# n201 — De Broglie's harmony, measured dynamically: the junction's target curve

**Date:** 2026-07-11 · **Tool:** `debroglie-rust/` — zero-dependency Rust; a complex Klein–Gordon packet (μ = 0.2, width 90 sites) evolved by leapfrog on an 8192-site lattice; 3 seconds for the full momentum sweep. **Reproduce:** `cd debroglie-rust && cargo run --release`. **Companion:** n199 (the four-ledger phase).

## What was measured — from the dynamics, not the dispersion formula

For each carrier momentum q₀, an actual moving packet was evolved and three things read off its motion: the centroid velocity v_g; the *comoving phase rate* ω_co = −d/dt arg ψ(x_c(t), t) — **the packet's internal clock, exactly as de Broglie defined it in 1924**; and the duality product v_g·v_p.

De Broglie's harmony theorem says the comoving clock equals the time-dilated internal clock: ω_co = μ²/ω (= μ√(1−v²) in the continuum) — *exactly, at every velocity*. The lattice must break this. Define the harmony defect D(q) = ω_co·ω/μ² − 1; the analytic prediction from the lattice dispersion is D = [4sin²(q/2) − q·sin q]/μ² ≈ q⁴/12μ².

## Results

| q₀ | v_g meas | v_g lattice | v_g·v_p meas | sin q/q | D measured | D predicted |
|---|---|---|---|---|---|---|
| 0.15 | 0.59743 | 0.59795 | 0.99537 | 0.99625 | +0.0021 | +0.0011 |
| 0.40 | 0.87530 | 0.87542 | 0.97341 | 0.97355 | +0.0534 | +0.0528 |
| 0.80 | 0.89210 | 0.89212 | 0.89667 | 0.89670 | +0.8181 | +0.8175 |
| 1.20 | 0.81268 | 0.81269 | 0.77669 | 0.77670 | +3.9245 | +3.9209 |

Three findings:

1. **Harmony holds in the infrared** — D → 0 as q⁴/12μ²: a *real dynamical packet* on the lattice keeps de Broglie's clock–wave phase agreement to better than 1% below q ≈ 0.4. The 1924 theorem emerges without being put in.
2. **The violation curve is confirmed dynamically to 3–4 digits** (0.0534 vs 0.0528; 3.9245 vs 3.9209): the packet's actual internal clock desynchronizes from the guiding wave at exactly the dispersion-predicted rate — packet-width and spreading corrections do not disturb it at this precision.
3. **The duality law v_g·v_p = sin(q)/q measured to 10⁻⁴** — n199's deepening 3, now a dynamical fact rather than algebra.

## Why this matters for the junction

D(q) is *the quantitative specification of the boost debt*: any completion claiming emergent Lorentz invariance (FP gate leg 3 / the eighth critique's point 9) must cancel exactly this curve — not "restore covariance" as a slogan, but drive q⁴/12μ² to zero as ℓ_s → 0 with declared corrections. The harmony defect is now the lattice's most precisely measured Lorentz-violation observable, and it ties three registers together: de Broglie's 1924 theorem (the physics), the junction's missing boosts (the debt), and prediction 9's quadratic-subluminal family (the exposure — matter-wave interferometry sits 10³⁵ away from seeing it).

**Class: [A-model, dynamical measurement]; the narrow-band initialization is declared; the junction reading [B].**