**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. 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.
## 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≃Z48 with a non-split exact sequence0→Z24→Z48→Z2→0,so every bank label can be decomposed uniquely as K=2k+p with k∈Z24 and a hidden “parity fiber” p∈Z2. The decisive non-split fact is purely arithmetic: the **fine generator** (one step) K↦K+1 *necessarily toggles* p at every tick, whereas the **coarse generator** K↦K+2 preserves p and lives entirely in the kernel Z24. 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 Z24 but present in Z48. 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) Πp that averages over p. Because the fine generator is p-odd, its leading contributions are erased by Πp unless a small parity-even residue survives. We parametrize this residue by a single leakage parameter ε≥0. In the minimal two-derivative effective action for the leaf-observable mode ψ+, this yields a universal kinetic structureZt(+)=2εZt,Zx(+)=Zx,and hence a characteristic speedc2=2εZtZx∝ε−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 ε>0**. Moreover, the same knob controls the type of the field equation: for ε>0 the projected dynamics is strictly **hyperbolic** (a well-posed Cauchy problem with a finite causal cone), while at ε=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), ε>0 is exactly the condition for a healthy (ghost-free) time kinetic and a Hamiltonian bounded below at quadratic order; ε<0 corresponds to a ghost and is excluded. A major technical threat—interaction blow-ups after canonical normalization as ε→0—is addressed by introducing an **ε-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 ε→0, so the ε=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 εZt with spatial stiffness Zx, making cone-splitting perturbations irrelevant or non-admissible in the infrared. Because Πp 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 ε, 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 Z48 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 (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.
## Abstract This audit note upgrades the dynamics/kinematics lane of the R8 two-tunnels Higgs-terminal proposal. The earlier formulation treated the minimal Z48 leakage coordinates p48, epsilon48, q48, the dressed contraction factor lambda-, and the terminal-osmosis credit 5q48 as structurally related but not generated by a single evolution map. Version v0.3 corrects two audit-critical issues: the grain-size mismatch between the half-step angle pi/24 and the full-passage angle pi/12, and the distinction between an unconditional traced channel and a post-selected no-click Kraus branch. The corrected statement is: one full hidden-channel passage is two half-steps plus one deck flip, U_{pi/12}=D R_{pi/24}^2. The half-step supplies the elementary leakage coordinate p48=sin^2(pi/24) and the survival ledger q48=-log cos^2(pi/24), while the full passage supplies the dressed coherence multiplier lambda-=-cos(pi/12)=2p48-1. Thus p48, q48, and lambda- are linked, but they are not the same readout of the same single-step measurement. The note makes the trace/post-selection distinction explicit. Tracing out the witness gives an unconditional dephasing channel with coherence multiplier lambda-. Conditioning on the no-click witness outcome gives the non-unitary Kraus branch diag(1,lambda-). These two dynamics must not be conflated. The terminal-osmosis term 5q48 is therefore meaningful only if each Z6-synchronized Coxeter layer corresponds to an elementary half-step survival event under a declared branch-weight convention. The note also adds the D1.1 Mobius-strip conveyor register patch. The relevant topology is not a Mobius transformation of the Riemann sphere, but Mobius-strip-type non-orientable transport of the non-split parity fibre. This gives a candidate R8-native location for the fresh witness/register, but does not prove it. The required condition is write-once-until-settlement: no address reuse before per-cycle erasure. If this fails, memory can return and Markov laws may show revivals near n=48,96,... The final status is: algebra exact, grain corrected, trace/post-selection typed, dynamics bridge candidate, governance open.
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.
# 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.
Quantum tunneling is conventionally modeled as transmission through a classically forbidden region due to the nonzero tail of the wavefunction, leading to the familiar WKB scaling T ~ exp[-2∫κ(x)dx]. In this note we reformulate the same phenomenon through a minimal coupled-channel scattering mechanism that is naturally expressible in the language of the Crystalline Axiverse: a local “projected” sector (Z24) embedded in a larger internal space (Z48), where part of the state can traverse a barrier through a complementary internal subchannel and subsequently re-project back into the visible sector. The intent is not to replace standard quantum mechanics, but to provide an explicit, calculable toy mechanism showing how an internal governance constraint (a discrete Z24 grid) can be kinematically compatible with tunneling-like phenomena without invoking any exotic “vacuum softening”. We consider a two-component stationary Schrödinger problem Ψ(x) = (ψV, ψH)ᵀ with a piecewise-constant 2×2 Hermitian potential matrix. The visible channel V encounters a barrier core of height V0 and width a, while the hidden channel H may see a reduced core barrier VH,core. Mixing between channels is confined to two narrow “coupler” layers at the left and right edges of the core, with coupling strength g and a relative interferometric phase Δφ. In the Axiverse interpretation, Δφ is an internal holonomy (an interferometric valve) that controls constructive or destructive reconversion from the hidden virtual path back into the visible projection. We emphasize that in this stationary setting Δφ is a geometric/mixing phase (holonomy), not a dynamical time delay. Using an exact transfer-matrix propagator exp(ML) for each segment, we compute reflection and transmission amplitudes and enforce flux conservation. For a representative working point (dimensionless units with ħ²/2m = 1): E = 1.0, V0 = 6.0, a = 1.2, coupler width ℓ = 0.3, and g = 5.0, the single-channel baseline visible transmission is Tv0 = 0.010367. When both channels are open asymptotically, the coupled-channel mechanism yields strong enhancement of visible transmission (up to 11.83× for VH,core = 0.0 at the optimal Δφ), with transmitted flux splitting between V and H in a manner consistent with partial re-projection. As VH,core is increased the enhancement decreases smoothly but remains substantial (for VH,core = 3.0 the enhancement is ≈ 5.45× and the visible fraction rises above 0.51). The strongest result is obtained when the hidden channel is closed asymptotically (V_H,∞ > E), so that H carries no asymptotic flux and acts as a purely virtual intermediate state. In this “closed hidden sector” regime, the mechanism can enhance visible transmission by approximately 70.65× at the optimal holonomy phase (Δφ* = 2π), while still satisfying flux conservation over open channels. This demonstrates, in the most conservative sense, that a hidden internal subspace can mediate tunneling as a virtual path without modifying the visible barrier itself. To harden interpretation against numerology, we quantify the “cost” of discretizing Δφ on a fixed governance grid Δφ_n = 2πn/N. For VH,core = 2.0, the continuum optimum at E = 1.0 is φ* ≈ 4.43907 with Tv,max ≈ 0.073557. Restricting Δφ to the Z24 grid yields Tv ≈ 0.073555 at n = 17 (Δφ = 2π·17/24), giving an efficiency η24 = Tv(Z24)/Tv(continuum) ≈ 0.999976. A direct control with N = 23 is also highly efficient (η23 ≈ 0.999137), demonstrating that the mechanism does not “prove” the number 24 but is permissive of it: any sufficiently dense grid (N ≳ 20) incurs negligible penalty because the interference resonances are broad-band rather than high-Q. We report an energy-drift table showing how the optimal discrete bin index shifts with energy (E = 0.6–1.4), while efficiencies remain near unity for both N = 23 and N = 24. Finally, a holonomy-blur (phase diffusion) Monte Carlo shows that noise-averaged efficiencies remain high (≈0.90 even for σ = 0.8 rad) and that the mean advantage of N = 24 over N = 23 under blur is small (O(10^-4)), consistent with alignment effects rather than any arithmetic uniqueness. Overall, this Appendix-T toy model provides an existence proof of compatibility: if a fundamental Z24 governance constraint exists in the Crystalline Axiverse framework, it is kinematically allowed to coexist with coupled-channel tunneling mechanisms without fine-tuning. The model suggests a clean experimental mapping to superconducting circuits (e.g., an asymmetric SQUID analogue where an internal bypass arm and flux-controlled holonomy tune reconversion), while remaining fully consistent with standard coupled-channel scattering physics. ### Abstract v.2 We revisit and harden a two-channel, piecewise-constant quantum tunneling toy model in which an internal holonomy phase acts as an interferometric “valve” controlling visible-channel transmission. The system is formulated as coupled-channel scattering for a two-component wavefunction Ψ=(ψ_V,ψ_H)^T under a Hermitian 2×2 potential that is diagonal in the core barrier (width a) and mixed only within thin coupling layers (width ℓ) on either side. The physically relevant control parameter is the relative holonomy Δφ=φ_R−φ_L between the left and right couplers, which governs constructive/destructive V→H→V interference at the exit. We implement an exact segment propagator Y(x+L)=exp(ML)Y(x), solve boundary matching for scattering amplitudes, and certify numerical stability by explicit flux conservation (machine-precision closure in sampled grids). At the canonical working point (E,V0,a,ℓ,g)=(1,6,1.2,0.3,5) in units ħ²/2m=1, the single-channel baseline (g=0) yields T_V0=0.01036713563683. With both channels open asymptotically, optimizing over Δφ reproduces the open-channel enhancement reported in the original note: for V_H,core=0 the visible transmission reaches T_V⋆≈0.122668 at Δφ⋆≈3.322667, corresponding to an 11.832× boost over T_V0, while the transmitted flux partitions between channels with a phase-dependent visible fraction T_V/(T_V+T_H). In the closed-channel regime (V_H,∞>E), the hidden channel serves as a virtual intermediate state and produces a much larger enhancement, with T_V,max≈0.732437 (≈70.65× over baseline) near Δφ=2π. Robustness under holonomy diffusion is assessed using a deterministic wrapped-Gaussian expectation over precomputed T_V(Δφ), reproducing the blur table (e.g. ≈69.38× at σ=0.4 and ≈65.53× at σ=0.8) without Monte Carlo variance. A central question is whether a discrete “governance” rule on Δφ can be imposed without reintroducing fine tuning. We define the governance efficiency η_N(E) as the best discrete-bin transmission over Δφ∈{2πn/N} relative to the continuum optimum. At (E,V_H,core)=(1,2), we reproduce near-lossless governance for Z24 (η_24=0.999976) and Z23 (η_23=0.999137). Hardening against energy drift over E∈{0.6,0.8,1.0,1.2,1.4} reveals a binding point at E=0.8 where η_24 drops to 0.997641, motivating a data-driven drift gate η_min≳0.997. An extended N-scan over N∈[6,48] exhibits a sharp “cliff” (e.g. Z12 gives η_12(E=1)=0.988401), while Z24 and Z48 remain near-unity at the canonical point. **Proposition (Minimum Viable Divisor-of-48 Governance).** Let the admissible governance grids be restricted by the Z48 architecture to divisors of 48 (N|48). Let the drift-hardened gate require η_min≡min_{E∈{0.6,0.8,1.0,1.2,1.4}} η_N(E) ≥ 0.997. Then the smallest admissible N that passes the gate is N=24. (Numerically, the first N in [6,48] satisfying the drift gate is N=18, but it is excluded by the Z48 divisor filter; among N|48, the minimum viable choice is N=24.) These results establish a compact, auditable benchmark: the enhancement mechanism is reproducible, the discretization cost is quantified by explicit efficiency gates, and the stability of the solver is verified by flux closure. The model is intended as a constrained toy laboratory for “crystalline” phase governance and does not by itself supply a microphysical derivation of the Z48→Z24 restriction; rather, it demonstrates how such a restriction, when imposed, can be tested and hardened against energy drift and phase diffusion in a transparent gate-based framework.