How light moves, and where the wave form comes from

A photon is the smallest piece of light; this page follows one photon.
The textbook wave form ψ = A·ei(kx − ωt), I = |Σψ|², and its intensity pattern come out of the VMS math and patterns below:
λ = S₀/Ad, k = 2π/λ = 2πAd/S₀ (S₀ locked to ħ at the electron), and ψ is the sum over routes Σ √weight·e2πiS/S₀, weight the product of the route’s shares (this page draws it, red, over the counted shape, grey).

The textbook wave form has no ħ. It takes the wavelength as given. VMS gives the wavelength:
one step ℓ = S₀/Ad, the distance in which the display action grows by one S₀ (one full turn of what the textbook calls the phase), is the wavelength λ.
That is where ħ lives: S₀ = ħ. (Compare de Broglie’s pλ = h = 2πS₀: with λ = S₀/Ad the display action ∫Ad ds stands where ∮p·dl stands, and 2πAd stands where p stands.)

Here is how. A photon hides a patch of space as it goes, its display area Ad = πr². The size of that patch decides what the photon can do at every pinch (caustic): a big patch has few ways through, a small patch has many. Along the route the patch adds up to display action S = ∫Ad ds, and the photon moves in steps of ℓ = S₀/Ad with S₀ = ħ: big patch, short steps; small patch, long steps. On a straight route S = Ad·x, so 2πS/S₀ = kx with k = 2πAd/S₀ = 2π/ℓ. That is the angle the textbook calls the phase, and one step is the wavelength λ. The step belongs to the photon and is the same in empty space. The pinches belong to the space, wherever it happens to be squished; their spacing has nothing to do with the wavelength.

A pinch (caustic) is where two fronts of bent space cross and the lattice is squeezed for a while (see the pinch figure). There a neighbouring route is an option while its extra action is small, ΔS = πr²·ΔL ≲ S₀, and its amplitude is e−ΔS/S₀, its share the square, e−2ΔS/S₀. The grey shape at E is where the photon can end up: the average over many, and the textbook envelope. The red curve is the pattern for just these few example caustics.

So a bigger display area takes fewer, sharper turns and lands in a narrow pattern; a smaller display area spreads wide. That is the wavelength dependence that emerges as the wave form, and it came from the display area and the caustics, not from a wave form itself.

small photon: r = 1 r₁, Ad = πr² = 1 A₁, one step λ = S₀/Ad = 1.00 λ₁, 7 routes inside ΔS ≲ S₀shares at each pinch ∝ e^(−2ΔS/S₀), ΔS = πr²ΔL: 4% / 11% / 22% / 27% / 22% / 11% / 4% (sum 100%)B4%11%22%27%22%11%4%C4%11%22%27%22%11%4%D4%11%22%27%22%11%4%AEred: added as amplitudesgrey: countedwhere it couldhave endedmedium photon: r = 1.5 r₁, Ad = πr² = 2.25 A₁, one step λ = S₀/Ad = 0.44 λ₁, 5 routes inside ΔS ≲ S₀shares at each pinch ∝ e^(−2ΔS/S₀), ΔS = πr²ΔL: 6% / 24% / 40% / 24% / 6% (sum 100%)B6%24%40%24%6%C6%24%40%24%6%D6%24%40%24%6%AEred: added as amplitudesgrey: countedwhere it couldhave endedbig photon: r = 3 r₁, Ad = πr² = 9 A₁, one step λ = S₀/Ad = 0.11 λ₁, 3 routes inside ΔS ≲ S₀shares at each pinch ∝ e^(−2ΔS/S₀), ΔS = πr²ΔL: 11% / 78% / 11% (sum 100%)B11%78%11%C11%78%11%D11%78%11%AEIn the big-photon row the red (amplitudes added) and grey (routes counted) differ;that is an artefact of the drawing(three pinches, three exits each, 27 routes, so the phases don’t average);in the world, with many caustics and continuous options, the two agree;fringes appear only when one regular caustic lines all the display actions up,like a standing razor slit.red: added as amplitudesgrey: countedwhere it couldhave ended

Everything on this page is computed from five lines of the framework.

Ad = πr²    S = ∫Ad ds    2πS/S₀ (the textbook’s phase), S₀ = ħ, closure ∮p·dl = 2πnS₀    ℓ = S₀/Ad = λ    route amplitude ∝ e−ΔS/S₀, share = its square

Display area, display action, closure (one S₀ of display action is one full turn of the textbook’s phase), one step (the wavelength), and the route amplitude. Only the ratio S/S₀ enters; S₀ is the one scale.

At a pinch (caustic). The photon can leave on a nearby route instead of the entering path. A nearby route costs extra action ΔS. If that extra is less than about one S₀, the route is an option; the less extra it costs, the bigger its share of the odds:

option if ΔS = πr²·ΔL ≲ S₀
amplitude ∝ e−ΔS/S₀, share ∝ e−2ΔS/S₀
shares at one pinch add to 100%

ΔL is how much longer the tilted route is than the straight one on the way to the next pinch. Line width and the printed percentage are these shares.

Three sizes shown using one rule. Every pinch offers the same seven exit directions, 3.5° apart. A tilted exit costs extra action ΔS = πr²·ΔL. A bigger photon pays r² times more for the same tilt, so fewer of the seven stay under S₀:

r = r₁: 7 routes
r = 1.5 r₁: 5 routes
r = 3 r₁: 3 routes
step ℓ = λ = S₀/Ad: 1 : 0.44 : 0.11

The turn at a pinch is drawn as spread over one step λ (this page’s reading of the linearised route), so a big photon corners and a small photon bends.

The grey shape at E. Every route from A to E, one option at each of B, C and D (7³, 5³, 3³ routes), counted with its weight:

weight = shareB · shareC · shareD
grey(E) = Σroutes weight

The red curve at E. The same routes, added as amplitudes instead of counted:

ψ(E) = Σroutes √weight · e2πiS/S₀
I = |ψ|²

S is the route’s display action; dividing by S₀ counts it in turns, and 2πS/S₀ is the angle the textbook calls the phase.
On a straight route S = Ad·x, so e2πiS/S₀ = eikx with k = 2πAd/S₀ = 2π/λ.
Inside the window the routes differ by up to one S₀ of display action; added as amplitudes they give the fringes of the textbook pattern inside the grey shape, its envelope.

Chosen for the drawing, not from the framework. Three pinches at 22%, 50% and 70% of the way from A to E (uneven on purpose to show random caustics).
Seven candidate tilts 3.5° apart. The three sizes 1 : 1.5 : 3, chosen so that each photon’s outermost surviving route sits right at the window edge (ΔS grows as (r·tilt)², and r times the last tilt is 3 in every row). S₀ placed at the small photon’s outermost option. Which option the photon takes at each pinch (middle, one up, middle; the same in every row). The shares at B, C and D are multiplied as independent choices: no memory carried from one pinch to the next.
The route counts 7, 5, 3 are relative to that S₀ placement; the r² scaling between them is the framework’s.
The red curve and grey shape are smoothed by half a tilt step, because the drawn routes are 3.5° apart while real routes are continuous.

Sources (vms-institute.org/theory). Proposed Mathematical Bridge for the Standard Model and General Relativity: Display Area to Display Action (Ad = πr², S = ∫Ad ds, ratios only), Open Path to Photon Sector (caustics mandate optionality; linearisation gives □ψ = 0). Mathematical Bridge Math Appendix: Caustic Formation and Ray Mapping (J = 0, Airy form near a fold), Loop Action Connection and the electron anchor (S₀ = ∮Ad ds, ∮p·dl = 2πnS₀, pλ = 2πS₀, S₀ = ħ). Particle Mechanics Math Appendix: Decays and Widths from Action Gaps (amplitude e−ΔS/S₀, probability e−2ΔS/S₀). VMS Closure Math Primer (optional paths, forced bottleneck, action gap ΔS/S₀). Generator: gen_one_light_HS_LOCKED_v5.py (locked 2026-09-12).