Tunneling provides a particularly clean microscopic setting in which the amplitude, PhaSe, projective geometry, and relational-time sectors of sPNP can be examined together. Strong spatial variation of a quantum state need not coincide with physical transport, and spatial change of an internal quantum state need not by itself define a clock.
In the kinematic regime, the wavefunctional obeys ordinary linear quantum dynamics, iħ ∂ₜΨ = ĤΨ, with Ψ = R eⁱˢ/ħ. The resolved Jacobi–Fisher geometry is constructed from this state at fixed Ψ. Transmission and reflection probabilities, resonances, interference structure, and asymptotic scattering observables therefore remain those generated by the underlying linear quantum dynamics. The sPNP construction supplies a relational geometry for the amplitude and PhaSe/current structure carried by the same quantum state. This is the fixed-Ψ geometric closure distinguished above from reflexive dynamical closure.
Evanescence as Resolved Amplitude Geometry
Let ρ(x) = |Ψ(x)|² and define the amplitude score sₓ = ∂ₓ ln ρ. At finite relational resolution Q₀, the amplitude sector is represented by the resolved second moment Fᴿ,(Q₀)ₓₓ(x) = ρ_Q₀(x)⁻¹ ∫dy K_Q₀(x,y) ρ(y) sₓ(y)², where K_Q₀ is the intrinsic heat kernel associated with the relational geometry. This is the one-dimensional reduction of the Q₀-resolved Fisher construction: nearby score directions are sampled across the relational resolution cell rather than identified with a bare pointwise tensor.
For an approximately uniform forbidden region, R(x) ∝ e⁻κˣ, where κ² = 2m(V − E)/ħ². Hence ρ(x) ∝ e⁻²κˣ and sₓ = −2κ, giving Fᴿ,(Q₀)ₓₓ = 4κ² = 8m(V − E)/ħ². Because the ideal exponential has constant score, finite-Q₀ averaging leaves this result unchanged away from the barrier boundaries. The forbidden region therefore becomes increasingly stiff in its amplitude geometry as V − E grows. This belongs to the sPNP Distinction geometry: κ measures the spatial rate of amplitude change, whereas physical transport requires the additional PhaSe/current structure of Ψ.
A smooth turning region makes this interpolation explicit. Locally take V(x) − E ≈ Fx and define the turning length ℓₜ = [ħ²/(2mF)]¹ᐟ³. The decaying solution is R(x) ∝ Ai(ξ), with ξ = x/ℓₜ, and the local Fisher contribution becomes Fₓₓ = 4ℓₜ⁻²[Ai′(ξ)/Ai(ξ)]². It remains finite at the turning point and approaches Fₓₓ → 4κ² in the forbidden WKB region. Finite Q₀ replaces this pointwise structure by its covariant neighborhood average and smooths the transition further.
Tunneling therefore realizes the sequence amplitude profile → local score → Q₀-resolved Fisher tensor → relational geometry. A classically forbidden region can carry strong amplitude geometry even when its ordinary phase current is small.
Amplitude Geometry, PhaSe, and Physical Change
The separation becomes richer when the tunneling system possesses internal, spin, orbital, or transverse quantum structure. Write locally Ψₐ(X) = R(X)eⁱˢ⁽ˣ⁾/ħuₐ(X), with u†u = 1. R supplies the amplitude geometry, while S and the variation of u supply the gauge-covariant PhaSe/current structure. The normalized internal state defines a quantum ray whose horizontal derivative is D_Iu = (1 − |u⟩⟨u|)∂_Iu, and the corresponding quantum geometric tensor is 𝒬_IJ = ⟨D_Iu|D_Ju⟩. Its real symmetric part gᴽ_IJ = Re 𝒬_IJ measures projective separation between nearby quantum rays, while its imaginary antisymmetric part carries the complementary symplectic structure associated with Berry curvature and geometric transport.
A spatially changing internal state need not therefore represent temporal evolution. Variation can arise from Hamiltonian evolution, differential attenuation, spatial filtering, geometric transport, or correlations with other configuration variables. The Fubini–Study metric measures how much the ray changes; its orientation relative to Hamiltonian flow determines what kind of change has occurred.
Projective Clock Compatibility
Let Wᴵ denote a proposed configuration-space clock direction and define its horizontal projective tangent by D_Wu ≡ WᴵD_Iu. For the longitudinal coordinate in the coupled-waveguide example below, W = ∂ₓ. The autonomous Hamiltonian tangent of the internal subsystem is X_H ≡ −(i/ħ)(H_int − ⟨H_int⟩)u. A configuration variable can parameterize internal Hamiltonian time only when its projective variation follows X_H up to a real reparameterization, X_H = aD_Wu with real a.
This motivates the gauge-invariant clock-compatibility scalar C_clock[W] = [Re⟨D_Wu|X_H⟩]²/[⟨D_Wu|D_Wu⟩⟨X_H|X_H⟩]. Under the local projective redundancy u → eⁱχu, both D_Wu and X_H acquire the same overall phase, so the inner products and norms entering C_clock are unchanged. Cauchy–Schwarz gives 0 ≤ C_clock ≤ 1, with C_clock = 1 when the proposed configuration change follows the Hamiltonian projective direction up to a real change of parameter.
The complementary Kähler component may be defined as C_K[W] = [Im⟨D_Wu|X_H⟩]²/[⟨D_Wu|D_Wu⟩⟨X_H|X_H⟩], with C_⊥ = 1 − C_clock − C_K. Thus C_clock + C_K + C_⊥ = 1, with each term non-negative. C_clock measures alignment with real Hamiltonian evolution, C_K measures alignment with its complex-structure-conjugate direction, and C_⊥ measures the part of Hamiltonian evolution not represented by either.
The triplet (C_clock, C_K, C_⊥) therefore supplies a local decomposition of a proposed relational clock direction. Projective compatibility is a necessary clock condition rather than the whole clock construction: a physical clock must also vary appropriately along the actual relational history and possess sufficient dynamical autonomy. Tunneling supplies a simple two-level case with C_⊥ = 0; whether the full triplet provides a useful clock-quality diagnostic in higher-dimensional quantum sectors remains open.
Coupled-Waveguide Tunneling
The coupled-waveguide experiment of Sharoglazova et al. provides a concrete two-mode realization. Population transfer between transverse optical modes was used as an internal timing variable, with the known transverse coupling J₀ supplying the reference rate. Spatial population growth along x was then converted into an inferred nondirectional speed. In the deeply evanescent regime, the reported relation approaches v = ħκ/m = √[2(V − E)/m] (Sharoglazova et al., 2025).
The propagating regime can be represented by the transverse state u_P(x) = cos(δkx)|m⟩ + i sin(δkx)|a⟩. The transverse double-well dynamics have coherent Rabi evolution of the form cos(J₀t)φₘ + i sin(J₀t)φₐ, with J₀ fixed by the splitting of the two lowest transverse eigenstates (Beck et al., 2026). For the corresponding two-level Hamiltonian H_int = −ħJ₀σₓ, after removal of the irrelevant mean energy, Dₓu_P = δk[−sin(δkx)|m⟩ + i cos(δkx)|a⟩] while X_H = J₀[−sin(δkx)|m⟩ + i cos(δkx)|a⟩]. Hence X_H = (J₀/δk)Dₓu_P, so the proportionality is real and (C_clock, C_K, C_⊥) = (1,0,0). The spatial sequence of transverse quantum rays is therefore the same sequence generated by internal Hamiltonian evolution, with longitudinal position providing a real reparameterization of time.
In the propagating approximation J₀/δk = ħk̄/m = v_g, so this projective conversion rate coincides with the ordinary longitudinal group velocity. This agrees with the broader result of Beck et al. that the transverse double-well dynamics can operate approximately as an autonomous clock in the propagating regime (Beck et al., 2026).
The evanescent state has a different structure. With κ̄ = (κ₊ + κ₋)/2 and δκ = (κ₋ − κ₊)/2, the normalized transverse part of the ideal stationary state can be written u_E(x) = [cosh(δκx)|m⟩ + sinh(δκx)|a⟩]/√cosh(2δκx). Writing tan θ = tanh(δκx) gives Dₓu_E = δκ sech(2δκx)e_θ, whereas the same transverse Hamiltonian gives X_H = iJ₀ sech(2δκx)e_θ. Thus X_H = i(J₀/δκ)Dₓu_E, so the two tangents have proportional norms but a purely imaginary rather than real relation. The resulting decomposition is (C_clock, C_K, C_⊥) = (0,1,0).
The evanescent spatial change is therefore not merely weakly aligned with Hamiltonian time. In the ideal two-level limit it is aligned with the corresponding Kähler-conjugate direction. The comparison is consequently propagating: (C_clock,C_K,C_⊥) = (1,0,0) and evanescent: (C_clock,C_K,C_⊥) = (0,1,0). The same local quantum-state direction is involved, but its relation to Hamiltonian evolution differs: propagating variation is Hamiltonian-aligned, whereas ideal evanescent variation is Kähler-conjugate.
This gives a geometric expression of the central result of Beck et al. Their analysis finds that the speed inference assumes transverse inter-waveguide tunneling remains an autonomous clock whose timescale is set solely by J₀; this approximation works in the propagating regime but fails in the evanescent regime because longitudinal and transverse structure are no longer dynamically separable in the required way (Beck et al., 2026).
Finite-Bandwidth Evanescence
A physical evanescent wave packet is not exactly stationary. Beck et al. emphasize that time-dependent packets possess a nonzero Bohmian velocity even in the forbidden region, although this velocity remains orders of magnitude smaller than the speed inferred from the transverse-population procedure (Beck et al., 2026). In their narrow-band two-mode approximation, write the relative longitudinal factor schematically as q(x,t) = A(x,t) − iB(x,t), where A parametrizes differential amplitude filtering and B parametrizes the induced relative-phase variation.
For this normalized two-level family, direct evaluation of the quantum geometric tensor gives the Fubini–Study line element ds²_FS = sech²(2A)(dA² + dB²). The spatial projective tangent consequently has the form Dₓu = AₓD_Au + BₓD_Bu, while the internal Hamiltonian tangent lies along the phase direction. It follows that C_clock = Bₓ²/(Aₓ² + Bₓ²), C_K = Aₓ²/(Aₓ² + Bₓ²), and C_⊥ = 0. The stationary limit Bₓ → 0 therefore reproduces (C_clock,C_K) → (0,1).
Finite bandwidth introduces a small Hamiltonian-aligned component into a spatial variation that remains predominantly Kähler-conjugate. This is a geometric decomposition of the ordinary linearly evolved quantum state in the kinematic regime; it is not an additional physical rotation imposed by sPNP.
In the narrow-band weak-splitting approximation, the same phase component is related to the small longitudinal Bohmian current calculated by Beck et al. Defining the large evanescent projective scale as v_ev ≡ J₀/δκ = ħκ̄/m, the leading-order relation becomes Bₓ/Aₓ ≈ −v_B/v_ev. Therefore, when |v_B| ≪ v_ev, one obtains C_clock ≈ (v_B/v_ev)² and C_K ≈ 1 − (v_B/v_ev)². The exact stationary result C_clock = 0 is thus approached continuously as the physical longitudinal current tends to zero. This correspondence is derived here from the narrow-band two-mode approximation of Beck et al.; it is not assumed to hold for arbitrary quantum states.
Projective Rate Decomposition
The same geometry separates three rate-like quantities: v_norm ≡ ‖X_H‖/‖Dₓu‖, v_real ≡ Re⟨Dₓu|X_H⟩/‖Dₓu‖², and v_K ≡ Im⟨Dₓu|X_H⟩/‖Dₓu‖². They obey v_real² + v_K² ≤ v_norm², with equality when the relevant Hamiltonian motion is contained in the local complex direction generated by Dₓu.
For the propagating two-level state, v_norm = v_real = ħk̄/m and v_K = 0. For ideal stationary evanescence, v_norm = v_K = ħκ̄/m while v_real = 0. For the finite-bandwidth evanescent packet, the narrow-band weak-splitting calculation instead gives v_norm ≈ ħκ̄/m, v_K ≈ ħκ̄/m, while, to leading order, v_real ≈ v_B.
The large scale ħκ̄/m ≈ √[2(V − E)/m] therefore remains present in the projective geometry without automatically becoming the longitudinal transport rate. In the evanescent regime it lies predominantly in the Kähler-conjugate component, whereas the much smaller real Hamiltonian-aligned component tracks the Bohmian longitudinal current in this controlled approximation. Sharoglazova et al. measure the corresponding large nondirectional scale through transverse populations, whereas Beck et al. argue that the temporal interpretation of that population variation fails in the evanescent regime (Sharoglazova et al., 2025; Beck et al., 2026).
This gives a natural geometric interpretation of why a procedure sensitive to the magnitude of internal-state change can return a quantity proportional to ħκ/m even when the actual longitudinal current is much smaller. The norm scale remains genuine structure of the quantum state; its orientation relative to Hamiltonian flow determines whether that scale can serve as a real time-to-distance conversion rate.
The relation v_real ≈ v_B is established here only to leading order in the narrow-band two-mode approximation. It is not presently a general identity between the projective rate decomposition and Bohmian velocity.
Two-Mode Amplitude Geometry
The amplitude sector can be examined independently of this projective orientation. For the ideal two-channel evanescent state, the longitudinal Fisher contribution decomposes as Fₓₓ = [−2κ̄ + 2δκ tanh(2δκx)]² + 4δκ² sech²(2δκx). The first term tracks the changing total evanescent envelope, while the second arises from redistribution between the transverse components.
The amplitude geometry therefore separates overall attenuation from internal mode restructuring without assigning either contribution a longitudinal velocity. Nor does failure of the internal clock require a large mixed Fisher component Fₓy. In the weak-overlap limit Fₓy can remain small even while x fails to parameterize autonomous transverse Hamiltonian evolution. The relevant structure is therefore not simply off-diagonality of the amplitude metric, but the relation between amplitude geometry and the orientation carried by the PhaSe/projective sector.
When Q₀ is small compared with the longitudinal variation length and transverse mode width, the resolved tensor approaches this local two-mode expression. Finite Q₀ supplies the corresponding covariant neighborhood average.
Clock Compatibility and Subsystem Autonomy
Projective clock compatibility and subsystem autonomy are related but separate questions. For an actual relational trajectory X(t), define the conditional projective tangent schematically as T_cond = P_u(∂ₜ + Ẋᴵ∂_I)u, where P_u = 1 − |u⟩⟨u|. The autonomous internal Hamiltonian tangent remains X_H, and the residual R_aut = T_cond − X_H measures failure of the conditional subsystem to evolve solely under H_int.
Thus C_clock asks whether variation along a proposed configuration coordinate can locally represent Hamiltonian time, whereas R_aut asks whether the corresponding conditional subsystem actually evolves autonomously under its internal Hamiltonian. Beck et al. make this autonomy issue central: in the evanescent regime the transverse conditional dynamics remain affected by the longitudinal degree of freedom, whereas the propagating regime admits the approximate autonomous Rabi-clock behavior required by the experimental inference (Beck et al., 2026).
A complete relational clock therefore involves more than local projective alignment. It also requires an appropriate physical history and sufficient subsystem autonomy for the selected degree of freedom to provide a stable temporal reference.
Relational Resolution and Gaussoherence
The intrinsic Q₀ resolution used to construct Fᴿ,(Q₀) should be distinguished from Gaussoherence as a dynamical contraction of unresolved structure. The Relational Kernel K_Q₀⁽ᴳ⁾ = exp(−Q₀²L_G) defines the finite neighborhood over which amplitude geometry is resolved. Evaluating this tensor does not by itself physically evolve or decohere Ψ. An isolated coherent tunneling state can therefore possess finite-Q₀ relational geometry while retaining its coherent quantum structure.
In the present sPNP architecture, Gaussoherence instead concerns dynamical contraction of modes that become unresolved relative to the surviving physical sector. The existing construction distinguishes contracted unresolved modes from coherent horizontal modes that may remain marginal.
The coupled-waveguide system makes the relational meaning of “transverse” especially clear. A laboratory-transverse degree of freedom can itself participate in a coherent physical clock. Gaussoherence therefore cannot classify a mode as contractible merely from its coordinate orientation. The relevant separation is between surviving coherent horizontal structure and relational directions that become unresolved.
The three operations are consequently distinct: intrinsic Q₀ resolution → construction of resolved geometry; Gaussoherence → contraction of unresolved relational structure; and spacetime projection → lower-dimensional representation of surviving upstream structure. These operations can be governed by closely related heat-kernel geometry without representing the same physical process.
Kinematic and Reflexive Tunneling
In the kinematic regime, tunneling provides a microscopic test of how an ordinary coherent quantum state is represented in sPNP geometry. The amplitude determines resolved Fisher structure; PhaSe supplies current and projective orientation; the quantum geometric tensor encodes the metric and symplectic structure of projective state space; and the associated complex structure relates gradient-like and Hamiltonian-like directions. Jacobi reduction then supplies the scalar relational parametrization along the physical history.
The coupled-waveguide system makes these roles explicit. A large projective rate can exist without being a real transport rate. In ideal propagation the spatial internal-state variation is Hamiltonian-aligned; in ideal evanescence it is Kähler-conjugate. Finite bandwidth supplies a small Hamiltonian-aligned component whose leading behavior in the Beck et al. narrow-band approximation tracks the actual longitudinal Bohmian current.
The stronger reflexive regime begins when the state-dependent geometry is fed back into the quantum generator, Ψ → Ĝ[Ψ] → Ĥ[Ĝ,Ψ] → Ψ. Only then can the tunneling state itself depart from the ordinary linear solution, potentially modifying transmission probabilities, resonance structure, delay observables, or other measurable quantities. Such a calculation requires an explicit state-dependent operator together with the corresponding conditions of positivity or ellipticity, self-adjointness, continuity, and equivariance.
Tunneling therefore supplies two complementary probes of sPNP. Kinematically, it tests the consistency of amplitude geometry, PhaSe, projective change, and relational time on a fully coherent quantum state. Reflexively, it provides a controlled setting in which state-dependent quantum geometry can eventually be tested for genuine dynamical departures from the standard theory.
Relational Clock Selection The projective clock criterion does not replace the Jacobi–Eisenhart construction of relational time. PhaSe—the gauge-covariant phase/current and holonomy sector—supplies directed relational change, while Jacobi–Eisenhart reduction selects and normalizes a scalar clock parametrization along the resulting physical history. PhaSe is therefore richer than time: coherent orbital, rotational, transport, and internal-holonomy structure can survive as physical change without itself constituting the scalar clock direction. This is the same separation used in the broader sPNP time construction. The triplet (C_clock, C_K, C_⊥) instead asks whether a chosen subsystem can provide a faithful local reading of that relational evolution. For a candidate clock direction W, C_clock ≈ 1 means that its projective change is locally aligned with autonomous Hamiltonian evolution; C_K > 0 measures change along the corresponding Kähler-conjugate direction; and C_⊥ > 0 measures Hamiltonian evolution not represented by either. These quantities characterize projective clock compatibility rather than defining the underlying scalar time. A usable internal clock additionally requires appropriate progression along the actual relational history and sufficient subsystem autonomy. With the conditional tangent T_cond = P_u(∂ₜ + Ẋᴵ∂I)u and autonomy residual R_aut = T_cond − X_H, small R_aut means that the selected subsystem approximately follows its own Hamiltonian evolution rather than remaining strongly parametrically driven by other degrees of freedom. Clock quality therefore involves at least three logically separate properties: projective compatibility, suitable progression along the Jacobi history, and approximate dynamical autonomy. The coupled-waveguide system illustrates the separation. In the propagating regime, C_clock ≈ 1 and the transverse state approximately undergoes autonomous Rabi evolution, allowing longitudinal position to parameterize its internal change. In ideal evanescence, (C_clock,C_K,C⊥) = (0,1,0): the internal state changes strongly with position, but its change is Kähler-conjugate rather than Hamiltonian-time aligned. Finite bandwidth introduces a small Hamiltonian-aligned component associated with the actual longitudinal current, without by itself restoring an autonomous clock. or Relational Clock Selection The projective clock criterion does not replace the Jacobi–Eisenhart construction of relational time. PhaSe—the gauge-covariant phase/current and holonomy sector—supplies directed relational change, while Jacobi–Eisenhart reduction selects and normalizes a scalar clock parametrization along the resulting physical history. PhaSe is therefore richer than time: coherent orbital, rotational, transport, and internal-holonomy structure can survive as physical change without itself constituting the scalar clock direction. This is consistent with the broader sPNP separation between PhaSe and time. Distinctions.docx The triplet (C_clock, C_K, C_⊥) instead tests whether a chosen subsystem can provide a faithful local reading of that relational evolution. C_clock ≈ 1 means that its projective change is locally aligned with autonomous Hamiltonian evolution; C_K > 0 measures change along the corresponding Kähler-conjugate direction; and C_⊥ > 0 measures Hamiltonian evolution not represented by either. These quantities characterize clock compatibility rather than defining the underlying scalar time. A useful internal clock additionally requires suitable progression along the actual Jacobi history and approximate subsystem autonomy. With T_cond = P_u(∂ₜ + Ẋᴵ∂I)u and R_aut = T_cond − X_H, small R_aut indicates that the selected subsystem approximately follows its own Hamiltonian evolution rather than remaining strongly driven by other degrees of freedom. Projective compatibility, progression along the physical history, and dynamical autonomy are therefore separate aspects of clock quality rather than individually sufficient conditions. The coupled-waveguide system illustrates the separation. In the propagating regime, C_clock ≈ 1 and the transverse subsystem approximately undergoes autonomous Rabi evolution, allowing longitudinal position to parameterize its internal change. In ideal evanescence, (C_clock,C_K,C⊥) = (0,1,0): the internal state changes strongly with position, but along the Kähler-conjugate rather than Hamiltonian-time direction. Finite bandwidth introduces a small Hamiltonian-aligned component associated with the actual longitudinal current without by itself restoring an autonomous clock.
