up in §11.2.
• Answers to the objections. R. Gambini, R. A. Porto, J. Pullin, "A relational solution to the problem of time in quantum mechanics and quantum gravity: a fundamental mechanism for quantum decoherence", New J. Phys. 6, 45 (2004), arXiv:gr-qc/0402118, computed conditional probabilities relative to a realistic quantum clock and found that the evolution is then not exactly unitary: pure states decohere through a Lindblad-type equation $\dot\rho = -i[H, \rho] - \sigma[H, [H, \rho]]$ whose only Lindblad operator is the Hamiltonian. With S. Torterolo they combined conditional probabilities with Dirac observables ("evolving constants"), which, in their words, "overcomes the objections levied by Kuchař" ("Conditional probabilities with Dirac observables and the problem of time in quantum gravity", Phys. Rev. D 79, 041501 (2009), arXiv:0809.4235). V. Giovannetti, S. Lloyd, L. Maccone, "Quantum time", Phys. Rev. D 92, 045033 (2015), arXiv:1504.04215, included the measuring apparatus and its memory in the timeless state and reproduced the correct statistics of sequential measurements at different times. P. A. Höhn, A. R. H. Smith, M. P. E. Lock, "Trinity of relational quantum dynamics", Phys. Rev. D 104, 066001 (2021), arXiv:1912.00033, proved that three formulations of relational dynamics are equivalent — the relational observables of Dirac quantization, the Page–Wootters conditional states, and a relational Heisenberg picture obtained by symmetry reduction — using covariant clock POVMs, which also cover non-ideal clocks; with this equivalence they showed objection (2) to be mistaken and resolved (3) in a way that "does not invoke approximations, ideal clocks or ancilla systems", treating (1) in a companion paper. Their equivalence was established for clocks whose Hamiltonian has a continuous non-degenerate spectrum. Standing: within the relational literature objections (2) and (3) are regarded as answered, on the judgment of these authors. For UHM: Definition 3.1 uses single-time conditioning only, so two-time statements such as the "temporal entanglement" of §9.4 need one of these answers, which the page does not use. The equivalence of §6 (T-53a) matches the constructions through their common label set — its stratificational leg, Lemma 6.3, rested on the fact that any two free transitive $\mathbb{Z}_7$-sets are isomorphic and is now retracted, because the stratal depth is not such a set — whereas the trinity is an equivalence of the full relational dynamics; §6 is the weaker kind of statement. Gambini, Porto and Pullin are also a precedent for dissipation out of relational time: their dissipator comes from the imperfection of the clock and has the Hamiltonian as Lindblad operator (dephasing in the energy basis), while UHM's dissipator has the pointer projectors as Lindblad operators and is not derived from the clock (§2.4).
• Experiments. E. Moreva, G. Brida, M. Gramegna, V. Giovannetti, L. Maccone, M. Genovese, "Time from quantum entanglement: an experimental illustration", Phys. Rev. A 89, 052122 (2014), arXiv:1310.4691: two polarization-entangled photons; the polarization of one serves as a clock with two readings, the other photon evolves in birefringent plates; an "observer" who reads the clock photon sees the other evolve, a "super-observer" who measures only global properties finds the joint state static. E. Moreva, M. Gramegna, G. Brida, L. Maccone, M. Genovese, "Quantum time: experimental multitime correlations", Phys. Rev. D 96, 102005 (2017), arXiv:1710.00707: two-time correlations in the Page–Wootters picture and a violation of a Leggett–Garg inequality for the internal observer, with the clock encoded in the position of a photon. Standing: the authors call these illustrations — they show the conditional-probability structure in a prepared state, not the absence of an external time. For UHM: the clocks used had two readings (2014) or a position register (2017); the seven-level O-register has not been realised in any experiment.
• Clock ambiguity. A. Albrecht, A. Iglesias, "The clock ambiguity and the emergence of physical laws", Phys. Rev. D 77, 063506 (2008), argued that different choices of the clock subsystem make one timeless state describe different laws. C. Marletto, V. Vedral, "Evolution without evolution and without ambiguities", Phys. Rev. D 95, 043510 (2017), arXiv:1610.04773, answered that a good clock is a subsystem with many distinguishable states that ideally does not interact with the rest, $H = H_C \otimes I + I \otimes H_R$, and that if such a split yields unitary evolution with a time-independent Hamiltonian it is unique. Standing: a published answer, valid under its stated assumption of a non-interacting clock. For UHM: the constraint couples the O-register to the rest through $H_{\text{int}}$ (Property 2) and the clock has seven states, so this uniqueness argument does not cover it; the corpus rests the uniqueness of the clock register on its own $G_2$ argument (T-87, step 3).
• Interacting clocks. A. R. H. Smith, M. Ahmadi, "Quantizing time: interacting clocks and systems", Quantum 3, 160 (2019), arXiv:1712.00081: when the constraint contains a clock–system interaction, the conditional state obeys a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced by a self-adjoint integral operator. For UHM: §3.6 writes a local effective Hamiltonian $H_{\text{eff}}(\tau) = H_{6D} + \langle\tau|H_{\text{int}}|\tau\rangle_O$ and §3.4 a local equation "plus corrections", while the cohesive-closure statement T-186(b) asserts that the Page–Wootters evolution is exact, with no $O(H_{\text{int}})$ correction. The corpus had not reconciled either with the time-nonlocal form; §3.4 and §3.6 now mark the local generator as an approximation, and T-186(b) is withdrawn.
• Quantum reference frames. F. Giacomini, E. Castro-Ruiz, Č. Brukner, "Quantum mechanics and the covariance of physical laws in quantum reference frames", Nat. Commun. 10, 494 (2019), arXiv:1712.07207: reference frames attached to quantum systems transform into one another by a "superposition of coordinate transformations", and entanglement and superposition are frame-dependent. Standing: an active programme; Höhn, Smith and Lock extend it to changes of temporal frame (2021, §VII). For UHM: the O-register is a temporal reference frame, and the corpus fixes one such frame without defining a change of clock; statements about entanglement between the clock and the rest, including the "temporal entanglement" of §9.4, hold relative to that frame.

11.2 The finite periodic clock, Unruh–Wald, and the arrow

• The Salecker–Wigner–Peres clock: prior art for the O-clock. H. Salecker, E. P. Wigner, "Quantum limitations of the measurement of space-time distances", Phys. Rev. 109, 571–577 (1958); A. Peres, "Measurement of time by quantum clocks", Am. J. Phys. 48, 552–557 (1980). The standard finite quantum clock: $d$ equally spaced levels $H = \sum_{n=0}^{d-1} n\omega\,|E_n\rangle\langle E_n|$, period $T_0 = 2\pi/\omega$, time states obtained from the energy states by the discrete Fourier transform, which rotate into one another every $T_0/d$ and return after $d$ steps; between ticks a time state spreads over its neighbours, and no finite clock satisfies the canonical commutation relation with its Hamiltonian. M. P. Woods, R. Silva and J. Oppenheim call it "the model of choice for finite-dimensional clocks" ("Autonomous quantum machines and finite-sized clocks", Ann. Henri Poincaré 20, 125–218 (2019), arXiv:1607.04591, Appendix B). The O-clock of §3.5–3.6 — $H_O = \omega_0 \sum_{k=0}^{6} k\,|k\rangle\langle k|$, Fourier clock states $|\tau_n\rangle$, cyclic shift $V_O$, chronon $\delta\tau = 2\pi/(7\omega_0)$ — is exactly this clock with $d = 7$. The discreteness of its readings, the cyclic group $\mathbb{Z}_d$ and the resolution $T_0/d$ are properties of every such clock, so the "discreteness of internal time" listed in §9.4 is a property of the chosen clock rather than an independent test of UHM; what is specific to UHM is the claim that the clock has exactly seven levels and is the O-dimension.
• Unruh–Wald against the O-clock: no collision at the level of the clock. $H_O$ has the finite spectrum $\{0, \omega_0, \ldots, 6\omega_0\}$, bounded below and above, so the Unruh–Wald theorem applies: no observable of the O-register correlates monotonically with time. The corpus does not claim otherwise — §10.3 states that no function of the tick $\tau$ alone can be monotone. Giovannetti, Lloyd and Maccone (2015, §C) spell out the consequence for Page–Wootters: a non-periodic time needs an unbounded Hamiltonian, "a system with finite global energy will have periodic evolution", and in that case "except as an approximation internal observers will not be able to use a Schrödinger equation".
• Periodic clocks: where the collision lies. L. Chataignier, P. A. Höhn, M. P. E. Lock, F. M. Mele, "Relational dynamics with periodic clocks", New J. Phys. 28, 034504 (2026), arXiv:2409.06479, extended the trinity to periodic clocks and proved that relational observables relative to a periodic clock are invariant only if the quantity itself is periodic — "counting winding numbers does not lead to invariant observables relative to the periodic clock" — and that the dynamics relative to a periodic clock is "necessarily periodic" in all three formulations, Page–Wootters included; a system that evolves periodically relative to a periodic clock can still evolve monotonically relative to an aperiodic one. Höhn, Smith and Lock (2021, §III A) had already noted that with a periodic clock one must keep track of its winding number to follow a non-periodic evolution. This bears on UHM in two places:
  1. The arrow of §10 is not supplied by the O-clock. The corpus places the arrow correctly off the cyclic tick: the tick $\tau \in \mathbb{Z}_7$ carries no arrow, and the arrow is the growth of the stratal depth $n \in \mathbb{N}$, the cumulative count of ticks with $\tau = n \bmod 7$ (§10.3). That count is exactly a winding number of the O-clock, which by the result above is not an invariant relational observable relative to it; and the functionals of Theorem 10.1 are monotone in the parameter $t$ of the Lindblad semigroup, which in relational terms is an aperiodic clock outside the 42-dimensional Page–Wootters state. The arrow therefore presupposes an aperiodic time that the O-clock cannot provide, and the corpus does not construct its physical carrier.
  2. Dissipative evolution in the O-clock's own time is impossible. §9.1 and T-53b stated that the conditional states $\Gamma(\tau)$ obey the full equation with $\mathcal{D}$ and $\mathcal{R}$; that statement is retracted, and T-53b now holds relative to the depth register of §11.4 instead of the O-clock (it was conditional on an assumed aperiodic time parameter in an intermediate version). Relative to a clock of period seven ticks, $\Gamma(\tau + 7) = \Gamma(\tau)$. A Lyapunov functional that never increases along the flow and is stationary only at the attractor (Theorem 10.1) cannot be periodic unless it is constant, so relative to the O-clock the conditional states would all coincide with a stationary state. Nontrivial dissipative dynamics needs the aperiodic parameter of item 1.

The corpus's own description — the cyclic tick carries no arrow, the arrow lives in the depth — is thus consistent. The collision was with the claims that the dissipative dynamics and the arrow are derived from the Page–Wootters clock itself (§9.1, T-53b), which the periodic-clock result and the Unruh–Wald theorem rule out for a seven-level clock; these claims are retracted (§9.1), and the dynamics and the arrow are stated relative to an aperiodic carrier, which §11.4 constructs.

11.3 Carriers of the aperiodic parameter: options

Status of this subsection

Proven [T] (proofs below, witnesses in website/scripts/check_core_numbers.py): in a world with pure point spectrum no clock whose readings are the orbit of a unitary group — continuous or discrete, periodic or not — supports dissipation on its whole orbit, and every Page–Wootters conditional dynamics is linear in the global state up to normalisation. Verdict: the O-clock, the stratal depth without a register, and composite O-clocks with incommensurate frequencies do not carry the parameter $t$ of T-53b; an ideal clock register with an infinite environment carries it approximately (Davies limit); the depth register of §11.4 — a finite chain of readings, not an orbit — carries it exactly, and T-53b holds there as a theorem [T]. An intermediate version of this box, of the same day, stated as a verdict that only the ideal clock with an infinite environment makes T-53b true, as [C], and left the choice of carrier to the author [Pr]; that verdict is superseded by §11.4.

T-53b asks three things of its time parameter: (R1) the dynamics in it is not periodic; (R2) a Lyapunov functional decreases strictly along it (Theorem 10.1); (R3) the nonlinear regenerative term $\mathcal{R}$ acts in it.

[T] Lemma (almost periodicity). Let the clock readings be the orbit of a unitary group — $|t\rangle = e^{-iH_C t}|0\rangle$, $t \in \mathbb{R}$, or $|n\rangle = V^n|0\rangle$, $n \in \mathbb{Z}$ — and let the joint generator of the clock and everything it is correlated with have pure point spectrum; in particular let the world be finite-dimensional, the clock periodic or not. Then every conditional expectation value is an almost periodic function of the reading, and a non-increasing almost periodic function is constant. Hence (R2) fails for every such clock on its whole orbit. (An earlier wording said "relative to any clock"; it is narrowed to clocks whose readings form a group orbit. The readings of a finite chain, §11.4, are not an orbit — the step from the last reading leads nowhere — and there (R2) holds on the whole chain.)

Proof. An expectation value is $\sum_{j,k} c_{jk}\,e^{i(E_j - E_k)t}$ (or $\sum_{j,k} c_{jk}\,e^{i(\theta_j - \theta_k)n}$ for the eigenphases $\theta_j$ of $V$), a trigonometric polynomial, or a uniform limit of such sums in the pure-point case; both are almost periodic in the sense of Bohr, on $\mathbb{R}$ or on $\mathbb{Z}$. For an almost periodic $f$ and every $\varepsilon > 0$ the $\varepsilon$-almost periods are relatively dense. If $f(t_1) < f(t_0)$ for some $t_1 > t_0$, take $\varepsilon < f(t_0) - f(t_1)$ and an $\varepsilon$-almost period $T > t_1 - t_0$: then $t_0 + T > t_1$ and $f(t_0 + T) > f(t_1)$, so $f$ is not non-increasing. $\blacksquare$ This is the mechanism of the quantum recurrence theorem (P. Bocchieri, A. Loinger, Phys. Rev. 107, 337 (1957)). Witness: a closed seven-level system with an eight-level environment and a random Hamiltonian, started in a pure product state: the relative entropy $D(\rho_S\,\|\,I/7)$ falls from $\log 7 = 1.946$ to $0.232$ for $t \le 200$ and afterwards rises again by up to $0.474$.

[T] Lemma (conditional dynamics is linear). In the Page–Wootters construction the conditional state at the clock reading $t$ is $\rho_S(t) = \mathrm{Tr}_E[(\langle t| \otimes I)\,\Gamma_{\text{total}}\,(|t\rangle \otimes I)]/p(t)$ — a linear map of the global state followed by normalisation, and $p(t)$ is constant for an ideal clock that does not interact with the rest. For interacting clocks the equation becomes time-nonlocal (Smith–Ahmadi 2019, §11.1) and stays linear in the global state. Hence $\mathcal{R}$ is not the conditional dynamics of any single constraint for all initial states: the map from the initial state to the conditional history is linear, the regenerative flow is not. It can enter as a law postulated in $t$; each of its solutions separately is a conditional history of a constraint fitted to that solution (§11.4, Theorem 11.3); or it can enter as an effective equation — for example the mean-field limit of many exchangeable systems (H. Spohn, "Kinetic equations from Hamiltonian dynamics: Markovian limits", Rev. Mod. Phys. 52, 569 (1980)). Where $p(t)$ varies, the conditional dynamics is normalised-linear, the Rembieliński–Caban class, which by physics correspondence, §8.7 cannot carry the viability gate either.

Proof. Conditioning, partial trace and the dynamics of the global state are linear; only the division by $p(t)$ is not. $\blacksquare$

The candidates.

Carrier	(R1) aperiodic	(R2) dissipation	(R3) $\mathcal{R}$	Verdict for T-53b
O-clock $\tau \in \mathbb{Z}_7$	no: period seven ticks (Chataignier–Höhn–Lock–Mele 2026)	no	no	[✗]
Stratal depth $n$	not an observable relative to the O-clock: a winding number, which is not an invariant relational observable relative to a periodic clock (§11.2)	only through a register that records it	only through that register	not a carrier by itself; recorded in a register it is the carrier of the last row
Composite O-clocks with incommensurate frequencies $\omega_m$	yes: the orbit $(\omega_1 t, \ldots, \omega_M t)$ on the torus never closes	no, by the almost-periodicity Lemma: two clocks with $\omega = 1, \sqrt 2$ never return exactly, yet at $t \approx 2\pi \cdot 70$ the clock state overlaps its initial state to $0.9987$	no	[✗]; the readings form $C(\mathbb{T}^M)$ — by Kronecker's theorem the orbit is dense in the torus — not $C_0(\mathbb{R})$, so T-118 is not obtained this way either
Ideal clock register ($H_C = \hat p$ on $L^2(\mathbb{R})$, spectrum $\mathbb{R}$; or a counter on $\ell^2(\mathbb{Z})$)	yes	only together with an infinite environment with continuous spectrum, in a Markov limit that gives $\mathcal{L}_0$ approximately (E. B. Davies, "Markovian master equations", Commun. Math. Phys. 39, 91 (1974))	postulated (Lemma above)	approximate: $\mathcal{L}_0$ only in the weak-coupling limit, with an infinite environment; superseded by the last row
Semigroup parameter $t$ taken as primitive	yes	as postulated	as postulated	the same as the ideal clock: by the trinity of Höhn, Smith and Lock (2021) the conditional dynamics relative to an ideal clock is the Schrödinger picture in $t$
Depth register: readings $0 < 1 < \cdots < N$ as a chain, realised positionally in $M = \lceil\log_7(N+1)\rceil$ O-registers, with a two-holon environment $\mathbb{C}^{49}$ and a Feynman–Kitaev constraint (§11.4)	yes: no reading follows $N$, the chain is not a group orbit	yes, exactly at every reading, for the linear part as a state-independent law (Theorem 11.1)	along each solution (Theorem 11.3)	[T]: T-53b holds, in a world of dimension $343(N+1)$

Cost of the ideal clock (the carrier that the intermediate version of this subsection called the only one that works). (i) An ideal clock has a Hamiltonian unbounded below, so no ground state; by Unruh and Wald (1989) no clock with a Hamiltonian bounded below reads time monotonically. (ii) Its finite approximations — the quasi-ideal clocks of M. P. Woods, R. Silva and J. Oppenheim, whose back-reaction is exponentially small in the clock dimension $d$ at a linear cost in energy (Ann. Henri Poincaré 20, 125 (2019)) — are finite, hence periodic, and by the almost-periodicity Lemma support dissipation only over a finite window; exact T-53b needs $d \to \infty$. (iii) The environment must be infinite, and $\mathcal{L}_0$ is then a weak-coupling approximation, not an exact law. (iv) The time of the dynamics is not emergent from the O-dimension: the O-clock becomes a kinematic label read off $t$ ($n = \lfloor t/\delta\tau \rfloor$, $\tau = n \bmod 7$), and the arrow of Theorem 10.1 is monotone in $t$, as it is already stated. (v) $\mathcal{R}$ is postulated in $t$, and the measurement reading of physics correspondence, §8.7 has to be fixed separately. At this carrier T-118 holds by Gelfand duality of the clock spectrum, $A_{\text{time}} \cong C_0(\mathbb{R})$ — true, but assumed rather than obtained as a limit of O-clocks.

The three options of the intermediate version, and what became of them. That version left the author a choice [Pr]: (a) accept the ideal clock, equivalently the semigroup parameter, as a primitive of the dynamics, with T-53b and T-118 [C at an ideal clock with an infinite environment]; (b) keep the dynamical time emergent from finite registers and prove T-53b as a finite-window theorem with explicit error; (c) add a counter register $\ell^2(\mathbb{Z})$ recording the depth $n$ as a new infinite-dimensional axiom. Option (b) is now proved, in a stronger form than asked: the error at the readings is zero, not merely small (§11.4). Option (c) is not needed — every finite window is carried by a finite chain inside the O-registers of finitely many holons — and its limit is not a better carrier: on $\ell^2(\mathbb{N})$ the Feynman–Kitaev constraint is the Laplacian of the half-line, whose kernel contains no normalisable vector, so the history state of an infinite register is not a state. Option (a) remains available as a reading of the continuum limit (Theorem 11.5), but it is no longer an assumption that T-53b needs.

11.4 The depth register: an exact finite carrier

Status of this subsection

Proven [T] (Theorems 11.1–11.5; witnesses in website/scripts/check_core_numbers.py). The aperiodic parameter of T-53b has a finite carrier: a register that records the stratal depth $n$ (§10.3) as an ordered chain $0 < 1 < \cdots < N$, not as a cycle. Relative to it the dissipative semigroup is the Page–Wootters conditional dynamics exactly at every reading, in a world of dimension $7 \cdot 49 \cdot (N+1)$, and the arrow holds on the whole recorded history; the full flow with $\mathcal{R}$ is reproduced along each of its solutions; the continuum parameter is the scaling limit of the readings, with an explicit error, and its algebra is $C_0(\mathbb{R})$. Not obtained: a monotone functional on the whole orbit of any clock whose readings form a group orbit (Lemma of §11.3), and $\mathcal{R}$ as a conditional law independent of the state (second Lemma of §11.3). Assumption, as for every Page–Wootters statement: that the world is in a timeless state of the kind constructed — the constraint half of A5 (T-87, step 4) — is not derived here; the theorems say that such a finite timeless world exists and what it yields.

Set-up. Fix $N \geq 1$. The system is a holon, $\mathcal{H}_S = \mathbb{C}^7$. The environment is two further holons, $\mathcal{H}_E = \mathbb{C}^7 \otimes \mathbb{C}^7 = \mathbb{C}^{49}$, with a fixed reference vector $|0\rangle_E$. The depth register is $\mathcal{H}_C = \mathbb{C}^{N+1}$ with orthonormal readings $|0\rangle, \ldots, |N\rangle$; the reading $|n\rangle$ records that $n$ steps have occurred. It sits inside the O-registers of $M = \lceil \log_7(N+1) \rceil$ holons in positional notation, $|n\rangle = |\tau_1\rangle \otimes \cdots \otimes |\tau_M\rangle$ with $n = \sum_m \tau_m 7^{m-1}$: the lowest digit $\tau_1 = n \bmod 7$ is the O-tick of §2–§3, and the higher digits count its windings — the quantity that relative to the O-clock alone is not an invariant observable (§11.2). Given unitaries $U_1, \ldots, U_N$ on $\mathcal{H}_S \otimes \mathcal{H}_E$, the constraint is of Feynman–Kitaev type,

$$\hat C = \sum_{n=1}^{N} \Big[ \big(|n\rangle\langle n| + |n-1\rangle\langle n-1|\big) \otimes 1 \;-\; |n\rangle\langle n-1| \otimes U_n \;-\; |n-1\rangle\langle n| \otimes U_n^\dagger \Big],$$

and the conditional state at reading $n$ is that of Definition 3.1, $\rho_S(n) = \mathrm{Tr}_E \langle n|\Gamma_{\text{total}}|n\rangle / p(n)$. The constraint is R. P. Feynman's clock ("Quantum mechanical computers", Found. Phys. 16, 507 (1986)), used by A. Yu. Kitaev (A. Yu. Kitaev, A. H. Shen, M. N. Vyalyi, Classical and Quantum Computation, AMS 2002, ch. 14), read as a discrete Page–Wootters constraint by J. R. McClean, J. A. Parkhill and A. Aspuru-Guzik ("Feynman's clock, a new variational principle, and parallel-in-time quantum dynamics", PNAS 110, E3901 (2013)) and studied as a system–time history state for a finite clock by A. Boette and R. Rossignoli ("History states of systems and operators", Phys. Rev. A 98, 032108 (2018)). What is added here is the environment and the dilation of step (i) below, which turn the unitary history into a dissipative one.

Theorem 11.1 (Exact dissipative dynamics relative to a finite chain) [T]

Let $\Phi_t = e^{t\mathcal{L}}$ be any CPTP semigroup on $\mathbb{C}^7$, $\Delta t > 0$ and $N \geq 1$. There are unitaries $U_1, \ldots, U_N$ on $\mathcal{H}_S \otimes \mathcal{H}_E$, depending on $\mathcal{L}$, $\Delta t$ and $n$ but not on any state, such that:

(a) $\hat C \geq 0$; $\ker \hat C = \{ \sum_n |n\rangle \otimes W_n \chi / \sqrt{N+1} : \chi \in \mathcal{H}_S \otimes \mathcal{H}_E \}$ with $W_n := U_n \cdots U_1$, of dimension $343$; the eigenvalues of $\hat C$ are $2 - 2\cos(\pi k/(N+1))$, $k = 0, \ldots, N$, each $343$-fold, so the gap above the kernel is $4\sin^2(\pi/(2(N+1)))$.

(b) For every state $\rho_0$ of the holon, $\Gamma_{\text{total}} := \mathcal{W}\big(|u\rangle\langle u| \otimes \rho_0 \otimes |0\rangle\langle 0|_E\big)\mathcal{W}^\dagger$, with $\mathcal{W} = \sum_n |n\rangle\langle n| \otimes W_n$ and $u = (1, \ldots, 1)/\sqrt{N+1}$, satisfies $\hat C\,\Gamma_{\text{total}} = 0$, $p(n) = 1/(N+1)$ and

$$\rho_S(n) = \Phi_{n\Delta t}(\rho_0) \qquad \text{exactly, for } n = 0, 1, \ldots, N.$$

(c) If $\mathcal{L} = \mathcal{L}_0$ is unital and primitive (Theorem 10.1), then along the readings $D(\rho_S(n)\,\|\,I/7)$ is non-increasing and the purity $P(\rho_S(n))$ is strictly decreasing, unless $\rho_0 = I/7$.

Proof. (i) Dilations. For each $n$ the map $\Phi_{n\Delta t}$ is CPTP on $\mathbb{C}^7$; its Choi matrix is $49 \times 49$, so it has Kraus operators $K_{n,1}, \ldots, K_{n,49}$ (some possibly zero) with $\sum_j K_{n,j}^\dagger K_{n,j} = 1$. The map $V_n \psi = \sum_j K_{n,j}\psi \otimes |j\rangle_E$ is an isometry $\mathbb{C}^7 \to \mathbb{C}^7 \otimes \mathbb{C}^{49}$ (W. F. Stinespring, Proc. Amer. Math. Soc. 6, 211 (1955)); completing its seven image vectors to an orthonormal basis gives a unitary $W_n$ with $W_n(\psi \otimes |0\rangle_E) = V_n \psi$. Put $W_0 = 1$ and $U_n := W_n W_{n-1}^\dagger$. Then $U_n \cdots U_1 = W_n$ and $\mathrm{Tr}_E\, W_n(\rho \otimes |0\rangle\langle 0|)W_n^\dagger = \sum_j K_{n,j}\rho K_{n,j}^\dagger = \Phi_{n\Delta t}(\rho)$ for every $\rho$.

(ii) Kernel and spectrum. Since $W_n^\dagger U_n W_{n-1} = 1$, conjugation by the unitary $\mathcal{W}$ gives $\mathcal{W}^\dagger \hat C\, \mathcal{W} = \Lambda \otimes 1$, where $\Lambda = \sum_{n=1}^{N} (|n\rangle - |n-1\rangle)(\langle n| - \langle n-1|)$ is the Laplacian of the path with $N+1$ vertices. Its quadratic form is $\sum_n |x_n - x_{n-1}|^2 \geq 0$, vanishing only on constant vectors, and its eigenvalues are $2 - 2\cos(\pi k/(N+1))$. This is (a).

(iii) Conditional states. $\langle n|\Gamma_{\text{total}}|n\rangle = W_n(\rho_0 \otimes |0\rangle\langle 0|)W_n^\dagger/(N+1)$, of trace $1/(N+1)$; by (i) its normalised partial trace over $E$ is $\Phi_{n\Delta t}(\rho_0)$. With (ii), $\mathrm{supp}\,\Gamma_{\text{total}} \subseteq \ker \hat C$, so $\hat C\,\Gamma_{\text{total}} = 0$. This is (b).

(iv) Monotonicity. $\Phi := \Phi_{\Delta t}$ is CPTP and unital, and $\rho_S(n+1) = \Phi(\rho_S(n))$ by the semigroup law. Monotonicity of relative entropy with $\Phi(I/7) = I/7$ gives the first claim. For the purity, the Kadison–Schwarz inequality for unital completely positive maps (M.-D. Choi, Illinois J. Math. 18, 565 (1974)) gives $\Phi(X)^2 \leq \Phi(X^2)$ for Hermitian $X$; taking the trace, $\mathrm{Tr}\,\Phi(X)^2 \leq \mathrm{Tr}\,X^2$. Hence $f(t) := \mathrm{Tr}\,(\Phi_t(\rho_0) - I/7)^2 = P(\Phi_t\rho_0) - 1/7$ is non-increasing in $t$. It is also real-analytic in $t$, since the entries of $e^{t\mathcal{L}_0}$ are entire functions. If $f(n\Delta t) = f((n+1)\Delta t)$, then $f$ is constant on that interval, hence on $[0, \infty)$; primitivity gives $f(t) \to 0$, so $f \equiv 0$ and $\rho_0 = I/7$. $\blacksquare$

Why this does not contradict §11.2–§11.3. The almost-periodicity Lemma needs readings that form the orbit of a unitary group; here the step $|n-1\rangle \mapsto |n\rangle$ is a partial isometry, and no reading follows $N$. The periodic-clock result of Chataignier, Höhn, Lock and Mele concerns clocks with a $U(1)$ action, which the chain does not carry. The Unruh–Wald theorem concerns correlation with an external Schrödinger parameter; here there is none — the readings are the time. The price is visible: the recorded history has a first and a last reading. And relative to the lowest digit alone, the O-tick $\tau_1 = n \bmod 7$, the conditional state is the average of $\Phi_{n\Delta t}(\rho_0)$ over $n \equiv \tau_1 \pmod 7$: the O-clock still reads no arrow.

Theorem 11.2 (Periodic registers: the arrow breaks at the wrap, and only there) [T]

Close the chain into a cycle $\mathbb{Z}_{N+1}$ by adding the edge $N \to 0$ with $U_0 := W_N^\dagger$. The state $\Gamma_{\text{total}}$ of Theorem 11.1 also satisfies the cyclic constraint and gives the same conditional states. Along the cycle $\rho_S(0) \to \cdots \to \rho_S(N) \to \rho_S(0)$ every functional that decreases on the first $N$ steps increases on the wrap step. No periodic register does better: a function on a cycle cannot decrease on every step. For the seven readings of one O-clock the best possible is six decreasing steps of seven.

Proof. The holonomy around the cycle is $U_0 U_N \cdots U_1 = W_N^\dagger W_N = 1$, so $\mathcal{W}$ conjugates the cyclic constraint to the cycle Laplacian $\otimes 1$, whose kernel is again spanned by $u$; the conditional states are computed as in Theorem 11.1. The rest is the fact that the increments of a function around a cycle sum to zero. $\blacksquare$

This is the finite form of the periodic-clock result (§11.2): the dynamics relative to a period-$(N+1)$ register is periodic, and the fraction of steps on which the arrow fails is $1/(N+1)$, attained.

Theorem 11.3 (The full flow along each solution) [T]

Let $\Gamma(t)$, $t \in [0, N\Delta t]$, solve the full equation of §9.1 with $\mathcal{R}[\Gamma] = c(\Gamma)\,(\rho_*(\Gamma) - \Gamma)$, $c = \kappa\, g_V \geq 0$ and $\rho_* = \varphi(\Gamma)$ a state, both continuous along the solution. Put $\mathcal{K}_s(X) := c(\Gamma(s))\,\big(\rho_*(\Gamma(s))\,\mathrm{Tr}\,X - X\big)$ and $\mathcal{L}_s := \mathcal{L}_0 + \mathcal{K}_s$. Then:

(a) each $\mathcal{L}_s$ is a Lindblad generator, and the propagators $\Phi_n := \mathcal{T}\exp\int_{(n-1)\Delta t}^{n\Delta t} \mathcal{L}_s\,ds$ are CPTP with $\Phi_n(\Gamma((n-1)\Delta t)) = \Gamma(n\Delta t)$;

(b) the construction of Theorem 11.1, with $W_n$ a dilation of $\Phi_n \circ \cdots \circ \Phi_1$, yields a finite timeless world whose conditional states are exactly $\Gamma(n\Delta t)$ at every reading $n = 0, \ldots, N$.

The constraint of (b) depends on the solution; for a nonlinear $\mathcal{R}$ no constraint independent of the solution does this (second Lemma of §11.3).

Proof. (a) $\mathcal{K}_s = c\,(\Psi_s - \mathrm{id})$ with the replacement channel $\Psi_s(X) = \rho_*\,\mathrm{Tr}\,X$, and $e^{h c(\Psi - \mathrm{id})} = e^{-hc}\,\mathrm{id} + (1 - e^{-hc})\,\Psi$ (since $\Psi^2 = \Psi$) is CPTP for $h, c \geq 0$; a sum of Lindblad generators is one. The time-ordered exponential of a continuous family of Lindblad generators is a limit of products $e^{h\mathcal{L}_{s_k}}$ of CPTP maps, and the CPTP maps form a closed set. Because $\mathrm{Tr}\,\Gamma(s) = 1$, $\mathcal{K}_s(\Gamma(s)) = \mathcal{R}[\Gamma(s)]$, so $\Gamma$ solves the linear equation $\dot X = \mathcal{L}_s X$; uniqueness for linear equations with continuous coefficients gives $\Phi_n(\Gamma((n-1)\Delta t)) = \Gamma(n\Delta t)$. (b) Step (i) of the proof of Theorem 11.1 uses only that the maps it dilates are CPTP. $\blacksquare$

Every Lyapunov statement about the solution — the H-theorem of T-261, the monotonicities of Theorem 10.1 — therefore holds at the readings of the register, because the readings reproduce the solution.

Theorem 11.4 (Continuum error with an explicit constant) [T]

In Theorem 11.1 let $t \in [0, N\Delta t]$ and $n(t) = \lfloor t/\Delta t \rfloor$. Then

$$\big\|\rho_S(n(t)) - \Phi_t(\rho_0)\big\|_1 \leq \Delta t\, \|\mathcal{L}\|_{1\to1}, \qquad \|\mathcal{L}\|_{1\to1} \leq 2\|H\| + 2\sum_k \|L_k\|^2$$

for $\mathcal{L} = -i[H, \cdot] + \sum_k \big(L_k \cdot L_k^\dagger - \tfrac12\{L_k^\dagger L_k, \cdot\}\big)$. For the full flow of Theorem 11.3 the same holds with $\|\mathcal{L}_0\|_{1\to1} + 2\max\kappa$. Hence a window $[0, T]$ is carried, with error at most $\varepsilon$ in trace norm at every $t$, by a finite world of dimension

$$d = 343\,(N+1), \qquad N = \lceil T\,\|\mathcal{L}\|_{1\to1}/\varepsilon \rceil ,$$

that is, $\varepsilon(d) \leq 343\, T\, \|\mathcal{L}\|_{1\to1}/(d - 343)$; at the readings themselves the error is zero, and there is no recurrence inside the window.

Proof. $\Phi_t\rho_0 - \Phi_s\rho_0 = \int_s^t \mathcal{L}\,\Phi_r\rho_0\,dr$ with $0 \leq t - s < \Delta t$, and $\|\Phi_r\rho_0\|_1 = 1$. For the bound on $\|\mathcal{L}\|_{1\to1}$: $\|HX - XH\|_1 \leq 2\|H\|\,\|X\|_1$, $\|L X L^\dagger\|_1 \leq \|L\|^2\|X\|_1$ and $\tfrac12\|\{L^\dagger L, X\}\|_1 \leq \|L\|^2\|X\|_1$. For the full flow $\|\dot\Gamma\|_1 \leq \|\mathcal{L}_0\|_{1\to1} + c\,\|\rho_* - \Gamma\|_1 \leq \|\mathcal{L}_0\|_{1\to1} + 2\kappa$, since $g_V \leq 1$. $\blacksquare$

Theorem 11.5 (The time algebra in the scaling limit; T-118) [T]

Measure the readings in a macroscopic unit: $t_k = (k - m)\,\Delta t$, $k = 0, \ldots, N$, where $\Delta t$ is the chronon in that unit and $m$ the reading taken as origin; the time algebra of the register is its diagonal algebra $A_N = C(\{t_k\}) \cong \mathbb{C}^{N+1}$. Take a sequence of registers with $\Delta t_N \to 0$, $m_N\Delta t_N \to \infty$ and $(N - m_N)\Delta t_N \to \infty$. Then:

(a) for every $R$, eventually every point of $[-R, R]$ lies within $\Delta t_N/2$ of a reading: the reading sets converge to $\mathbb{R}$ in the pointed Hausdorff sense;

(b) the sampling maps $s_N(f) = (f(t_k))_k$ are $*$-homomorphisms $C_0(\mathbb{R}) \to A_N$ with $\|s_N f\| \to \|f\|_\infty$ for every $f \in C_0(\mathbb{R})$; hence $s = (s_N)_N$ is an injective isometric $*$-homomorphism $C_0(\mathbb{R}) \to \prod_N A_N / \bigoplus_N A_N$.

With the origin at the first reading ($m_N = 0$) the same construction gives $C_0([0, \infty))$: the recorded time has a beginning, and $\mathbb{R}$ is the limit seen from readings far from both ends.

Proof. (a) The readings form a grid of mesh $\Delta t_N$ covering $[-m_N\Delta t_N, (N - m_N)\Delta t_N]$, which eventually contains $[-R, R]$. (b) Evaluations are $*$-homomorphisms, and $\|s_N f\| \leq \|f\|_\infty$. A function $f \in C_0(\mathbb{R})$ attains $|f|$'s maximum at some $x_0$; by (a) some reading $t_k$ is within $\Delta t_N/2$ of $x_0$ eventually, so $\|s_N f\| \geq \|f\|_\infty - \omega_f(\Delta t_N/2)$, where $\omega_f$ is the modulus of uniform continuity of $f$. The norm on $\prod_N A_N/\bigoplus_N A_N$ is $\limsup_N \|s_N f\| = \|f\|_\infty$. $\blacksquare$

The contrast with the other clocks of §3.8 and §11.3 is in the shape of the reading set: summed O-clocks fill a circle of fixed circumference, composite O-clocks with incommensurate frequencies fill a torus, the depth register fills a line. In the positional realisation, $N + 1 = 7^M$ readings take $M$ holons, so the window is $7^M - 1$ chronons; in a macroscopic unit $T_{\text{unit}}$ the limit of Theorem 11.5 is reached with $\Delta t = \delta\tau/T_{\text{unit}} \to 0$ and $7^M \delta\tau / T_{\text{unit}} \to \infty$.

Routes that were examined and not used. (1) Davies weak coupling (1974) gives $\mathcal{L}_0$ only in the limit of vanishing coupling with an infinite environment, and its error constants for a finite environment are not explicit; the construction above is exact. (2) Quasi-ideal clocks (Woods–Silva–Oppenheim 2019) have readings that form a group orbit on a finite space, so by the almost-periodicity Lemma any dissipation relative to them holds only on a finite window, and only approximately; they are not needed. (3) Poincaré recurrence times of a finite environment depend on Diophantine properties of its spectrum and admit no computable bound of the form $\varepsilon(d)$ uniform in the Hamiltonian, so they cannot give a theorem with explicit constants. (4) An infinite register $\ell^2(\mathbb{N})$ as the limit of the tower of strata: its Feynman–Kitaev constraint has no normalisable kernel (option (c) of §11.3), so it is not a carrier but only the limit of carriers. (5) Deriving the chain register from the axioms: not achieved. The register is built from holons, but that the world's timeless state is of the constructed kind is the constraint assumption of A5, as for the O-clock.

11.5 The de Sitter observer algebra and the holon tower

Since 2022 there is a precise sense in which an observer changes the algebra of a gravitating region. Quantum fields in a region have an algebra of type III₁, which has no trace and no finite entropy; adding an observer with a clock and imposing the gravitational constraint turns it into an algebra of type II, which has a trace, density matrices and entropies. In the static patch of de Sitter space the result is of type II₁: the trace is finite and there is a state of maximal entropy. Two points of contact with UHM follow. The observer is a clock — a Page–Wootters reference frame (§3, §11.1) — and the maximal-entropy state of the holon is its trace $I/7$. This subsection lists what the literature establishes, then proves what can be proved about the relation (T-348), and says what does not follow.

Terms. A von Neumann algebra is an algebra of bounded operators closed under adjoints and weak limits; a factor is one whose centre is the scalars. Type I factors are full matrix algebras $B(\mathcal H)$; a type II₁ factor is infinite-dimensional and has a finite trace, normalised to $\mathrm{Tr}\,1 = 1$; type $\mathrm{II}_\infty$ has a trace that is only semifinite; type III has no trace. A factor is hyperfinite (equivalently injective) if it is the weak closure of an increasing union of finite-dimensional algebras. The entropy relative to the trace of a state with density $\rho$ is $S = -\mathrm{Tr}\,\rho\ln\rho$; in a II₁ factor $S \leq 0$ with equality only for $\rho = 1$.

What the literature establishes.

• Chandrasekaran, Longo, Penington, Witten (CLPW). "An algebra of observables for de Sitter space", JHEP 02 (2023) 082, arXiv:2206.10780. Theorem within their model ($G_N \to 0$, one observer on a geodesic). The matter algebra $\mathcal A$ of the static patch is of type III₁ and its only operators invariant under the static-patch Hamiltonian $H$ are scalars. The observer has Hamiltonian $H_{\text{obs}} = q \geq 0$ on $L^2(\mathbb R_+)$ ("it is physically sensible to assume that the energy of the observer is non-negative"); the constraint is $H + q = 0$. Without $q \geq 0$ the invariant algebra is the crossed product of $\mathcal A$ by its modular flow, of type $\mathrm{II}_\infty$ (Takesaki duality); the projection $\Theta(q)$ onto $q \geq 0$ cuts it to a type II₁ factor $\hat{\mathcal A}$ with $\mathrm{Tr}\,1 = 1$. The state of maximal entropy is the Bunch–Davies state times the thermal distribution $p(q) = \beta_{\text{dS}} e^{-\beta_{\text{dS}} q}$ of the observer's energy, and entropies equal the generalised entropy up to a state-independent constant. Replacing $q \geq 0$ by $q \geq m$ keeps type II₁. Their §2.4 adds: a hyperfinite II₁ factor is isomorphic to the Murray–von Neumann algebra of infinitely many almost maximally mixed qubits; local algebras of quantum field theory "are believed to be always hyperfinite", so "we expect" $\hat{\mathcal A}$ to be that algebra. Their footnote 8: a clock whose energy is bounded below cannot tell time perfectly.
• Jensen, Sorce, Speranza (JSS). "Generalized entropy for general subregions in quantum gravity", JHEP 12 (2023) 020, arXiv:2306.01837. Argument at $G_N \to 0$. A spatially compact region with an observer gives type II₁ and an entropy-maximising state; a region containing an asymptotic boundary gives type $\mathrm{II}_\infty$. Their examples include a ball in flat space with conformal matter: type II₁ is not tied to $\Lambda > 0$.
• Related constructions. S. Ali Ahmad, R. Jefferson, SciPost Phys. Core 7, 020 (2024), arXiv:2306.07323 (crossed products by the modular flow for subregions of general theories give type $\mathrm{II}_\infty$); J. Kudler-Flam, S. Leutheusser, G. Satishchandran, Phys. Rev. D 111, 025013 (2025), arXiv:2309.15897 (for dynamical black holes the entropy of the dressed algebra is the generalised entropy); T. Faulkner, A. J. Speranza, arXiv:2405.00847 (the generalised second law from the crossed product); C.-H. Chen, G. Penington, arXiv:2406.02116 (slow-roll inflation and evaporating Schwarzschild–de Sitter give type $\mathrm{II}_\infty$ factors, with no state of maximal entropy). Arguments within their models; none of them derives $\Lambda$ or a coupling constant.
• De Vuyst, Eccles, Höhn, Kirklin (DEHK). "Gravitational entropy is observer-dependent", arXiv:2405.00114. Theorems within the model. The observers of CLPW are quantum reference frames, and the CLPW description of states is the Page–Wootters one ("PW = CLPW"). The type and the entropy depend on the observer: a clock is ideal if its energy spectrum is the whole line, and then the algebra stays $\mathrm{II}_\infty$; a non-ideal clock with energy bounded below gives II₁. A periodic clock (a harmonic oscillator) is an incomplete frame: its isotropy group is $\mathbb Z$, and it resolves only properties of the field that are themselves periodic.
• Fewster, Janssen, Loveridge, Rejzner, Waldron. "Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory", Commun. Math. Phys. 406, 19 (2025), arXiv:2403.11973. Theorem. If the field has a KMS state at some $\beta > 0$, the frame-invariant algebra has a semifinite trace; if the frame also has a KMS weight at the same temperature, the trace is finite; precise sufficient conditions for a type II₁ factor.
• Witten. "A background independent algebra in quantum gravity", JHEP 03 (2024) 077, arXiv:2308.03663. Proposal. The algebra along an observer's worldline, with the Hartle–Hawking state as universal state of maximal entropy; with only de Sitter vacua of different $\Lambda$ the definition gives sensible results. $\Lambda$ enters as a label of the vacua, not as an output.
• Operator-algebra results used below. F. J. Murray, J. von Neumann, Ann. Math. 44, 716 (1943): the approximately finite II₁ factor is unique. A. Connes, Ann. Math. 104, 73 (1976): every injective II₁ factor with separable predual is isomorphic to it, $R$. U. Haagerup, Acta Math. 158, 95 (1987): uniqueness of the injective III₁ factor. H. Araki, E. J. Woods, Publ. RIMS 4, 51 (1968): types of infinite tensor products of matrix algebras (ITPFI). A. Connes, E. Størmer, J. Funct. Anal. 28, 187 (1978): in a III₁ factor any two faithful normal states are approximately unitarily equivalent. D. Buchholz, C. D'Antoni, K. Fredenhagen, Commun. Math. Phys. 111, 123 (1987): the split property makes local algebras hyperfinite. R. Verch, Lett. Math. Phys. 29, 297 (1993), and C. J. Fewster, Lett. Math. Phys. 105, 1633 (2015), arXiv:1501.02682: the split property for the free Klein–Gordon field and for locally covariant theories in curved spacetime.

Theorem T-348 (The holon tower and the de Sitter observer algebra) [T]

Let $A_M = M_7(\mathbb C)^{\otimes M}$ be the algebra of the O-registers of $M$ holons — the carrier of the depth register of §11.4 — with the normalised trace $\mathrm{tr}_M = \mathrm{Tr}/7^M$, embedded by $A_M \to A_{M+1}$, $x \mapsto x \otimes 1$ (one holon added).

(a) The tower is $R$. $\mathrm{tr}_{M+1}(x \otimes 1) = \mathrm{tr}_M(x)$; the union $A_\infty$ is the UHF algebra of type $7^\infty$, and its closure in the representation of $\mathrm{tr}_\infty$ is a hyperfinite factor of type II₁, hence $R$. For a state $\rho$ on $A_M$, $S_{\mathrm{tr}}(\rho) = S(\rho) - M\ln 7 = -D(\rho\,\|\,I/7^M) \leq 0$, with equality only at $I/7^M$; restricting to $A_{M-1}$ does not lower $S_{\mathrm{tr}}$, and for a normal state $\omega$ of $R$, $S_{\mathrm{tr}}(\omega) = \lim_M S_{\mathrm{tr}}(\omega|_{A_M})$. The number 7 survives only before closure: projections of $A_M$ have traces $k/7^M$, never $1/2$, while $R$ has projections of every trace in $[0, 1]$, and $\bigotimes(M_2, \mathrm{tr}_2)$ gives the same $R$. If the added holon is read as the lowest digit of the depth register, reading $n \in \{0, \ldots, 7^M - 1\}$ marks the point $n/7^M$ of $[0, 1)$, and the trace of the projection onto the readings in $[a, b)$ differs from $b - a$ by at most $2/7^M$: on the readings the trace becomes Lebesgue measure — the uniform weight $p(n) = 1/(N+1)$ of the history state of Theorem 11.1.

(b) CLPW is hyperfinite, under a named condition. If the matter net in the Bunch–Davies representation has the split property along an increasing sequence of regions exhausting the static patch — established for the free massive Klein–Gordon field (Verch 1993; Fewster 2015) — then $\mathcal A$, $\mathcal A \rtimes_\sigma \mathbb R$ and $\hat{\mathcal A} = \Theta(q)(\mathcal A \rtimes_\sigma \mathbb R)\Theta(q)$ are injective, and $\hat{\mathcal A} \cong R$.

(c) What the isomorphism carries. Under (b) there is an isomorphism $\Phi: R \to \hat{\mathcal A}$. It is not unique (two differ by an automorphism of $R$), and every such $\Phi$ carries $\mathrm{tr}$ to $\mathrm{Tr}$: the maximal-entropy state of CLPW corresponds to the product state $\bigotimes I/7$, and $S_{\hat{\mathcal A}}(\omega) = S_{\mathrm{tr}}(\omega \circ \Phi)$ for every normal state. Nothing else is carried: no $\Phi$ is singled out that maps $A_M$ to operators with a place or a time, and the observer's Hamiltonian, the modular flow of the Bunch–Davies state and the geometry of the patch are not transported.

(d) Only finitely many holons are alive. In every normal state $\omega$ of $R$ the reduced state $\rho_n$ of the $n$-th holon tends to $I/7$ in trace norm, so only finitely many holons are viable ($P > 2/7$). If $L \geq 1$ of them are viable, $S_{\mathrm{tr}}(\omega) < -L\,D^*$ with $D^* = \ln 7 - S(\lambda^*) = 0.34406$ nat, $\lambda^* = \big((1+\sqrt6)/7,\ (6-\sqrt6)/42 \times 6\big)$. An infinite product $\bigotimes\rho$ of one living state is not normal on $R$: it generates an Araki–Woods factor of type III — $\mathrm{III}_\lambda$ when all eigenvalue ratios of $\rho$ are integer powers of one $\lambda \in (0,1)$, III₁ otherwise. Type II₁ arises only for $\rho = I/7$, whose purity $1/7$ lies below the window.

(e) A finite clock gives a type I algebra. Let $\mathcal A$ be a factor on which $H$ generates a flow with a KMS state at $\beta > 0$ and $\mathcal A^H = \mathbb C$ (the assumption of CLPW), and let the clock Hamiltonian $H_c$ have pure point spectrum — the O-clock $H_O = \omega_0\,\mathrm{diag}(0, \ldots, 6)$, or the depth register at any finite $N$. Then the part of $\mathcal A \otimes B(\mathcal H_c)$ invariant under $H + H_c$ is $\bigoplus_E \mathbb C \otimes B(\ker(H_c - E))$: type I, with no operator of the field in it.

(f) No $\Lambda$ and no $\kappa$. The type and the trace of $\hat{\mathcal A}$ are the same for every $\beta_{\text{dS}} > 0$ (every $\Lambda > 0$) and every observer mass ($q \geq m$); the tower gives the same $R$ for every factor dimension $d \geq 2$; JSS obtain II₁ for compact regions in flat space. So no invariant of the identification fixes $\Lambda$, its sign, or the regeneration rate $\kappa$, which (a)–(d) never use (cf. T-346). A finite truncation $7^M = e^{S_{\text{dS}}}$, $S_{\text{dS}} = 3\pi/(\Lambda\ell_P^2) = 3.263 \times 10^{122}$ at $\Lambda = 1.1056 \times 10^{-52}\ \mathrm{m}^{-2}$, gives $M = 1.677 \times 10^{122}$ — a strictly monotone reparametrisation of $\Lambda$, not a derivation, because nothing in UHM fixes $M$.

Proof. (a) $\mathrm{Tr}(x \otimes 1) = 7\,\mathrm{Tr}\,x$. Every tracial state of $M_{7^M}(\mathbb C)$ is $\mathrm{tr}_M$, so $A_\infty$ has exactly one tracial state; a tracial state whose representation had a non-trivial centre would split into two different traces, so the closure is a factor. The trace is faithful and normal on it, the factor is infinite-dimensional, so it is of type II₁, and as the closure of an increasing union of matrix algebras it is hyperfinite; Murray–von Neumann uniqueness gives $R$. The density of $\rho$ relative to $\mathrm{tr}_M$ is $7^M\rho$, so $-\mathrm{tr}_M(7^M\rho\ln 7^M\rho) = S(\rho) - M\ln 7$. Monotonicity is that of relative entropy under the partial trace (G. Lindblad, Commun. Math. Phys. 40, 147 (1975)), and the limit is the martingale property of Araki's relative entropy along an increasing net with dense union (M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer 1993, ch. 5). A projection of $A_M$ has trace $\mathrm{rank}/7^M$, and $7^M$ is odd. The readings $n/7^M$ in $[a, b)$ number $7^M(b - a)$ up to $\pm 2$. The argument uses $d = 7$ nowhere except in the arithmetic.

(b) By additivity $\mathcal A = (\bigcup_n \mathcal A(O_n))''$. Split: between $\mathcal A(O_n) \subset \mathcal A(O_{n+1})$ sits a type I factor $N_n$, so $\mathcal A = (\bigcup_n N_n)''$; the closure of an increasing union of injective algebras is injective (Buchholz–D'Antoni–Fredenhagen). A crossed product of an injective algebra by the amenable group $\mathbb R$ is injective; a corner $pMp$ of an injective $M$ is injective (compose the expectation onto $M$ with $x \mapsto pxp$). $\hat{\mathcal A}$ is a II₁ factor (CLPW) on a separable Hilbert space; Connes' theorem gives $\hat{\mathcal A} \cong R$.

(c) $\mathrm{Tr} \circ \Phi$ is a normalised normal trace on $R$, and a II₁ factor has only one. CLPW show that their maximal-entropy state is $\hat a \mapsto \mathrm{Tr}\,\hat a$, and entropies are defined through densities relative to the trace, which $\Phi$ preserves. Two isomorphisms differ by an automorphism of $R$; the group of automorphisms moves every finite-dimensional subalgebra $A_M$ to other copies of $M_{7^M}$, so the image of $A_M$ has no invariant meaning.

(d) Let $\omega = \mathrm{tr}(d\,\cdot)$. $A_\infty$ is dense in $L^1(R, \mathrm{tr})$: pick $d_k \in A_k$, $\mathrm{tr}\,d_k = 1$, $\|d - d_k\|_1 \to 0$. For $x$ in a factor $n > k$ the product trace gives $\mathrm{tr}(d_k x) = \mathrm{tr}(x)$, so $\|\rho_n - I/7\|_1 \leq 2\|d - d_k\|_1 \to 0$, and $P(\rho_n) \to 1/7$. For the bound, take $K$ larger than every viable index: $D(\omega\|\mathrm{tr}) \geq D(\omega|_{A_K}\|\mathrm{tr}_K) = \sum_{n \leq K} D(\rho_n\|I/7) + C \geq \sum_{\text{viable}} D(\rho_n\|I/7)$, with $C \geq 0$ the total correlation. At fixed purity the entropy is maximal on one dominant eigenvalue plus equal others (P. Harremoës, F. Topsøe, IEEE Trans. Inf. Theory 47, 2944 (2001)), and this maximum falls as $P$ grows, so $D(\rho\|I/7) > D^*$ when $P > 2/7$. Two product states $\bigotimes\rho_n$, $\bigotimes\tau_n$ are quasi-equivalent only if $\prod_n F(\rho_n, \tau_n) > 0$ (D. Bures, Trans. Amer. Math. Soc. 135, 199 (1969)); for a living $\rho$, $F(\rho, I/7) < 0.85505$ (at fixed purity the fidelity with $I/7$ is largest on the same extremal spectrum, $0.85505$ at $P = 2/7$; checked numerically to $10^{-15}$), so the product with $\bigotimes I/7$ diverges. The type is that of the Araki–Woods classification: for identical factors the asymptotic ratio set is the closed subgroup generated by the eigenvalue ratios.

(e) Write an invariant element as a block matrix $(x_{EE'})$ over the eigenspaces of $H_c$. Invariance means $\sigma_t(x_{EE'}) = e^{i(E - E')t}x_{EE'}$ entrywise, where $\sigma_t = \mathrm{Ad}\,e^{itH}$. For $E = E'$ the entries lie in $\mathcal A^H = \mathbb C$. For $\mu = E - E' \neq 0$ an eigen-operator $x$ has $x^*x, xx^* \in \mathcal A^H$, so if $x \neq 0$ it is a multiple of a unitary $u$ with $\sigma_t(u) = e^{i\mu t}u$; the KMS condition $\psi(u^*\sigma_{i\beta}(u)) = \psi(uu^*)$ gives $e^{-\beta\mu} = 1$, a contradiction. So the off-diagonal blocks vanish.

(f) (b)–(c) hold verbatim for every $\beta_{\text{dS}} > 0$ and every $m$ (CLPW §2.5), and (a) for every $d$. The numbers are arithmetic; $M \mapsto \Lambda = 3\pi/(\ell_P^2 M\ln 7)$ is strictly decreasing. $\blacksquare$

Status. (a), (d), (f) are theorems [T]. (b) and (c) are theorems under the split property, which is proved for free massive matter and assumed for interacting matter; (e) is a theorem under the assumption $\mathcal A^H = \mathbb C$ that CLPW themselves use. CLPW already expected the isomorphism of (c), with qubits in place of holons; what T-348 adds is the holon tower as the concrete approximating chain, the list of what an isomorphism can and cannot carry, the count of living holons (d), the verdict on finite clocks (e), and the no-go for $\Lambda$ and $\kappa$ (f). Witnesses: check_core_numbers.py (holon_tower_trace_entropy_is_nonpositive_and_monotone, living_holon_costs_at_least_0344_nats_below_the_trace, lambda_as_a_holon_count_is_a_reparametrisation).

Physical reading [I]. Empty de Sitter space, the state of maximal entropy, corresponds to the tower in which every holon is at $I/7$: the dead background, below the window. Excitations are normal states, and by (d) they carry finitely many living holons, each costing more than $0.344$ nat of entropy below the maximum; in CLPW a particle of energy $E$ at the centre of the patch costs $\beta_{\text{dS}}E$. The comparison is an interpretation: (c) gives no map from holons to particles. The CLPW condition $q \geq 0$ has one UHM counterpart, $H_O \geq 0$; by (e) it is not enough, because the step from type III to type II needs a clock with continuous energy spectrum, and every finite clock of UHM has pure point spectrum; the limit of the depth register does have continuous spectrum bounded below (T-352, §11.6).

Open [Pr]. (1) A UHM clock with continuous spectrum bounded below. The scaling limit of Theorem 11.5 is a continuum of readings (times), not of energies. Answered by T-352 (§11.6) at a stated premise: the Feynman–Kitaev constraint of the depth register itself, in the limit $N \to \infty$, has purely absolutely continuous spectrum $[0, 4\hbar/\delta\tau]$, and as the observer's clock it gives a type II₁ factor; an earlier sentence here, that such a spectrum is a choice of level spacing and not a consequence of the construction, is superseded. (2) Whether the relational algebras of such registers converge to $\hat{\mathcal A}$. (3) Whether anything in UHM fixes the level $M$; without it $\Lambda$ stays an input, as in Witten's proposal.

11.6 Direction and the continuous clock

The foundation of UHM is an ∞-topos $\mathrm{Sh}_\infty(\mathcal C)$, read in homotopy type theory, where every identification is invertible: its objects are ∞-groupoids. The dynamics is not invertible: a dissipative channel has no inverse among channels, the depth of §10.3 grows, and T-348 left open a clock with continuous energy spectrum bounded below. This subsection asks what of the arrow a symmetric foundation can carry, where the direction lives, and whether UHM has the clock that T-348 lacked. It lists what the literature establishes, then proves T-352.

What the literature establishes.

• Directed type theory, synthetic ∞-categories. E. Riehl, M. Shulman, "A type theory for synthetic ∞-categories", Higher Structures 1, 147 (2017), arXiv:1705.07442. Theorems. Homotopy type theory with a directed interval and extension types; Segal types (unique composites up to homotopy), Rezk types (local univalence), covariant families and a dependent Yoneda lemma. Semantics: Reedy fibrant bisimplicial sets, where Segal and Rezk types are Segal spaces and complete Segal spaces. U. Buchholtz, J. Weinberger, "Synthetic fibered (∞,1)-category theory", Higher Structures 7 (2023), arXiv:2105.01724; J. Weinberger, "Strict stability of extension types", arXiv:2203.07194 (the system "has semantics in simplicial objects of an ∞-topos").
• Directed univalence. M. Z. Weaver, D. R. Licata, "A constructive model of directed univalence in bicubical sets", LICS 2020, 915–928. D. Gratzer, J. Weinberger, U. Buchholtz, "Directed univalence in simplicial homotopy type theory", arXiv:2407.09146 (theorem: a universe of discrete types whose homomorphisms are functions, built in triangulated type theory); "The Yoneda embedding in simplicial type theory", LICS 2025, 127–142, arXiv:2501.13229; "The ∞-category of ∞-categories in simplicial type theory", arXiv:2602.02218 (straightening–unstraightening inside the theory). E. Cavallo, E. Riehl, C. Sattler, "Directed univalence for simplicial objects in an ∞-topos", arXiv:2607.02420 (July 2026; theorem: directed univalence holds in simplicial objects of any ∞-topos).
• Other directed type theories. D. R. Licata, R. Harper, "2-dimensional directed type theory", MFPS 2011, ENTCS 276, 263 (2011); A. Nuyts, Towards a directed homotopy type theory based on 4 kinds of variance, MSc thesis, KU Leuven (2015); P. R. North, "Towards a directed homotopy type theory", MFPS 2019, ENTCS 347 (2019), arXiv:1807.10566 (a hom-type former interpreted in $\mathbf{Cat}$); J. Neumann, T. Altenkirch, "Synthetic 1-categories in directed type theory", TYPES 2024, LIPIcs 336, 7, arXiv:2410.19520 (a directed-J rule, variance by modalities; a first step toward a directed higher observational type theory).
• Model-independent ∞-category theory and internal categories. E. Riehl, D. Verity, Elements of ∞-Category Theory, Cambridge 2022 (∞-cosmoi: one formal theory for quasi-categories, complete Segal spaces and their fibred and internal variants). L. Martini, "Yoneda's lemma for internal higher categories", arXiv:2103.17141, and L. Martini, S. Wolf, "Internal higher topos theory", arXiv:2303.06437: ∞-categories internal to an arbitrary ∞-topos are simplicial objects of it satisfying the Segal and completeness conditions.
• Directed spaces. M. Grandis, Directed Algebraic Topology, Cambridge 2009, arXiv:math/0111048 for part I (d-spaces, fundamental categories in place of fundamental groupoids, motivated by concurrency and by portions of space-time). Theorems within the framework; no directed analogue of a Grothendieck topos serves as a foundation there.
• Time operators and positive energy. W. Pauli, Handbuch der Physik 24/1 (1933), p. 140 (argument); in rigorous form: a self-adjoint $H$ and a projection-valued time observable covariant under $e^{-isH}$ form a system of imprimitivity of $\mathbb R$, and the Mackey–Stone–von Neumann theorem gives $\sigma(H) = \mathbb R$ (P. Busch, M. Grabowski, P. J. Lahti, Operational Quantum Physics, Springer 1995, ch. III), so a Hamiltonian bounded below admits only a covariant positive operator valued time. R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, AMS 1934 (functions supported on a half-line have Fourier transforms in the Hardy space $H^2$). H. Wold (1938) and B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland 1970 (isometries: the Wold decomposition and minimal unitary dilations). P. D. Lax, R. S. Phillips, Scattering Theory, Academic Press 1967 (translation on the line with outgoing subspace $L^2(\mathbb R_+)$).

What remains open in the literature. Directed type theory has models in simplicial objects of every ∞-topos, and directed univalence holds there (2026). It does not replace the ∞-topos: its types are interpreted inside one, as category objects. There is no established "directed topos" as a stand-alone foundation, and, beyond Grandis' motivating examples, no link in this literature between directed paths and physical time; Chapter 17 of the mathematics corpus names that link as a hypothesis [H].

Theorem T-352 (The directed depth register and its continuous clock) [T]

Let $[N]$ be the readings $0 < 1 < \cdots < N$ of the depth register (§11.4) as a category, with one arrow $m \to n$ exactly when $m \leq n$; $\mathbb N$ in the limit. Let $\mathbf{Chan}_7$ be the category whose objects are the states of a holon and whose arrows are CPTP maps.

(a) The arrow lives in a category, not in a groupoid. The maximal subgroupoid of $[N]$ is discrete ($N+1$ objects, identities only); its ∞-groupoid completion is contractible (the nerve has Euler characteristic $\sum_{k=0}^{N}(-1)^k\binom{N+1}{k+1} = 1$). On an ∞-groupoid every function that does not increase along paths is constant on components. The history of Theorem 11.1 is a functor $\Gamma: [N] \to \mathbf{Chan}_7$, $n \mapsto \rho_S(n)$, $(m \leq n) \mapsto \Phi_{(n-m)\Delta t}$; for $\mathcal L_0$ unital and primitive no arrow $\Phi_t$, $t > 0$, is invertible in $\mathbf{Chan}_7$, and $D(\cdot\,\|\,I/7)$ turns $\Gamma$ into a functor to the poset $(\mathbb R_{\geq 0}, \geq)$.

(b) Sharp readings with a first reading have no Hamiltonian. (i) On $\ell^2(\mathbb N)$ the step $S|n\rangle = |n+1\rangle$ is an isometry with $SS^* = 1 - |0\rangle\langle 0|$, purely non-unitary; $\sigma(S)$ is the closed unit disc and every $|\lambda| < 1$ is an eigenvalue of $S^*$, with eigenvector $\sum_n \lambda^{n}|n\rangle$; no unitary on $\ell^2(\mathbb N)$ extends $S$, and its minimal unitary dilation is the bilateral shift on $\ell^2(\mathbb Z)$, with spectrum the unit circle. (ii) In the continuum limit with the origin at the first reading (Theorem 11.5, $C_0([0,\infty))$), the translations $(T_tf)(x) = f(x-t)\,\mathbf 1_{x \geq t}$ of $L^2(\mathbb R_+)$ are $T_t = e^{-itP}$ with $P = -i\,d/dx$ on $\{f \in H^1(\mathbb R_+) : f(0) = 0\}$; $P$ is closed and symmetric with deficiency indices $(1, 0)$, has no self-adjoint extension, and $\sigma(P) = \{\lambda : \mathrm{Im}\,\lambda \leq 0\}$. (iii) On $L^2(\mathbb R)$, no nonzero $\psi$ is supported in $[0, \infty)$ while its Fourier transform is supported in $[0, \infty)$. So a clock whose sharp readings have a first reading and are shifted by its own evolution has no self-adjoint energy; its unitary dilation has energy spectrum $\mathbb R$ — the ideal clock that keeps the algebra of type $\mathrm{II}_\infty$ (DEHK) — and Pauli's theorem, in the rigorous form cited above, denies sharp covariant readings to an energy bounded below.

(c) The clock the constraint already contains. The dressed constraint of Theorem 11.1(a) is the path Laplacian $\Lambda_N$. Its eigenvalues $E_k = 2 - 2\cos(\pi k/(N+1))$ are exactly the quantiles $k/(N+1)$ of the arcsine law $F(E) = \tfrac{2}{\pi}\arcsin(\sqrt E/2)$ on $[0, 4]$, so the normalised counting measure of the spectrum is within $1/(N+1)$ of $F$ in Kolmogorov distance, attained. In the limit, $\Lambda_\infty = 2 - S - S^* - |0\rangle\langle 0|$ on $\ell^2(\mathbb N)$ is bounded and self-adjoint; the cosine transform $(\mathcal C x)(k) = \sqrt{2/\pi}\,\sum_n x_n \cos((n+\tfrac12)k)$ is a unitary $\ell^2(\mathbb N) \to L^2([0, \pi])$ carrying $\Lambda_\infty$ to multiplication by $4\sin^2(k/2)$. Hence $\sigma(\Lambda_\infty) = [0, 4]$, purely absolutely continuous and simple, with density of states $1/(\pi\sqrt{E(4-E)})$; $0$ is not an eigenvalue. For every $\psi$ and every $K$, the probability of a reading $\leq K$ in $e^{-is\Lambda_\infty}\psi$ tends to $0$ as $s \to +\infty$ and as $s \to -\infty$.

(d) Type II₁ from the depth register. In the setting of CLPW — a type III₁ factor $\mathcal A$ on $\mathcal H$ whose modular flow is generated by $H$, KMS at $\beta > 0$ — take as the observer's clock $\ell^2(\mathbb N)$ with Hamiltonian $H_c = \varepsilon\Lambda_\infty$, $\varepsilon > 0$ ($\varepsilon = \hbar/\delta\tau$ for a chronon $\delta\tau$). The operators of $\mathcal A \otimes B(\ell^2(\mathbb N))$ invariant under $H + H_c$ form an algebra unitarily equivalent to $\Pi(\mathcal A \rtimes_\sigma \mathbb R)\Pi$, $\Pi = \mathbf 1_{[0, 4\varepsilon]}(q)$, with $q$ the observer energy of CLPW. It is a factor of type II₁; in the normalisation where CLPW's projection onto $q \geq 0$ has trace $1$, $\mathrm{Tr}\,\Pi = 1 - e^{-4\beta\varepsilon}$, and its state of maximal entropy is the Bunch–Davies state times the thermal law $\beta e^{-\beta q}$ of the observer energy restricted to $[0, 4\varepsilon]$ and renormalised. Under the split property of T-348(b) it is isomorphic to $R$. At every finite $N$ the same construction gives type I (T-348(e)): the passage from type I to type II₁ happens exactly at $N = \infty$, where the history state stops being normalisable (§11.3, option (c)).

(e) What is not fixed. The type and the trace structure in (d) are the same for every $\varepsilon > 0$ and every $\beta > 0$; the only new scale is the bandwidth $4\varepsilon = 4\hbar/\delta\tau$, and nothing in UHM fixes $\delta\tau$.

Proof. (a) An arrow $m \to n$ of $[N]$ has an inverse only if $n \to m$ exists too, that is $m = n$. The ∞-groupoid completion of a category is the geometric realisation of its nerve, and a category with an initial object ($0$) has contractible nerve (D. Quillen, "Higher algebraic K-theory I", LNM 341, 85 (1973), §1). The non-degenerate $k$-simplices of the nerve of $[N]$ are the chains of $k+1$ distinct readings, $\binom{N+1}{k+1}$ of them, and $\sum_k (-1)^k\binom{N+1}{k+1} = 1 - (1-1)^{N+1} = 1$. On an ∞-groupoid every path $x \rightsquigarrow y$ has an inverse, so $f(x) \geq f(y) \geq f(x)$. Functoriality of $\Gamma$ is the semigroup law $\Phi_s\Phi_t = \Phi_{s+t}$. A channel with a channel inverse is a unitary conjugation (standard; proved in full through Kraus representations as Lemma 17.1 of Chapter 17 of Part XVIII of the mathematics-foundations corpus), which preserves purity; by Theorem 11.1(c) $\Phi_t$ lowers the purity of every $\rho_0 \neq I/7$, so it has no inverse. Monotonicity of relative entropy under channels (G. Lindblad, Commun. Math. Phys. 40, 147 (1975)) with $\Phi_t(I/7) = I/7$ gives the functor to $(\mathbb R_{\geq 0}, \geq)$.

(b) (i) $S^*S = 1$ and $SS^*$ omits $|0\rangle$, so $S$ is not onto and no unitary on the same space agrees with it; $S^{*n} \to 0$ strongly, so $S$ has no unitary part (Wold). $S^*\sum_n \lambda^{n}|n\rangle = \lambda\sum_n\lambda^{n}|n\rangle$, a vector of squared norm $1/(1 - |\lambda|^2)$; so the open disc lies in the spectrum of $S^*$, hence in that of $S$, and $\|S\| = 1$ closes it. The minimal unitary dilation is Sz.-Nagy–Foias, ch. I. (ii) $\frac{d}{dt}T_tf = -f'$, and $T_tf$ vanishes on $[0, t)$, which forces $f(0) = 0$ on the domain. $P^* = -i\,d/dx$ on $H^1(\mathbb R_+)$; $P^*f = if$ has the solution $e^{-x} \in L^2$, $P^*f = -if$ only $e^{x} \notin L^2$, so the indices are $(1, 0)$ and von Neumann's theorem excludes a self-adjoint extension. $P^*f = \lambda f$ has the solution $e^{i\lambda x}$, square-integrable exactly when $\mathrm{Im}\,\lambda > 0$; for such $\lambda$ the range of $P - \bar\lambda$ is not dense, so $\bar\lambda \in \sigma(P)$; for $\mathrm{Im}\,\lambda > 0$, $\|(P - \lambda)f\| \geq \mathrm{Im}\,\lambda\,\|f\|$ and the range is dense, so $\lambda \notin \sigma(P)$. (iii) If $\psi$ vanishes on $(-\infty, 0)$, its Fourier transform is the boundary value of a Hardy-space function (Paley–Wiener); a nonzero $H^2$ function cannot vanish on a set of positive measure of the boundary (F. and M. Riesz). The last sentence is (ii), the dilation of (i), and Pauli's theorem as cited above.

(c) $\Lambda_N$ has diagonal $(1, 2, \ldots, 2, 1)$ and $-1$ on the off-diagonals. For $x_n = \cos((n+\tfrac12)k)$: in the interior $2x_n - x_{n-1} - x_{n+1} = (2 - 2\cos k)\,x_n$; at the first site $x_0 - x_1 = \cos\tfrac k2 - \cos\tfrac{3k}2 = 2\sin k\sin\tfrac k2 = (2 - 2\cos k)\,x_0$; at $n = N$ the condition is $\sin((N+1)k) = 0$, $k = \pi j/(N+1)$. $F(2 - 2\cos\theta) = \tfrac2\pi\arcsin(\sin\tfrac\theta2) = \theta/\pi$, so $F(E_k) = k/(N+1)$. The functions $\sqrt{2/\pi}\cos((n+\tfrac12)k)$, $n \geq 0$, are the eigenfunctions of $-d^2/dk^2$ on $[0, \pi]$ with $f'(0) = 0 = f(\pi)$, an orthonormal basis; so $\mathcal C$ is unitary, and the first computation gives $\mathcal C\Lambda_\infty\mathcal C^* = 4\sin^2(k/2)$. This function is continuous and strictly increasing on $[0, \pi]$ from $0$ to $4$; the change of variable $E = 4\sin^2(k/2)$, $dE = 2\sin k\,dk = \sqrt{E(4-E)}\,dk$, identifies $\Lambda_\infty$ with multiplication by $E$ on $L^2([0,4], dE/\sqrt{E(4-E)})$, absolutely continuous and of multiplicity one. For the escape, $\langle n|e^{-is\Lambda_\infty}\psi\rangle = \int_0^4 e^{-isE} g_n(E)\,dE$ with $g_n \in L^1$, which tends to $0$ as $|s| \to \infty$ (Riemann–Lebesgue lemma); sum over $n \leq K$.

(d) By (c) there is a unitary $V: \ell^2(\mathbb N) \to L^2([0, 4\varepsilon], dq)$ with $V H_c V^* = q$ (the density of states changes only the unitary). $1 \otimes V$ carries the operators invariant under $H + H_c$ to those invariant under $H + q$ on $\mathcal H \otimes L^2([0, 4\varepsilon])$. With $\Pi = \mathbf 1_{[0,4\varepsilon]}(q)$ on $L^2(\mathbb R)$, $\mathcal A \otimes B(L^2([0,4\varepsilon])) = \Pi(\mathcal A \otimes B(L^2(\mathbb R)))\Pi$. The invariant part of $\mathcal A \otimes B(L^2(\mathbb R))$ under $H + q$ is the crossed product $\mathcal A \rtimes_\sigma \mathbb R$, generated by $e^{ipH}ae^{-ipH}$ and $q$ (CLPW §3; M. Takesaki, Acta Math. 131, 249 (1973)); $\Pi$ belongs to it, and an operator $y = \Pi y\Pi$ is invariant exactly when it lies in $\Pi(\mathcal A \rtimes_\sigma \mathbb R)\Pi$. Since $\mathcal A$ is of type III₁, the crossed product is a factor of type $\mathrm{II}_\infty$ (Takesaki, loc. cit.), whose trace restricted to functions of $q$ is $f \mapsto \int f(q)\,\beta e^{-\beta q}\,dq$ in CLPW's normalisation — the observer distribution of their maximal-entropy state. So $\mathrm{Tr}\,\Pi = \int_0^{4\varepsilon}\beta e^{-\beta q}\,dq = 1 - e^{-4\beta\varepsilon}$, finite and nonzero; the corner of a factor by a nonzero finite projection is a factor of type II₁, and its normalised trace is the stated state. A corner of an injective algebra is injective, so T-348(b) and Connes' theorem give $R$. The finite case is T-348(e), since $\Lambda_N$ has pure point spectrum; $\ker\Lambda_\infty = 0$ by (c).

(e) Nothing in (a)–(d) depends on the values of $\varepsilon$ and $\beta$ beyond their positivity. $\blacksquare$

Numbers. The quantile identity $F(E_k) = k/(N+1)$ and the Kolmogorov distance $1/(N+1)$ hold to $10^{-12}$ for $N = 6, 48, 342, 2400$ ($N + 1 = 7, 49, 343, 2401$: the windows of $1$ to $4$ holons), and the cosine vectors diagonalise $\Lambda_N$ with residual below $2 \times 10^{-13}$. For $N = 4000$ and the start $|0\rangle$, the probability of a reading $\leq 10$ is $0.377$ at $s = 10$, $0.0175$ at $s = 200$ and $0.00233$ at $s = 1500$, and the mean reading grows as $\langle n\rangle/s \to 4/\pi = 1.273$ readings per unit of $s$ ($1.2729$ at $s = 1500$), while the weight beyond $n = 2.05\,s$ (the maximal velocity is $2$) is below $10^{-12}$ at $s = 1500$. The Euler characteristic of the nerve of $[N]$ is $1$ for $N = 1, 6, 48$. Witness: check_core_numbers.py (depth_register_constraint_is_a_continuous_clock_bounded_below, depth_readings_form_a_category_with_contractible_groupoid_completion).

Status. (a)–(c) and (e) are theorems [T]. (d) is a theorem in the model of CLPW (their $\mathcal A$, $H$, $\beta$, $G_N \to 0$) once the observer's clock is the depth register with Hamiltonian $\varepsilon\Lambda_\infty$; that identification is a premise, not a consequence of the axioms — as in §11.4, the constraint half of A5 is assumed, and here in addition that the constraint operator itself is the clock's energy. $R$ in (d) needs the split property, as in T-348(b).

What this changes.

• For the foundation. The arrow of T-53b and T-53c is a functor out of a category, $[N] \to \mathbf{Chan}_7$, followed by a monotone functional; by (a) no ∞-groupoid built from the readings can carry it — in the maximal subgroupoid there is nothing to be monotone along, and the completion is a point. It needs no new foundation: Rezk's classifying diagrams of $[N]$ and of $\mathbf{Chan}_7$ are complete Segal spaces (C. Rezk, Trans. Amer. Math. Soc. 353, 973 (2001)), which is where the Riehl–Shulman types live, and by Cavallo–Riehl–Sattler directed univalence holds in simplicial objects of any ∞-topos, so also over $\mathrm{Sh}_\infty(\mathcal C)$. The directed half that Chapter 17 of the mathematics corpus calls missing is thus available inside the symmetric foundation, as category objects; that the corpus should be rewritten in simplicial type theory is not claimed, and the identification of this direction with experienced time stays [H] there.
• For T-53b and T-118. No status changes. T-53b is the statement that $\Gamma$ is a functor and that $D$ decreases along it. Of the two limits of Theorem 11.5, $C_0([0, \infty))$ is the directed one: by (b)(ii) its translations are isometries without a Hamiltonian; $C_0(\mathbb R)$ is its two-sided dilation, whose translation generator has spectrum $\mathbb R$.
• For T-348. Open question (1) is answered at the premise of (d): UHM has a clock with continuous spectrum bounded below, and it is not a choice of level spacing — it is the $N \to \infty$ limit of the Feynman–Kitaev constraint that already carries the dynamics, with spectrum $[0, 4\hbar/\delta\tau]$ and the arcsine density. The type I verdict of T-348(e) for finite registers and the type II₁ of (d) are two sides of one fact: the constraint gains continuous spectrum exactly when its history state stops being normalisable. The difference from CLPW's clock is the upper edge: the observer energy is cut at $4\hbar/\delta\tau$, which changes the trace of the projection by $e^{-4\beta\hbar/\delta\tau}$ and not the type.

What does not follow. The time reading of the continuous clock is not sharp in the energy picture: by (b)(iii) and Pauli's theorem no covariant sharp reading exists for $\Lambda_\infty$, and the readings $|n\rangle$ are not shifted by $e^{-is\Lambda_\infty}$; a packet spreads at up to $2$ readings per unit of $s$, and the direction of the readings is the same for $s \to +\infty$ and $s \to -\infty$ — the unitary group does not know the arrow, the half-line of readings does. The theorem fixes neither $\delta\tau$ nor $\beta$, hence neither $\Lambda$ nor $\kappa$ (T-348(f), T-346).

Open [Pr]. (1) Whether the relational algebras of the finite registers, of type I, converge to the type II₁ algebra of (d) in a sense that carries the trace (T-348, open (2)). (2) Whether anything in UHM fixes the chronon $\delta\tau$, the only scale in (d). (3) A formulation of the holon dynamics in simplicial type theory in which $\mathbf{Chan}_7$ is a Rezk type over $\mathrm{Sh}_\infty(\mathcal C)$ and T-53c becomes an internal statement.

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