This exploratory preprint studies a controlled quantum-scattering toy model with an additional harmonically confined coordinate w. It compares a fixed-width one-dimensional barrier, the exact shrinking observable slice V(x,0), and the full coupled (x,w) Hamiltonian. The identical-slice control shows that the non-monotonic transmission turn is not unique to an additional coordinate, while the coupled calculation produces a quantitatively distinct transmission curve and a converged transverse-broadening signal. The action-coordinate hypothesis is S_E = f(w), with A = S_E / ħ. The original eleven-point parameter sweep gives a monotonic nonlinear locus when A is plotted against the center broadening Δw_rms. A new matched-state stress test independently changes longitudinal energy or transverse confinement and retunes the barrier to reproduce Δw_rms = 0.08, 0.18, and 0.35. The corresponding WKB actions spread by 50.2%, 31.1%, and 22.7% relative to the baseline; the exact-scattering exponent agrees, and a refined-grid calculation is stable. Thus a universal one-variable proxy A = F(Δw_rms) fails without refitting. This result does not falsify an otherwise unspecified microscopic relation S_E = f(w), because Δw_rms is an observable moment rather than the coordinate or tunneling path w. It shows that broadening alone is insufficient and that a richer state description or path-level definition is required. The work is non-peer-reviewed and does not claim evidence that nature has an accessible fourth spatial dimension. The supplement contains the calculation scripts, CSV data, numerical metadata, convergence checks, and figures; AI assistance is disclosed in the manuscript.
This exploratory preprint studies a controlled quantum-scattering toy model with an additional harmonically confined coordinate w. It compares a fixed-width one-dimensional barrier, the exact shrinking observable slice V(x,0), and the full coupled (x,w) Hamiltonian. The identical-slice control shows that the non-monotonic transmission turn is not unique to an additional coordinate, while the coupled calculation produces a quantitatively distinct transmission curve and a converged transverse-broadening signal. The action-coordinate hypothesis is S_E = f(w), with A = S_E / ħ. The original eleven-point parameter sweep gives a monotonic nonlinear locus when A is plotted against the center broadening Δw_rms. A new matched-state stress test independently changes longitudinal energy or transverse confinement and retunes the barrier to reproduce Δw_rms = 0.08, 0.18, and 0.35. The corresponding WKB actions spread by 50.2%, 31.1%, and 22.7% relative to the baseline; the exact-scattering exponent agrees, and a refined-grid calculation is stable. Thus a universal one-variable proxy A = F(Δw_rms) fails without refitting. This result does not falsify an otherwise unspecified microscopic relation S_E = f(w), because Δw_rms is an observable moment rather than the coordinate or tunneling path w. It shows that broadening alone is insufficient and that a richer state description or path-level definition is required. The work is non-peer-reviewed and does not claim evidence that nature has an accessible fourth spatial dimension. The supplement contains the calculation scripts, CSV data, numerical metadata, convergence checks, and figures; AI assistance is disclosed in the manuscript.
This non-peer-reviewed exploratory preprint tests whether a coupled quantum Hamiltonian with one additional spatial coordinate can produce barrier transmission distinct from ordinary one-dimensional quantum mechanics when both calculations share exactly the same observable slice V(x,0). A prescribed self-similar Gaussian barrier is compared using three calculations: fixed-width 1D, shrinking identical-slice 1D, and full coupled (x,w) scattering with harmonic confinement. The shrinking 1D slice is itself non-monotonic, so an upward turn is not uniquely extra-dimensional. The full model nevertheless gives a quantitatively different curve, with a sampled minimum near V0=18 and T=9.87e-3 at V0=64 versus 6.09e-9 for the identical slice. This is a reproducible coupled-channel result within a mathematical toy Hamiltonian, not evidence that nature has an accessible fourth spatial dimension. The barrier family is prescribed rather than derived from a constitutive field law. The record includes the manuscript and a supplement containing code, complete numerical results, metadata, and convergence checks. ChatGPT and Sarvam AI assisted with brainstorming and language development. ChatGPT also assisted with literature triage, control design, numerical implementation and auditing, and drafting. AI systems are not authors. Chaman Prakash Kanth remains responsible for the manuscript, assumptions, metadata, and publication decision.
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.
This paper develops a phase-centered tunneling framework that is deliberately positioned as a bridge theory rather than a claimed final unification of quantum theory and relativity. The starting point is a PWD-inspired background phase field whose gradient defines a null causal skeleton. The missing ingredient is the forbidden-region sector: under a barrier, real classical trajectories fail, the eikonal must complexify, and attenuation becomes unavoidable. The central move of the paper is therefore to place a minimal complex-action fluctuation sector on top of the null background, while keeping a strict separation between the background phase gradient Kµ = ∇µΘ and the physical fluctuation momentum Pµ = ∇µS − qAµ. An auxiliary null direction defines a screen-space geometry in which attenuation can be written as a covariant closure without opening a new causal cone.The conceptual claim is narrow but strong. The relativistic contact is not a literal identification between an imaginary Lorentz factor and an imaginary tunneling momentum. It is the structural statement that both relativistic edge cases and forbidden-region quantum dynamics require a complex phase-action description once a naive real-trajectory picture collapses. This immediately yields a1local-universality/global-selectivity principle: the local Airy/Stokes machinery near simple turning points remains standard, whereas genuinely PWD-selective content should first appear in globally organized multi-turning-point observables such as resonance shifts, linewidth modulation, and phase-memory effects.The manuscript is tested at three levels. First, exact one-dimensional Schrodinger and Klein-Gordon barriers recover the expected large-thickness attenuation slope and the expected dominance of the imaginary under-barrier wavevector, establishing consistency rather than novelty. Second, the paper formulates an explicit empirical protocol for deciding whether a putative PWD signal is real: penalized model comparison, parameter identifiability, holdout stability across neighboring resonances or geometry sweeps, digitization-perturbation robustness, and negative controls that should return vanishing correction coefficients. Third, the paper includes exact source-level operational comparisons on two publisher-plate target-corpus panels aligned with the theory’s predicted first battlefields: a Kobos linewidth panel and a Schmidt geometry/asymmetry panel. In both cases, a minimal phase-organization proxy outperforms the corresponding baseline under RMSE, MAE, AIC, and BIC. These results do not yet constitute the final low-parameter direct δΦPWD /δAPWD theory fit, but they move the proposal beyond a purely future-facing benchmark plan.The resulting picture is disciplined. The framework isolates a mathematically coherent contact surface between null causal organization and forbidden-region attenuation, embeds that contact surface in a screen-supported complex-action EFT, and identifies precise observables on which the theory can fail. What remains open is equally explicit: the attenuation sector is still a leading-order effective closure rather than a completed microscopic derivation, higher-order coefficients remain to be fixed, and decisive empirical status requires direct low-parameter fits that survive robustness and holdout tests. The strongest defensible claim is therefore that a falsifiable bridge framework has been constructed and partially anchored at the exact source level, not that a completed final unification has already been achieved.
This educational technical note derives the leading semiclassical tunneling exponent for a one-dimensional particle in a static electrostatic potential from the local Dirac dispersion relation, and shows how the Schrödinger WKB result follows in the low-energy limit. It also introduces a hypothetical parameter lambda multiplying the quadratic relativistic term and compares the corresponding exponents for a defined rectangular barrier. The standard Dirac-WKB derivation reproduces established relativistic quantum mechanics; it is not presented as a new physical law. The lambda modification is exploratory and has no independent theoretical derivation or experimental support in this note. The numerical comparison is a sensitivity calculation for an illustrative 10 fm rectangular barrier with constant Delta = 0.1 MeV and mu*c^2 = 1118 MeV. It is not a realistic D-T interaction model or a prediction of a fusion reaction rate. The note does not claim cold fusion, anomalous fusion enhancement, or reactor-relevant performance. A one-particle Dirac equation does not describe the complete D-T nuclear collision, which involves composite nuclei, strong-interaction channels, spin, and plasma/environment effects. The cited 2026 D-D metallic-hydride study concerns ion-beam and electrochemical loading conditions and does not validate the hypothetical lambda model or establish self-sustaining fusion. AI assistance: ChatGPT assisted with literature triage, numerical calculations, equation review, and drafting. Chaman Prakash Kanth is responsible for the scientific assumptions, calculations, manuscript, and metadata.
This open research notebook develops a falsification-first route from the Flatland analogy to testable additional-coordinate models in quantum tunnelling. It separates three independent pipelines: predictions derived from one fixed four-spatial-dimensional model, residuals extracted from calibrated measurements without invoking w, and ordinary-physics controls. The primary constitutive hypothesis is S_E=f(w), where S_E is dimensionful Euclidean action and A=S_E/hbar is the dimensionless tunnelling exponent. No linear law is assumed, and the coordinate w is not identified with RMS transverse broadening. Existing controlled calculations establish a coupled-channel effect inside a specified (x,w) Hamiltonian, while matched-broadening stress tests show that center broadening alone does not determine action. Status: open, non-peer-reviewed working paper. The notebook does not report evidence for a fundamental fourth spatial dimension. It provides a pre-registered, no-refit framework for deciding what future observations would count against or in favor of the model. Related control study: https://doi.org/10.5281/zenodo.22071952
We study finite-size adiabatic state preparation on Boolean hypercubes using graph-local drivers built from sector/path coordinates related to monotone Gray-code representatives. The construction is not presented as a new all-n Gray-code existence theorem; rather, it provides finite representatives, explicitly checked through the cases used in the numerical experiments, for testing problem-dependent graph-local drivers. For ordinary diagonal-cost transverse-field annealing, the ordering does not yield a robust advantage, and we include this negative result as a baseline. For non-diagonal target Hamiltonians whose geometry is expressed in the same sector/path coordinates, hybrid drivers combining sector, path-window, and small transverse-field components can substantially improve the final ground-state fidelity in centered barrier instances. Reproduction runs from the accompanying code confirm a representative centered original-window barrier value of approximately 0.9799 for the fixed-control hybrid parameters (w,α,ε)=(8,0.50,0.15), while also showing that the improvement is target-class dependent. Randomized and ablation controls indicate that the dominant contribution is the sector-preserving skeleton, with strict one-bit completion acting as a secondary refinement. We provide code, finite certificates, CSV files, validation logs, and reproduction scripts to make the finite-size claims traceable.