## AbstractMany effective frameworks begin by *assuming* a 3+1D Lorentzian “leaf” with a universal causal cone, and then focus on *protecting* that cone (e.g., ensuring all propagating sectors share the same limiting speed). This work targets the missing “Level −1” question: **why does a finite causal cone exist at all, and why is it stable and universal**, before any higher-level universality clause is invoked? We propose a minimal structural mechanism in which **time dynamics is not primitive** but arises as the **residual cost of a projection**. The bank-level organization is taken to be a cyclic group \(G \simeq \mathbb{Z}_{48}\) with a non-split exact sequence\[0 \to \mathbb{Z}_{24} \to \mathbb{Z}_{48} \to \mathbb{Z}_2 \to 0,\]so every bank label can be decomposed uniquely as \(K = 2k + p\) with \(k \in \mathbb{Z}_{24}\) and a hidden “parity fiber” \(p \in \mathbb{Z}_2\). The decisive non-split fact is purely arithmetic: the **fine generator** (one step) \(K \mapsto K+1\) *necessarily toggles* \(p\) at every tick, whereas the **coarse generator** \(K \mapsto K+2\) preserves \(p\) and lives entirely in the kernel \(\mathbb{Z}_{24}\). We then adopt a single interpretive postulate (“time-from-fine-step”): physical time translation on the leaf is identified with the fine generator, because it is exactly the structure absent from \(\mathbb{Z}_{24}\) but present in \(\mathbb{Z}_{48}\). This closes the key gap: **time translation is unavoidably \(p\)-odd**, i.e. it inherently traverses the hidden channel. Leaf observables are assumed to be **blind to the parity fiber**, implemented by a non-invertible projection (coarse-graining) \(\Pi_p\) that averages over \(p\). Because the fine generator is \(p\)-odd, its leading contributions are erased by \(\Pi_p\) unless a small parity-even residue survives. We parametrize this residue by a single leakage parameter \(\varepsilon\ge 0\). In the minimal two-derivative effective action for the leaf-observable mode \(\psi_+\), this yields a universal kinetic structure\[Z_t^{(+)} = 2\varepsilon Z_t,\qquad Z_x^{(+)}=Z_x,\]and hence a characteristic speed\[c^2 = \frac{Z_x}{2\varepsilon Z_t}\;\propto\;\varepsilon^{-1}.\]The numeric value of \(c\) is not the point (it can be absorbed into units); the point is structural: **a finite, positive, universal cone exists iff \(\varepsilon>0\)**. Moreover, the same knob controls the type of the field equation: for \(\varepsilon>0\) the projected dynamics is strictly **hyperbolic** (a well-posed Cauchy problem with a finite causal cone), while at \(\varepsilon=0\) it degenerates to an **elliptic constraint** (Poisson/Laplace type) with no finite propagation cone. In this precise PDE sense, **causality is emergent**: it appears discontinuously when leakage becomes nonzero. Stability is tied to the same mechanism. In the minimal admissible operator class (two derivatives, no exotic higher-derivative instabilities), \(\varepsilon>0\) is exactly the condition for a healthy (ghost-free) time kinetic and a Hamiltonian bounded below at quadratic order; \(\varepsilon<0\) corresponds to a ghost and is excluded. A major technical threat—interaction blow-ups after canonical normalization as \(\varepsilon\to 0\)—is addressed by introducing an **\(\varepsilon\)-grading** (a governance-consistent operator filtration): each insertion of the projected mode and each time derivative carries a leakage cost, and the admissible operator space is closed under loops. This yields a sufficient inequality ensuring that no coupling diverges as \(\varepsilon\to 0\), so the \(\varepsilon=0\) limit is a well-defined constrained theory rather than a pathological collapse. Finally, we articulate two complementary explanations for **universality and IR stability of the cone**: (i) *governance locking*—a “one-clock” principle in which all time-kinetic operators must be generated through the same fine-step/leakage channel, suppressing species-dependent cones; and (ii) *RG locking*—the same admissible, graded structure correlates the renormalization of \(\varepsilon Z_t\) with spatial stiffness \(Z_x\), making cone-splitting perturbations irrelevant or non-admissible in the infrared. Because \(\Pi_p\) is a coarse-graining map, it intrinsically destroys micro-information about the parity fiber. When this is combined with a standard retarded/Markovian limit (integrating out hidden degrees of freedom), the reduced leaf dynamics becomes a **semigroup** rather than a group—providing the natural substrate for thermodynamic irreversibility and entropy production. In this sense, the arrow of time can be framed as “the direction along which the projection loses micro-information,” while the existence and health of time dynamics itself is “what survives” that projection. This note is an internal-consistency construction: it is intended to be referee-resistant at the level of algebra, PDE type, and EFT naturalness, while remaining agnostic about which UV completion (if any) Nature chooses. Open problems include a concrete microscopic derivation of \(\varepsilon\), its stabilization, and explicit computations of dissipative (odd-in-frequency) terms generated by tracing over the hidden channel. # Why Light Has a Speed Limit: Time as the Price of Hiding a Bit**Plain-language abstract:** Most physics frameworks assume spacetime already comes with time and a universal speed limit. This note asks a simpler “why”: what if time is not a basic coordinate, but a faint residue of a hidden bookkeeping channel that the world we observe cannot fully access? Imagine microscopic states come in two interleaved channels, while our macroscopic description averages over one hidden bit. In a non-split cyclic structure, the smallest possible step inevitably flips that hidden bit; if you identify that smallest step with a “tick,” then time-translation necessarily passes through what the observable description throws away. Pure time motion would disappear—unless a small leakage survives the averaging. That same leakage simultaneously makes cause-and-effect finite (no instantaneous propagation) and keeps the dynamics stable (no runaway negative-energy behavior). The numerical value of the speed of light is a matter of units; the structural claim is that a single, universal causal cone appears whenever the leakage is nonzero. This is a conceptual mechanism meant to motivate “why there is a finite speed limit and time at all,” not a claim that Nature must realize this specific model. ## Short Abstract (Addendum / v1.1) This update adds a concise “Limitations and Responses” section to clarify the exact status of the framework’s key assumptions and the scope of its claims. In particular, we (i) reframe the ε‑grading as an explicit operator-admissibility/closure hypothesis motivated by projection structure and technical naturalness (rather than a proven UV theorem), (ii) spell out the discrete→continuum step as a standard IR gradient-expansion of the projected quadratic kernel (not a claim that \(\mathbb{Z}_{48}\) is a spacetime lattice), and (iii) position the thermodynamic discussion as a consistency layer based on coarse-graining/open-system logic, with a clear program for tightening the link via an explicit influence-functional calculation. The algebraic and PDE-structural results remain unchanged; the revision improves interpretability and reviewer robustness by making all “leaps” and their falsifiers explicit.
AbstractMany effective frameworks begin by assuming a 3+1D Lorentzian “leaf” with a universal causal cone, and then focus on protecting that cone (e.g., ensuring all propagating sectors share the same limiting speed). This work targets the missing “Level −1” question: why does a finite causal cone exist at all, and why is it stable and universal, before any higher-level universality clause is invoked? We propose a minimal structural mechanism in which time dynamics is not primitive but arises as the residual cost of a projection. The bank-level organization is taken to be a cyclic group (G \simeq \mathbb{Z}{48}) with a non-split exact sequence[0 \to \mathbb{Z}{24} \to \mathbb{Z}{48} \to \mathbb{Z}2 \to 0,]so every bank label can be decomposed uniquely as (K = 2k + p) with (k \in \mathbb{Z}{24}) and a hidden “parity fiber” (p \in \mathbb{Z}2). The decisive non-split fact is purely arithmetic: the fine generator (one step) (K \mapsto K+1) necessarily toggles (p) at every tick, whereas the coarse generator (K \mapsto K+2) preserves (p) and lives entirely in the kernel (\mathbb{Z}{24}). We then adopt a single interpretive postulate (“time-from-fine-step”): physical time translation on the leaf is identified with the fine generator, because it is exactly the structure absent from (\mathbb{Z}{24}) but present in (\mathbb{Z}{48}). This closes the key gap: time translation is unavoidably (p)-odd, i.e. it inherently traverses the hidden channel. Leaf observables are assumed to be blind to the parity fiber, implemented by a non-invertible projection (coarse-graining) (\Pi_p) that averages over (p). Because the fine generator is (p)-odd, its leading contributions are erased by (\Pi_p) unless a small parity-even residue survives. We parametrize this residue by a single leakage parameter (\varepsilon\ge 0). In the minimal two-derivative effective action for the leaf-observable mode (\psi+), this yields a universal kinetic structure[Z_t^{(+)} = 2\varepsilon Z_t,\qquad Z_x^{(+)}=Z_x,]and hence a characteristic speed[c^2 = \frac{Z_x}{2\varepsilon Z_t};\propto;\varepsilon^{-1}.]The numeric value of (c) is not the point (it can be absorbed into units); the point is structural: a finite, positive, universal cone exists iff (\varepsilon>0). Moreover, the same knob controls the type of the field equation: for (\varepsilon>0) the projected dynamics is strictly hyperbolic (a well-posed Cauchy problem with a finite causal cone), while at (\varepsilon=0) it degenerates to an elliptic constraint (Poisson/Laplace type) with no finite propagation cone. In this precise PDE sense, causality is emergent: it appears discontinuously when leakage becomes nonzero. Stability is tied to the same mechanism. In the minimal admissible operator class (two derivatives, no exotic higher-derivative instabilities), (\varepsilon>0) is exactly the condition for a healthy (ghost-free) time kinetic and a Hamiltonian bounded below at quadratic order; (\varepsilon<0) corresponds to a ghost and is excluded. A major technical threat—interaction blow-ups after canonical normalization as (\varepsilon\to 0)—is addressed by introducing an (\varepsilon)-grading (a governance-consistent operator filtration): each insertion of the projected mode and each time derivative carries a leakage cost, and the admissible operator space is closed under loops. This yields a sufficient inequality ensuring that no coupling diverges as (\varepsilon\to 0), so the (\varepsilon=0) limit is a well-defined constrained theory rather than a pathological collapse. Finally, we articulate two complementary explanations for universality and IR stability of the cone: (i) governance locking—a “one-clock” principle in which all time-kinetic operators must be generated through the same fine-step/leakage channel, suppressing species-dependent cones; and (ii) RG locking—the same admissible, graded structure correlates the renormalization of (\varepsilon Z_t) with spatial stiffness (Z_x), making cone-splitting perturbations irrelevant or non-admissible in the infrared. Because (\Pi_p) is a coarse-graining map, it intrinsically destroys micro-information about the parity fiber. When this is combined with a standard retarded/Markovian limit (integrating out hidden degrees of freedom), the reduced leaf dynamics becomes a semigroup rather than a group—providing the natural substrate for thermodynamic irreversibility and entropy production. In this sense, the arrow of time can be framed as “the direction along which the projection loses micro-information,” while the existence and health of time dynamics itself is “what survives” that projection. This note is an internal-consistency construction: it is intended to be referee-resistant at the level of algebra, PDE type, and EFT naturalness, while remaining agnostic about which UV completion (if any) Nature chooses. Open problems include a concrete microscopic derivation of (\varepsilon), its stabilization, and explicit computations of dissipative (odd-in-frequency) terms generated by tracing over the hidden channel. # Why Light Has a Speed Limit: Time as the Price of Hiding a BitPlain-language abstract: Most physics frameworks assume spacetime already comes with time and a universal speed limit. This note asks a simpler “why”: what if time is not a basic coordinate, but a faint residue of a hidden bookkeeping channel that the world we observe cannot fully access? Imagine microscopic states come in two interleaved channels, while our macroscopic description averages over one hidden bit. In a non-split cyclic structure, the smallest possible step inevitably flips that hidden bit; if you identify that smallest step with a “tick,” then time-translation necessarily passes through what the observable description throws away. Pure time motion would disappear—unless a small leakage survives the averaging. That same leakage simultaneously makes cause-and-effect finite (no instantaneous propagation) and keeps the dynamics stable (no runaway negative-energy behavior). The numerical value of the speed of light is a matter of units; the structural claim is that a single, universal causal cone appears whenever the leakage is nonzero. This is a conceptual mechanism meant to motivate “why there is a finite speed limit and time at all,” not a claim that Nature must realize this specific model. ## Short Abstract (Addendum / v1.1) This update adds a concise “Limitations and Responses” section to clarify the exact status of the framework’s key assumptions and the scope of its claims. In particular, we (i) reframe the ε‑grading as an explicit operator-admissibility/closure hypothesis motivated by projection structure and technical naturalness (rather than a proven UV theorem), (ii) spell out the discrete→continuum step as a standard IR gradient-expansion of the projected quadratic kernel (not a claim that (\mathbb{Z}_{48}) is a spacetime lattice), and (iii) position the thermodynamic discussion as a consistency layer based on coarse-graining/open-system logic, with a clear program for tightening the link via an explicit influence-functional calculation. The algebraic and PDE-structural results remain unchanged; the revision improves interpretability and reviewer robustness by making all “leaps” and their falsifiers explicit.
**Abstract.** We present an explicit, minimal two‑channel mechanism that turns an order‑48 bank tick into a leaf‑level leakage parameter in the Bank/IRIS framework. Starting from the affine Z₄₈ holonomy that produces the primitive phase ω₂₄ = e^{2πi/24}, we model one fine tick as a swap‑plus‑phase operator U acting on a hidden Z₂ fiber (a parity doublet). The unitary driver closes exactly after 48 ticks (U^{48} = I) while swapping the hidden fiber at each step. Assuming leaf observables are fiber‑blind (p‑blind), we compute the per‑tick projection leakage as the transition probability from the visible even channel |+⟩ to the hidden odd channel |−⟩, yielding p = sin²(π/24). The odds‑ratio parameter appearing in the time‑kinetic kernel is then fixed as ε = p/(1−p) = tan²(π/24). We provide a compact dictionary γ = cos(2θ), p = (1−γ)/2, ε = (1−γ)/(1+γ), showing that sin² and tan² forms are reparametrizations of the same channel rather than competing claims. Finally, we state the correct consistency condition behind “clean restart” over 48 ticks: stroboscopic closure of the unitary driver together with repinning/reprojection at cycle boundaries, rather than reversal of coarse‑grained dissipation. This resolves a key modeling ambiguity and supplies a shared bridge behind multiple numerical hints, while keeping any mapping to specific Standard Model structures explicitly model‑dependent.
# Abstract — Complex Structure from a Non‑Split Fine Step, and Born Weights as Operational Closure We present a compact extension of the ε‑Projection (“time-as-leakage”) framework that targets a separate foundational “why”: why quantum theory is naturally formulated with complex amplitudes, and what minimal additional assumption fixes the Born weights once a complex structure is present. The starting point is the non‑split cyclic clock architecture (e.g., a Z48 lift with kernel Z24), whose fine generator implements an unavoidable toggle+carry action on labels K=2k+p. After Fourier transforming along the kernel direction k, each momentum sector reduces to an explicit 2×2 unitary block g(q) satisfying the exact projective identity g(q)2=eiq1. This makes the phase degree of freedom structural rather than decorative: the minimal faithful representation requires a rotation plane with an operator J obeying J2=−1, i.e., a complex structure (equivalently, complex amplitudes or a real 2D formulation with SO(2)≅U(1) phase rotations). We then give a worked IR/coarse-graining interpretation showing how repeated discrete fine steps admit a continuous, norm-preserving evolution of the form U(t)≈e−JHefft (or e−iHefft in complex notation), with Heff self-adjoint. Complex structure alone does not yet specify probabilities. To close that gap at the “weights” level (without claiming to solve the full measurement problem), we state a minimal operational assumption: noncontextual additivity of probabilities over mutually exclusive outcomes represented by projectors (or, more generally, POVM effects). Under standard theorems (Gleason-type results, with POVM extensions covering the two-dimensional case), this assumption forces probabilities to take the trace form p(P)=Tr(ρP), yielding the Born weights p(P)=∥Pψ∥2 for pure states. The combined message is that (i) projective phase structure and a complex J arise naturally from the non‑split fine-step clock, and (ii) once one adopts the standard operational identification of “events” with subspaces/effects and requires consistent noncontextual additivity, the Born weights follow as the unique probability calculus. Open work is then cleanly separated: deriving measurement dynamics (records, effective irreversibility) from the projection/open-system sector rather than postulating it.
### Abstract (Part One: Crystalline Time → Quantum Kinematics) We present Part One of a two-part programme that connects a discrete, non-split “crystalline time” framework to the safest core of quantum kinematics, with every nontrivial statement backed by simulation certificates. The aim of Part One is intentionally narrow: we build (i) a geometric lexicon for phases and symplectic structure, (ii) a minimal and assumption-transparent route to the Schrödinger equation via unitary time translation (Stone’s theorem), and (iii) the Heisenberg uncertainty principle via symplectic commutators and the Robertson inequality. Born weights and the Dirac operator are deferred to Part Two, where the structural and interpretive inputs are heavier. Section 1 introduces the geometric dictionary: “phases” are treated as holonomies of a U(1) connection on a complex line bundle, and classical mechanics is formulated on phase space as a symplectic manifold (T*Q, ω = dp ∧ dq). On the framework side, crystalline time is encoded by a non-split extension 0 → Z24 → Z48 → Z2 → 0 with the decomposition K = 2k + p (k ∈ Z24, p ∈ Z2). The fine tick K → K+1 necessarily toggles p, while the coarse tick K → K+2 acts as k → k+1 and preserves p. The leaf observer is p-blind; we model this by a projection/coarse-graining Πp over the Z2 fiber, yielding an effective open-system dynamics on the leaf. Time-kinetic support is regulated by bridge kernels S and J through Kt = Zt(S + εJ), with ε controlling whether the leaf sector is hyperbolic (ε > 0), constrained/elliptic (ε = 0), or ghost-like (ε < 0). All of these statements are verified by discrete operator checks at machine precision. Section 2 derives Schrödinger from one minimal physical principle: strongly continuous unitary time translations U(t). By Stone’s theorem, U(t) = exp(−iHt/ħ), giving iħ ∂t|ψ(t)⟩ = H|ψ(t)⟩. We then show how this “safe tower” is realized inside the framework: define the leaf embedding E+ (pure-even state), the leaf projection P+ (extract ψ+), and the fine-step unitary U on Z48. The cadence-dependent leaf amplitude map A_eff(m) = P+ U^m E+ exhibits an exact even/odd dichotomy (confirmed by closed-form derivation and machine-precision verification). For even cadences m = 2r (i.e., N|48 with m = 48/N even), A_eff(m) is a unitary kernel shift S^r: purity remains 1 and entropy remains 0. For odd cadences m = 2r+1, A_eff(m) = S^r (I+S)/2 is contractive: average retention Tr(A†A)/24 = 0.5, a near-zero singular mode appears, and entropy is produced under the projected density-matrix channel. Numerically, starting from a pure-even state, one fine tick generates substantial odd support (||ψ−|| = 0.6146). A divisor-of-48 scan shows that N=48 (m=1) and N=16 (m=3) produce identical entropy/purity trajectories (up to translation), while all even cadences (e.g., N=24,12,8,6) remain unitary and entropy-free. We further connect the odd-tick projected map to a continuum generator: in the long-wavelength, diffusive scaling limit the comoving odd-tick dynamics converges to ∂t φ = D ∂x^2 φ, with D = Δx^2/(8Δt). This diffusion limit is validated numerically with relative L2 errors ≈ 9.3×10^−6 (smooth profile) and ≈ 1.0×10^−5 (delta profile) at t ≈ 0.01. Finally, the ε-controlled cone speed c^2 = Zx/(2εZt) is confirmed by right-moving packet propagation with v_meas matching c_pred to machine precision for ε ∈ {1, 0.3, 0.1, 0.03}. Section 3 derives uncertainty from symplectic structure. The symplectic backbone motivates the canonical commutator [x, p] = iħ, with discrete phase-space realizations through Weyl pairs. The Robertson inequality ΔAΔB ≥ (1/2)|⟨[A, B]⟩| then yields the Heisenberg bound ΔxΔp ≥ ħ/2. On the leaf kernel Hilbert space, a Gaussian wavepacket saturates the bound numerically: ⟨[Q, P]⟩ ≈ (−9.37×10^−17) + 0.9999999998486 i, ΔQΔP ≈ 0.4999999999337, and (1/2)|⟨[Q, P]⟩| ≈ 0.4999999999243. In parallel, the unitary translation group built from the kernel momentum operator satisfies ||U(t+s) − U(t)U(s)|| ≈ 2.06×10^−14 and ||U(Δx) − S|| ≈ 2.17×10^−14, and a finite-difference Schrödinger check yields a relative residual ≈ 1.09×10^−6 at dt = 10^−6. Part One therefore closes a reproducible bridge: (i) crystalline time plus a p-blind projection yields an exact cadence-dependent split into unitary (even) and contractive/entropic (odd) leaf dynamics, (ii) the unitary kernel sector realizes Stone’s theorem and Schrödinger evolution explicitly, and (iii) the symplectic backbone yields commutators and uncertainty with quantitative saturation. Part Two will address Born weights and the Dirac operator, where the required axioms/structures are heavier and must be handled with the same “Court” discipline.
1. This note records a claim-bounded theorem-target route from the R^8 non-split Z48/Z24 Bank/Leaf scaffold toward quantum kinematics and record stability. It does not claim that R^8 derives quantum mechanics from finite arithmetic alone, solves the measurement problem, derives hbar, predicts a numerical collapse threshold, or supplies a complete stochastic record dynamics. The narrower claim is that an interleaved 48/24 fine-step coordinate exposes projective phase in its complexified character decomposition; a faithful real local calculus for generic projective phase naturally uses a two-dimensional phase plane, equivalently a complex structure J^2 = -I, once an effective real Hilbert carrier is admitted. Given such a complex Hilbert carrier and standard operational projector/POVM additivity, Born weights follow by standard representation theorems. The record problem is then reformulated as a stability and settlement problem: a stable Leaf record is a branch sector whose phase debt has become unrecoverable at the Leaf level through leakage, decoherence, and coarse-graining stability. The resulting pointer-basis and single-outcome mechanisms are theorem targets, not completed physical derivations. 2. This note records a claim-bounded theorem-target response to the phase-per-information conversion problem in the R^8 Bank/Leaf scaffold. The coefficient χ_{φI}, if it exists as an effective scalar, should not be introduced as a primitive naked constant. Nor should it be identified with a direct contraction such as Tr(F h^{-1}) or Tr(FG). When F is a curvature two-form and h is a symmetric information metric, such a direct contraction is naturally trace-trivial, skew, or convention-dependent rather than positive. The safer theorem target is a positive phase-information stress tensor K_{μν} = <F_{μα}, h^{αβ} F_{νβ}>_g, where F is Bank phase curvature, h_{μν} is a positive Leaf information metric on the record or connector base, h^{μν} = (h^{-1})^{μν}, and <·,·>_g is an invariant positive inner product on the compact physical phase sector. Effective scalar quantities such as χ_{φI}^{eff} are to be read from generalized eigenvalues or other invariants of the pair (K,h), with the readout map left open. Integer carry is a separate topological object, defined by phase winding, normalized flux, Chern/Hopf number, or spectral-flow integrality under appropriate geometric conditions; it is not stress magnitude. This note does not derive collapse, a CP record-selection instrument, a numerical threshold, or a universal phase-per-nat constant. 3. This note records a claim-bounded theorem-target route for selecting the Leaf information metric h_{μν} in the R^8 phase-information response architecture. It does not claim that the correct metric h*, the phase-per-information coefficient χ_{φI}, the record threshold ε_rec, a Bank throttle law, or a CP record-settlement instrument has been derived. The narrower proposal is that h* should not be chosen by taste or by fitting a desired threshold. It should be selected, if at all, by compatibility with exact governance, normalized monotone quantum-information geometry, phase-information stress viability, and bounded throttle clearability. The previous Phase-Information Response Court replaced the rejected primitive scalar Tr(F h^{-1}) with the positive theorem-target stress tensor K^{(h)}_{μν} = <F_{μα}, h^{αβ}F_{νβ}>_g. The present note asks which metric field h_ρ can make the generalized spectrum of the pair (K_h, h) both alive and non-runaway while keeping the information debt clearable by a bounded Bank response. The result is a concrete metric-sensitivity protocol comparing normalized monotone candidates such as SLD/Bures, Wigner-Yanase, and Kubo-Mori. The key methodological patches are built in: h is treated as a metric field evaluated at a record state, spectra are computed through the generalized pair (K_h, h), monotonicity is required for unconditional CPTP coarse-graining rather than arbitrary postselected CP maps, curvature ensembles must be normalized independently of the tested metric, and metric selection is Pareto-first rather than a hidden weighted scalar fit. The note remains a PASS/YELLOW theorem-target architecture: it defines a disciplined court for h* without pretending that h*, χ_{φI}, ε_rec, or record dynamics have already been solved.
COS-CNS develops the causal-dynamics module of the Collapsing-Structure (COS) program. The manuscript studies a verifiable class of nonunitary, discrete-time dynamics on structures equipped with a causal partial order, focusing on operational no-signaling, finite-speed influence propagation, and scheduling robustness. The framework is formulated at the level of completely positive trace-preserving channels, instruments, local algebras, dependency neighborhoods, and locally implementable causal admissibility filters. It carefully separates unconditional branch-ignored dynamics from conditional trajectory or postselected descriptions, so that no-signaling is defined through operational marginal statistics rather than through selected branches. The main results apply to a restricted and auditable class of local, filtered channels satisfying explicit locality, support-propagation, causal-filtering, and scheduling/confluence assumptions. Within this class, the paper formulates no-signaling criteria, a strict discrete causal-cone statement, and robustness targets for approximate or leakage-affected dynamics. These results are not claimed to hold for arbitrary COS microdynamics or for nonlocal global projectors. COS-CNS also identifies failure modes in which the assumptions break down, including nonlocal Kraus support, nonlocal causal filtering, deterministic branch control, and nonconfluent scheduling. These counterexamples clarify why the locality and admissibility conditions are necessary for an operational causality minimum. The release includes a technical supplement with reproducibility protocols, numerical demonstrations, algorithmic details, supplementary proofs, negative controls, and audit conventions. The numerical material illustrates no-signaling tests, influence-cone diagnostics, and scheduling-robustness checks under controlled benchmark conditions. COS-CNS therefore serves as the causality and finite-speed-influence verification layer for COS-QD-like discrete open-system dynamics, while treating the connection to relativistic microcausality, Lorentz symmetry, and continuum causality as a restricted and future-facing interface rather than a completed derivation.
This paper constitutes the fourth installment of the Mapping the Time research program and provides explicit toy models that concretely instantiate the abstract framework developed in the preceding papers. Building on the kinematic emergence of time established in Paper 1, the tri-level projective architecture introduced in Paper 2, and the dynamical constraints derived in Paper 3, this work constructs low-dimensional, fully explicit examples that realize all structural requirements simultaneously. The goal is not to introduce new theoretical claims, but to demonstrate that the framework is operationally consistent and mathematically realizable. The paper presents two hierarchical toy models that explicitly implement: A tri-level structure consisting of an Extended Volumetric Substrate (EVS), Observed Spacetime (OS), and a Reduced Informational Manifold (RIM); Two strictly non-invertible projections linking these levels; A volumetric spatial coordinate in the EVS that projects to the temporal coordinate in observed spacetime; Projection-compatible dynamics defined on the extended level; Emergent irreversibility at the observed and informational levels despite reversibility at the extended level; A clear distinction between the abstract evolution parameter and observed time. The central results demonstrate that: All constraints derived in Papers 1–3 can be satisfied simultaneously within simple, explicit models; Structural irreversibility arises purely from projective constraints, without invoking thermodynamics, entropy, or fundamental time asymmetry; The induced dynamics at the observed and informational levels naturally form semigroups, even when the underlying dynamics remain reversible; The mechanism is robust under changes in coordinate choices and fiber geometry. Importantly, the analysis remains strictly structural and pedagogical: No physical interactions or forces are modeled; No empirical claims are made; No new postulates are introduced beyond those already established in the trilogy; The models are illustrative rather than physically realistic. By providing explicit realizations, this paper closes the foundational phase of the Mapping the Time program. It validates the internal coherence of the framework and clarifies how the abstract results of the previous papers can be instantiated concretely, preparing the ground for future work on physically motivated models or applications. Keywords: emergent time, toy models, projective time, non-invertible projections, tri-level structure, projection-compatible dynamics, structural irreversibility, pre-dynamical physics, spacetime emergence, information loss, mathematical physics, differential geometry, foundations of time, theoretical physics.