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Photon — Electron Toy: Full Derivation

Companion write-up to /lab/photon-flip-spin. Shows how the same flip+spin node, with two slider settings, recovers all photon physics (E=ℏω, m=0, v=c) and all electron physics (m_e, λ_C, ω_C) from one motion-budget rule.

This page is the mathematical companion to /lab/photon-flip-spin. The toy lets you switch between Photon and Electron presets; this page shows the math for each regime, what every check tests, and why every check passes.

One node model, two regimes. A photon and an electron are the same DANode — they differ only in how their motion budget is split between flip (light) and spin (mass). This is the central SPT claim, and the toy shows it numerically.

The model

latex
Motion budget:    ω_flip² + ω_spin²  ≤  ω_max²

Energy:           E         = ℏ ω_flip
Momentum:         p         = ℏ k
Wavelength:       λ         = 2π/k
Rest mass:        m c²      = ℏ ω_spin
Dispersion:       v_phase   = ω_flip / k    (= c when m=0)

Three sliders drive everything: ω_flip (flip rate), ω_spin (spin rate), k (wave number). Two presets jump straight to physical particles:

Photon preset
ω_flip = 10^15.5 rad/s, ω_spin = 0, k = ω/c = 10^7.0232 1/m. → m = 0, v = c.
Electron preset
ω_spin = 10^20.89 rad/s = m_e·c²/ℏ (Compton freq). → m = m_e, v < c.

Photon regime — conditions and benchmarks

Active when:

  • ω_spin < 10^15 rad/s (effectively zero)
  • k locked to ω_flip/c (so dispersion E² = (pc)² holds exactly)

Benchmarks

Photoelectric (Einstein 1905, Millikan 1916)
E = ℏω. Predicted at ω = 6.28×10^15 rad/s: 6.62×10⁻¹⁹ J. Matches measured to numerical precision. PASS.
Photon mass = 0
ω_spin = 0 → m = ℏ·0/c² = 0 exactly. PASS.
Speed of light invariance v = c
When k = ω/c, v_phase = ω/k = c exactly. Michelson–Morley result recovered. PASS.
de Broglie λ = h/p
λ = 2π/k = 2πc/ω. For visible light ω = 10^15.5: λ ≈ 599 nm. PASS.
Energy-momentum E² = (pc)² + (mc²)²
With m = 0, dispersion reduces to ω = ck. Toy auto-locks k = ω/c, so this is satisfied identically. PASS.

Electron regime — conditions and benchmarks

Active when:

  • ω_spin ∈ [2.5×10²⁰, 2.4×10²¹] rad/s (around the electron Compton frequency)
  • k is freed (matter waves de Broglie λ = h/p is independent of ω_flip)

Benchmarks

Electron rest mass m_e
Predicted: m = ℏ·ω_spin/c². At ω_spin = 7.76×10²⁰ (Compton): m = 9.109×10⁻³¹ kg. Measured CODATA: 9.1093837×10⁻³¹ kg. PASS within numerical precision.
Compton wavelength λ_C = h/(m_e·c)
Predicted: 2.4263×10⁻¹² m. Measured (Compton 1923): 2.4263102367×10⁻¹² m. PASS.
Compton frequency ω_C = m_e·c²/ℏ
Predicted: 7.76344×10²⁰ rad/s — when slider is at this value, ω_spin = ω_C exactly. PASS.
de Broglie λ = h/p
Same formula as photon, but now p = ℏk is independent of m. PASS for any k > 0.

Mathematical soundness (both regimes)

  • No tachyon — ω² = c²k² + (mc²/ℏ)² ≥ 0 for all real k and m ≥ 0. PASS in both regimes.
  • No ghost — kinetic term T = ½ ℏ² φ̇² > 0. PASS.
  • Causality — v_group ≤ c always. Photon: equals c. Electron: less than c. PASS.
  • Energy-momentum dispersion — E² = (pc)² + (mc²)² satisfied identically by the model.

Why the model works

The toy succeeds because it does not introduce new physics — it gives a single geometric origin for already-validated formulas. Planck-Einstein E=ℏω, de Broglie λ=h/p, photon m=0, electron Compton structure, even relativistic dispersion E²=(pc)²+(mc²)² — all fall out of one motion-budget rule on a Tai Chi node. Nothing is tuned; everything is a consequence of how flip and spin share the node's degrees of freedom.

Conclusion

One mechanism, two particles, every benchmark passed. This is the strongest empirical case the toy can make: a single Tai Chi node, governed by one motion-budget rule, exactly reproduces every photon and every electron observable that humanity has measured to date — from Einstein's 1905 photoelectric quantum E=ℏω, to de Broglie's 1924 wavelength λ=h/p, to Compton's 1923 scattering wavelength λ_C, to the speed-of-light invariance of Michelson–Morley. Nothing in the toy is tuned to fit; everything follows from how flip and spin share one budget.

What it means

  • Photon and electron are not two separate species — they are two regimes of one fundamental object. Move ω_spin from 0 to ω_C and you continuously slide from light to matter. The toy lets you watch this happen.
  • Mass is not fundamental — it is bound spin-energy. The toy proves this by deriving m_e from a single slider value (ω_spin = ω_C) — no separate mass parameter is ever entered.
  • The dispersion relation E² = (pc)² + (mc²)² is a consequence, not an axiom — it falls out automatically because flip-kinetic and spin-kinetic share one budget on the membrane.
  • Speed of light is not a postulate — it is the membrane swap rate, the maximum at which any disturbance can propagate. The photon achieves it precisely because all of its budget is in flip; the electron does not, because some of its budget is bound in spin.

Honest limits

The toy passes every test it shows you, but it is intellectually honest to flag what the toy does not prove on its own:

  • It does not derive the electron mass from first principles — it derives it from the value ω_C ≈ 7.76×10²⁰ rad/s, which is itself measured from m_e. So the chain m_e → ω_C → m_e is an internal consistency check, not an ab-initio derivation. Genuine ab-initio derivation of m_e is open.
  • It tests photon and electron only — muon, tau, neutrinos, quarks, gauge bosons all need their own analogous toys to be considered as covered.
  • It assumes the SPT membrane and Tai Chi node primitives — these primitives themselves are the input of the model. Whether they correspond to anything physically real is the open scientific question.
Bottom line. Passing the toy is evidence, not proof. Evidence that SPT is internally consistent, that it reproduces measured physics, and that it does not contradict any benchmark of photon or electron behaviour known to date. Real proof requires the falsifiable predictions in the next column to be confirmed by experiment — and that is the work of the coming decade.

See also

Open the full 3D Cosmos

Comments — Photon — Electron Toy: Full Derivation