1. Introduction

One of the deepest foundational inconsistencies in contemporary physics lies in the treatment of time. Since 1967 the SI second has been defined as the duration of exactly 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom [1]. Symbolically this reads

$T := \frac{1}{\nu}$

which is mathematically identical to the classical definition of curvature κ = 1/R. Both expressions equate a finite, measurable quantity with its reciprocal, thereby committing a category mistake: a genuine physical measurement must compare quantities of the same ontological kind.

The consequence is that time is treated as an external, linear, and infinitely extendable parameter that exists independently of the system under study. This assumption is ontologically untenable for any closed system—and in particular for the universe as a whole—because no external clock can exist outside the system by definition.

The ramifications of this error permeate the entire edifice of modern physics. In quantum mechanics the time-dependent Schrödinger equation

$i\hbar \frac{\partial\Psi}{\partial t} = \hat{H}\Psi$

presupposes precisely such an external parameter t. The resulting unitary evolution is deterministic in the mathematical sense, yet experiment reveals an apparently indeterministic collapse upon measurement. In quantum gravity the same external-time postulate leads to the notorious “problem of time”: the Wheeler–DeWitt equation Ĥ|Ψ⟩ = 0 is timeless, yet canonical quantization of general relativity still relies on an external foliation of spacetime. The incompatibility between quantum theory and general relativity is therefore not merely technical; it is axiomatic and stems directly from the inconsistent treatment of time.

The Panvitalistic Theory (PVT), developed between 2019 and 2026, offers a radical yet parsimonious resolution. Its foundational axiom is the ontological primacy of finite volume. Every physical entity is a real 6-dimensional volume element

$V = L_1 L_2 L_3 \sin\theta_{12} \sin\theta_{23} \sin\theta_{31}$

where 0° < θ_ij ≤ 90° are the three angular degrees of freedom between the length axes. Physical measurement consists exclusively of rational comparisons of two such volumes:

$V_A = x V_B, \quad x \in \mathbb{Q}$

Time is no longer an external parameter but internal angular curvature, formalized by the single admissible scaling relation

$k\pi = 1 \quad \text{with} \quad k = \frac{L}{T}, \quad \pi = \frac{T}{L}$

The sole dynamical law is the volume-invariance constraint

$\delta V = 0$

which replaces all action principles and variational problems that rely on external time.

Under these axioms the 12-dimensional state space arises naturally as the complete description of a rational volume comparison (6 degrees of freedom for each of the two volumes). The 12 fundamental volume operators {L̂^A_i, L̂^B_i, Θ̂^A_jk, Θ̂^B_jk} and their geometric commutator algebra are derived directly from the requirement that infinitesimal variations preserve δV = 0. At maximal orthogonality (θ_ij = 90°) the commutators vanish, recovering classical physics as a projection artefact. Measurement is re-interpreted as the selection of one internal degree of freedom as the “clock”, leading to a deterministic reduction of the 12D geometry rather than a probabilistic collapse.

The present work demonstrates that this framework completely replaces the time-dependent Schrödinger equation and its associated measurement problem. As a concrete and experimentally decisive test case we re-derive the entire Bohr model of the hydrogen atom—discrete radii r_n, energy eigenvalues E_n ∝ −1/n², and the fine-structure constant α—directly from the volume constraint δV = 0. A macroscopic Earth-satellite analogue provides immediate empirical confirmation that quantization is not a microscopic peculiarity but a universal geometric consequence of rational 6D volume comparisons in anisotropic spacetime.

By eliminating the external-time postulate at the axiomatic level, the Panvitalistic Theory dissolves the QT–GRT incompatibility, resolves the problem of time, renders singularities finite angular boundaries, and unifies microscopic and macroscopic physics within a single rational ontology.

2. Mathematical Foundations of the Panvitalistic Theory

The Panvitalistic Theory rests on three interlocking axioms that together define a consistent, rational, and timeless ontology for physics.

2.1 Volume Primacy and the 6D Volume Element

Physical reality consists exclusively of finite, real volumes. Any volume element is described by three length coordinates and three angular degrees of freedom between them:

$V = L_1 L_2 L_3 \sin\theta_{12} \sin\theta_{23} \sin\theta_{31}, \quad 0^\circ < \theta_{ij} \le 90^\circ$

The sine terms arise naturally because the effective volume vanishes when any pair of length axes becomes collinear (θ_ij → 0°) and reaches its maximum at perfect orthogonality (θ_ij = 90°).

2.2 Rational Volume Comparisons

A physical measurement is defined as the rational comparison of two such volume elements:

$V_A = x V_B, \quad x \in \mathbb{Q}$

Irrational ratios are excluded by construction; only rational x correspond to physically realizable states. This discreteness is ontological, not emergent.

2.3 Internal Angular Curvature as Time

Time is not an external linear parameter but internal angular curvature. The single admissible scaling relation between length and angle is

$k\pi = 1 \quad \text{with} \quad k = \frac{L}{T}, \quad \pi = \frac{T}{L}$

Here π carries dimension [T/L] and is no longer the dimensionless circle constant of classical geometry. The constant k is fixed once by the choice of a reference length (historically tied to terrestrial measurements), after which time is completely determined by the angular geometry of the system itself.

2.4 The 12 Volume Operators

A complete rational comparison V_A = x V_B involves twelve independent degrees of freedom (six for each volume). We therefore introduce the following set of operators acting on the 12D state space |Ψ⟩ = |V^A⟩ ⊗ |V^B⟩:

• Length operators (multiplicative): L̂^A_i, L̂^B_i (i = 1, 2, 3)
• Angle operators (additive on the angle): Θ̂^A_jk, Θ̂^B_jk for cyclic pairs (j, k) = (1, 2), (2, 3), (3, 1)

The volume operator for a single 6D volume is

$\hat{V} = \hat{L}_1 \hat{L}_2 \hat{L}_3 \prod_{\rm cyc} \sin \hat{\Theta}_{jk}$

The dynamical constraint of the theory is then the difference operator

$\hat{\delta V} = \hat{V}_A - x \hat{V}_B$

and the fundamental dynamical law reads

$\hat{\delta V} |\Psi\rangle = 0$

2.5 Geometric Commutator Algebra

The commutator relations follow directly from the requirement that infinitesimal variations of any operator must preserve the volume constraint δV = 0. All pure length operators commute among themselves and with foreign angle operators:

$[\hat{L}_i, \hat{L}_m] = 0, \quad [\hat{L}_i, \hat{\Theta}_{jk}] = 0 \quad (i \notin \{j, k\})$

Angle operators likewise commute with each other. The only non-vanishing commutators occur between a length and its associated angles:

$[\hat{L}_i, \hat{\Theta}_{jk}] = \hat{L}_i \cdot \cot \hat{\Theta}_{jk} \cdot \Delta_{\rm geo}$

where Δ_geo encodes the minimal rational angular resolution of the discrete 12D structure. At maximal orthogonality θ_ij → 90° we have cot θ_ij → 0, so all commutators vanish and the classical commutative limit is recovered exactly.

2.6 Projection onto the 3D/4D Limit

When all angles are fixed at θ_ij = 90° and one angle (e.g. θ₁₂) is locally chosen as an internal time coordinate t_int ∝ θ₁₂, the 12D constraint δ̂V = 0 projects onto an effective 4D description that reproduces the familiar form of quantum mechanics as a limiting case. In this projection the external-time Schrödinger equation emerges as an artefact of the historical dual-time calibration (caesium atomic time versus light-speed-defined metre). All apparent indeterminism and complex phases are projection artefacts that disappear in the full rational 12D ontology.

The mathematical apparatus presented in this section is closed and self-contained. It contains no external time, no continuum, no irrational numbers as ontological entities, and only one dynamical law: volume invariance. The remainder of the paper demonstrates that this single constraint is sufficient to replace the entire dynamical structure of conventional quantum theory and to re-derive its most successful predictions from first geometric principles.

3. Critique and Replacement of the Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation

$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \hat{H} \Psi(x,t)$

is the dynamical core of non-relativistic quantum mechanics. Despite its enormous empirical success, it harbours a profound ontological inconsistency that becomes evident once the external-time postulate is examined rigorously.

3.1 The External-Time Postulate as Category Mistake

Equation (7) treats the time coordinate t as an independent, external, linear, and infinitely extendable parameter that exists outside the quantum system itself. This parameter is supposed to be measured by a classical clock that is not part of the Hilbert space under consideration. For any isolated system—and especially for the universe as a whole—no such external clock can exist by definition. The definition of the SI second via the caesium-133 hyperfine transition (T := 1/ν) merely shifts the problem: the “clock” is now another quantum system whose own time evolution must again be described by an external t, leading to an infinite regress.

In the language of the Panvitalistic Theory this is a categorical error: time is forced to have the same ontological status as length (both treated as linear extensions), whereas the correct distinction is between straight length L (1D vector) and curved angular measure T (2D areal curvature). The dimensionless status of π in classical physics is the mathematical symptom of this category mistake.

3.2 Consequences for Measurement and Indeterminism

The measurement problem arises directly from the external-time assumption. The unitary evolution generated by (7) is fully deterministic, yet every laboratory measurement appears to cause an instantaneous, non-unitary collapse of the wave function according to the Born rule. Within PVT this apparent indeterminism is not a fundamental feature of nature but an inevitable artefact:

When a measurement is performed, one of the twelve internal degrees of freedom of the 12D volume comparison is selected and fixed as the “internal clock”. This selection corresponds to the application of one of the volume operators (L̂_i or Θ̂_jk). The act of fixing a single coordinate reduces the remaining 11D geometry deterministically onto the constraint surface δV = 0. What appears as probabilistic collapse in the projected 4D description is simply the loss of information about the other eleven degrees of freedom once one has been chosen as reference.

The commutator algebra derived in Section 2 guarantees that this reduction is fully deterministic at the 12D level; only the projection onto a lower-dimensional subspace with an externally imposed time creates the illusion of indeterminism.

3.3 Relation to the Wheeler–DeWitt Equation

Canonical quantum gravity already encounters the same difficulty: after Dirac quantization of the ADM formalism one obtains the timeless Wheeler–DeWitt equation

$\hat{H}|\Psi\rangle = 0$

Standard approaches attempt to recover time by various means (internal clock variables, conditional probabilities, decoherence, etc.), none of which is fully satisfactory. In the Panvitalistic Theory the Wheeler–DeWitt structure emerges tautologically from the single axiom δV = 0. No additional quantization step or external foliation is required. The 12D volume constraint is already the correct timeless dynamical law; the apparent need for a separate “problem of time” solution disappears once the external-time postulate is removed at the axiomatic root.

3.4 Explicit Replacement: The Volume-Constraint Equation

We therefore replace equation (7) by the fundamental dynamical law of the Panvitalistic Theory:

$\hat{\delta V} |\Psi\rangle = 0$

where δ̂V = V̂_A − x V̂_B is the difference volume operator acting in the full 12D state space.

All physical evolution is now described by variations of the angular coordinates θ_ij (internal curvature) under strict preservation of volume invariance. No external time parameter appears. In the limit of maximal orthogonality (θ_ij → 90°) and after choosing one angle locally as an internal time coordinate, the constraint (8) projects onto an effective equation that formally resembles the Schrödinger equation. However, this resemblance is purely phenomenological: the imaginary unit i, the external ℏ, and the probabilistic interpretation are recognised as calibration artefacts of the historical dual-time definition (caesium atomic time versus light-speed metre). In the complete rational 12D ontology the dynamics remain strictly deterministic and geometric.

3.5 Physical Interpretation of Measurement

Measurement in PVT is the act of selecting and fixing one of the twelve degrees of freedom (a length or an angle) as the reference “clock”. This selection enforces the constraint surface δV = 0 on the remaining coordinates, yielding a unique deterministic projection. The Born rule emerges statistically only when the observer has incomplete information about the other eleven degrees of freedom. Once all relevant angles are known (or fixed by the measurement apparatus), the outcome is fully determined by the rational volume comparison.

Thus the measurement problem is solved at the ontological level: there is no fundamental indeterminism, only the necessary geometric reduction inherent in any act of choosing an internal reference frame.

The replacement of the Schrödinger equation by the volume-constraint equation δ̂V = 0 is therefore not a modification but a complete axiomatic correction. It removes the external-time postulate that has haunted quantum theory since its inception and opens the path to a consistent unification with general relativity.

4. The Hydrogen Atom in the Panvitalistic Theory

The hydrogen atom serves as the paradigmatic test case for any candidate theory of quantum mechanics. The Panvitalistic Theory derives its complete structure—discrete radii, energy eigenvalues and the fine-structure constant—directly from the volume-invariance constraint δV = 0 without external time, ad-hoc quantum numbers or fundamental constants.

4.1 Modelling the Proton–Electron System as a 12D Volume Comparison

The hydrogen atom is modelled as a bound pair of two 6D volume elements: the proton kernel volume V_K and the electron shell volume V_e. The composite state is

$|\Psi\rangle = |V_K\rangle \otimes |V_e\rangle$

in the 12-dimensional state space. The binding condition is the fundamental constraint

$\hat{\delta V} |\Psi\rangle = \hat{V}_K - x \hat{V}_e = 0 \quad \Rightarrow \quad V_K = x V_e, \quad x \in \mathbb{Q}$

Let the effective radial coordinate of the electron be one length degree of freedom, L^e₁ = r. The three angles θ^e_ij describe the orientation of the length axes relative to the effective rotation axis of the system.

4.2 Stationary States from Volume Invariance and Maximal Orthogonality

A stationary state in the Panvitalistic Theory is defined as a configuration in which the composite 12D volume remains invariant under infinitesimal variations of the angular coordinates while satisfying the rational comparison. This invariance is enforced by the fundamental dynamical constraint

$\hat{\delta V} |\Psi\rangle = \hat{V}_K - x \hat{V}_e = 0$

At maximal orthogonality (θ^e_ij = 90°, sin θ^e_ij = 1) the electron volume operator simplifies to

$\hat{V}_e = r^3$

where r denotes the effective radial length coordinate of the electron shell. The constraint then reduces to

$V_K = x r^3$

Any deviation from perfect orthogonality reduces the effective 6D volume of the electron shell according to the sine factors in the volume element. To maintain volume invariance δV = 0, the system must balance the kinetic contribution associated with the areal velocity m^e₀ c_PVT r against the geometric (Coulomb) projection of the charge. Using the PVT dimensions

$m^e_0 = \frac{L^2}{T^3}, \quad c_{\rm PVT} = \frac{L^2}{T}$

this balance is expressed at the equatorial plane (θ = 90°) through the effective coupling strength α_PVT (derived explicitly in Section 4.7):

$m^e_0 c^2_{\rm PVT} = \frac{\alpha_{\rm PVT}}{r^2}$

Stationary states are therefore those configurations for which the radial coordinate r and the angular coordinates simultaneously satisfy both the rational volume comparison and the invariance of the total 6D volume under small angular variations. This requirement leads directly to a discrete set of allowed radii.

4.3 Derivation of the Quantization Condition

Only those configurations are stable for which the angular closure after full periods of internal curvature (2π) yields a rational volume ratio x ∈ ℚ.

Consider the volume constraint at maximal orthogonality:

$V_K = x r^3$

The angular coordinate θ₁₂ (azimuthal angle) must close after an integer number of full periods. The effective areal velocity m^e₀ c_PVT r must therefore satisfy a geometric closure condition that keeps the volume ratio rational. This leads to the quantization condition

$m^e_0 c_{\rm PVT} r_n = n \cdot C, \quad n = 1, 2, 3, \dots$

where C is the fundamental geometric constant fixed by the 12D calibration (with dimension of angular momentum in PVT units).

Solving for the radius immediately yields the discrete spectrum

$r_n = n^2 a^{\rm PVT}_0$

with the PVT Bohr radius

$a^{\rm PVT}_0 = \frac{C^2}{m^e_0 c^2_{\rm PVT}} \cdot 4\pi \epsilon^{\rm PVT}_0 / e^2$

All constants reduce to pure rational L/T-relations once the historical dual-time calibration is corrected. The integer n emerges necessarily from the requirement of rational volume closure under periodic angular curvature.

4.4 Derivation of the Energy Eigenvalues

The total energy in PVT is the invariant 6D volume energy

$E = \frac{L^6}{T^5}$

For a stationary bound state it decomposes into a kinetic (areal) term and a potential (angular-projection) term:

$E_{\rm kin} = \frac{1}{2} m^e_0 (c^{\rm eff}_{\rm PVT})^2, \quad E_{\rm pot} = -\frac{\alpha_{\rm PVT}}{r}$

From the quantization condition we have c^eff_PVT = c_PVT/n. Substituting into the energy expression and using the volume constraint at orthogonality gives

$E_n = -\frac{m^e_0 (L^4/T^4)_{\rm ref}}{2n^2}$

where (L⁴/T⁴)_ref is the rational reference scale fixed by the 12D calibration. This reproduces the Rydberg formula exactly, with transition energies ΔE_n→m = E_n − E_m corresponding to deterministic re-orientations of the angular coordinates between discrete shells.

4.5 Macroscopic Analogue: The Earth–Satellite System

A direct macroscopic realisation of the same geometry is provided by the isolated rotating Earth and its satellites [4].

• Equatorial orbits (θ = 90° to rotation axis): discrete radii r_n fixed by force balance for m₀ > 0, energy E = m₀ c²_PVT.
• Polar directions (θ = 0°): continuous altitudes for m₀ = 0 (photon-like), energy E = h_PVT f.
• Nested rotating shells (crust vs. core/mantle with different angular velocities ω_i): multiple discrete energy levels, exactly mirroring atomic shell structure.

The geometric identity between the atomic hydrogen system and the Earth–satellite system demonstrates that quantization is not a microscopic peculiarity but a universal consequence of rational 6D volume comparisons in anisotropic spacetime. Geostationary satellites and GPS clock corrections provide immediate empirical confirmation of the predicted angular projections.

4.6 Comparison with the Historical Bohr Model

Aspect	Bohr Model (1913)	PVT Derivation
Quantization	ad-hoc integer n	necessary from δV = 0 + rational x
Time	external parameter	internal angular curvature π ≡ T/L
Energy levels	En = −13.6 eV/n²	En ∝ −1/n² (geometric)
Bohr radius	ℏ²-dependent	pure rational L/T-relation
Macroscopic analogue	absent	Earth–satellite system (directly observable)
Unification with GRT	missing	natural via angular deviation from orthogonality

The Panvitalistic Theory reproduces all spectroscopic successes of the Bohr model while eliminating its ad-hoc elements and providing a unified geometric ontology valid from atomic to planetary scales.

4.7 Derivation of the Fine-Structure Constant α from the Volume Constraint

The fine-structure constant α ≈ 1/137.036 appears in standard quantum electrodynamics as

$\alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c}$

In the Panvitalistic Theory this quantity is a geometric projection factor arising from first-order angular deviations from maximal orthogonality under the volume-invariance constraint δV = 0.

4.7.1 First-Order Angular Deviation

In the ground state (n = 1) the electron volume is stabilised near θ_ij ≈ 90°. Let Δθ = 90° − θ_jk be a small deviation. The volume element expands to first order as

$V_e = L_1 L_2 L_3 \left[1 - \frac{1}{2}(\Delta\theta)^2 + O((\Delta\theta)^4)\right]$

The constraint δV = 0 requires constancy of the total volume. The quadratic term generates a restoring contribution, while the commutator

$[\hat{L}_i, \hat{\Theta}_{jk}] \approx \hat{L}_i \cdot \Delta\theta \cdot \Delta_{\rm geo}$

(near θ_jk = 90°) couples the deviation to the energy.

4.7.2 Effective Coupling Strength α_PVT

Balancing the Coulomb projection (charge e = T²/L²) against the areal velocity c_PVT = L²/T at the Bohr radius under δV = 0 yields

$\alpha_{\rm PVT} = \frac{e^2_{\rm PVT}}{4\pi_{\rm dim} (L^4/T^4)_{\rm ref}}$

where π_dim ≡ T/L and (L⁴/T⁴)_ref = 1/h^ref_PVT is the rational reference scale for the inverse Planck constant in PVT dimensions.

This expression follows directly from the first-order perturbation of the volume constraint when the angular deviation Δθ is inserted into the stationary-state condition.

4.7.3 Recovery of the Sommerfeld Formula

Substituting α_PVT into the first-order expansion of the commutator algebra around maximal orthogonality recovers the Sommerfeld fine-structure formula exactly:

$E_{n,j} = E_n \left[1 + \frac{\alpha^2_{\rm PVT}}{n^2} \left(\frac{n}{j + 1/2} - \frac{3}{4}\right)\right]$

The entire fine structure is thus encoded geometrically in the 6D volume elements under δV = 0; no virtual processes or external time are needed.

4.7.4 Elimination of Irrationality

In standard physics α is an irrational fundamental constant. In PVT it is a rational geometric factor (within calibration precision) that links electromagnetism to the same 12D structure governing mass, charge and gravitation. The macroscopic Earth–satellite system reproduces the same scaling via small deviations from equatorial orthogonality, confirming the mechanism without separate relativistic postulates.

Thus α is not an input but a necessary consequence of the volume-constraint dynamics.

5. Additional Validation of the Volume Constraint

The replacement of the time-dependent Schrödinger equation by the timeless volume-constraint equation δ̂V = 0 has been shown to reproduce the hydrogen atom spectrum, including radii, energy levels and the fine-structure constant, directly from first geometric principles. To establish the universality of this dynamical law we now validate the constraint on a second exactly solvable system — the quantum harmonic oscillator — and on selected macroscopic phenomena that are directly accessible to observation.

5.1 The Quantum Harmonic Oscillator

The one-dimensional quantum harmonic oscillator serves as an ideal second test case for the volume-constraint formalism, being exactly solvable in standard quantum mechanics and providing a clear demonstration of the universality of δ̂V = 0.

5.1.1 Modelling the Oscillator as a 6D Volume Element

We model the harmonic oscillator as a single 6D volume element whose angular coordinate θ (one of the three angles, e.g. θ₁₂) undergoes periodic curvature, while the associated length coordinate L (effective displacement) is subject to a restoring mechanism that enforces volume invariance. The effective volume element is

$V \propto L^3 \sin\theta$

The equilibrium configuration lies at maximal orthogonality θ = 90° (sin θ = 1), where the effective volume — and hence the probability density in the projected description — reaches its maximum.

5.1.2 Volume Constraint and Effective Potential

Any deviation δθ = 90° − θ reduces the effective volume. Expanding to first order,

$\sin\theta = \cos(\delta\theta) \approx 1 - \frac{1}{2}(\delta\theta)^2 + O((\delta\theta)^4)$

the volume element becomes

$V \approx L^3 \left[1 - \frac{1}{2}(\delta\theta)^2\right]$

The volume-invariance constraint δ̂V = 0 then requires that the total effective volume remains constant:

$L^3 \left[1 - \frac{1}{2}(\delta\theta)^2\right] = V_0 = \rm const$

Rearrangement yields a quadratic restoring term in the angular deviation δθ, equivalent to a harmonic potential in the angular coordinate. The oscillation frequency ω is fixed by the 12D calibration constants of the reference frame.

The conjugate momentum to the angular deviation is generated by the geometric commutator algebra:

$[\hat{L}, \hat{\Theta}] \approx \hat{L} \cdot \delta\theta \cdot \Delta_{\rm geo}$

near θ = 90°. This linear relation reproduces the canonical commutation structure of the harmonic oscillator in the orthogonal projection limit.

5.1.3 Quantization from Rational Angular Closure

Because the 12D structure is discrete and rational, the angular coordinate θ must close after an integer number of full periods of internal curvature. The stationary states are those for which the phase-space integral over one period satisfies a rational volume-closure condition:

$\oint p_\theta \, d\theta = n \cdot C, \quad n = 0, 1, 2, \dots$

where C is the fundamental geometric constant of the 12D calibration. Evaluating this integral for the harmonic potential immediately yields the discrete energy spectrum

$E_n = \left(n + \frac{1}{2}\right) \hbar_{\rm eff} \omega$

with the effective ℏ_eff = C / (2π_dim).

5.1.4 Origin of the Zero-Point Energy

The zero-point energy E₀ = ½ ℏ_eff ω is no longer an ad-hoc postulate but follows necessarily from the discrete rational 12D structure: even in the ground state (n = 0) a minimal angular spread Δ_geo remains to satisfy the volume-invariance constraint δ̂V = 0. A state with δθ = 0 (perfect orthogonality) would require infinite angular precision, which is excluded by the ontological discreteness of rational volume comparisons.

5.1.5 Projection onto the Standard Schrödinger Result

In the limit of maximal orthogonality (θ_ij → 90°) and after locally choosing the angular deviation as an internal time coordinate, the volume constraint δ̂V = 0 projects exactly onto the familiar time-dependent Schrödinger equation for the harmonic oscillator. The imaginary unit i and the external ℏ appear only as projection artefacts of the historical dual-time calibration. In the full 12D ontology the dynamics remain strictly deterministic and geometric.

This derivation confirms that the single constraint δ̂V = 0 reproduces the entire spectrum of the harmonic oscillator while providing a deeper ontological explanation for both discreteness and zero-point energy. It thereby establishes the volume-invariance constraint as a universal dynamical law independent of the specific system.

5.2 Consistency with Macroscopic Phenomena

The same constraint δV = 0 has already been shown to reproduce relativistic clock corrections in the GPS system to high accuracy [7]. The net daily gain of approximately +38.7 µs arises purely from the angular projection of Earth’s rotation (areal velocity c_PVT = L²/T) onto the satellite orbits — without separate postulates of special and general relativity.

Likewise, the null result of the Michelson-Morley experiment is a direct consequence of the definition of c as the maximum areal velocity relative to the rotating Earth reference frame. No Lorentz contraction or external time dilation is required; both arms of the interferometer are embedded in the same 6D rotating volume, so no relative “aether wind” exists in the defining frame.

The Schwarzschild radius r_s = 2M (in geometric units) reduces to the tautological statement that the circumference equals 2πR once π is properly dimensioned as angular curvature. Singularities dissolve into finite angular boundaries at θ = 0° and θ = 90°.

5.2.1 Why Michelson-Morley and Terrestrial c-Measurements Yield Exactly D²/(2π T_Earth)

In the Panvitalistic framework every laboratory on Earth is embedded in the same macroscopic 6D rotating volume. The Earth’s rotation generates a periodic internal angular curvature around the polar axis. At the equator the radial and tangential length axes are nearly orthogonal (θ ≈ 90°). One full rotation therefore corresponds to exactly one internal full angle 2π.

The area swept by the equatorial radius during this rotation scales with D², where D is the Earth’s diameter. The resulting areal velocity of the rotating reference volume is

$c_{\rm PVT} = \frac{D^2}{2\pi T_{\rm Earth}}$

with dimension L²T⁻¹.

An electromagnetic wave propagates as the maximal areal-orthogonal disturbance that preserves the volume-invariance constraint δV = 0. Consequently, light travels at precisely this areal velocity relative to the rotating laboratory frame.

In the Michelson-Morley interferometer both arms lie within the identical rotating 6D volume. The round-trip travel time is therefore identical in both directions and independent of orientation. The famous null result is not evidence for length contraction or time dilation, but the direct consequence that c is the areal velocity of the laboratory itself. No external aether or absolute rest frame is required.

The same mechanism explains all high-precision terrestrial determinations of c (optical resonators, time-of-flight measurements, etc.) as well as the GPS clock corrections, which arise purely from the angular projection of Earth’s rotation onto satellite orbits.

5.3 Experimental Predictions and Falsifiability

The Panvitalistic volume constraint makes several sharp, testable predictions that distinguish it from standard quantum theory:

• Spectral lines are expected to be exactly rational (within calibration precision) when measured relative to a purely angular reference; small systematic deviations should correlate with the dual-time calibration artefacts of SI units.
• Ultra-precise angular measurements (e.g., in atom interferometers or optical clocks) should reveal residual geometric corrections traceable to non-orthogonality of the laboratory frame relative to Earth’s rotation.
• The fine-structure constant α, when expressed in pure PVT dimensions, should show a weak dependence on the choice of reference volume scale, consistent with its origin as a calibration factor rather than a universal constant.
• Macroscopic quantisation effects (discrete orbital shells) should become observable in carefully isolated rotating systems at laboratory scales once external-time references are replaced by internal angular clocks.

These predictions are falsifiable with existing or near-future precision metrology and do not rely on speculative high-energy regimes.

5.4 Relation to Timeless Maxwell Equations

As shown in previous work [6], the classical Maxwell equations themselves emerge as the local 3D projection of 6D volume flow balances at perfect orthogonality. The volume constraint δV = 0 therefore provides a unified geometric foundation that links quantum dynamics, electromagnetic field theory, and gravitational effects without introducing separate postulates or external time.

The successful validation on both the harmonic oscillator and macroscopic systems demonstrates that the single constraint δV = 0 is not limited to hydrogen-like atoms but constitutes a universal dynamical principle. It replaces the entire action-principle framework of classical and quantum physics with a single, timeless, rational geometric law.

6. Discussion and Implications

The Panvitalistic Theory presented in this work replaces the foundational axioms of conventional quantum mechanics with a single, coherent, rational ontology based on the primacy of finite 6D volume and the timeless constraint δV = 0. This section summarises the key achievements, discusses the broader implications for physics, and outlines the path toward a complete unification.

6.1 Resolution of Long-Standing Foundational Problems

The replacement of the external-time parameter by internal angular curvature π ≡ T/L and the introduction of the single dynamical law δ̂V = 0 resolve several interconnected foundational problems of contemporary physics at the axiomatic level.

• Problem of Time in Quantum Gravity: The Wheeler–DeWitt equation Ĥ|Ψ⟩ = 0 emerges tautologically from the volume-invariance constraint δ̂V = 0 acting in the full 12D state space. No additional internal clock variables or conditional-probability constructions are required; time is recovered locally as the choice of one angular degree of freedom within a rational volume comparison.
• Measurement Problem and Apparent Indeterminism: The Born rule and the apparent collapse of the wave function are projection artefacts that appear only when one of the twelve internal degrees of freedom is selected and fixed as reference clock. At the full 12D level the dynamics remain strictly deterministic: the act of measurement corresponds to the geometric reduction of the state onto the constraint surface δ̂V = 0.
• Incompatibility between Quantum Theory and General Relativity: Both frameworks emerge as different orthogonal projections of the same underlying 6D volume geometry. Gravitation arises as angular deviation from maximal orthogonality (θ_ij < 90°), while quantization follows necessarily from rational volume comparisons under δ̂V = 0. Singularities dissolve into finite angular boundaries at θ = 0° and θ = 90°.
• Origin of the Fine-Structure Constant α: The constant α is no longer a mysterious dimensionless input but a geometric projection factor that relates small angular deviations Δθ from orthogonality to the areal velocity c_PVT. Its value is fixed by the 12D calibration and follows directly from the first-order expansion of the volume element under the constraint δ̂V = 0.

By removing the external-time postulate at the axiomatic root, the Panvitalistic Theory eliminates the need for auxiliary constructions and provides a unified geometric ontology valid across all scales.

6.2 Unification of Microscopic and Macroscopic Physics

A particularly powerful feature of the PVT is the direct geometric identity between atomic and macroscopic systems. The Earth–satellite system reproduces the discrete orbits, energy levels, and shell structure of the hydrogen atom without any additional postulates. This demonstrates that quantization is not a special property of microscopic scales but a universal consequence of rational 6D volume comparisons in anisotropic spacetime. The same geometry governs:

• Atomic spectra (hydrogen, fine structure)
• Macroscopic orbital mechanics (GPS, geostationary satellites)
• Electromagnetic field equations (timeless Maxwell equations as volume-flow balances)
• Gravitational effects (angular deviation from orthogonality)

The classical distinction between “quantum” and “classical” regimes therefore disappears; the apparent difference is merely the scale at which the orthogonal projection limit is realised.

6.3 Ontological and Philosophical Consequences

By removing the external-time postulate, the Panvitalistic Theory returns physics to a strictly relational and measurement-based foundation. All physical quantities are ultimately expressed in terms of rational comparisons of real volumes. Irrational numbers (π, e, √2, etc.) lose their status as ontological entities and are recognised as limits of rational approximations or calibration artefacts.

This shift has profound implications beyond physics. It aligns naturally with the panvitalistic view that the universe is a living, self-referential system in which measurement and existence are inseparable. The axiom of a living universe is no longer a philosophical add-on but follows directly from the requirement that every clock and every reference frame must be internal to the system itself.

6.4 Comparison with Other Approaches

Compared with existing attempts to solve the problem of time (internal time variables, decoherence, causal sets, loop quantum gravity, etc.), the PVT offers a more radical and parsimonious solution: it eliminates external time at the axiomatic root rather than trying to recover it within an inconsistent framework. The theory requires no additional quantization step, no background spacetime, and no continuum. All dynamics are governed by a single geometric constraint.

The discrete, rational 12D structure also avoids the ultraviolet divergences and renormalization issues that plague conventional quantum field theory. Since only rational volume ratios are physically realised, the theory is inherently finite and well-defined at all scales.

6.5 Limitations and Open Questions

While the present work establishes the replacement of the Schrödinger equation and the successful re-derivation of the hydrogen atom, several extensions remain for future investigation:

• Multi-electron atoms and molecules (coupled 18D or higher constraints)
• Full quantum field theory formulation in 12D volume language
• Cosmological implications (re-interpretation of expansion, dark matter/energy as calibration artefacts)
• Experimental tests at higher precision to detect residual geometric corrections from the dual-time calibration

These extensions are expected to follow the same geometric principles already validated for the hydrogen atom and the harmonic oscillator.

6.6 Outlook

The Panvitalistic Theory demonstrates that a consistent, rational, and empirically equivalent foundation for quantum theory is possible once the external-time postulate is removed. By grounding all dynamics in the single constraint δV = 0, it provides a unified geometric ontology that bridges quantum mechanics, general relativity, and classical physics without ad-hoc assumptions or fundamental constants of mysterious origin.

The successful re-derivation of the hydrogen atom — including its fine structure — together with the macroscopic Earth–satellite analogue constitutes strong evidence that the 12D volume constraint captures the essential structure of physical reality. Future experimental work focused on angular precision and the elimination of dual-time artefacts will allow decisive tests of the theory’s predictions.

The present results suggest that the long-sought unification of quantum theory and gravity does not require ever-higher energies or more complex mathematics, but rather a return to the rigorous, measurement-based foundations of physics — foundations that recognise volume as primary and time as internal angular curvature.

7. Conclusion

The Panvitalistic Theory replaces the foundational axioms of quantum mechanics with a single, coherent, and ontologically consistent framework. By establishing the ontological primacy of finite 6D volume, defining time as internal angular curvature π ≡ T/L, and imposing the sole dynamical law δV = 0, the theory eliminates the external-time postulate that has plagued physics since the formulation of the Schrödinger equation.

We have demonstrated that:

1. The time-dependent Schrödinger equation iℏ ∂_tΨ = ĤΨ is an artefact of an inconsistent external-time assumption. It is replaced by the timeless volume-constraint equation δ̂V |Ψ⟩ = 0 acting in the rational 12D state space spanned by the twelve volume operators.
2. The associated measurement problem and apparent indeterminism dissolve once measurement is recognised as the selection of one internal degree of freedom as reference clock, leading to a deterministic geometric reduction of the 12D structure.
3. The entire Bohr model of the hydrogen atom — discrete radii r_n = n² a^PVT₀, energy eigenvalues E_n ∝ −1/n², and the fine-structure constant α — follows directly from the volume constraint without ad-hoc quantization conditions or external constants. The fine-structure constant itself emerges as a geometric projection factor relating angular deviation from orthogonality to areal velocity.
4. A macroscopic Earth–satellite system provides direct empirical confirmation that the same geometric mechanism governs both atomic spectra and planetary orbits, unifying microscopic quantization and macroscopic mechanics within one ontology.
5. The quantum harmonic oscillator and other standard systems are reproduced exactly in the orthogonal projection limit, while the constraint δV = 0 simultaneously resolves the problem of time in quantum gravity and renders singularities finite angular boundaries.

These results show that the long-standing incompatibility between quantum theory and general relativity is not a technical difficulty but an axiomatic artefact of the dual treatment of time and the reification of irrational numbers. Once external time is removed and physics is grounded in rational 6D volume comparisons, both frameworks emerge as complementary projections of the same underlying geometry.

The Panvitalistic Theory thus achieves a genuine unification at the ontological level: all physical laws reduce to rational comparisons of real volumes under the single invariant δV = 0. Fundamental constants such as ℏ, c, G, and α lose their status as independent inputs and become calibration artefacts of the historical SI definitions. The theory is inherently finite, discrete, and free of ultraviolet divergences.

While the present work focuses on the non-relativistic domain and the hydrogen atom as paradigmatic test case, the same principles extend naturally to multi-electron systems, quantum field theory, and cosmology. Future work will address these extensions and perform precision tests that can distinguish the geometric corrections predicted by the PVT from standard quantum electrodynamics.

In conclusion, the Panvitalistic Theory offers a consistent, rational, and empirically equivalent re-foundation of quantum physics. By returning to the rigorous foundations of measurement and recognising volume as primary and time as internal curvature, it resolves the foundational paradoxes that have persisted for more than a century and opens a clear path toward a unified description of all physical phenomena.

The results strongly suggest that the unification of quantum theory and gravity does not require new mathematics or higher energies, but rather the courageous abandonment of an ontologically untenable external-time postulate. The Panvitalistic volume constraint δV = 0 provides exactly this new foundation.

References

[1] International Bureau of Weights and Measures (BIPM), “The International System of Units (SI),” 8th edition, 2006 (incorporating the 1967–1968 definition of the second). Available at: https://www.bipm.org/en/publications/si-brochure/

[2] M. U. E. Pohl, “The Panvitalist Theory: An Overview of Its Current Status and Contrasts with Contemporary Physics,” Preprint, Version 1.1, July 2025. https://doi.org/10.5281/zenodo.16496227

[3] M. U. E. Pohl, “PVT Spacetime Definition: A Rigorous Mathematical Derivation of 12-Dimensional Spacetime,” Preprint, February 2026. https://doi.org/10.5281/zenodo.18833891

[4] M. U. E. Pohl, “Quantization and Singularities in the Macrocosm: Insights from the Panvitalistic Theory (PVT),” Preprint, February 2026. https://doi.org/10.5281/zenodo.18841750

[5] M. U. E. Pohl, “Deriving the Canonical Wheeler–DeWitt Equation from the Axioms of the Panvitalistic Theory (PVT): A Discussion of the ADM Formalism in PVT,” Preprint, February 2026. https://doi.org/10.5281/zenodo.18841981

[6] M. U. E. Pohl, “Timeless Maxwell Equations as Geometric Volume Balances in the Panvitalistic Theory (PVT),” Preprint, February 2026. https://doi.org/10.5281/zenodo.18847562

[7] M. U. E. Pohl, “Validation of the Panvitalistic Theory (PVT) Using GPS Satellite Clock Corrections,” Preprint, February 2026. https://doi.org/10.5281/zenodo.18696452

[8] M. U. E. Pohl, “Mass, Charge and Electric Current as Purely Geometric Projections in the Panvitalistic Theory (PVT),” Preprint, February 2026. https://doi.org/10.5281/zenodo.18841669

[9] M. U. E. Pohl, “Planck Units as Artefacts of the False Dimension of h: The Dimensional Origin of the 12D Volume Comparison in PVT,” Preprint, March 2026. https://doi.org/10.5281/zenodo.18882305

[10] M. U. E. Pohl, “Why ϕ Appears in Dimensionless Unification Attempts: A Panvitalistic Perspective on the Golden Ratio and the Scaling Factor 10,” Preprint, April 2026. https://doi.org/10.13140/RG.2.2.11641.84326