This article develops a unified theory of the w-axis, linking the mass domain (\(Q^2\)) and the charge domain (\(Q^3\)) through the square root of Planck momentum \(Q\). We investigate the geometric origin of the difference in orbital periods between gravitational systems (Article 3: Orbital Mechanics) and atomic systems (Article 4: Atomic Orbitals), demonstrating that the scaling shift from \(r_{\alpha}\) to \(r_{\alpha}^2\) is a consequence of the dimensional contribution of the third wave-axis (\(z/w\)). By modelling particles as an intersection of standing waves of \(Q\), we derive the \(Q^5\) monopole structure and provide a first-principles derivation of the Ampere \(A\).
The fundamental bridging constant in this model is \(Q\), defined as the square root of normalized Planck momentum. This \(Q\) provides the direct link between the Mass domain and the Charge domain (see Article 6.):
\[ Q = \sqrt{\frac{m_P c}{2\pi}} \approx 1.019113422 \left( \text{kg m/s} \right)^{1/2} \]The "W-axis" represents the non-integer domain in the simulation and acts as the "unzipped" state of these integers. While the spatial domain sees only the collapsed results (\(Q^2\)), the W-axis permits the interaction of the underlying wave components (\(Q^3, Q^2 \times Q^3 = Q^5\)).
Definition (W-Axis): The W-axis is the orthogonal geometric direction in which the non-integer (\(\sqrt{\text{int}}\)) domain manifests. Physically, it corresponds to the polarization/helicity degree of freedom of the electromagnetic field.
A central thesis of this synthesis is that our current physics observes a derived version of the fine-structure constant. While standard physics utilizes \(\alpha \approx 1/137.036\), we theorize that the true fundamental constant is \(r_{\alpha}\):
\[ r_{\alpha} = \sqrt{\frac{2}{ \alpha}} \approx 16.556... \]Because we are embedded in the Mass Domain (the 2D spatial plane), we only observe the squared projection of this constant:
\[ r_{\alpha}^2 \approx 274.1198... \]This "squared visibility" is a recurring theme in the dual-domain model: just as polarization intensity is measured as \(Q^2\), the observed coupling in our spatial plane is the squared projection of the higher-dimensional scaling \(r_{\alpha}\). This duality explains the divergence in orbital periods between gravitational and atomic scales.
We define the Planck Momentum \(p_P = m_P c\). In the dual-domain model, this momentum is the product of the interaction of the two primary wave-centers. The fine-structure constant \(\alpha\) represents the ratio of the electron's charge-domain coupling to the mass-domain coupling. If we postulate that the vacuum is a resonant cavity where \(2\) units of wave-action are distributed over a scale \(R\), and the resulting coupling efficiency is \(\alpha\), we find:
\[ \alpha = \frac{2}{R^2} \implies R = \sqrt{\frac{2}{\alpha}} \]Thus, \(r_{\alpha}\) is the geometric radius of the vacuum's standing wave cavity required to produce the observed coupling constant \(\alpha\).
The difference between gravitational and atomic regimes is expressed through their respective orbital periods \(T\):
\[ T_{\text{grav}} = 2\pi r \cdot r_{\alpha} \cdot n \] \[ T_{\text{atom}} = 2\pi r \cdot (r_{\alpha}^2) \cdot n \]We theorize that this divergence arises from the dimensional depth of the interaction with Planck momentum \(Q\):
We model the vacuum as an intersection of standing waves of \(Q\):
| Wave | Domain | Center | Plane | Scaling |
|---|---|---|---|---|
| Wave 1 | Mass | \((+1, 0, 0)\) | \(xy\)-plane | \(r_{\alpha}\) |
| Wave 2 | Mass | \((-1, 0, 0)\) | \(xy\)-plane | \(r_{\alpha}\) |
| Wave 3 | Charge | \((0, 0, 0)\) | \(zw\)-plane | \(r_{\alpha}\) |
The core idea here is that mass-domain motion is governed by a two-wave coincidence process, while charge-domain motion requires an additional (approximately independent) coincidence on the \(w\)-axis. The resulting scaling is therefore not a strict deterministic period in general, but an expected waiting time (or, in a deterministic simulator, a design constraint on the update rules).
Let each standing wave be represented by a phase variable on the circle,
\[ \phi_i(t)\in \mathbb{T} := \mathbb{R}/2\pi\mathbb{Z},\qquad \phi_i(t)=\omega_i t+\phi_{i,0}\;(\mathrm{mod}\;2\pi), \]for \(i\in\{1,2,w\}\), where \((1,2)\) are the two planar waves (Mass domain) and \(w\) is the orthogonal wave (Charge domain). A ``meeting at the origin'' is defined by a phase window of width \(\varepsilon\):
\[ E_i(\varepsilon)=\{\,t:\ |\mathrm{wrap}(\phi_i(t))|<\varepsilon\,\}, \]where \(\mathrm{wrap}(\cdot)\) returns the principal value in \((-\pi,\pi]\).
Remark (deterministic vs statistical time). If the \(\omega_i\) are commensurate and \(\varepsilon\to 0\), then exact simultaneous meetings are governed by a Diophantine/LCM-type condition. In contrast, if the dynamics mix phases (or we work at finite tolerance \(\varepsilon\)), it is natural to model meetings as approximately independent events and compute expected waiting times. The present article uses this latter interpretation, because it is the one that produces robust scaling laws and connects directly to measurable rates.
Let the simulation advance in discrete Planck ticks \(k\in\mathbb{N}\). Define the planar (mass-domain) coincidence event
\[ E_{\mathrm{mass}}(k) := \bigl(E_1(\varepsilon)\cap E_2(\varepsilon)\bigr)\ \text{occurs at tick }k, \]and the \(w\)-axis coincidence event
\[ E_{w}(k) := E_{w}(\varepsilon)\ \text{occurs at tick }k. \]Assume the following calibrated hypothesis:
(H1) Calibration: At the chosen tolerance/resolution, the probability that the planar coincidence occurs at a given tick satisfies \(\mathbb{P}(E_{\mathrm{mass}}(k))\approx 1/r_{\alpha}\).
(H2) \(w\)-axis symmetry: The \(w\)-axis coincidence has the same marginal rate, \(\mathbb{P}(E_{w}(k))\approx 1/r_{\alpha}\).
(H3) Approximate independence: \(E_{\mathrm{mass}}(k)\) and \(E_{w}(k)\) are approximately independent at the tick scale: \(\mathbb{P}(E_{\mathrm{mass}}(k)\cap E_w(k))\approx \mathbb{P}(E_{\mathrm{mass}}(k))\,\mathbb{P}(E_w(k))\).
Let \(\tau_2\) be the waiting time (in ticks) to the next planar coincidence and \(\tau_3\) the waiting time to the next triple coincidence:
\[ \tau_2:=\min\{k\ge 1: E_{\mathrm{mass}}(k)\},\qquad \tau_3:=\min\{k\ge 1: E_{\mathrm{mass}}(k)\cap E_w(k)\}. \]Under (H1)--(H3) the tickwise success probabilities are
\[ p_2 \approx \frac{1}{r_{\alpha}},\qquad p_3 \approx \frac{1}{r_{\alpha}^2}, \]so (to leading order) \(\tau_2\) and \(\tau_3\) are geometric waiting times with
\[ \mathbb{E}[\tau_2]\approx r_{\alpha},\qquad \mathbb{E}[\tau_3]\approx r_{\alpha}^2. \]Interpretation. In this form, the scaling shift from \(r_{\alpha}\) (Mass domain) to \(r_{\alpha}^2\) (Charge domain) is a direct consequence of adding one additional, approximately independent coincidence constraint. This provides a rigorous version of the intuitive statement: ``atomic motion is slower because it must wait for the \(w\)-axis meeting.''
Dimensional coupling. The interaction intensity is governed by the total dimensional coupling of the domains. \(\text{Momentum Tensor:}\quad Q^{2} \otimes Q^{3} \cong Q^{2+3} = Q^{5}\) The unified node is therefore a \(Q^5\) overlap:
\[ (Q^2)_{\text{Mass}} \times (Q^3)_{\text{Charge}} \;\leadsto\; Q^5. \]This \(Q^5\) monopole acts as a geometric sentinel, ensuring that every transition in the atom follows the same "unzipped" address-space path.
This section sharpens the photon discussion by defining the photon as a wave-state curvature excitation on the \(w\)-axis, and then interpreting absorption/emission as a change of constraints on the allowed phase-coherent orbital trajectories.
Let \(\xi(x)\in \mathbb{C}^2\) be a normalized spinor field, \(\xi^{\dagger}\xi=1\), representing the local wave-state orientation. Define a \(U(1)\) gauge potential and curvature by
\[ a_{\mu}(x) := -i\,\xi^{\dagger}(x)\,\partial_{\mu}\xi(x),\qquad F_{\mu\nu}(x) := \partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}. \]Within this model, a photon is identified with a localized propagating perturbation \((\delta a_{\mu},\delta F_{\mu\nu})\) supported on the hypersphere surface (pure wave-state, no mass point-state). In vacuum, the propagation condition is taken to be the source-free field equation
\[ \partial^{\nu} \delta F_{\mu\nu}=0, \]together with a null (surface-propagating) kinematic constraint consistent with lateral motion on the expanding hypersphere.
An orbital transition is modeled as a temporary photon--orbital hybrid state in which the orbital geometry (the \(n\)-scaffold) and the photon ``information layer'' (angular momentum bookkeeping) are simultaneously active. In the coincidence framework above, a transition step requires:
Therefore the rate of charge-domain orbital progress is suppressed by a factor \(\sim 1/r_{\alpha}\) relative to purely mass-domain progress, and the characteristic waiting time scales as \(\mathbb{E}[\tau_3]\sim r_{\alpha}^2\).
In the present synthesis, the photon carries the minimal wave-state momentum packet at the \(Q^2\) level, while the bound orbital structure is a \(Q^3\)-enabled configuration. Absorption/emission is then the controlled conversion
\[ (Q^2)_{\text{photon packet}}\ \longleftrightarrow\ (Q^3)_{\text{orbital constraint}}\quad\text{via the $w$-axis gate}, \]with the \(Q\)-bridge providing the common momentum scale. This is the precise sense in which the electron (possessing dual-domain citizenship) mediates ``dimensional momentum exchange'' (via the photon) between the Mass (integer) and Charge (non-integer / \(w\)-axis) domains.
The discrete nature of the simulation is governed by a fundamental step count for a single orbital cycle, \(\alpha_{calc}\). We discover that this constant is not an arbitrary input, but is derived directly from the \(r_{\alpha}\) scaling:
\[ \alpha_{calc} = 2\pi \cdot r_{\alpha}^4 \approx 471,964 \]This \(r_{\alpha}^4\) scaling arises from the product of the two interacting domains. Since the Mass Domain operates on a cycle of \(r_{\alpha}^2\) (the observed projection), and the Charge Domain also requires a full \(r_{\alpha}^2\) synchronization cycle, the total address space for a complete resonant interaction is the product of their periods:
\[ \text{Total Steps} \propto (r_{\alpha}^2)_{\text{Mass}} \times (r_{\alpha}^2)_{\text{Charge}} = r_{\alpha}^4 \]This confirms that the simulation step count is not arbitrary but is the total geometric surface area of the dual-domain interaction.
The Ampere \(A\) is the macroscopic manifestation of the quintic momentum interaction, linking the mass and charge domains:
\[ A = \frac{128\pi^3 Q^3}{m_P^3 r_{\alpha}^2} \approx 0.29722125623 \times 10^{25} \]Here, \(r_{\alpha}^2\) (the observed mass-domain scaling) acts as the denominator for the 3D charge component \(Q^3\), effectively normalizing the monopole interaction for spatial measurement. Ampere formula derivation is given in Article 6.
Drawing from the "Mathematical Electron" extended theory (Article 7), we can now specify the internal structure of the particle in this unified framework. The particle is not a point-mass but a temporal oscillation between a point-state and a wave-state. The "3-Wave" interaction identifies the core configuration of these states:
The \(Q^5\) monopole represents the unzipped interaction node where the 2D Mass domain (\(Q^2\)) and the 3D Charge domain (\(Q^3\)) overlap. This intersection is the "guard-rail" for the simulation, ensuring that every transition in the atom follows the same geometric path. See Article 7 for a full definition of the \(DDD\), \(DUU\) notation.
The "W-axis" is not just an abstract coordinate but the geometric representation of the extra scaling factor \(r_{\alpha}\) required to transition from mass-domain interactions (\(Q^2\)) to charge-domain interactions (\(Q^3\)). This synthesis shows that the "numbers of the atom" are encoded in the dimensionality of the vacuum's momentum structure.
Article 5 (this article) treats the photon primarily as a momentum-transfer packet in the \(Q\)-bridge language (using only \(Q=\sqrt{m_Pc/(2\pi)}\)), so that transitions were modelled as essentially a transfer of Planck-momentum bookkeeping. Article 7 extends this by proposing that the electron wave-state contains an internal monopole/quark factorization (\(DDD\), \(DUU\)) built from the same geometric primitives. Since one of the aims of the series is low Kolmogorov complexity, the natural question is whether the photon can be represented by a closely related composite of the same primitives, so that particles and photons are ``made of the same substance'' and differ mainly by boundary conditions.
Bridge principle: \(Q\) is the SI embedding of the geometric \(\Omega\) carrier. In the MLTA formulation (Article 7), the square-root momentum object is \(P=\Omega\), and the SI quantity \(Q\) plays the same role once the appropriate scalars are reinstated. In this sense, Article 5's \(Q\)-based thought experiments can be read as tracking the same carrier degree of freedom as \(\Omega\), but without resolving the internal \(AL/AV\) monopole structure.
From \(Q^2Q^3=Q^5\) to \(\Omega^2\Omega^3=\Omega^5\). The core algebraic motif of this article is \(Q^2 \times Q^3 \;\cong\; Q^5\), interpreted as the overlap of mass-domain (\(Q^2\)) and charge-domain (\(Q^3\)) wave content at a unified node. In the geometric language of Article 7 the corresponding identity is \(\Omega^2 \times \Omega^3 \;=\; \Omega^5\), because the kinematic objects scale as \(\Omega^2\) (via \(V\) and \(L\)) while the electromagnetic amplitude scales as \(\Omega^3\) (via \(A\)). Hence the \(\Omega^5\) object is precisely the monopole block that appears in the quark-like constructions: \(D \equiv AL,\; U \equiv AV,\;\Rightarrow\; AL \propto \Omega^5,\; AV \propto \Omega^5\). This reframes the \(Q^5\) ``unzipped interaction node'' as the dimensional statement: a monopole block is the product of kinematic depth and electromagnetic phase content.
Photon primitive as a neutral, scalar-free monopole composite. Article 7 motivates a photon candidate built from the same blocks as the electron/positron sector:
\[ \gamma \;\equiv\; DDU \;=\; (AL)^2(AV). \]Using the unit-number assignments of the MLTA rule set (Article 7), \(\theta(AL)=-10,\quad \theta(AV)=+20\), we obtain \(\theta(\gamma)=2(-10)+20=0\), so \(\gamma\) is unitless in the unit-number algebra. Moreover, because the quark blocks carry non-cancelling scalars individually but cancel in this triplet, \(\gamma\) is also scalar-free. Finally, since \(AL\) and \(AV\) are each \(\Omega^5\) blocks, we have \(\gamma \;\propto\; (\Omega^5)^3 \;=\; \Omega^{15}\), which ties the photon primitive directly to the recurring base-15 residue observed throughout the dimensionless cancellation sector.
Hybrid interpretation: photon--orbit states as \(\gamma\) under boundary constraints. The photon-orbital hybrid hypothesis can now be stated more precisely:
A free photon corresponds to a propagating \(\gamma\) excitation (pure wave-state) on the hypersphere surface, whereas an ``orbital photon'' corresponds to the same \(\gamma\) excitation subject to a closed/helical boundary constraint imposed by the local charged geometry.
Thus the difference between ``radiation'' and ``orbit'' is not a change of internal primitives, but a change in allowed trajectories of the same wave-state object. This preserves low descriptive complexity: the model reuses the same short ``program'' (the \(\Omega^5\) blocks) and varies only the constraint set.
Standing-wave closure (thought experiment). In the bound/orbital regime, the hybrid state is required to close after an integer number of winding cycles,
\[ 2\pi R \;=\; n\lambda, \]where \(\lambda\) is the observed wavelength of the hybrid excitation. The frequency is fixed by the usual energy accounting, \(u=\frac{E}{h},\quad \lambda=\frac{c}{u}\), but in this framework \((c,h)\) are not new inputs: they are already derived from the same MLTA objects and scalars. Therefore this equation acts as a geometric quantization condition on the allowed \(\gamma\) trajectories, consistent with the coincidence-gated update picture developed earlier in this article.
Recombination view of \(e^-e^+\) two-photon emission. A further (speculative) benefit of introducing \(\gamma\) is that the observed two-photon final state in electron--positron annihilation can be interpreted as a recombination of the same monopole blocks: \(e^- \sim DDD,\; e^+ \sim DUU,\;\Longrightarrow\; DDD+DUU \;\to\; (DDU)+(DDU)\;=\;\gamma+\gamma\). This is not offered as a replacement for QED, but as a geometric accounting identity consistent with charge neutrality and momentum conservation (the two photons emerge with opposite propagation directions in the center-of-mass frame). In the present language, ``annihilation'' corresponds to the release of two \(\gamma\) excitations from a temporarily bound photon--orbital hybrid configuration.
Kolmogorov/MDL note. The purpose of introducing \(\gamma\) is compression: photons and particles are both constructed from the same \(\Omega^5\) monopole blocks. No new constants, unit-number rules, or degrees of freedom are introduced; only the boundary/trajectory constraints change. This preserves the guiding principle of the series that the simplest consistent geometric rule set should generate the widest range of observed phenomena.
[1] Macleod, Malcolm J. "The Programmer God, are we in a simulation?" theprogrammergod.com
[2] Macleod, M.J. Programming Planck units from a virtual electron: a simulation hypothesis. Eur. Phys. J. Plus 133, 278 (2018). https://doi.org/10.1140/epjp/i2018-12094-x
[3] Macleod, Malcolm J., 1. Planck unit scaffolding to Cosmic Microwave Background correlation https://www.doi.org/10.2139/ssrn.3333513
[4] Macleod, Malcolm J., 2. Relativity as the mathematics of perspective in a hyper-sphere universe https://www.doi.org/10.2139/ssrn.3334282
[5] Macleod, Malcolm J., 3. Gravitational orbits from n-body rotating particle-particle orbital pairs https://www.doi.org/10.2139/ssrn.3444571
[6] Macleod, Malcolm J., 4. Geometrical origins of quantization in H atom electron transitions https://www.doi.org/10.2139/ssrn.3703266
[7] Macleod, Malcolm J., 5. Atomic Transitions via a Photon-Orbital Hybrid https://www.doi.org/10.13140/RG.2.2.10680.20487
[8] Macleod, Malcolm J., 6. Do these anomalies in the physical constants constitute evidence of coding? https://www.doi.org/10.2139/ssrn.4346640
[9] Macleod, Malcolm J., 7. Geometric Origin of Quarks, the Mathematical Electron extended https://www.doi.org/10.13140/RG.2.2.21695.16808
[10] Macleod, Malcolm J., 8. Holographic Emergence in the Simulation Hypothesis https://www.doi.org/10.13140/RG.2.2.20919.28320