The Udoy-Choi Framework: Formalization of a 4D Informational Rhythm Tick(T4) and Its Operational Signatures in N-Quanta Discrete-Time Models

The Udoy-Choi Framework: Formalization of a 4D Informational Rhythm Tick(T4) and Its Operational Signatures in N-Quanta Discrete-Time Models Mohd Ubaidur Rahman Udoy Seungyoung Choi October 30, 2025 Abstract We formalize Tick(T4)as a 4-dimensional informational rhythm within the QTIP framework and examine how its operation manifests in the real domain through observable traces. It is to be noted that not a measurable temporal unit yet, within the framework of modern physics, nor equivalent to the Planck time defined in conventional physics, but a rhythmic condition generating a three-dimensional temporal frame observable as a momentary state; the sequence of such frames constitutes what we perceive as the continuum of time. In this paper, we demonstrate QTIP can be modeled in N-quanta discrete-time simulations. In this work, we: 1. Describe Tick(T4) as a four-dimensional rhythm generating state-frame updates. 2. Map its operational effect to discrete-time increments in N-quanta models. 3. Aim to derive theoretical bounds on possible observable traces using energetic and information-theoretic arguments (Margolus–Levitin, Landauer). 4. Provide a theoretical estimate of the real-domain manifestation of these traces. 5. Discuss potential observational signatures, including cosmological effects such as the Hubble tension, atomic clocks, and quantum phase deviations. This framework links a hidden informational rhythm to measurable discrete-time phenomena, offering a novel pathway to test QTIP concepts empirically. 1 1 Introduction The fundamental nature of time remains intriguing and central to physics. In QTIP cosmology, Tick(T4) is a 4-dimensional informational rhythm that governs how information is processed and transitions between states in a higher-dimensional domain. Tick(T4) itself is not a temporal unit, but rather the rhythmic condition through which time is produced and expressed in the observable 3D universe. This idea aligns with the discrete-time perspective proposed in the N-Quanta Model (Udoy, M. U. R. 2025); DOI: https://doi.org/10.5281/zenodo.17148556), which treats physical evolution as occurring through quantized informational updates at the Planck scale. The N-Quanta framework provides convenient coarse-graining formulas (Planck volume, discrete time step τ, system volume definitions via covariance ellipsoids, and a diagnostic number Ntotal) that we adopt and connect to the Tick(T4). Although Tick(T4) represents a concept fundamentally different from the Planck scale, the Planck scale is employed here for the purpose of linking with modern physics, serving only as a reference to characterize the residual traces of Tick(T4). The term “Planck scale” mentioned in this paper is used solely as a computational reference scale and should not be interpreted as representing the fundamental origin or lower bound of the Tick(T4) rhythm. According to the QTIP interpretation, the Planck time and Planck length correspond to the observable boundary conditions of the real domain, whereas the actual origin of the informational rhythm is determined by higher-order rhythm conditions in the imaginary domain. Therefore, throughout the text, the term “Planck scale” designates not a physical absolute limit, but rather the lower boundary of observable temporal expression within the real domain. 2 Axioms and Definitions We adopt three working axioms for Tick(T4): (A1) Informational Rhythm: Tick(T4) defines the interval between representation processes in 4D; ontologically primitive but not physically measurable. (A2) Universality: The rhythm is uniform across 4D space and invariant under QTIP symmetries. (A3) Update Semantics: Each Tick(T4) corresponds to a global informational update: Sn→ Sn+1.(1) 2 Parametrization of Real-Domain Manifestation (Trace): Ttrace 4=k tP, tP=rℏG c5≈5.39 ×10−44 s, k > 0 (2) Here, Ttrace 4is the apparent shadow of Tick(T4) in the real domain, not Tick(T4) itself. The phrase “QTIP symmetry transformations” does not refer to a mathematical symmetry group. Rather, it denotes the self-maintenance condition by which the Tick(T4) rhythm sustains the universe’s informational consistency. Despite temporal progression, this condition preserves the consistency across Tick(T4)’s, that is, the invariance of the Tick(T4) interval. From the perspective of QTIP cosmology, although this process is not expressed within the real domain, the universe must fully compute all absolute information residing in the imaginary domain at every Tick(T4). As a consequence, the tick interval remains necessarily invariant. Therefore, the term “QTIP symmetry” in this context does not signify morphological or mathematical symmetry, but a physical invariance condition ensuring that the Tick(T4) interval remains constant throughout the universe’s process of self-maintenance. 3 Mapping T4to N-Quanta Models Discrete time increments in N-quanta frameworks can be expressed as integer multiples of Tick(T4)’s real-domain trace: τ=n Ttrace 4, n ∈Z+.(3) Discrete derivatives are then replaced accordingly: ∆nT trace 4f(t) = f(t+nTtrace 4)−f(t) nTtrace 4 .(4) This connects the hidden rhythm of Tick(T4) to the operational structure of discrete-time physics. 4 QTIP / N-Quanta coarse-graining formulas To connect Tick(T4) with operational discrete-time models, we borrow and restate the main coarse-graining definitions used in the N-Quanta framework: Planck cell volume: VP=ℓ3 P,(5) where ℓPis the Planck length. 3 Discrete time steps: in N-Quanta, the operational time steps are taken as integer multiples of a base quanta τ: tk=k τ, τ =n Ttrace 4(possible operational choice).(6) Instantaneous system volume: for an N-particle system sampled at step tkwe define a coarse-grained system volume Vsys(tk) (e.g. via covariance ellipsoids, convex hulls, or Voronoi partitions). Then the dimensionless granularity diagnostic Ntotal is defined by Ntotal(tk) = Vsys(tk) VP ,(7) which counts how many Planck cells compose the instantaneous system volume at Planck scale, a key diagnostic step in the N-Quanta framework Discrete derivative (N-Quanta style): the forward finite difference at step τis ∆τf(t)≡f(t+τ)−f(t) τ,(8) and, when mapping to Tick(T4), we replace τ7→ nTtrace 4where appropriate. These definitions provide the operational link between Tick(T4)’s hidden rhythm and measurable N-Quanta diagnostics. 5 Illustrative Model: Discrete Frame Generation through Tick(T4) To visualize the function of Tick(T4), consider the conceptual diagram supplied by the QTIP collaborator (figure). Each Tick(T4) event corresponds to the transition between two informational states: CHANGE(n) : Sn→ Sn+1, and the display/projection operator Dmaps the collection of local patch states to the single observable 3-D frame Fn: Fn=D{Sα,n}α∈L. Only Fnis observable at the nominal physical present associated with Tick(T4)n. Observers perceive the sequence {Fn}, producing the impression of continuous time. 4 Figure 1: Schematic of Tick(T4) operation: each Tick(T4) triggers an informational update, producing discrete 3D frames Fnthat, when sequenced, are perceived as continuous time. Only the currently updated frame is observable in 3D. 6 Theoretical Bounds on Trace 6.1 Margolus–Levitin Bound Ttrace 4≳πℏ 2Epatch .(9) 6.2 Landauer Bound Ttrace 4≳πℏ 2IkBTln 2.(10) 6.3 Combined Bound Ttrace 4≳max πℏ 2Epatch ,πℏ 2IkBTln 2.(11) 6.4 Estimation of Real-Domain Trace (Step-by-step derivation) We now present the full stepwise algebraic and numeric derivation for the estimate Ttrace 4≈ 8.5×10−44 s. Step 1 — Start from the Margolus–Levitin (M–L) bound. The M–L theorem provides a lower bound on the time to evolve to an orthogonal quantum state using energy 5 E: tmin ≥πℏ 2E. Identifying the minimal apparent update interval with this bound and setting E=Epatch gives Ttrace 4≳πℏ 2Epatch . Step 2 — Choose the patch energy. We adopt the modeling choice Epatch =EP, the Planck energy, defined by EP=rℏc5 G. Step 3 — Substitute EPinto the M–L expression. Ttrace 4≈πℏ 2EP =πℏ 2 1 rℏc5 G . Step 4 — Algebraic simplification. Rewriting the radical: ℏ √ℏ=√ℏ, so Ttrace 4=π 2rGℏ c5. Recognize the Planck time tP: tP=rGℏ c5. Thus Ttrace 4=π 2tP. Step 5 — Numeric evaluation (constants and multiplication). Use standard CODATA values (digits shown explicitly): 6 ℏ= 1.054571817 ×10−34 J·s, G= 6.67430 ×10−11 m3kg−1s−2, c= 2.99792458 ×108m s−1. Compute Planck time: tP=rℏG c5=s(1.054571817 ×10−34)(6.67430 ×10−11) (2.99792458 ×108)5. Evaluating the numeric expression (digit-by-digit or using a calculator) yields tP≈5.39124644666 ×10−44 s. Multiply by π 2: π 2≈1.57079632679, so Ttrace 4=π 2tP≈1.57079632679 ×5.39124644666 ×10−44 s. Carrying out the multiplication, Ttrace 4≈8.46855011526 ×10−44 s. Step 6 — Round and present final estimate. Rounded to two significant figures (order-of-magnitude presentation): Ttrace 4≈8.5×10−44 s. This corresponds to the dimensionless scale factor. k=π 2≈1.5707963, in the parametrization Ttrace 4=k tP. Caveats. The numeric estimate depends directly on the modeling choice Epatch =EP. Using a different energy budget or including additional physical constraints (e.g., Landauer thermodynamic limits) will change the estimate. The M–L bound provides a lower limit; actual effective projection times could be longer depending on detailed dynamics. 7 7 Observational Constraints and Possible Traces Delayed or jittered processing within Tick(T4) may produce observational effects. Modeling the projection times as tn=n Ttrace 4+εn,(12) where εnis aggregate jitter coming from local patch delays, we obtain observable phase deviations for a mode of angular frequency ω: Φn=ωεn.(13) This leads to decoherence or phase noise scaling with ω, suggesting that high-frequency quantum oscillators and precision clocks are sensitive probes. For wave propagation, discretization modifies dispersion relations; for a discrete linear update one finds corrections to group velocity vg(E) which lead to time-of-flight delays scaling with energy — the usual signature searched for in high-energy astrophysical tests (GRBs, pulsars). A concrete macroscopic example is the Hubble tension: small systematic projection offsets varying with redshift or cosmic scale could bias distance ladder calibrations and expansion inferences. Parameterizing an average projection offset ¯ε(z) allows one to test whether data prefer a small Tick(T4)-induced systematic. 8 Local Causal Update Architecture Spacetime can be modeled as a lattice of Planck-scale causal patches indexed by α(coarsegrained cells of volume VP). The local causal update rule reads Sα,n+1 =Uα {Sβ,n}β∈N (α),(14) with projection to the observable frame via D. Aggregating patches gives the system volume Vsys(tk) that enters the N-Quanta diagnostic Ntotal. 9 Integration into N-Quanta Equations Replacing τby nTtrace 4in the N-Quanta difference operators yields: ϕ(t+nTtrace 4)−ϕ(t) nTtrace 4 =F[ϕ(t)].(15) 8 For n= 1 we recover the standard forward finite difference expansion: ϕ(t+Ttrace 4)−ϕ(t) Ttrace 4 =˙ ϕ(t) + Ttrace 4 2¨ ϕ(t)+O((Ttrace 4)2).(16) Using the N-Quanta coarse-graining diagnostics (e.g. Ntotal =Vsys/VP) allows one to study how discrete corrections scale with system size and coarse-graining. 10 Notation and Definitions •ℏ: Reduced Planck constant (1.0545718 ×10−34 J·s). •G: Gravitational constant (6.674 ×10−11 m3/kg·s2). •c: Speed of light (3 ×108m/s). •tP: Planck time = pℏG/c5. •ℓP: Planck length. •VP: Planck cell volume = ℓ3 P. •EP: Planck energy = pℏc5/G. •k: Dimensionless proportionality factor linking Ttrace 4to tP. •Ttrace 4: Apparent temporal trace of Tick(T4) in the observable domain. •Sα,n: Information state of causal patch αat update n. •Fn: Observable 3-D frame at tick n,Fn=D({Sα,n}). •τ: Operational discrete time increment in N-Quanta (τ=nTtrace 4in mapping). •Ntotal: Coarse-grained number of Planck cells in system volume: Ntotal =Vsys/VP. •Uα: Local update operator for patch α. •F[ϕ]: Discrete evolution operator acting on field ϕ. •Epatch: Energy budget of a causal patch. •I: Information processed per update (bits). •kB: Boltzmann constant. •T: Temperature (Kelvin). 9