# Route R2, light as a phase wave of the quons' circling: the converged computation

2 November 2026, isla session, about 07:02 EST, for Doug Westmoreland (project lead: direction and review). Pre-registered in papers/Lock_R2_phase_wave.md (committed as b831998 at 06:54:56 EST, before the run at 06:56:38). Scripts: sim/lepton/phase_wave.py (the locked run, 6 s) and phase_wave_post.py (additions written after seeing the output, disclosed in the lock); outputs phase_wave_out.txt and phase_wave_post_out.txt. The preliminary version of 06:31 EST is kept as phase_wave_v1.py. Labels: [A] Aspden's own, [D] derived, [M] our modelling choice, [P] proposed, [S] standard physics. Independently checked: sim/lepton/lepton_check8/REPORT.md (a Fable agent with no part in the work, own code throughout; finished 07:45 EST); its corrections are folded in below, each marked with its finding number, and listed in §9.

## 1. The question

Route R2 (Improvement_directions_2026-11-02.md §4) asks whether light could be a slow modulation of the phase of the quons' circling rather than a displacement wave. The preliminary sum of 06:31 EST suggested that the phase mode is unstable along the circling axis and gapped across it, but its cubic-shell sums were not converged. This page does the computation properly and answers two questions: is the synchronous circling unstable to phase modulations along its axis, and can a phase wave travel at c in any direction.

## 2. The model, stated in full [M]

1. **Lattice.** Simple cubic, spacing d, a charge e on every site [A]. No continuum term: the continuum is uniform, and a quon kept on a circle of fixed radius has the same continuum energy at every phase [D].
2. **Motion.** Quon i moves on a circle of radius r = 0.3028737692 d about its own site, in the plane perpendicular to the circling axis zL = [001], with phase φ_i(t) = Ωt + u_i(t). The radius is held rigid: the quon is a bead on a ring. This is our choice [M]. It is the most permissive reading of "let the phase vary from cell to cell": the phase is a free coordinate and nothing else moves. Aspden's own restoring force makes the orbit an isochronous harmonic one (Part 1A §5); what that changes is in §7.
3. **Inertia.** The quon's mass m₀ on the ring: m₀r² per quon for the phase.
4. **Forces.** Coulomb forces between quons only, expanded to second order in the phase deviations u with the fast common phase Ωt kept explicitly.
5. **Units.** Ω² = 8πe²/(m₀d³) [A: POC 7.5 with both frames moving, as Part 1A reads it]. A wave at c has ω²/Ω² = (kd)²(c/Ωd)² = (kd)²(2r/d)² = 0.36689 (kd)², since c = 2rΩ [A]; at 511 keV, kd = 1.651 and this line gives exactly 1. (The first version of this page, the lock's item 9 and the script had the line inverted, (kd)²/0.36689, a factor 7.43 too high; corrected after check F1. Only the "wave at c" comparisons changed; no lattice number did.)

**How it works, in words.** All quons circle in step. Give the phases a ripple from cell to cell. Averaged over a turn, two quons stacked one above the other along the axis sit closest when they are in phase and farthest apart when they are opposite, so like charges prefer opposite phases along the axis. Two quons side by side in the plane keep their distance when in phase and swing between closer and farther when opposite, and the average of 1/distance is larger for the swing, so side by side they prefer the same phase. The synchronous state is therefore a saddle of the averaged energy: stable to ripples across the axis, unstable to ripples along it. The computation below makes that quantitative, with the exact long-wave limits and the fast 2Ω term that the average leaves out.

## 3. The equation of motion [D]

The Coulomb energy of a pair is e²/|s_ij + r c(φ_j) − r c(φ_i)| with c(φ) = (cos φ, sin φ, 0). Its mixed second derivative at equal phases, −∂²E/∂φ_i∂φ_j, is

  κ_ij(t) = e²r²[3(ŝ·τ)²/s³ − 1/s³], τ = (−sin Ωt, cos Ωt, 0),

and the diagonal second derivative is κ_ij − e²r(s_ij·c)/s³, the second term coming from the quon's inward sag ½ru² toward its site (the first version had the sign of κ wrong here; check F4; the sign of the sag term depends on the orientation chosen for s_ij, and it drops out either way). Summed over the lattice, the extra diagonal term is e²r c·Σ_j s_ij/s_ij³ = 0 (the field at a site from all other sites vanishes by inversion symmetry), so the linear equation is m₀r²ü_i = −Σ_j κ_ij(t)(u_i − u_j). With H_αβ(s) = (3s_αs_β − s²δ_αβ)/s⁵, the dipole kernel,

  κ_ij(t) = e²r²[−½H_zz − ½(H_xx − H_yy) cos 2Ωt − H_xy sin 2Ωt](s_ij).

Its time average is the preliminary page's K_ij = e²r²(½ − 3/2 cos²θ_ij)/s³, which is half the energy of two dipoles e·r along the axis. The term at frequency Ω has dropped out exactly; what remains is a constant and a 2Ω part. The script checks the formula against direct second differences at 25 (s, φ) pairs: largest discrepancy 5.6 × 10⁻⁸.

For a plane wave u_i = u_k e^{ik·s_i}, with D_αβ(k) = Σ_{j≠0} H_αβ(s_j)e^{ik·s_j} and D(0) = 0 (every cubic shell of a cubic lattice sums to zero),

  ü_k/Ω² = −[A(k) + B_c(k) cos 2Ωt + B_s(k) sin 2Ωt] u_k, A = D_zz/16π, B_c = (D_xx − D_yy)/16π, B_s = D_xy/8π.

A is the secular dispersion, ω²/Ω² = A(k). The full equation is a Hill equation; its exact linear stability comes from the Floquet multipliers μ over one period π/Ω: unstable if |μ| > 1, growth rate Ω ln|μ|/π; for stable modes the frequency is Re ω = Ω arccos(tr M/2)/π. In units of Ω the whole problem is a pure lattice number: both the stiffness and the inertia scale as r², so r cancels.

**The long-wave limit [S].** D(k) is the lattice sum that gives the Lorentz local field. For a cubic lattice its k → 0 limit is 4π(δ_αβ/3 − k̂_αk̂_β): direction-dependent, not zero, although D(0) itself is zero. So

  ω²/Ω² → 1/12 − ¼ cos²θ_k (θ_k the angle of k to the circling axis):

−1/6 along the axis, +1/12 across it, zero on the cone cos²θ_k = 1/3 (54.7°). The uniform phase shift (k = 0 exactly) costs nothing; the limit depends on direction, as for the LO–TO splitting of an ionic crystal.

## 4. The lattice sums and their checks

D(k) is summed by Ewald's method [S] with splitting parameter η = 2/d, 13³ real-space and 13³ reciprocal terms. Checks required by the lock, all from phase_wave_out.txt:

| Check | Required | Found |
|---|---|---|
| (a) independence of η (1.5 against 2.5) | 10⁻⁸ | 7 × 10⁻¹⁵ |
| (b) agreement with a direct cubic-shell sum, N = 60, kd ≥ 1 | 10⁻⁴ | 5 × 10⁻⁶ for k = (1.2, 0.8, 0.5); **0.03–0.06 (1%) for k along a lattice axis, (1, 0, 0) and (0, 0, 2): not met as written** |
| (c) trace D(k) = 0 | exact | 10⁻¹⁵ |
| (d) k → 0 limit 4π(I/3 − k̂k̂) at kd = 10⁻³ | | 3 × 10⁻⁷ in four directions |
| D(0) = 0 | | 1.4 × 10⁻¹⁴ (direct, N = 20) |

**Check (b) for axis-aligned k, settled after the run (disclosed).** The direct sum is conditionally convergent, and for k along a cube face normal the surface term does not average out: the cubic-shell value at k = (1, 0, 0) oscillates with N (4.223, 4.405, 4.319, 4.330, 4.368, 4.307, 4.368 for N = 20 to 80) around the Ewald value 4.3401. With a Gaussian window exp(−(s/L)²) in place of the hard cube, the direct sum approaches the Ewald value as 1/L² (differences −0.38, −0.17, −0.098, −0.063, −0.044 for L = 8, 12, 16, 20, 24; extrapolated in 1/L² from L = 20 and 24: 4.3399 against 4.3401). So the Ewald value is the limit of the direct sum, and the lock's 10⁻⁴ target for a hard cubic truncation was not achievable for a conditionally convergent sum; the locked requirement is reported as not met, and the windowed extrapolation replaces it. The check (F8) confirmed the Ewald value by three methods to 10⁻⁶, derived the windowed deviations analytically as −8π/(kL)² (which reproduces −0.38 to −0.044 exactly), and identified the hard-cube oscillation as a 1/(kN) edge artefact: the Ewald value is the physically right one for a plane-wave modulation of an infinite lattice.

## 5. Results

Secular ω²/Ω² = A(k), the Floquet growth rate (0 means stable) and, for stable modes, the Floquet frequency. Full tables of 26 wavevectors per direction are in phase_wave_out.txt; the wave at c would need 0.3669 (kd)². kd = π is the zone boundary along [001] and [100] only (π√2 along [110] and [101], π√3 along [111]; check F3).

| k direction | kd = 0.01 | 0.11 | 0.46 | 0.95 | 1.95 | π |
|---|---|---|---|---|---|---|
| **[001] along the axis**, secular A | −0.16667 | −0.16675 | −0.16804 | −0.17213 | −0.18450 | −0.19273 |
| growth rate / Ω (Floquet = secular here, exact) | 0.408 | 0.408 | 0.410 | 0.415 | 0.430 | **0.439** |
| **[100] across the axis**, A | +0.08333 | +0.08337 | +0.08402 | +0.08606 | +0.09225 | +0.09636 |
| Floquet: growth / frequency / Ω | 0 / 0.303 | 0 / 0.303 | 0 / 0.305 | 0 / 0.309 | 0 / 0.321 | 0 / 0.329 |
| **[110] across the axis**, A | +0.08333 | +0.08337 | +0.08403 | +0.08614 | +0.09340 | +0.10241 |
| Floquet: growth / frequency / Ω | 0 / 0.303 | 0 / 0.303 | 0 / 0.304 | 0 / 0.305 | 0 / 0.311 | 0 / 0.321 |
| **[101] 45° from the axis**, A | −0.04167 | −0.04169 | −0.04201 | −0.04307 | −0.04670 | −0.05120 |
| growth / Ω: Floquet (secular) | 0.199 (0.204) | 0.200 (0.204) | 0.200 (0.205) | 0.203 (0.208) | 0.211 (0.216) | 0.220 (0.226) |
| **[111] the magic angle**, A | 0 | 0 | 0 | 0 | 0 | 0 (exact, §5.3) |
| Floquet: growth / frequency / Ω | 0 / 0.059 | 0 / 0.059 | 0 / 0.058 | 0 / 0.054 | 0 / 0.039 | 0 / 0.018 |
| wave at c needs A = 0.3669 (kd)² | 0.000037 | 0.00442 | 0.0783 | 0.330 | 1.39 | 3.62 |

(The Floquet frequencies at the 0.46, 0.95 and 1.95 columns were first read off at kd = 0.5, 1 and 2; corrected per check F2. The last row was 7.43 times too high in the first version; check F1.)

### 5.1 Along the circling axis: unstable, exactly

For k along [001] the 2Ω coefficients vanish identically (the lattice's 4-fold symmetry about the axis; the script finds |B_c|, |B_s| < 10⁻¹⁶), so the secular equation is the exact linear equation there. ω² is negative at every wavelength, from −Ω²/6 at long wavelength to −0.1927 Ω² at the zone boundary. The fastest-growing modulation is the zone-boundary one, adjacent layers in antiphase, with growth rate 0.439 Ω: an e-folding in 2.3 radians of the circling, about a third of a turn. The angular scan at kd = 0.05 reproduces 1/12 − ¼cos²θ_k to 10⁻⁵ at every angle. Every direction within 54.7° of the axis is unstable at long wavelength ([101] grows at 0.20–0.22 Ω; the 2Ω term lowers its growth rate by 2% from the secular value); at the zone boundary the unstable region reaches about 57° from the axis (check F10).

### 5.2 Across the axis: gapped, not a wave at c

For k perpendicular to the axis the secular ω² goes to +Ω²/12 as k → 0, a gap of 0.289 Ω; the exact Floquet frequency is 0.303 Ω at long wavelength (the 2Ω term, with amplitude 0.25 Ω², raises it by 5%) and 0.32–0.33 Ω at the zone boundary. The mode is stable (|μ| = 1 to 10⁻¹³) and almost dispersionless: its frequency changes by 8% across the whole zone. In energy, ħ × 0.303 Ω = 155 keV. Nothing goes to zero with k, so there is no speed to compare with c; the value a wave at c would need, 0.3669 (kd)², is 2,300 times smaller than the gap at kd = 0.01 and 38 times larger at kd = π. The gapped branch crosses the light line near kd = 0.5, where its group velocity is 0.01 c (check F1).

### 5.3 The magic cone: soft, not light-like

On the cone cos²θ_k = 1/3 the long-wave gap vanishes. Along [111] it vanishes at every k, exactly: for k on a 3-fold axis n the tensor D(k) is invariant under the 3-fold rotation, so D = aI + bnn; the trace is zero, so 3a + b = 0 and D_zz = a + b/3 = 0 (the script finds |D_zz| < 10⁻¹⁶ at all 26 wavevectors). So along [111] the secular stiffness is zero at every wavelength: no restoring force and no wave, not a slow one. The full equation there is a pure parametric oscillation driven by the 2Ω term, at 0.059 Ω for long waves, falling to 0.018 Ω at the zone boundary; its frequency does not vanish with k, so it is not a wave either. Elsewhere on the cone (in the x–z plane) ω² is zero to 10⁻⁵ up to kd = 0.5, −5 × 10⁻⁵ at kd = 1, −0.0012 at 2 and −0.0108 at kd = π, where the Floquet multiplier is 1.28 (growth 0.078 Ω): soft, and weakly unstable at short wavelength. The check (F10) notes that although the secular ω² is slightly negative from kd ≈ 0.2 on, the Floquet multiplier stays at 1.000000 up to kd = 2: the 2Ω term holds the mode there.

## 6. Against the lock's expectations

| # | Expected | Found | |
|---|---|---|---|
| 1 | [001]: → −1/6, negative everywhere, most negative near the zone boundary (−0.15 to −0.30), growth 0.4–0.55 Ω, exact | −1/6 → −0.1927 at the zone boundary; growth 0.408–0.439 Ω; exact | met |
| 2 | across the axis: → +1/12, stable, Mathieu a = 1/12, q = 1/8 stable | +1/12; stable; the 2Ω term lifts the gap to 0.303 Ω | met |
| 3 | [111]: acoustic, speed 0.1–0.5 c | ω² = 0 identically along [111] by symmetry; no wave of any speed; a k-independent parametric oscillation at 0.06 Ω | **missed**: the branch is not acoustic |
| 4 | [101]: → −1/24, weakly unstable | −1/24; growth 0.20 Ω | met |
| 5 | k = 0 a zero mode; direction-dependent limits | as expected | met |
| 6 | R2 fails under this model | fails | met |

The lock's check (b) was not met as written for axis-aligned k (§4), and was replaced after the run by the windowed extrapolation; that replacement is disclosed in the lock. The lock's item 9 carried the inverted light line (F1); its speed formula in the same item is right and implies the corrected line.

## 7. The second reading: Aspden's isochronous orbit [D, algebra only]

In Aspden's own model the quon is not a bead on a ring. The lattice as a whole circles in a harmonic, isotropic well at its own natural frequency (Part 1A §5: "every radius has the same frequency"; Aspden: the two frames are "capable of disturbance without change of the periodic time parameter"). The model used here, one quon in an isotropic well at Ω with a torque −Ku from its neighbours' phases, is our idealisation [M] (check F5: a single quon displaced on its own sees a well at Ω²/6, Part 1A §4.4, so "at Ω" is the collective mode's frequency, not a single quon's). For such an orbit the phase is not a slow coordinate. Linearising the radius ξ and the phase u of one quon about the circle:

  ξ̈ = 2rΩu̇,  d/dt(r²u̇ + 2rΩξ) = −(K/m₀) u,

whose characteristic equation is ω²[ω² − 4Ω² − K/(m₀r²)] = 0. The slow root is ω = 0 whatever K is, with eigenvector (ξ, u) = (1, 0): a slow torque on the phase changes the orbit's radius, not its phase. The double zero root is a Jordan block (check F6): a steady phase offset u drives the radius linearly in time, ξ̇ = −Ku/(2rΩm₀), and leaves the phase alone. The conclusion holds more widely than this model (check F5): once the radius is free, the phase is not an independent coordinate at all, and a phase ripple of wavevector k is a superposition of displacement waves at Ω ± ω whatever the lattice's modes are. (A tangential push on an isochronous orbit grows the amplitude and leaves the phase alone; in a harmonic potential a constant force moves the centre and a force rotating with the particle enlarges the circle.) The other root is near 2Ω. So under Aspden's own restoring force a "phase wave" cannot exist as a slow mode of the phases at all. What exists is Model L's lattice waves riding on the circling (the harmonic-potential theorem makes the circling a passenger: the pair distances are unchanged by it, so the relative motions are exactly the static lattice's phonons, whatever r is). A phase ripple of wavevector k is then the beat of a lattice wave at k against Ω, with frequency ω_ph(k) − Ω: gapped where the branch is acoustic, and quadratic in k where it is optical; never linear in k, and never c. R2 fails under this reading as well, and the instability of §5.1 does not arise in it: the circling's stability is then the static lattice's, which Model L found wanting for the simple cubic lattice for other reasons (HANDOFF §3).

## 8. Reading

1. **Is the circling's phase mode unstable along its axis?** Under the model locked here, yes, and exactly: every phase ripple with a wavevector within 54.7° of the circling axis grows, fastest for adjacent layers in antiphase, at 0.439 Ω. Coulomb forces between the quons do not hold Aspden's synchronism; along the axis they work against it. The synchronism must come from something else: Aspden's own isochronous orbit, under which the phases are not free (§7), or a rule outside the lattice's mechanics. That is the new statement about Aspden's lattice, and it is conditional on the bead-on-a-ring model.
2. **Can a phase wave be light?** No, under either reading. In the locked model there is no branch that goes to zero with k except on one cone, and there it has no speed; across the axis the mode is gapped at 0.30 Ω (155 keV) and nearly dispersionless; along the axis it does not oscillate at all. Light's speed is isotropic to about 10⁻¹⁸ (Nagel et al. 2015, Nat. Commun. 6, 8174, abstract read at arXiv 1412.6954; the first version said 10⁻¹⁷ without a source, check F7); this mode is not even qualitatively isotropic. Under Aspden's own orbit dynamics there is no slow phase mode to begin with. And a phase wave would still be a lattice wave, with no wavelength below 2d, so it could not be a gamma ray in any case (Improvement_directions §2 item 3).
3. **What this does not say.** It does not say Aspden's lattice flies apart: that depends on which model of the quon's orbit is right, and under his own isochronous orbit the question becomes Model L's phonon stability. It does not test the continuum's own waves (route R3), nor light as the field with a transparent lattice (route R1), which is the route left. It also leaves out the magnetic forces between charges moving at c/2, which are of order (v/c)² = 1/4 of the Coulomb forces and outside the model (check F8).
4. **Pass or fail by the lock's rule.** Fail. R2 is closed under both readings. The night's preliminary reading stands; what is new is that the limits are exact, the instability is exact along the axis, the [111] branch is not acoustic but identically soft, and the second reading closes the route without a lattice sum.

## 9. Independent check

sim/lepton/lepton_check8/REPORT.md, finished 07:45 EST; folded in here at 07:50 EST. The checker rebuilt everything with its own code: its own Ewald sum, a layer-by-layer method using no 3D Ewald at all, windowed direct sums with an analytic 1/L² term, its own monodromy integrator and an independent Hill-matrix eigenproblem.

- **Verdict:** the computation is right and the conclusion stands. Every lattice sum, growth rate, gap and Floquet frequency reproduces. The lock's chronology is proven from git and the file times (commit 06:54:56, script born 06:56:13, output 06:56:45); the post-run additions are disclosed and change no locked number.
- **Corrections folded in:** F1 (Major, number only): the "wave at c" line was inverted by 7.43 in this page, the lock's item 9, the script and Improvement_directions §4; the "300 times smaller / 280 times larger" sentence is now "2,300 / 38". F2: three Floquet frequencies in the table were read at the wrong kd. F3: kd = π is the zone boundary only along [001] and [100]. F4: a sign in the diagonal second-derivative term (it sums to zero). F5: the isochronous-well model of §7 is ours [M], and its conclusion holds more widely. F6: the zero root is a Jordan block. F7: the isotropy of c is 10⁻¹⁸ with a source, not 10⁻¹⁷. F8: the Ewald value confirmed three ways; magnetic forces named as outside the model. F9: phase_wave_post.py's header time was an estimate (07:05) ahead of the clock (06:59); fixed. F10: the unstable region reaches 57° at the zone boundary; the 2Ω term holds the cone up to kd = 2.
- **Not checked by it:** r/d and Ω² as inputs from Part 1A; whether Model L includes the on-site spring; the isotropy bound at source (read by the R1 check the same morning); the magnetic coupling.