Paying for Time: A Non‑Split Z₄₈ Projection Mechanism for Finite Causality, Stability, and an Emergent Arrow

## 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