Complementary Continuous-Discrete Time, Chronon Layering and Temporal Folding

Abstract

Within the framework of a discrete-time chronon model, we consider a dual description of physical time. In this description, macroscopic time is a continuous parameter, while a microscopic integer chronon index labels elementary updates of the system. On this basis, a hierarchy of temporal layers ${\mathrm{Ch}}_N$ (Chronon) is introduced. The simple layers $\mathrm{Ch}2$, $\mathrm{Ch}3$ and $\mathrm{Ch}4$ are analysed, and it is shown that they naturally support $U\left(1\right)$ (Unitary group), $SU\left(3\right)$ (Special Unitary group) and a pair-locked $SU\left(2\right)$ (Special Unitary group) symmetry, respectively. Special attention is paid to the twelve-slot layer $\mathrm{Ch}12$. This layer is the minimal one which simultaneously separates partitions into four triads and three quartets. For $\mathrm{Ch}12$, we demonstrate that the intersection of the corresponding commutants in ${\mathbb{C}}^3\otimes {\mathbb{C}}^4$ reproduces the Standard Model gauge algebra $SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ and the pattern of hypercharges and anomaly cancellation. The appearance of three fermion generations is interpreted in terms of three inequivalent embeddings of a triad into the dodecad which preserve the quartet structure. Possible connections of the chronon dynamics with the hierarchy of masses (via Floquet-type quasi-energies), with dark sectors associated with misaligned layers, and with a temporal interpretation of entanglement are briefly discussed on a qualitative level. These questions are formulated as open problems for further study.

1. Introduction

The problem of reconciling General Relativity (GR), where time is a dynamical coordinate on a Lorentzian manifold [1,2], with Quantum Field Theory (QFT), where time appears as an external parameter for unitary evolution, is still open. Different approaches to quantum gravity and to the microscopic structure of spacetime are actively discussed in the literature. Among them, one can mention Loop Quantum Gravity and spin-foam models [3,4,5,6], Causal Set Theory [7,8,9,10,11], Causal Dynamical Triangulations [12,13,14,15], as well as the idea of asymptotic safety [16] and various scenarios where gravity is an emergent macroscopic phenomenon [17,18,19,20,21,22].

In parallel, there exist many attempts to understand the internal symmetries and matter content of the Standard Model directly from some underlying discrete or algebraic structure [23,24,25]. In such works, the gauge group and the representation content are expected to follow from the construction and not to be postulated by hand. In the present paper, we do not try to compete with these approaches on the level of dynamics. Our aim is closer to kinematics and to show that a very simple discrete time organisation of microscopic evolution is already sufficient to encode several characteristic features of the Standard Model spectrum.

We consider a dual description of time. On the one hand, there is a continuous macroscopic time t, which appears in the usual way in effective field theory. On the other hand, there is a discrete chronon index $n\in \mathbb{Z}$ describing a fundamental update step of the total system. The microscopic dynamics on the base layer (Ch1) is assumed to be a stochastic process on a hyperfinite Borel structure [26,27,28]. Coarse-graining over N steps defines an effective layer ${\mathrm{Ch}}_N$ with N internal temporal “slots”. The algebraic structure of the effective Hamiltonian on a given layer is restricted by permutations of these slots which preserve the coarse-grained measure.

It turns out that already the first non-trivial layers are important. The layers $\mathrm{Ch}2$, $\mathrm{Ch}3$ and $\mathrm{Ch}4$ support $U\left(1\right)$, $SU\left(3\right)$ and a pair-locked $SU\left(2\right)$ structures, respectively. The least common multiple $N=12$ is the smallest period which allows us to accommodate simultaneously triadic and quartic patterns. In the dodecad layer $\mathrm{Ch}12$, we obtain an effective Hilbert space ${\mathcal{H}}_{12}\simeq {\mathbb{C}}^{12}\simeq {\mathbb{C}}^3\otimes {\mathbb{C}}^4$, and the intersection of the commutants of triad and quartet permutations in this space leads to the gauge algebra $SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$. In the same combinatorial picture, the cancellation of gauge anomalies reduces to simple integer relations, and three generations of fermions correspond to three inequivalent embeddings of a triad into the dodecad which preserve the quartet partition.

The chronon hierarchy naturally suggests to use a Floquet-type language for describing the coarse-grained evolution [29,30,31,32,33,34]. In such a description, quasi-energies provide a convenient way to parameterise mass scales and possible small periodic corrections to the effective metric. However, in the present work, we restrict ourselves to qualitative comments in this direction and do not attempt a quantitative phenomenological analysis. The same remark applies to the possible interpretation of entanglement as a manifestation of temporal correlations between different layers.

Scope and Status of Claims

The core results of this work are kinematic: starting from a discrete chronon update index and the coarse-grained hierarchy of layers ${\mathrm{Ch}}_N$, we constrain the algebra of the effective stroboscopic dynamics by slot-permutation invariances and compute commutants/intersections in finite-dimensional slot spaces. The identification with $SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ refers to the global internal symmetry algebra of the effective theory on $\mathrm{Ch}12$; we do not derive a UV completion nor the full gauge-boson dynamics here. Remarks on gravity, dark sectors and entanglement (Section 9 and Section 10) are qualitative and stated as open problems.

The paper is organised as follows. In Section 2 we introduce the basic definitions of the chronon hierarchy and the notion of a layer ${\mathrm{Ch}}_N$. In Section 3 and Section 4 we analyse the algebraic structure of the layers $\mathrm{Ch}2$, $\mathrm{Ch}3$ and $\mathrm{Ch}4$. Section 5 is devoted to the special role of the dodecad layer $\mathrm{Ch}12$. In Section 6 and Section 7 we show how the Standard Model gauge algebra, charges and generation structure arise in this framework. Section 8 contains brief remarks about Higgs and neutrino sectors. In Section 9 and Section 10 we discuss the effective field-theoretic description, gravity and entanglement. Section 11 contains a summary of the results and a list of open problems.

2. Foundational Framework

The dual-time framework is constructed to ensure compatibility with standard QFT while introducing a discrete substructure.

2.1. Dual Representation and Matching

We assume distinguishable histories differ by integer multiples of action, $S(t_2,t_1)=m\hslash$, which introduces a characteristic microscopic timescale ${\tau }_1$ without imposing a global lattice on t. Here the “dual-time” terminology does not mean two physical clock times. There is one macroscopic time parameter t used in the effective field theory (EFT) description, while $n\in \mathbb{Z}$ is a microscopic update counter that allows us to define stroboscopic coarse-graining and the layers ${\mathrm{Ch}}_N$. The “quantum of action” assumption $S(t_2,t_1)=m\hslash$ is a modelling postulate that introduces a microscopic update scale ${\tau }_1$ (as a minimal phase/action increment) without imposing a rigid lattice on the macroscopic time t. It is not meant as a new Standard-Model field degree of freedom by itself. The microscopic evolution is described by a sequence of graphs $G_n=(V_n,E_n)$, where edges carry spin-action labels satisfying the local balance condition:

$$\sum _{e\in E_{\mathrm{in}}\left(v\right)}\mathbf{J}\left(e\right)=\sum _{e\in E_{\mathrm{out}}\left(v\right)}\mathbf{J}\left(e\right).$$

(1)

To link the discrete and continuous sectors, we impose a matching condition. Every chronon edge e corresponds to a smooth embedding ${\iota }_e:[0,1]\to M$ into the Lorentzian manifold. The endpoints are constrained to lie within a tolerance $\delta \ll {\tau }_1$ of the integer time steps $n{\tau }_1$. This ensures that the causal order defined by the discrete updates is causally consistent with the geometric time t.

2.2. Borel Structure and Renormalization

To rigorously define the layers, we employ the formalism of descriptive set theory. Let $X=Y^{\mathbb{Z}}$ be the configuration space of chronon labels. The shift operator T generates an orbit equivalence relation $E_T$, which is known to be hyperfinite. This property allows $E_T$ to be approximated by finite relations $E_n$, providing a mathematical basis for our layering hypothesis.

We define the factor map

$${\pi }_N:X\to X,{\left({\pi }_{N}x\right)}_k=x_{kN}.$$

(2)

This map induces a coarse-graining of the measure $\mu \mapsto {\mu }_N=\mu \circ {\pi }_N^{-1}$.

Definition (Temporal Layer ${\mathrm{Ch}}_N$ and Slot Space)

Given the base system $(X,\mathcal{B},\mu ,T)$, the block map ${\pi }_N$ induces ${\mathcal{B}}_N={\pi }_N^{-1}\left(\mathcal{B}\right)$, ${\mu }_N=\mu \circ {\pi }_N^{-1}$ and the stroboscopic dynamics $T^N$. We refer to the corresponding factor description as the temporal layer ${\mathrm{Ch}}_N$ (sampling at resolution $\tau \left(N\right)=N{\tau }_1$). A single ${\mathrm{Ch}}_N$ cell contains N internal slots. Its internal slot Hilbert space is $H_N\simeq {\mathbb{C}}^N$, on which slot permutations act naturally. For the rigorous measure-theoretic construction, see Appendix B.

The full hierarchy of different layers is shown on

Table 1. Layer hierarchy, minimal assumptions, and the resulting internal symmetry structure.

Layer	Minimal Structural Input (Assumption/Model Choice)	Slot Space	Resulting Internal Symmetry (Interpretation)
${\mathrm{Ch}}_2$	Two-slot cell, global phase invariance of $H_{\mathrm{eff}}^{\left(2\right)}$	$H_2\simeq {\mathbb{C}}^2$	$U\left(1\right)$ (seed for hypercharge)
${\mathrm{Ch}}_3$	Full slot equivalence, commutant constraint, isotropic/degenerate triplet sector	$H_3\simeq {\mathbb{C}}^3$	$SU\left(3\right)$ (color sector, modulo phase)
${\mathrm{Ch}}_4$	Pair partition + pair-locking (coherent doublets)	$H_4\simeq {\mathbb{C}}^4$	$SU\left(2\right)$ (weak isospin from locked doublet)
${\mathrm{Ch}}_{12}$	Simultaneous triad/quartet invariances, intersection of commutants in adapted factorization	$H_{12}\simeq {\mathbb{C}}^3\otimes {\mathbb{C}}^4$	$SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$, charges/anomalies; 3 compatible triad embeddings

.

Physically, this represents a Renormalization Group (RG) transformation ${\mathcal{R}}_N$. (Assumption.) A temporal layer ${\mathrm{Ch}}_N$ is dynamically distinguished if ${\mu }_N$ is a fixed point/attractor of the temporal RG flow. The effective dynamics on a layer ${\mathrm{Ch}}_N$ is governed by a stroboscopic unitary operator

$$U_N\approx \prod _{k=1}^N{U}_1^{\left(k\right)},$$

(3)

which allows us to define an effective Hamiltonian $H_{\mathrm{eff}}^{\left(N\right)}$. The symmetry of this Hamiltonian is constrained by the permutations of the N internal slots that leave the coarse-grained measure invariant.

3. Spin-Action Structure and Elementary Layers

The hierarchy begins with the elementary loop $\mathrm{Ch}1$, carrying a single quantum of action. The first non-trivial structures emerge at $N=2$ and $N=3$.

The Doublet ($\mathrm{Ch}2$) This layer consists of two slots. The effective Hamiltonian $H_{\mathrm{eff}}^{\left(2\right)}=\delta {\sigma }_3+\kappa {\sigma }_1$ describes a two-level system. The canonical commutation relation $[x_a,p_b]=i\hslash {\delta }_{ab}$ implies a minimal $(2+1)$-dimensional kinematics. Crucially, the Hamiltonian commutes with the global phase shift $exp\left(i\alpha \right){\mathbb{1}}_2$, identifying the internal symmetry group as $G\left(\mathrm{Ch}2\right)=U\left(1\right)$. This Abelian factor is the seed for the hypercharge.

The Triplet ($\mathrm{Ch}3$): Coarse-graining over three steps yields a three-slot cell with $H_3\simeq {\mathbb{C}}^3$. Assuming full statistical equivalence of the slots (invariance under a transitive slot-permutation action), the effective Hamiltonian is constrained to lie in the commutant of the corresponding permutation representation on $H_3$. At a maximally isotropic fixed point (or within a fully degenerate triplet sector), the continuous unitary symmetry is enhanced to $U\left(3\right)$ (and hence to $SU\left(3\right)$ modulo an overall phase). A short explicit toy example of the commutant/degeneracy mechanism is given in Appendix B.

These elementary symmetries, $U\left(1\right)$ from the doublet and $SU\left(3\right)$ from the triplet, are the algebraic building blocks. Their combination requires a higher-order layer that can accommodate both structures simultaneously.

4. The $\mathrm{Ch}\mathbf{4}$ Layer and Pair-Locked $\mathbf{SU}\left(\mathbf{2}\right)$

The layer $\mathrm{Ch}4$ (${\tau }_4=4{\tau }_1$) introduces a new structural element which is the pair-locking mechanism. The four slots naturally partition into two pairs, $(1,3)$ and $(2,4)$, inherited from the underlying $\mathrm{Ch}2$ blocks. Imposing symmetry under the permutation P that exchanges these pairs will force the Hamiltonian to be block-diagonal. A stronger physical condition is pair-locking, where the two doublets evolve coherently. This is formalized by requiring $[H_{\mathrm{eff}}^{\left(4\right)},\mathcal{L}]=0$, where $\mathcal{L}$ swaps the pairs. The resulting Hamiltonian takes the tensor product form:

$$H_{\mathrm{eff}}^{\left(4\right)}=A\otimes {\mathbb{1}}_2.$$

(4)

The operators ${\sigma }_i\otimes {\mathbb{1}}_2$ commute with this structure, generating an $SU\left(2\right)$ algebra. Thus, $G\left(\mathrm{Ch}4\right)=SU\left(2\right)$. This derivation suggests that the weak isospin symmetry is not fundamental in the same sense as color, but is an emergent property of pair-correlated chronon dynamics. The effective geometry of this layer, after integrating out relative degrees of freedom, retains the $(2+1)$ Lorentzian signature.

5. The Dodecad: Structure and Stability of $\mathrm{Ch}\mathbf{12}$

We now turn to the central result of this framework. We seek the minimal temporal layer capable of sustaining the symmetries of the lower layers ($\mathrm{Ch}2$, $\mathrm{Ch}3$, $\mathrm{Ch}4$) in a unified structure. Intermediate layers like $\mathrm{Ch}6$ and $\mathrm{Ch}9$ fail to produce the Standard Model group uniquely. $\mathrm{Ch}6$ yields independent triadic and doublet sectors [$SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$], but lacks the geometric locking to enforce a single gauge structure. $\mathrm{Ch}9$ leads to an excessive $SU\left(3\right)\times SU\left(3\right)$ symmetry.

The layer $\mathrm{Ch}12$ (${\tau }_{12}=12{\tau }_1$) is unique. It admits simultaneous partitions into four triads and three quartets. Concretely, labeling the 12 slots by $k\in {\mathbb{Z}}_{12}$, the triads can be taken as $\{i,i+4,i+8\}$ for $i=0,1,2,3$, while the quartets can be taken as $\{a,a+3,a+6,a+9\}$ for $a=0,1,2$. Equivalently, writing $k=4a+i$ with $a\in \{0,1,2\}$ and $i\in \{0,1,2,3\}$ makes the compatibility with both partitions explicit and motivates the adapted identification $H_{12}\simeq {\mathbb{C}}^3\otimes {\mathbb{C}}^4$ used in the commutant intersection. The effective Hamiltonian must satisfy:

$$[H_{\mathrm{eff}}^{\left(12\right)},S_3]=0\mathrm{and}[H_{\mathrm{eff}}^{\left(12\right)},S_4]=0,$$

(5)

where $S_3$ and $S_4$ generate triad and quartet permutations. The algebraic intersection of their commutants in ${\mathcal{H}}_{12}\simeq {\mathbb{C}}^3\otimes {\mathbb{C}}^4$ is generated by ${\lambda }^a\otimes {\mathbb{1}}_4$, ${\mathbb{1}}_3\otimes {\sigma }_i$, and the global phase. This yields precisely the group:

$$G\left(\mathrm{Ch}12\right)=SU\left(3\right)\times SU\left(2\right)\times U\left(1\right).$$

(6)

Dynamical Stability: The preference for $N=12$ is not merely combinatorial but dynamical. In the language of coupled oscillators, a stable periodic orbit requires phase locking of subharmonics [35]. $N=12$ is the first “highly composite number” that supports the simultaneous locking of modes $2\omega$ (doublet), $3\omega$ (triplet), and $4\omega$ (quartet). An effective Landau functional for the chronon phases, ${\mathcal{V}}_N\sim {\sum }_{m|N}{\alpha }_m[1-cos\left(m\theta \right)]$, is minimized when N has a maximal density of divisors. This identifies $\mathrm{Ch}12$ as a deep local minimum of the vacuum energy, an attractor of the temporal RG flow.

6. Emergence of Gauge Symmetry and Charges

Having established the symmetry group, we define the physical charges as integer-valued projections on the slot indices. Let ${\rho }_k$ be the integer label of the k-th slot.

Hypercharge Y is the unique linear combination invariant under triad permutations:

$$Y={\displaystyle \frac{1}{6}}\sum _{i=0}^3{C}_i,$$

(7)

where $C_i$ are triad sums. The factor $1/6$ arises naturally from the normalization of the integer lattice.

Weak Isospin $T_3$ arises from the quartet structure, specifically the difference between the first and second pairs in the quartets:

$$T_3={\displaystyle \frac{1}{2}}\left[({\rho }_1+{\rho }_2)-({\rho }_3+{\rho }_4)\right].$$

(8)

Electric Charge follows as $Q=T_3+Y$. These definitions are not ad hoc. They are the only linear invariants consistent with the $\mathrm{Ch}12$ partitions. This implies that charge quantization is a direct consequence of the discrete slot structure.

7. Generation Structure and Anomaly Cancellation

The phenomenon of three fermion generations finds a topological explanation in the $\mathrm{Ch}12$ geometry. A fermion field corresponds to an embedding of a triadic pattern, which represents color, into the 12-slot cycle. There are four possible positions for a triad. However, we must impose compatibility with the quartet structure, that is with weak isospin. The subgroup of cyclic shifts preserving the quartets is $\Gamma =\{0,4,8\}$. The quotient space of triad embeddings under this subgroup consists of exactly three equivalence classes: ${\mathcal{P}}_0,{\mathcal{P}}_4,{\mathcal{P}}_8$. The fourth shift, ${\mathcal{P}}_3$, is incompatible with the quartet partition. Thus, there are exactly three inequivalent ways to embed a colored fermion into the dodecad vacuum. These correspond to the three generations ${\psi }_{\left(0\right)},{\psi }_{\left(1\right)},{\psi }_{\left(2\right)}$.

Crucially, this geometric construction enforces an anomaly cancellation. The integer identities associated with the triad and quartet partitions ensure that

$$\sum _X{Y}^{\left(X\right)}=0,\sum _X{\left[Y^{\left(X\right)}\right]}^3=0.$$

(9)

For clarity, in the standard left-handed Weyl convention (i.e., right-handed fields represented by left-handed conjugates), the resulting $U{\left(1\right)}_Y$ assignment matches the usual SM normalization, and one finds within one generation $\mathrm{Tr}Y=0$ and $\mathrm{Tr}{Y}^3=0$ explicitly; the mixed anomalies cancel for the same reason: triad and quartet contributions enter the relevant traces in cancelling combinations fixed by the partition invariants.

The mixed anomalies vanish similarly because weak doublets and color triplets contribute to the hypercharge sum in cancelling pairs. This renders the theory consistent at the quantum level without fine-tuning.

8. Higgs and Neutrino Sectors

The scalar sector is constrained by the decomposition of the internal Hilbert space ${\mathcal{H}}_{12}\simeq {\mathbb{C}}^3\otimes {\mathbb{C}}^4$. The quartet factor decomposes as $\mathbf{2}\oplus \mathbf{1}\oplus \mathbf{1}$ under $SU\left(2\right)$. The unique doublet component $\mathbf{2}$ is identified as the Higgs field H. Its potential $V\left(H\right)$ is the standard quartic invariant. For neutrinos, the triadic factor contributes a threefold degenerate subspace in the absence of mixing. This degeneracy explains why there are three light neutral modes. The mixing term $\delta H_{\mathrm{mix}}$ between the triad and quartet sectors lifts this degeneracy, generating small neutrino masses via a mechanism analogous to the seesaw, but purely structural. The suppression comes from the hierarchy of scales between the fundamental chronon and the effective layer period.

9. Effective Field Theory and Gravity

The coarse-graining of the chronon dynamics leads to an effective field theory (EFT) with the standard Lagrangian

$$\mathcal{L}={\mathcal{L}}_{\mathrm{gauge}}+{\mathcal{L}}_{\mathrm{ferm}}+{\mathcal{L}}_{\mathrm{H}}+{\mathcal{L}}_{\mathrm{Y}}.$$

(10)

The mass parameters in this EFT are determined by the quasi-energies of the Floquet Hamiltonian $H_F$.

In the gravitational sector, the metric $g_{\mu \nu }$ emerges as the statistical expectation value of chronon embeddings. In this sense, our description is naturally phrased in the language of covariant effective actions for gravity, where quantum and microscopic corrections are encoded in higher-derivative and nonlocal terms in the metric sector [36]. A novel prediction of this framework is the Floquet modulation of the metric:

$$g_{\mu \nu }\left(t\right)\approx {\overline{g}}_{\mu \nu }+{\epsilon }_{\mu \nu }cos\left({\displaystyle \frac{2\pi t}{{\tau }_{12}}}\right).$$

(11)

Any such modulation, if present, would have to be extremely suppressed and consistent with existing observational bounds. In the present paper, we do not compute ${\epsilon }_{\mu \nu }$ nor derive quantitative constraints; we only note this as a possible phenomenological direction for future study.

Furthermore, the framework offers a candidate for Dark Matter. Excitations in layers like $\mathrm{Ch}6$ or $\mathrm{Ch}9$ are “misaligned” with the $\mathrm{Ch}12$ gauge group. Their projections ${\Pi }_{b\to 12}$ are gauge-neutral but possess energy-momentum. Consequently, they interact only gravitationally. The smoothing effect of the projection naturally leads to “cored” density profiles ($\rho \sim {(1+r/r_c)}^{-2}$), potentially resolving the cusp-core problem in galactic halos.

10. Entanglement as Temporal Contextuality

We propose a reinterpretation of quantum entanglement based on the non-commutative geometry of the layers. The global condition for a closed history in the dodecad layer is $C^{12}=\mathbb{1}$, where C is the joint shift operator. This defines a global projector $P_{\mathrm{ctx}}$. Local observables $A_a\otimes \mathbb{1}$ do not necessarily commute with $P_{\mathrm{ctx}}$. This implies that “entanglement” is the manifestation of a global temporal constraint that prevents the assignment of local hidden variables. The violation of Bell inequalities is thus a result of temporal contextuality in the Kochen–Specker sense [37,38,39] rather than superluminal influence. The overlap integral $I_{AB}$ of the layer activation profiles quantifies the capacity of the system to sustain this contextuality against decoherence.

11. Discussion and Open Problems

The proposed chronon framework shows that a rather simple discrete-time organisation of microscopic dynamics can reproduce several structural properties of the Standard Model. The dodecad layer $\mathrm{Ch}12$ simultaneously contains four triads and three quartets, and this fact is sufficient to obtain the gauge algebra $SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$, the usual pattern of hypercharges and anomaly cancellation, and the existence of three fermion generations. Only three inequivalent triad embeddings remain compatible with the quartet partition, which we interpret as three generations. In this sense, the model is close in spirit to other algebraic constructions of the Standard Model, but the main organising principle here is the hierarchy of temporal layers.

At the same time, a number of important questions remain open.

First, although the chronon picture suggests that fermion and boson masses should be related to Floquet-type quasi-energies, in the present work no concrete mass ratios or mixing angles are calculated. The mixing Hamiltonian $\delta H_{\mathrm{mix}}$, which couples different sectors, is introduced only schematically and is effectively a free parameter. To speak about phenomenology, one needs at least one explicit example where $\delta H_{\mathrm{mix}}$ is derived from the microscopic dynamics and leads to realistic spectra.

Second, the ultraviolet completion of the theory is not constructed. The base dynamics on the fundamental layer (Ch1) is described only in general terms as a stochastic process on a hyperfinite Borel structure. We assume that this process has an attractor and that the effective description flows towards the dodecad layer $\mathrm{Ch}12$, but a direct derivation of such a flow is absent. A more detailed study of the Borel dynamics and of the conditions under which the hierarchy ${\mathrm{Ch}}_N$ is realised is required.

Third, the idea to identify certain “shadow” layers (for example $\mathrm{Ch}6$ and $\mathrm{Ch}9$) with dark sectors is attractive, since excitations on these layers are neutral with respect to the effective gauge group but still gravitate. However, in the present work, only a qualitative argument is given that such layers may produce approximately cored profiles. A realistic comparison with astrophysical data would require to compute relic densities and halo profiles in a fully dynamical model.

We do not address inflation or late-time accelerated expansion in this work; the “misaligned layer” remark is only a qualitative dark-matter-motivated possibility (SM-neutral but gravitating excitations). A concrete particle/cosmological model (spectrum, stability, relic abundance) requires specifying the microscopic update dynamics and is left for future work.

Finally, small periodic corrections to the effective metric, which appear naturally in the chronon picture, are expected to be strongly suppressed. Nevertheless, they can in principle lead to weak frequency dependence in the propagation of gravitational waves or to scale dependence of the effective Newton constant. Existing experimental and observational bounds on Lorentz invariance and equivalence principle violations may therefore place rather strict constraints on the fundamental time scale ${\tau }_1$ [40,41,42]. A careful analysis of these constraints is left for future work.

12. Conclusions

In this paper, a chronon model with a dual description of time has been considered. The starting point is a microscopic discrete-time dynamics with an integer chronon index and a coarse-graining procedure which produces a hierarchy of temporal layers ${\mathrm{Ch}}_N$. Each layer contains N internal slots, and the symmetries of the effective Hamiltonian are restricted by permutations of these slots which preserve the coarse-grained measure.

It has been shown that the simple layers $\mathrm{Ch}2$, $\mathrm{Ch}3$ and $\mathrm{Ch}4$ already reproduce structures similar to $U\left(1\right)$, $SU\left(3\right)$ and a pair-locked $SU\left(2\right)$. The special role belongs to the dodecad layer $\mathrm{Ch}12$, which is the minimal layer containing simultaneously triads and quartets. In this case, the effective Hilbert space can be written as ${\mathcal{H}}_{12}\simeq {\mathbb{C}}^3\otimes {\mathbb{C}}^4$, and the intersection of the commutants of triad and quartet permutations leads to the gauge algebra $SU{\left(3\right)}_C\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$. Within the same combinatorial scheme the standard hypercharge assignments and the cancellation of anomalies follow from simple integer relations, and three fermion generations correspond to three inequivalent embeddings of a triad in the dodecad which preserve the quartet structure.

Several more speculative elements of the construction have also been indicated. The layered chronon dynamics naturally suggests a Floquet description of the effective evolution, in which mass scales are related to quasi-energies. Misaligned layers can play the role of dark sectors which interact only gravitationally, and entanglement can be interpreted as a consequence of temporal correlations between different layers. In the present work these topics are discussed only qualitatively and are formulated as directions for further research.

Thus, the main result of the paper is kinematical. A very simple discrete-time organisation of microscopic updates is sufficient to encode the Standard Model gauge algebra, the pattern of charges and anomalies, and the number of generations. Whether such a chronon hierarchy is really realised in Nature can only be decided by a more detailed dynamical analysis and by comparison with experiment, but even in its present form the construction may be useful as a compact way to organise several features of the Standard Model within a single combinatorial scheme.