Chapter 1: Substrate

1.2 Temporal Ontology

Defining time in a universe that does not yet possess entropy or clocks presents a distinct challenge. While imagining a universal metronome ticking in the background is tempting, we know that in a background-independent theory, no such external reference exists. We must strip time down to its barest function. We must identify the mechanism that distinguishes one state from the next. Without this separation, there is no cause and effect. There is only a static singularity of information where everything happens at once. To rely on a pre-existing temporal coordinate would be to assume the very thing we are trying to derive. We must build time from the ground up as a process of change.

Distinguishing between the logical sequence of updates that drives the system and the physical time eventually measured by observers within it becomes essential. We must separate the hand that moves the pieces from the experience of the pieces themselves. Ordering events requires a mechanism that operates independently of the geometry of spacetime because the assembly of spacetime has not yet occurred. We need a raw iterator. We need a counter that marks the progression of the computational process itself. This iterator acts as the heartbeat of the algorithm. It ensures that events occur in a definitive sequence even before the concept of duration exists.

Establishing a dual architecture for time resolves this difficulty by separating the iterator from the metric. We also confront the paradoxes inherent in an infinite past. If the universe had no beginning, the information required to describe the current state would be boundless. This would violate the finiteness criterion we just established. We demonstrate that the timeline must be bounded in the past to avoid physical contradictions like the Grim Reaper paradox. Therefore, we define a temporal domain that initiates at zero and advances by integer steps. This boundary condition is not merely a philosophical preference but a logical necessity for a constructive theory. A program cannot run if it never starts.

---

1.2.1 Postulate: Dual Time Architecture

Separation of Emergent Physical Time from Fundamental Logical Time through a Dual-Time Architecture

The temporal structure of the physical theory is constituted by two distinct, orthogonal, and non-interchangeable parameters:

1. Global Logical Time ($t_L$): The fundamental ordering parameter of state evolution. The domain of $t_L$ is strictly restricted to the set of non-negative integers $\mathbb{N}_0$. This parameter serves as the discrete iteration counter for the Universal Evolution Operator and is not subject to relativistic dilation or coordinate transformation.
2. Physical Time ($t_{phys}$): An emergent, continuous parameter derived from relational path lengths within the graph substrate. $t_{phys}$ is subordinate to $t_L$ and possesses geometric character, emerging only in the macroscopic limit.

The foundational postulate of this theory asserts that physical reality emerges as a secondary phenomenon rather than serving as a primary, self-subsistent entity; this assertion compels an immediate and total rupture with standard temporal formulation, thereby necessitating the complete rejection of all such formulations without any form of compromise or partial retention. In their place, the theory introduces a strict dual-time structure, wherein two distinct temporal parameters operate at orthogonal levels of ontological priority, each fulfilling precisely defined roles that preclude overlap or interchangeability.

This dual-time structure comprises the following two components, rigorously delineated to ensure no ambiguity arises in their application or interpretation:

• $t_{phys}$: This parameter emerges within the internal dynamics of the physical system itself; it is inherently relational, meaning its values derive solely from comparisons among events or states embedded within the system; it possesses a geometric character, aligning with the curved spacetime metrics of general relativity; it remains local in scope, applicable only to subsystems or observers confined to specific regions of the universe; it appears continuous in the effective macroscopic limit, where quantum discreteness averages out to yield smooth trajectories; and it becomes measurable exclusively through the agency of physical clocks, which are themselves constituents of the system and thus subject to the same emergent constraints.

• $t_L$: This parameter stands as the fundamental temporal scaffold upon which all physical emergence depends; it originates externally to the physical system, positioned at a meta-theoretical level that transcends the system's own dynamics; it manifests as strictly discrete, advancing only in integer increments without intermediate fractional values; it enforces an absolute ordering across the entirety of the universe's state sequence, providing a universal "before" and "after" that admits no exceptions or relativizations; it remains strictly unobservable from the vantage point of any internal state within the system, as no physical process can access or register its progression; and it functions solely as the iteration counter within the universal computation, tallying each discrete application of the evolution operator without contributing to the observable content of the states themselves.

This distinction between $t_{phys}$ and $t_L$ constitutes not an optional ornament or heuristic convenience but an indispensable structural necessity. It represents the sole known resolution capable of simultaneously accommodating the following five critical requirements of a viable physical theory:

1. Background independence, which demands that no fixed external arena preconditions the dynamics;
2. Finite information content, which prohibits unbounded informational resources at any finite stage;
3. Causal acyclicity, which ensures that the partial order of causation contains no closed loops;
4. Constructive definability, which mandates that all entities and processes arise from finite specifications;
5. The phenomenon of evolution, wherein states succeed one another and generate observable change.

Any attempt to merge or conflate these two temporal parameters would reintroduce at least one of the paradoxes afflicting prior formulations, such as the timeless stasis of the Wheeler-DeWitt constraint (Anderson, 2012).

---

1.2.2 Definition: Global Logical Time

Global Sequencer ($t_L$) as the Fundamental Iterator of State Evolution

$t_L \in \mathbb{N}_0$ constitutes the discrete, non-negative integer that systematically labels the successive global states of the universe as they arise under the repeated action of $\mathcal{U}$. Formally, this labeling traces the iterative progression of the universe's configuration through the following infinite but forward-directed chain:

$$U_0 \xrightarrow{\mathcal{U}} U_1 \xrightarrow{\mathcal{U}} U_2 \xrightarrow{\mathcal{U}} \dots \xrightarrow{\mathcal{U}} U_{t_L}$$

In this sequence, each application of $\mathcal{U}$ transforms the prior state $U_{t_L}$ into the subsequent state $U_{t_L + 1}$, preserving the necessary constraints while introducing the potential for structural evolution. $t_L$ thereby imposes a strict total order on the entire sequence of states, establishing an unequivocal precedence relation such that for any $i < j$, the state $U_i$ precedes $U_j$ without ambiguity or overlap. Consequently, $t_L$ emerges as the sole known parameter capable of distinguishing “before” from “after” at the most fundamental level of ontological description, serving as the primitive arbiter of temporal succession in the absence of any deeper or more elemental mechanism.

$\hat{H} \Psi = 0$ does not embody any intrinsic error in its formulation; rather, it stands as radically incomplete with respect to the full architecture of temporal dynamics. This equation accurately encodes the constraint that every valid state $U_{t_L}$ must satisfy, namely that $\hat{H}$ annihilates the wavefunction associated with that state, thereby enforcing the diffeomorphism invariance and constraint algebra inherent to background-independent theories. However, the equation remains entirely silent regarding the dynamical origin of the sequence itself, offering no mechanism to generate the progression from one constrained state to the next. The Global Sequencer rectifies this deficiency by supplying the missing dynamical rule: $\mathcal{U}$ acts to map any Wheeler–DeWitt-constrained state to another state that likewise satisfies the Wheeler–DeWitt constraint, ensuring that the constraint propagates invariantly across the entire sequence. As a direct consequence, the total wavefunction of the universe cannot be construed as a single, timeless entity $\Psi$ devoid of internal structure; instead, it manifests as an ordered history $\lbrace\Psi[U_{t_L}]\rbrace_{t_L=0}^\infty$, wherein the constraint $\hat{H} \Psi[U_{t_L}] = 0$ holds locally within logical time at every discrete step $t_L$, thereby reconciling the static constraint with the dynamical reality of succession.

1.2.2.1 Commentary: Ontological Status

Classification of the Sequencer Parameter as a Meta-Theoretical Operator

$t_L$ does not qualify as a physical observable, in the sense that no measurement protocol within the physical system can yield its value; no coordinate embedded within the spacetime manifold; no field propagating through the configuration space; no degree of freedom that varies independently within the dynamical variables of the theory; and no integral part of the substrate from which states are constructed. $t_L$ does not parametrize change within any state, instead, $t_L$ exists as a purely formal, meta-theoretical iteration counter, operating at a level of description that oversees and enumerates the computational steps without participating in their content or evolution. Its role parallels precisely the step number $n$ in a Conway’s Game of Life simulation, where $n$ merely indexes the generations of cellular updates without influencing the rules or states; or the renormalization scale $\mu$ in a holographic renormalization group flow, where $\mu$ parametrizes the coarse-graining hierarchy externally to the field theory itself; or the fictitious time $\tau$ employed in the Parisi–Wu stochastic quantization procedure, where $\tau$ drives the imaginary-time evolution as a non-physical auxiliary parameter; or the ontological time invoked in ’t Hooft’s Cellular Automaton Interpretation of quantum mechanics, where it discretely advances the hidden-variable substrate; or the unimodular time $\mathcal{T}$ introduced in the Henneaux–Teitelboim formulation of gravity, where $\mathcal{T}$ provides a global foliation parameter decoupled from local metrics. In each of these diverse frameworks (regardless of whether their respective authors have explicitly acknowledged the implication), an external, non-dynamical parameter covertly assumes the responsibility of generating succession, underscoring the ubiquity of such meta-temporal structures in foundational physical modeling.

1.2.2.2 Commentary: Computational Cosmology

Algorithmic Origins of Physical Law derived from Computational Universes

The operational nature of the Global Sequencer attains its most concrete and mechanistically detailed realization within the domain of discrete computational physics, particularly through the frameworks established by the Wolfram Physics Project (Wolfram, 2002); (Wolfram, 2020) and Gerard 't Hooft’s Cellular Automaton Interpretation (CAI) of Quantum Mechanics. These frameworks furnish the essential conceptual and mathematical machinery required to effect a profound transition in the conceptualization of time: from a passive geometric coordinate subordinated to the metric tensor, to an active algorithmic process that orchestrates the discrete unfolding of relational structures.

Within the Wolfram model, the instantaneous state of the universe deviates fundamentally from the paradigm of a continuous differentiable manifold; instead, it materializes as a spatial hypergraph (a vast, dynamically evolving network comprising abstract relations among a multitude of nodes, where edges encode the primitive causal or adjacency connections). In this representational scheme, the "laws of physics" transcend the rigidity of static partial differential equations imposed on continuous fields; they instead embody a set of dynamic Rewriting Rules, which prescribe transformations on local substructures of the hypergraph. The evolution of the universe proceeds precisely as the algorithmic process of exhaustively scanning the hypergraph for occurrences of predefined target sub-patterns (for instance, a pairwise relation denoted as $\{A, B\}$ conjoined with $\{B, C\}$) and systematically replacing each such occurrence with a prescribed updated pattern, such as $\{A, C\}$ augmented by $\{A, B\}$. This rewriting operation, when applied in parallel across all eligible sites, generates the progression of states.

In this context, the Global Sequencer discharges the function of the Updater, coordinating the synchronous execution of all applicable rewrites within a given iteration. Each complete cycle of pattern identification and substitution delineates an "Elementary Interval" of logical time, during which the hypergraph undergoes a unitary transformation under the collective rule set. Time, therefore, does not "flow" as a continuous fluid medium susceptible to infinitesimal variations; rather, it "ticks" forward through a series of discrete updating events, each demarcated by the completion of the rewrite phase. The cumulative history of these successive updates coalesces into the Causal Graph, a directed acyclic structure that traces the precedence relations among elementary events; from this graph, the familiar macroscopic structures of relativistic spacetime (such as Lorentzian metrics, light cones, and geodesic paths) eventually emerge as effective approximations in the thermodynamic limit of large node counts. The Sequencer itself operates analogously to the "CPU clock" in a computational architecture, imposing a rhythmic discipline on the rewrite process and thereby converting the latent potential encoded within the initial rule set into the manifest actuality of an unfolding state history, replete with emergent complexity and observable phenomena.

In a parallel vein, 't Hooft advances the position that the apparent indeterminism permeating standard formulations of Quantum Mechanics arises not as an intrinsic feature of nature but as an epistemic artifact stemming from the misapplication of continuous probabilistic superpositions to what is fundamentally a deterministic, discrete underlying mechanism. He delineates a sharp ontological distinction between the "Ontic State" (a precise, unambiguous configuration of binary bits (or analogous discrete elements) realized at each integer value of time $t$, constituting the bedrock reality inaccessible to direct measurement) and the "Quantum State," which serves merely as a statistical ensemble averaged over epistemic uncertainties, employed by observers whose instruments fail to resolve the granular updates of the ontic layer. Within this interpretive scheme, the universal evolution manifests as the action of a Permutation Operator $\hat{P}$, defined on the space of all possible ontic configurations and mapping this space onto itself in a bijective manner: $|\psi(t+1)\rangle = \hat{P} |\psi(t)\rangle$. This operator, by virtue of its discrete and exhaustive permutation of states, enacts precisely the role of the Global Sequencer: it constitutes the inexorable "cogwheel" mechanism that propels reality from one definite, ontically resolved configuration to the immediately succeeding one, thereby obviating any prospect of "timeless" stagnation or eternal superposition. The permutation ensures that succession occurs with absolute determinacy, aligning the discrete ticks of logical time with the emergence of quantum probabilities as mere shadows cast by incomplete observational access.

1.2.2.3 Commentary: Unimodular Gravity

Restoration of Unitarity by the Dynamical Cosmological Constant

Although computational models delineate the precise mechanism underlying the Global Sequencer, the physical justification for separating the Sequencer parameter ($t_L$) from the emergent geometric time ($t_{phys}$) draws robust and formal support from the theory of Unimodular Gravity (UMG), with particular emphasis on the canonical quantization framework developed by Henneaux and Teitelboim. This theoretical edifice extracts the concept of a global time parameter from the paralyzing "frozen formalism" endemic to standard General Relativity, wherein the diffeomorphism constraints render time evolution illusory.

In the canonical formulation of standard General Relativity, the cosmological constant $\Lambda$ enters the action as an immutable, fixed parameter woven into the fabric of the Einstein field equations, dictating the global curvature scale without dynamical variability. Unimodular Gravity fundamentally alters this paradigm by promoting $\Lambda$ to the status of a dynamical variable (more precisely, by interpreting it as the canonical momentum conjugate to an independent spacetime volume variable, often denoted as the total integrated 4-volume). This promotion establishes a canonical conjugate pair, $[\hat{\Lambda}, \hat{\mathcal{T}}] = i\hbar$, wherein the commutator encodes the quantum uncertainty inherent to non-commuting observables. Here, the Unimodular Time variable $\mathcal{T}$ assumes the role of the "position-like" coordinate, while $\Lambda$ functions as its "momentum-like" counterpart; given that $\Lambda$ governs the vacuum energy density permeating empty spacetime, its conjugate $\mathcal{T}$ correspondingly tracks the cumulative accumulation of 4-volume across the cosmological expanse, thereby furnishing a global, objective metric for the universe's elapsed "run-time" that transcends local gauge choices.

This canonical structure achieves the restoration of unitarity to the formalism of quantum cosmology, which otherwise succumbs to the atemporal constraints of general covariance. In the conventional approach to quantum gravity, $\hat{H}$ imposes a primary constraint demanding $\hat{H}\Psi = 0$ on the physical state space, thereby projecting the dynamics onto a subspace where time evolution vanishes identically and yielding the infamous frozen 'Block Universe,' in which all configurations coexist in a static, changeless totality devoid of intrinsic becoming (Rovelli & Smolin, 1990). By contrast, the incorporation of the dynamical time variable $\mathcal{T}$ within Unimodular Gravity perturbs the underlying constraint algebra, elevating the temporal progression to a first-class dynamical principle. The resultant equation of motion assumes the canonical form of a genuine Schrödinger equation parametrized by $\mathcal{T}$:

$$i \hbar \frac{\partial \Psi}{\partial \mathcal{T}} = \hat{H} \Psi$$

This evolution equation governs a state vector $|\Psi(\mathcal{T})\rangle$ that advances unitarily with respect to the affine parameter $\mathcal{T}$, preserving probabilities and inner products across increments in $\mathcal{T}$ while permitting the coherent accumulation of phases and amplitudes. The parameter $\mathcal{T}$ thereby incarnates the physical referent of the Global Sequencer within the gravitational sector: it operates in a "de-parameterized" mode, signifying its independence from the arbitrary local coordinate systems (or gauges) adopted by internal observers, who perceive only the relational $t_{phys}$ derived from light signals and rod-and-clock measurements.

This separation of temporal scales aligns seamlessly with the principles of Lee Smolin’s Temporal Naturalism, which systematically critiques the Block Universe ontology (characterized by the eternal, simultaneous existence of past, present, and future) as profoundly incompatible with the empirical reality of quantum evolution, wherein unitary transformations manifest genuine change and contingency. Smolin contends that time must occupy a fundamental ontological status, irreducible to an emergent illusion, and that the laws of physics themselves may undergo evolution across cosmological epochs, thereby demanding a dynamical framework capable of accommodating such variability. The Global Sequencer ($t_L$), when physically instantiated as the Unimodular Time ($\mathcal{T}$), delivers precisely this preferred foliation: it enforces a universal slicing of the state sequence that underwrites the reality of the present moment, while preserving the local Lorentz invariance experienced by inertial observers, who remain ensconced within their parochial geometric clocks and precluded from discerning the meta-temporal progression.

1.2.2.4 Commentary: Background Independence

Independence of the Sequencer from Emergent Geometric Foliations through Pre-Geometric Definitions

Precisely because $t_L$ resides at an external and non-dynamical stratum of the theory (untouched by the variational principles or symmetries governing the physical content), the entirety of the theory's physical articulation (encompassing the relational linkages, correlation functions, and entanglement architectures intrinsic to each individual state $U_{t_L}$) remains utterly independent of any preferred time slicing, foliation scheme, or presupposed background manifold structure. All observables within the theory, ranging from scalar invariants to tensorial quantities like the emergent metric tensor and its associated Riemann curvature, derive their definitions and values exclusively from the internal relational properties and covariance relations obtaining within each $U_{t_L}$, without recourse to extrinsic coordinates or auxiliary geometries. The Sequencer thus qualifies as pre-geometric in its essence: it inaugurates the genesis of geometric structures through the iterative application of relational updates, rather than presupposing their prior existence as a scaffold for dynamics, thereby upholding the stringent demands of manifest background independence characteristic of quantum gravity theories.

1.2.2.5 Commentary: Page-Wootters Comparison

Superiority of the Sequencer Mechanism due to the Elimination of Clock Decoherence

The canonical Page–Wootters mechanism, which posits the total wavefunction of the universe as an entangled superposition of clock and system degrees of freedom wherein subsystem evolution emerges conditionally from the global constraint, harbors three fatal defects that undermine its foundational viability as a complete resolution to the problem of time:

1. Ideal-clock assumption: In realistic physical implementations, any candidate clock subsystem inevitably undergoes decoherence due to environmental interactions, thereby entangling with the observed system and inducing non-unitary evolution that dissipates coherence and inner products violates the preservation of probabilities required for faithful timekeeping.

2. Multiple-choice problem: The partitioning of the total Hilbert space into a "clock" subsystem and a "system" subsystem admits a proliferation of inequivalent choices, each yielding distinct conditional evolution operators; these operators fail to commute or align, generating observer-dependent descriptions that lack universality and invite inconsistencies across different experimental contexts.

3. Absence of genuine becoming: The total state persists as an eternal, unchanging block configuration encompassing the entire history in superposition; what masquerades as "evolution" reduces to the computation of conditional probabilities within this preordained totality, precluding any ontological transition from potentiality to actuality and rendering change illusory.

$t_L$ obviates all three defects in a unified stroke, restoring a robust ontology of temporal becoming:

• The operative "clock" resides at the meta-theoretical level and thus achieves perfection by constructive fiat, immune to decoherence, entanglement, or operational failure.

• Uniqueness inheres in the Sequencer by design; no multiplicity of alternatives exists, as it constitutes the singular, canonical iterator governing the universal state sequence.

• The update process effected by the Sequencer qualifies as an objective physical transition, wherein uncomputed potential configurations crystallize into definite, actualized states through the deterministic application of $\mathcal{U}$, thereby instantiating genuine novelty and diachronic identity.

Internal observers, operating within the emergent physical time $t_{phys}$, reconstruct the Page–Wootters conditional probabilities as an effective, approximate description valid in the regime of weak entanglement and coarse-grained measurements; however, the foundational ontology embeds authentic evolution, wherein each tick of $t_L$ marks an irrevocable advance from one ontically distinct reality to the next (Page & Wootters, 1983); (Gambini, García-Pintos, & Pullin, 2023).

---

1.2.3 Lemma: Finite Information Substrate

Finiteness and Quadratic Boundedness of the Information Substrate

Let $t_L$ denote a finite logical time. Then the information content $S(U_{t_L})$ is strictly finite, and the growth of this content is bounded by a quadratic function of logical time, $S(U_{t_L}) \le \mathcal{O}(t_L^2)$.

1.2.3.1 Proof: Finite Information Substrate

Derivation of the Quadratic Entropy Bound via Inductive Branching

I. Setup and Assumptions

Let $\Omega_{t}$ denote the set of admissible physical states at logical time $t$. Let $S(U_{t}) = \log_2 |\Omega_{t}|$ quantify the information content.

The physical postulates impose the following growth constraints:

1. Finite Local Branching ($b$): The Finite Nature Hypothesis limits the update capacity of the substrate. The number of physically distinct successor states for any state $U$ is bounded by the local branching factor $b$ raised to the number of active sites.

   $$\forall U \in \Omega, \quad | \{ U' \mid U \xrightarrow{\mathcal{U}} U' \} | \le b^{s_t}$$

2. Holographic Surface Scaling ($\delta$): The Bousso Bound restricts the number of active degrees of freedom to the surface area of the causal graph. This area $s_t$ scales linearly with the radius in a discrete graph growing from a root.

   $$s_{t} \le \delta \cdot t \quad \text{where } \delta > 0$$

II. Derivation

The cardinality of the state space at step $t+1$ is bounded by the product of the previous cardinality and the successor count defined by the branching factor and active sites.

$$|\Omega_{t+1}| \le |\Omega_t| \cdot b^{s_t}$$

Logarithmic transformation converts this product into a summation for entropy calculation:

$$\log_2 |\Omega_{t+1}| \le \log_2 |\Omega_t| + \log_2(b^{s_t})$$ $$S(U_{t+1}) \le S(U_t) + s_t \log_2 b$$

Let $\Delta S_t = S(U_{t+1}) - S(U_t)$. Substitution of the Holographic Surface Scaling constraint yields the explicit bound:

$$\Delta S_t \le (\delta t) \log_2 b$$

III. Accumulation

The total entropy at time $T$ is the sum of the initial entropy and all incremental changes.

$$S(U_T) = S(U_0) + \sum_{t=0}^{T-1} \Delta S_t$$

The unique primordial vacuum at $t=0$ establishes the Base Case:

$$|\Omega_0| = 1 \implies S(U_0) = 0$$

Substitution of the derived bound for $\Delta S_t$ into the cumulative sum produces:

$$S(U_T) \le 0 + \sum_{t=0}^{T-1} (\delta t \log_2 b)$$

Factoring out the time-independent constants $C = \delta \log_2 b$ isolates the arithmetic series:

$$S(U_T) \le C \sum_{t=0}^{T-1} t$$

IV. Resolution and Conclusion

The arithmetic series evaluates via the standard summation formula with $n = T-1$:

$$\sum_{t=0}^{T-1} t = \frac{(T-1)((T-1)+1)}{2}$$ $$\sum_{t=0}^{T-1} t = \frac{(T-1)T}{2}$$ $$\sum_{t=0}^{T-1} t = \frac{T^2 - T}{2}$$

Substitution of this result back into the entropy inequality yields:

$$S(U_T) \le C \cdot \left( \frac{T^2 - T}{2} \right)$$ $$S(U_T) \le \frac{\delta \log_2 b}{2} (T^2 - T)$$

For $T > 1$, the quadratic term strictly dominates the linear term, such that $T^2 - T < T^2$. This dominance relation establishes the upper bound:

$$S(U_T) < \frac{\delta \log_2 b}{2} T^2$$

We conclude that the information content growth is bounded by a quadratic function of logical time:

$$S(U_{t_L}) \le \mathcal{O}(t_L^2)$$

This scaling holds for any locally finite, causally expanding graph.

Q.E.D.

---

1.2.4 Lemma: Backward Accumulation

Exclusion of Unbounded Past Direction

Assume the domain of the global logical time parameter $T$ extends to the infinite past. Then this unbounded configuration is excluded by the Finite Information Substrate (§1.2.3).

1.2.4.1 Proof: Backward Accumulation

Derivation of Contradiction via Entropy and Capacity Divergence

I. Setup and Assumptions

Let the temporal domain be unbounded in the past direction, denoted $T = \mathbb{Z}_{\le 0}$. Let the history of the universe be the infinite sequence of states $\mathcal{H} = \{ \dots, U_{-n}, \dots, U_{-1}, U_0 \}$.

II. Case A: Irreversible Dynamics

Let $\mathcal{U}$ be a dissipative operator satisfying the Second Law of Thermodynamics. Let $\Delta S_k = S(U_{k+1}) - S(U_k)$ denote the entropy production at step $k$.

1. Thermodynamic Positivity: For non-equilibrium evolution involving coarse-graining or erasure, the expected entropy production is strictly positive:

   $$\mathbb{E}[\Delta S_k] = \mu > 0$$

   The fluctuations are bounded by the Finite Information Substrate (§1.2.3):

   $$\text{Var}(\Delta S_k) = \sigma^2 < \infty$$

2. Cumulative Summation: The total entropy at the present $t=0$ is the accumulation of all prior productions. Let $S_n$ denote the sum over the past $n$ steps:

   $$S_n = \sum_{k=-n}^{-1} \Delta S_k$$

3. Probabilistic Divergence: We apply Chebyshev's Inequality to bound the deviation of the time-averaged entropy production from the mean $\mu$:

   $$\mathbb{P}\left( \left| \frac{S_n}{n} - \mu \right| > \epsilon \right) \leq \frac{\sigma^2}{n \epsilon^2}$$

   The limit $n \to \infty$ drives the probability of deviation to zero:

   $$\lim_{n \to \infty} \mathbb{P}\left( \left| \frac{S_n}{n} - \mu \right| > \epsilon \right) = 0$$

   This implies almost sure convergence of the sum to the linear growth trend:

   $$S(U_0) \approx \lim_{n \to \infty} n\mu = \infty$$

4. Contradiction: The divergence $S(U_0) \to \infty$ is excluded by the Finite Information Substrate (§1.2.3).

III. Case B: Reversible Dynamics

Let $\mathcal{U}$ be a strictly unitary (bijective) operator.

$$U_{t+1} = \mathcal{U}(U_t) \iff U_t = \mathcal{U}^{-1}(U_{t+1})$$

1. Injectivity of History: The requirement of a non-cyclic history implies injectivity of the mapping from time to state:

   $$\forall t_a, t_b \in T, \quad t_a \neq t_b \implies U_{t_a} \neq U_{t_b}$$

2. Information Preservation: In a deterministic reversible system, unitarity requires that the present state $U_0$ encode the unique trajectory of the past. Let $\Delta I_k$ denote the unique information bit distinguishing state $U_{-k}$ from any other state in the sequence:

   $$\Delta I_k \ge 1 \text{ bit}$$

3. Capacity Aggregation: The total information capacity required for $U_0$ to distinguish an infinite set of unique predecessors is the sum of these contributions:

   $$I(U_0) \ge \sum_{k=1}^{\infty} \Delta I_{-k}$$

   We evaluate the sum:

   $$I(U_0) \ge \sum_{k=1}^{\infty} 1 = \infty$$

4. Contradiction: An infinite information capacity $I(U_0) = \infty$ is excluded by the Finite Information Substrate (§1.2.3).

IV. Conclusion

Both dynamical regimes necessitate an infinite information content in the present state $U_0$ given an infinite past. We conclude that the temporal domain is bounded by a finite origin.

Q.E.D.

---

1.2.5 Lemma: Finite State Recurrence

Incompatibility of Infinite Past Duration with Strictly Finite Configuration Spaces

Assume the configuration space $\Omega$ possesses strictly finite cardinality. Then an infinite past trajectory necessitates a state recurrence that forms a closed causal loop, violating the axiom of Acyclic Effective Causality (§2.7.1).

1.2.5.1 Proof: Finite State Recurrence

Demonstration of Inevitable Causal Loops via the Dirichlet Principle

I. Setup and Assumptions

Let $\Omega$ denote the universal configuration space of admissible states. Assume the cardinality of this state space is strictly finite:

$$|\Omega| = N < \infty$$

II. The Infinite Past Hypothesis

Assume the temporal domain extends to the infinite past. Let the history of the universe correspond to a sequence of states indexed by non-positive logical time:

$$\mathcal{T} = (\dots, U_{-2}, U_{-1}, U_0)$$

Consider a finite subsequence of this history with length $N+1$:

$$\mathcal{T}_{sub} = (U_{-N}, \dots, U_0)$$

Let $T = \{-N, \dots, 0\}$ denote the set of time indices for this subsequence, such that $|T| = N + 1$.

III. Application of the Dirichlet Principle

Let $f: T \to \Omega$ define the mapping $f(t) = U_t$. Comparison of the domain cardinality $|T| = N + 1$ and the codomain cardinality $|\Omega| = N$ reveals that the mapping cannot be injective. The Dirichlet (Pigeonhole) Principle implies the existence of at least two distinct time indices $t_a, t_b \in T$ with $t_a < t_b$ such that the system occupies identical states:

$$U_{t_a} = U_{t_b}$$

IV. Deterministic Evolution and Cycle Formation

Let $\mathcal{U}$ denote the deterministic evolution operator satisfying $U_{t+1} = \mathcal{U}(U_t)$. The identity of the states $U_{t_a}$ and $U_{t_b}$ implies the identity of their successors:

$$\mathcal{U}(U_{t_a}) = \mathcal{U}(U_{t_b}) \implies U_{t_a+1} = U_{t_b+1}$$

Mathematical induction extends this identity to all subsequent steps $k \ge 0$, establishing $U_{t_a+k} = U_{t_b+k}$. The trajectory enters a periodic cycle of length $P = t_b - t_a$:

$$C = (U_{t_a}, U_{t_a+1}, \dots, U_{t_b-1})$$

This recurrence establishes the following closed causal structure:

$$U_{t_b-1} \to U_{t_b} \equiv U_{t_a}$$ $$U_{t_a} \to U_{t_a+1} \to \dots \to U_{t_b-1} \to U_{t_a}$$

V. Contradiction with Acyclicity

The existence of the cycle $C$ implies that the state $U_{t_a}$ constitutes a causal ancestor of itself ($U_{t_a} \prec U_{t_a}$). This transitive self-reference violates the axiom of Acyclic Effective Causality (§2.7.1). We conclude that an infinite past acyclic trajectory is incompatible with a strictly finite configuration space.

Q.E.D.

---

1.2.6 Lemma: Supertask Impossibility

Impossibility of Infinite Operation Sequences from Logical and Physical Non-Termination

The traversal of an infinite sequence of discrete computational steps to arrive at the present state $U_0$ constitutes a Supertask. The completion of a Supertask is physically undefined within the dynamical constraints of the theory, as it requires the execution of $\aleph_0$ operations in finite time or the existence of a completed infinity. Neither is permissible in a constructive ontology.

1.2.6.1 Proof: Supertask Limits

Formal Proof of Non-Termination via Turing Computability and Relativistic Constraints

I. Definition of the History Sequence

Let the history $\mathcal{H}$ be defined as the ordered set of computational operations $\mathcal{U}_i$ required to generate the present state $U_0$ from a precedent state. Under the hypothesis of an infinite past ($t \in \mathbb{Z}_{\le 0}$), the index set is the negative integers $\mathbb{Z}_{\le -1}$.

$$\mathcal{H} = \{ \dots, \mathcal{U}_{-3}, \mathcal{U}_{-2}, \mathcal{U}_{-1} \}$$

This set possesses the order type $\omega^*$ (the order of the negative integers), which is characterized by having a last element ($\mathcal{U}_{-1}$) but no first element.

II. The Supertask Constraint

For the state $U_0$ to be physically realized (to exist as the output of a computation), the entire sequence of operations in $\mathcal{H}$ must have been executed to completion. This implies the performance of a Supertask: an infinite number of discrete steps completed within the timeline prior to $t=0$.

III. Computational Undefinability (The Initialization Problem)

We model the physical universe as a State Machine $M = (S, \Sigma, \delta, s_0)$, where $s_0$ is the initial state.

1. Requirement: For any computation to proceed, the machine must be initialized in state $s_0$ at some time $t_{start}$.

2. Deficiency: In the sequence $\mathcal{H}$, for any hypothesized starting time $t_k$, there exists a prior operation $\mathcal{U}_{t_k-1}$ that was required to generate the input for $\mathcal{U}_{t_k}$.

   $$\forall k \in \mathbb{Z}, \quad \exists (k-1) \in \mathbb{Z} \quad \text{such that} \quad k-1 < k$$

3. Result: There is no time $t$ at which the machine $M$ could have been initialized.

   $$\text{Domain}(\mathcal{H}) \cap \{ t_{start} \} = \emptyset$$

   A computation with no initial state is mathematically undefined.

IV. Energy Divergence (The Resource Problem)

Let $\epsilon(op)$ be the energy cost of a single logical operation. By Landauer's Principle and the Margolus-Levitin theorem, any state transition takes a non-zero amount of energy and time.

$$\epsilon(op) \ge \epsilon_{min} > 0$$

The total energy $E_{total}$ dissipated to reach state $U_0$ is the sum over the infinite history:

$$E_{total} = \sum_{k \in \mathcal{H}} \epsilon(\mathcal{U}_k)$$

Since the sequence is infinite and the terms are bounded below by $\epsilon_{min}$:

$$E_{total} \ge \sum_{k=1}^{\infty} \epsilon_{min} = \lim_{n \to \infty} (n \cdot \epsilon_{min}) = \infty$$

An infinite energy dissipation implies that the universe must have exhausted all free energy (reached thermodynamic equilibrium) infinitely long ago. This contradicts the existence of the low-entropy, ordered state $U_0$ observed at the present.

Q.E.D.

1.2.6.2 Commentary: Collapse of Supertasks

Dynamical Instability of Infinite Computation due to General Relativistic Constraints

The logical impossibility inherent to an infinite past finds a precise physical counterpart in the phenomenon designated as the Gravitational Collapse of Supertasks, a dynamical instability wherein the machinery postulated to execute such a transfinite computation self-destructs under general relativistic backreaction. As demonstrated by Gustavo Romero in 2014, the apparatus required to perform an infinite sequence of operations (thereby "arriving" at the present from an eternal regress) inevitably succumbs to singularity formation prior to completion.

This collapse arises from the interplay of two inexorable physical limits, each amplifying the other's effects to catastrophic divergence:

1. Landauer’s Principle: Every irreversible logical operation, such as bit erasure or conditional branching in the Sequencer’s update rules, incurs a minimal thermodynamic cost of $E \geq k_B T \ln 2$ in dissipated heat (Landauer, 1991); (Bennett, 1982), where $T$ denotes the ambient temperature of the computational substrate. For an infinite sequence of steps, assuming a constant (or even diminishing) energy per operation $\epsilon > 0$, the cumulative energy expenditure integrates to $E_{total} = \sum_{k=-\infty}^{0} \epsilon_k \to \infty$, demanding an unbounded reservoir that no finite universe can supply without violating the first law of thermodynamics.

2. Heisenberg Uncertainty: To confine the infinite sequence within a finite elapsed coordinate time (or to "reach" the present from an eternal regress), the temporal allocation per step must contract to $\Delta t_k \to 0$ as $k \to -\infty$. The time-energy uncertainty relation $\Delta E \Delta t \geq \hbar / 2$ then mandates that energy fluctuations scale inversely: $\Delta E_k \geq \hbar / (2 \Delta t_k) \to \infty$. These fluctuations, manifesting as virtual particle-antiparticle pairs or vacuum polarization in quantum field theory, engender unbounded energy densities within the localized computing region.

Within the framework of General Relativity, localized energy concentrations serve as the gravitational source term in the Einstein field equations $G_{\mu\nu} = 8\pi G T_{\mu\nu}/c^4$; the accumulation of infinite total energy (or infinite density from quantum fluctuations) thus warps spacetime with ever-increasing curvature. The Schwarzschild radius $R_s = 2 G M / c^2$, where $M$ quantifies the enclosed mass-energy, swells without bound as $M \to \infty$. Inevitably, $R_s$ surpasses the physical extent of the computational domain (say, the horizon of the observable universe or the causal patch of the Sequencer), triggering the formation of an event horizon. Beyond this threshold, the system implodes into a black hole singularity, where geodesics terminate and information retrieval becomes impossible.

This inexorable collapse precludes the universe from "computing" an infinite history to manifest the present, as the requisite machinery gravitationally annihilates itself mid-task, prior to outputting a coherent "Now." The empirical persistence of a stable, non-singular present configuration (evidenced by the absence of horizon encirclement and the continuity of cosmic evolution) thus constitutes irrefutable proof that the past admits no infinite regress; the temporal domain must commence at a finite origin to evade such dynamical catastrophe.

---

1.2.7 Theorem: Temporal Finitude

Necessity of a Finite Temporal Origin demanded by the Logical Exclusion of Infinite Regress

The domain of Global Logical Time $t_L$ is strictly lower-bounded. There exists a unique initial state, designated $U_0$, which possesses no causal predecessor. The domain of $t_L$ is isomorphic to the set of non-negative integers $\mathbb{N}_0$, establishing a definite moment of genesis for the computational process.

1.2.7.1 Proof: Temporal Finitude

Formal Derivation of the Lower Bound on Logical Time via Reductio Ad Absurdum (§1.2.7)

I. The Infinite Hypothesis Let it be assumed, for the explicit purpose of demonstrating a contradiction, that the domain of Global Logical Time $t_L$ is unbounded in the past direction. This assumption implies that the set of temporal indices is isomorphic to the non-positive integers ($T_L \cong \mathbb{Z}_{\le 0}$), thereby asserting the existence of an infinite sequence of distinct antecedent states $\{\dots, U_{-2}, U_{-1}, U_0\}$.

II. The Constraint Chain The validity of this hypothesis is interrogated against the established lemmas of the theory:

1. Finite Information Density (Lemma §1.2.3): The system enforces a strict holographic bound on the information content of any state within the sequence. It is established that $S(U_t)$ must remain finite for all finite $t$. The assumption of an infinite past requires the current state to encode a history of infinite depth, which necessitates an information capacity that exceeds this finite bound.
2. Thermodynamic Divergence (Lemma §1.2.4): Under the condition of irreversible dynamics, an infinite past necessitates an unbounded accumulation of entropy production ($\Sigma \Delta S \to \infty$). This accumulation would result in a present state $U_0$ characterized by maximal entropy (Thermodynamic Equilibrium or Heat Death), a condition that stands in direct contradiction to the observed low-entropy configuration of the physical universe.
3. Causal Loops via Recurrence (Lemma §1.2.5): Under the condition of reversible dynamics within a state space of finite cardinality, an infinite temporal duration necessitates the occurrence of Poincaré recurrence ($U_t = U_{t+k}$). Such recurrence establishes closed causal loops, which constitute a direct violation of the Acyclicity axiom governing the causal graph.
4. Operational Non-Termination (Lemma §1.2.6): The logical traversal of an infinite sequence of operations to arrive at the present state $U_0$ constitutes a Supertask. The completion of such a task is computationally undefined, as it lacks a valid initialization condition, rendering the existence of $U_0$ logically impossible under constructive dynamical rules.

III. Convergence The assumption of an unbounded past generates inescapable contradictions under both thermodynamic and computational constraints. Whether the dynamics are reversible or irreversible, the hypothesis fails to yield a consistent physical model.

IV. Formal Conclusion Consequently, the temporal domain cannot be unbounded. There must exist a unique initial state $U_0$ such that for all integers $t < 0$, the state $U_t$ is undefined. The domain of Global Logical Time is isomorphic to the set of non-negative integers $\mathbb{N}_0$, thereby establishing a definite and absolute moment of genesis.

Q.E.D.

1.2.7.2 Commentary: Grim Reaper Paradox

Logical Necessity of Finite Temporal Origins demonstrated by the Grim Reaper Paradox

The assertion that the Global Sequencer demands a definite starting point ($t_L = 0$), precluding any infinite regress, garners unassailable logical reinforcement from the Grim Reaper Paradox (originally formulated by José Benardete and subsequently fortified through the analytic refinements of Alexander Pruss and Robert Koons). This paradox furnishes a formal, a priori proof for Causal Finitism, the foundational axiom decreeing that the historical trajectory of any causal system cannot extend to an actual infinity in the backward direction, as such an extension vitiates the chain of sufficient reasons.

Envision a hypothetical universe inhabited by a single victim, designated Fred, alongside a countably infinite ensemble of Grim Reapers $\{R_1, R_2, R_3, \dots\}$, each programmed with an execution protocol contingent on Fred's survival. The drama unfolds within the temporal interval spanning 12:00 PM to 1:00 PM, with assignments calibrated to converge supertask-wise:

• Reaper $R_1$ activates at precisely 1:00 PM, tasked with killing Fred should he remain alive at that instant.

• Reaper $R_2$ activates at 12:30 PM (midway to 1:00 PM), similarly conditioned on Fred's survival to that earlier threshold.

• In general, Reaper $R_n$ activates at the epoch $12:00 + (1/2)^{n-1}$ hours PM, executing the kill if Fred persists alive upon its arrival.

As the index $n$ ascends to infinity, the activation epochs form a convergent geometric series: $t_n = 12:00 + \sum_{k=1}^{n-1} (1/2)^k$ hours, with $\lim_{n\to\infty} t_n = 12:00$ PM approached asymptotically from the future side. This setup prompts two innocuous interrogatives concerning Fred's status at 1:01 PM, each exposing the paradox's barbed core:

1. Is Fred dead? Affirmative. Survival beyond 1:00 PM proves impossible, as Reaper $R_1$ (the coarsest sentinel) guarantees termination at or before that boundary; no prior reaper can avert this, and the ensemble collectively overdetermines the outcome.

2. Which Reaper killed him? Indeterminate by exhaustive elimination. Suppose, per absurdum, that Reaper $R_n$ effects the kill at $t_n$. This supposition entails Fred's aliveness immediately antecedent to $t_n$, permitting $R_n$'s conditional trigger. Yet Reaper $R_{n+1}$, stationed at $t_{n+1} = t_n - (1/2)^n$ hours (strictly prior), would have encountered that aliveness and preemptively executed, rendering $R_n$'s opportunity moot. This regress applies recursively: no finite $n$ sustains the supposition, as each defers to a denser predecessor.

The resultant impasse manifests a closed causal loop: the terminal effect (Fred's death) stands guaranteed by the infinite assembly, yet its proximal cause (the executing reaper) eludes identification within the countable set, dissolving into logical vacuity. The death precipitates as a "brute fact" (an occurrence destitute of mechanistic ancestry, flouting the Principle of Sufficient Reason by which every contingent event traces to a determinate precursor). This configuration unveils the Unsatisfiable Pair Diagnosis: the conjoined propositions of an infinite past and causal consistency prove jointly untenable, as the former erodes the latter into paradox. Since the ontology of physics presupposes causal consistency (insisting that each state $U_{t_L + 1}$ emerges as a well-defined function $f(U_{t_L}, \mathcal{U})$ of its antecedent and the evolution rule), we must excise the infinite past to preserve the chain's integrity. The Sequencer thus requires bounding below by a First Event, the uncaused cause ($U_0$) from which all subsequent effects descend with unambiguous pedigree, ensuring the historical manifold remains a tree-like arborescence rather than a gapped abyss.

The "Unsatisfiable Pair Diagnosis" (UPD), as articulated and defended by philosophers of time such as Alexander Pruss, reframes the perennial debate over temporal origins from speculative metaphysics to a logical trilemma. It diagnoses the paradoxes of infinite regress (exemplified by the Grim Reaper ensemble) not as idiosyncratic curiosities amenable to ad hoc dissolution, but as diagnostic indicators of a profound incompatibility between two axiomatic pillars that cannot coexist without mutual subversion.

1. The Logical Fork

The UPD compels a binary election between two elemental axioms, whose simultaneous affirmation generates inconsistency:

• Axiom A (Infinite Past): The temporal domain extends without lower bound, such that $t_L \in \mathbb{Z}_{\leq 0}$, admitting an actualized transfinite regress of prior states and events.

• Axiom B (Causal Consistency): The governance of physical events adheres to causal laws, encompassing local interaction Hamiltonians, the Markov property (future dependence solely on the present configuration), and the Principle of Sufficient Reason (every contingent occurrence admits a complete causal explication), thereby ensuring that effects inherit their necessity from identifiable antecedents.

2. The Conflict

Within the Grim Reaper tableau, endorsement of Axiom A (positing the actual existence of the infinite reaper sequence) precipitates the downfall of Axiom B. Fred's demise at or before 1:00 PM follows inexorably from the supertask convergence, yet the identity of the lethal agent proves logically inaccessible: it cannot devolve to Reaper 1 (preempted by $R_2$), nor to Reaper 2 (preempted by $R_3$), nor to any finite Reaper $R_n$ (preempted by $R_{n+1}$), exhausting the possibilities without resolution.

This lacuna births a "brute fact" (the death eventuates sans specific causal agency, an ex nihilo irruption unmoored from the dynamical laws). Under infinite regress, causality fractures into "gaps," wherein terminal effects manifest without proximal mechanisms, akin to spontaneous violations of unitarity or conservation. The infinite ensemble, while ensuring the outcome, dilutes responsibility across an uncompletable chain, rendering the causal narrative incomplete within the countable set.

3. The Priority of Physics

The discipline of physics dedicates itself to the elucidation of Causal Consistency, modeling phenomena through predictive functions that map initial data to outcomes via invariant laws. To countenance "uncaused effects" as a mere concession to the mathematical allure of an infinite past would eviscerate this enterprise: we could no longer assert that $U_{next}$ derives deterministically (or probabilistically) from $U_{current}$, inviting arbitrariness and undermining empirical falsifiability. The scientific method, predicated on reproducible causation, demands the rejection of brute facts in favor of explanatory closure.

Conclusion

Empirical scrutiny confirms the universe's obeisance to causal laws (Axiom B enjoys verificatory status through the success of predictive theories from quantum electrodynamics to general relativity), while the UPD attests the mutual exclusivity of A and B. Ergo, Axiom A must yield to falsehood.

The universe thus mandates a finite history, with the Global Sequencer initiating at $t_L = 0$ to forge an unbroken causal spine: every event traces, through finite recursion, to the First Event $U_0$, the axiomatic genesis beyond which no antecedents lurk. This finitistic resolution not only exorcises the Grim Reaper's specter but elevates the temporal ontology to a bastion of logical and physical coherence.

1.2.7.3 Diagram: The Grim Reaper Paradox

Visualization of Asymptotic Convergence within the Grim Reaper Paradox

┌───────────────────────────────────────────────────────────────────────┐
│              THE PARADOX OF THE INFINITE PAST (Grim Reaper)           │
└───────────────────────────────────────────────────────────────────────┘

   The Scenario: An infinite line of Reapers.
   If you survive Reaper n, Reaper n-1 kills you.

   Time:   12:00                                                   1:00
   Range:    [-------------------------------------------------------]

   Reaper    R(∞)... R(4)    R(3)        R(2)                    R(1)
   Location: |.......|.......|...........|.......................|
             ^       ^       ^           ^                       ^
             |       |       |           |                       |
   Trigger:  (...)  12:07   12:15       12:30                   1:00

   THE LOGICAL CRASH:
   1. You are dead at 1:01. (R1 ensures it).
   2. Who killed you?
      - Was it R1? No, you were dead before 1:00 (R2 killed you).
      - Was it R2? No, you were dead before 12:30 (R3 killed you).
      - Was it R(n)? No, R(n+1) killed you first.

   CONCLUSION:
   Effect (Death) exists without a Cause (Killer).
   Therefore: Infinite causal regress is impossible.

---

1.2.Z Implications and Synthesis

Temporal Ontology

Forcing the timeline to be finite cuts off the infinite regress. This ensures that every state possesses a definite causal ancestry traceable back to a singular origin. The conclusion is inescapable. "Becoming" is a discrete process. It is a sequence of state transitions that can be counted but not divided. This eliminates the possibility of a universe that has always existed. It grounds physics in a definite genesis where the first state acts as the uncaused cause of the computational chain. It implies that the history of the universe is a finite string of data. It is fully enumerable and logically bounded. This prevents the singularities associated with infinite pasts.

The logical clock $t_L$ emerges here not as a coordinate dimension that one can travel through. It emerges as the relentless driver of existence itself. It acts as the fundamental CPU cycle of the universe. It is an external iterator that processes the state transition function. This distinction is vital because it separates the act of change from the measurement of change. Physical time is the variable that appears in relativity equations and is measured by atomic clocks. It is an emergent property of the relations inside the graph and is subject to dilation and curvature. Logical time is the absolute ordering of the computation. It is immune to these relativistic effects. By separating these two concepts, we resolve the Problem of Time in quantum gravity. The universe has a heartbeat, but it is not a clock hanging on the wall of spacetime.

With the clock established, the nature of the object that evolves must be defined. We have secured the timing and the mechanism of the update cycle. However, a heartbeat requires a body to animate. Time cannot exist in a vacuum because it requires a state to transition from and to. We turn now to the definition of the spatial substrate. We must define the graph that serves as the memory of the system. We must define the canvas upon which this temporal iterator paints the history of the cosmos.

---