A photon is the smallest piece of light; this page follows one photon closed into a loop, and a second one standing in its wave, and derives what the high school page stated. The near side of the second loop rides the stronger wave, so it sets the size; the two facings set the sign. The drawing is the high school page’s; each numbered circle is a panel below, and each panel a step of the derivation.
Published documents used, and nothing else: the Mathematical Bridge (section 4) and its Math Appendix (Electromagnetism from Void Transport; Source Extension; Transport Circulation from Moving Sources; Magnetism from Moving Sources via Transport Calculus), with the Bridge’s section 3 behind the mass pages’ map; the Electromagnetism Narrative (What Is Charge; sections 1, 4, 6); the Electromagnetism Math Walk-Through (sections 1 and 2, Boxes 1 to 4) and Math Appendix (section 9); the Particle Mechanics Narrative (section 7) and Math Appendix (the closure phase); the Student Workbook (q = σe). Only ratios enter; S₀ is the one scale; ε₀, μ₀, ke and μeff are limit locks, not inputs.
A note on the mathematics. As on the previous page, the general solution is written in differential forms (A, K = dA, ⋆, Stokes) and the observer split of a 2-form into its time–space and space–space parts, which is the standard way of writing electromagnetism covariantly (Flanders, Differential Forms with Applications to the Physical Sciences, 1963; Frankel, The Geometry of Physics, 3rd ed., 2011); the retarded Green’s operator is textbook (Jackson, Classical Electrodynamics, 3rd ed., 1999, chapter 6). Nothing on this page needs more than that; the caustic asymptotics of the previous page enter only through the axis n̂.
The source on the right is the electron loop of the previous page, and the wave it throws off is that page’s null gravity wave, drawn by the same rule (release edge at drawn speed c/(1 − δ), head edge at c; the drawing’s own). Its contrast at radius R is the share of space missing there, from the mass pages’ map, which is their construction of two lines of the Bridge (section 3): “The loop acts like a geometric anchor that removes a volume of available transverse area from the surrounding space, fixing the missing volume”, and “This deficit propagates outward with profile Δg(r) ~ 1/r²”:
The Electromagnetism Narrative: “A ‘charged’ thing here is a rotating void-loop that’s constantly throwing off null-gravity waves (NGWs) ripples. Those ripples don’t pile anything up, no net space-density change. Another loop only feels something if it’s also spinning and its facing lines up with the ripple pattern. The effect is purely orientation-gated.”
Each particle is a loop of radius ƛe standing at R = 16.5 ƛe. Its near edge is at 15.5, its far edge at 17.5, and the wave is not the same at the two:
Those are the paired arrows on the drawing, at each particle’s two edges, on one scale. The near side always rides the stronger wave, so the near side sets the size of what happens; the kid page says it in five words, “closest wins”. What the size does not fix is which way: a difference is a number, not a direction.
The direction is the Mathematical Bridge’s line (section 4): “Relative orientation determines the sign of far-field interaction. The far-field again follows a 1/r² dependence, but its direction depends only on σ. Binary polarity is therefore a direct geometric consequence of orientation.”
On the drawing the upper particle turns opposite to the source and is pulled; the lower turns the same way and is pushed. The size of both is panel 2; the sign of each is this line. Nothing else is needed for the H page, and nothing else is claimed there.
Why facing matters at all is the Narrative’s surfer (section 1): “Think of the second loop as a surfer and the NGW pattern as ocean swell. The swell isn’t a ‘standing downhill’, it’s a traveling pattern. If the surfer angles the board into the swell the right way, the path bends and you ‘catch’ it. If you’re misaligned, the chops shove you back and forth but average out, no net drift. If you’re faced the other way, you get the opposite path change.” The one place the site writes facing as an equation is the Electromagnetism Math Appendix’s neutral-rotor test (section 9):
That equation is written for a neutral spinning body in the caustic plane, so it is quoted here for the shape of the facing dependence, not as the two-charge force; the two-charge law is panel 3. The Bridge gives the same handedness in its Waveform section: “reversing the loop’s rotation flips the side toward which the transverse push acts, setting the handed response of the force.”
The same loop has two far fields (matter page, step 11): its obscured volume makes a standing dent, gravity, always toward; its facing makes a travelling pattern, charge, with a sign. The Narrative (section 6): “Gravity behaves like a standing curvature gradient, a downhill. Anything with inertia ‘rolls’ the same way no matter how it’s turned. Electromagnetism here is different: there’s no standing downhill, just a traveling pattern, so the outcome depends completely on how you’re oriented. Right facing → push. Wrong facing → it cancels. Opposite facing → pull.”
And the Math Appendix’s remark on range: “Magnetism drops off faster than gravity because here the conserved flux is tied to transverse oscillations set by tension. Gravity, by contrast, is encoded in space curvature directly and dilutes differently.”
Now the general case, from the Mathematical Bridge Math Appendix. A closed loop is a transport that does not close perfectly, and that is a source: “Sources arise when Void transport fails to close perfectly (broken closures, defects, intersections), creating conserved transport currents.” Introduce the current 3-form J with dJ = 0; the transport equations and the charge:
“Thus Q is measured by the flux of ⋆K through any closed 2-surface surrounding the source. Conservation dJ=0 implies Q is independent of the particular surface chosen (as long as it encloses the same sources).” (Source Extension, 13). With E named as the time–space part of K, that is Gauss’s law: the charge inside any closed surface is the flux through it. The Workbook’s q = σe is this Q for one loop.
Apply d⋆ to K = dA with the source present, choose the relabelling ∇·A = 0, and the transport equation is the wave equation with a source; its solution is the retarded one (Math Appendix, Source Extension 14; Magnetism from Moving Sources 5):
“Solutions are obtained with the retarded Green’s operator of □, guaranteeing causal propagation at speed c.” Nothing a source does is felt before r/c: the wave of panel 1 is this solution for a rotating source, and the arrival of the wave at the particle in panel 2 is at c, as the H page says. These are the retarded potentials of the textbook, without the potentials having been postulated.
The force is in the action (Math Appendix, Source Extension 12). One action produces the source law and the coupling: “S[A; J] = (1/2σs) ∫ K ∧ ⋆K + ∫ A ∧ J, with K:=dA. Variation δS/δA = 0 gives d⋆K = J.” A second loop carries its own current J′, and the term ∫ A ∧ J′ is how the first loop’s transport acts on it. For a point charge that coupling is the minimal-coupling Lagrangian of the Walk-Through (section 2):
“Gauge choices (Coulomb/Lorentz) do not change observables.” The q in front of v·A − Φ is the σ of panel 3 in the textbook’s units, so the sign of the force is the relative orientation, as the Bridge says; the size is the wave at the particle, panel 2. That is the Lorentz force, recovered from the coupling term and nothing else.
Hold everything still. The source law’s time–space part becomes Poisson’s equation and the point source gives the potential and the force (Electromagnetism Math Walk-Through, section 2):
“Static point source solution of Poisson’s equation gives Coulomb’s law.” With q = σe, q₁q₂ is σ₁σ₂ e², and this is the Bridge’s F ∝ σ₁σ₂/r² with its scale set: the same 1/r², the same sign rule, now in coulombs and newtons. The H page’s two particles are this law at R = 16.5 ƛe. The 1/r² here is the static force from Poisson; the 1/r² of panel 7 is the radiated intensity of a moving source; the documents keep them apart and so does this page.
The H page draws still loops. Let the source move (Math Appendix, Transport Circulation from Moving Sources; Magnetism from Moving Sources via Transport Calculus): “When the source moves with velocity u (timelike worldline tangent), transport conservation requires that the flux lines ‘lean’ in the direction of motion. This tilt produces circulation of the transport 1-form A around the axis of motion.” The same source law, split against the motion, and the steady-current case:
“This is the transport analogue of the Biot–Savart law. It shows that moving sources generate circulating transport proportional to their current strength.” And: “The transport circulation law coincides with Ampère’s law with a material constant μeff set by A3.” Magnetism is not a second thing beside charge; it is the same transport seen from a source that moves, which is why a moving charge and a current loop act alike.
Everything above came from two lines, the geometric identity dK = 0 and the source law d⋆K = J. Split against an observer’s time direction (Math Appendix, Magnetism from Moving Sources, sections 1 to 3; boxed results), they are the four equations:
“(E,B) are not primitives but simply the decomposition of K into radial and circulatory parts relative to the source’s motion.” “If one elects to adopt conventional names at the end, identify the observer-split components of K with (E, B). Then Sections 2–3 reproduce the standard magnetic sector exactly.” The Narrative (section 5): “When you average these orientation-gated path nudges over time and over many loops, in smooth, weak-curvature conditions, the effective path law you recover is the same set engineers already use: the standard Maxwell equations.” That is the recovery this page set out to show: the push and pull of the drawing, and the whole of vacuum electromagnetism, from a closed loop’s orientation and the transport of the space it hides.
| Line | Status | Where it stands |
|---|---|---|
| F ∝ σ₁σ₂/r²; direction depends only on σ; reversing the rotation flips the side | published | Mathematical Bridge, 4 and Waveform (steps 3, 4). |
| the map R = √(r² − ƛe²), δ(R) = ƛe²/(R² + ƛe²); δ(15.5) = 4.2×10⁻³, δ(17.5) = 3.3×10⁻³, ratio 1.27 | the mass pages’ construction + computed | The map is the mass figures’ construction (thread 28) of two published lines, Mathematical Bridge section 3: a loop “removes a volume of available transverse area from the surrounding space, fixing the missing volume” (the map keeps πƛe² missing at every radius) and “Δg(r) ~ 1/r²” (its far field). Its exact form is not a line in any document. Values evaluated here and on the H page (steps 1, 2). |
| the wave’s width rule c/(1 − δ); contrast eased √δ; the particles at 16.5 ƛe; net-drift arrows as indicators | the drawing’s rule / chosen | Declared here and on the H page; not a line in any document. |
| a charged loop throws off null-gravity waves; another loop feels them only through its facing; the surfer; not a standing downhill | published | Electromagnetism Narrative, What Is Charge, 1, 6 (steps 1, 4, 5). |
| ⟨Δp⟩ ∝ ε·σ·cos θ, push / null / pull | published (rotor test) | Electromagnetism Math Appendix, 9, Eq. 9.4; quoted for the facing dependence, not as the two-charge force (step 4). |
| the near side sets the size, the facing sets the sign | reading | The map’s two δ values (published) beside the Bridge’s sign law (published); the sentence joining them is this page’s and the H page’s. |
| dJ = 0; dK = 0, d⋆K = J; Q[V] = ∮⋆K; ∇·E = ρ | published; recovers Gauss | Math Appendix, Source Extension 10 to 13; Magnetism 3 (step 6). |
| □A = 𝒮[J]; A = ∫Gret𝒮[J]; far zone 1/r, 1/r² | published; recovers retarded potentials | Math Appendix, Source Extension 14; Magnetism 5 (step 7). |
| S[A; J] with ∫A∧J; L = ½mv² + q v·A − qΦ ⇒ ma = q(E + v×B) | published; recovers Lorentz | Math Appendix, Source Extension 12; Walk-Through, 2, Box 3 (step 8). |
| ∇²Φ = −ρ/ε₀; Φ = keq/r; F = keq₁q₂/r² | published; recovers Coulomb | Walk-Through, 2, Box 4 (step 9). |
| q₁q₂ = σ₁σ₂e², so Coulomb is the Bridge’s line with its scale set | reading | q = σe (Workbook) put into Box 4; the two published lines side by side. |
| the lean; ∇×B − ∂E/∂t = j; ∮A = μeffIenc; Biot–Savart | published; recovers Ampère, Biot–Savart | Math Appendix, Transport Circulation 2, 3; Magnetism 3, 4, 6 (step 10). |
| dK = 0 ⇒ ∇·B = 0, ∇×E + ∂B/∂t = 0; d⋆K = J ⇒ ∇·E = ρ, ∇×B − ∂E/∂t = j; continuity; □A = 0, c | published; recovers Maxwell | Math Appendix, Magnetism boxed results; Walk-Through, Boxes 1 and 2 (step 11). |
| the wave’s 1/r² (step 7) and Coulomb’s static 1/r² (step 9) are two statements | published, kept apart | Radiated intensity of a moving source; static force from Poisson. |
| S₀ = ħ; ε₀, μ₀, ke, μeff | calibrations / limit locks | One scale and the acceptance locks the documents name; μeff “derived from A3’s finite tension Ts and σs”; never derived here. |
Chosen for the drawing, not from the framework. The two particles stand at R = 16.5 ƛe, a drawing choice, and their net-drift arrows are summary indicators, not computed magnitudes; the wave’s width rule (release edge at drawn speed c/(1 − δ), head edge at c) is the drawing’s own, as on the previous page, and its contrast is eased as √δ to print. In panels 6 to 11 the closed surface, the light cone, the field pictures, the current and its loops, and the plotted ranges are drawing choices; the curves and the two δ values are the stated functions evaluated.
Computed from those choices. δ(R) on 0 to 20 ƛe; δ(15.5) = 4.15×10⁻³, δ(17.5) = 3.25×10⁻³, ratio 1.27, difference 9.0×10⁻⁴, the arrows in panel 2 drawn proportional; ⟨Δp⟩ ∝ σ cos θ for both σ; Φ = −Gm/r; ±keq₁q₂/r² for the two signs.
Sources (vms-institute.org/theory). Proposed Mathematical Bridge: Loop Orientation to Electromagnetism (F ∝ σ₁σ₂/r²; direction depends only on σ); Waveform (reversing the rotation flips the side). Mathematical Bridge Math Appendix: Electromagnetism from Void Transport, 7 (the range remark); Source Extension, 10 to 17 (J, dJ = 0; d⋆K = J; S[A; J]; Q[V] = ∮⋆K; retarded □A = 𝒮[J]); Transport Circulation from Moving Sources, 1 to 5 (the lean; ∮A = μeffIenc); Magnetism from Moving Sources via Transport Calculus, 1 to 6 and boxed results (the 3+1 split of K; dK = 0 and d⋆K = J as the four equations; Biot–Savart; the retarded solution; naming at the end); Mathematical Bridge, section 3 (the fixed missing volume; Δg ~ 1/r²), of which the mass pages’ map is a construction. Electromagnetism Narrative: What Is Charge in VMS?; sections 1 (the surfer), 5 (the Maxwell limit), 6 (how this is not gravity). Electromagnetism Math Walk-Through: section 1 (Boxes 1 and 2), section 2 (minimal coupling, Lorentz force, Poisson, Coulomb, Boxes 3 and 4). Electromagnetism Math Appendix: section 9, Eq. 9.4. Particle Mechanics Narrative: section 7 (q = n·q₀). Electromagnetism Student Workbook: symbols (q = σe). Generator: gen_charge_force_proofs3d.py.