# n189 — The dark equivalence condition: do the windings fall like matter? (deepening n188)

**Date:** 2026-07-11 · **Question raised by n188's reframe:** if dark matter is frozen k-strain — topological states of the same field whose smooth sector is gravity — does it *fall* like matter? **Method:** derive the winding's gravitational coupling from the energy functional; verify numerically; extract the consistency condition. **Script:** in-session (128² lattice, tight-vortex proxy — declared).

## The derivation (three lines)

A phase-winding stores E_w = ½Σ_e k_e(dθ_e)² with circulation fixed by topology. Matter curves the coupling field: k_e = 1 + R·Φ(x_e), where R ≡ d ln k/dΦ is the Action's metric convention. Then to first order

**E_int(d) = R · E_w · Φ(d)** — verified numerically (E_int/Φ → R·E_w: 4.86/4.93 at R = 1, 9.72/9.87 at R = 2, converging with distance).

Three structural facts fall out:

1. **Windings gravitate automatically and Newtonianly** — no new coupling was added; the 1/r potential appears because the winding's stored energy sits in the same k that matter curves. "One field, two phases" is dynamically consistent at first order.
2. **The winding's passive gravitational mass is exactly R × (its stored energy).** Active = passive by the mutuality of the energy functional.
3. **Matter's coefficient is already fixed at 1** by the redshift mechanism (E(x) = m(1+Φ), the billing clock — mechanism 4).

## The dark equivalence condition

**Windings fall like matter iff R = 1** — iff the phase stiffness reads the same k(Φ) as the billing clock. The naive "k is the spatial metric" reading (g_ij ~ 1+2Φ) gives R = 2: **dark matter would fall twice as hard as baryons.** That is not a small anomaly — it is grossly excluded by the consistency of cluster dynamical masses with lensing masses (both read the same halos), by halo–baryon co-orbiting in every disk galaxy, and by the Bullet's lensing tracking the (dark) mass.

**So observation forces R = 1 today** — and this closes the loop opened in n187 in the strongest possible way: dark-matter phenomenology does not merely *promise* future information about parameters; it **already fixes an Action convention** — the k(Φ) normalization must be the clock normalization, not the spatial-metric one. A parameter the lattice could not pin, pinned by clusters.

## What this makes falsifiable

- **The registered consequence:** any confirmed DM/baryon free-fall ratio ≠ 1 (cluster dynamics-vs-lensing tension of the coherent kind, dark-EP tests via tidal streams) now maps to R ≠ 1 — one number, no freedom. Conversely the framework, with R pinned, predicts **exact dark–baryon equivalence at every scale** — no "dark charge" screening, no fifth-force segregation between sectors.
- **The residue:** the tight-loop proxy hides gradient corrections of order (core size/d)² — a short-range dark-EP anomaly formally present, utterly negligible at astrophysical scales; recorded so no one rediscovers it as a claim.

## Honest limits

The vortex is a proxy (tight 4-edge loop; the relaxed defect spreads); 2D; the inertial mass of a winding (m_i = E_w?) is *not* derived here — full equivalence needs m_grav = m_inert, and inertia of topological states is an open computation (the information-metric route, n141's machinery, is the natural attack). Until then the condition covers passive gravitational coupling only.

**Class: derivation + numeric [A-model]; the R = 1 pinning [B — observation forcing a convention]; dark–baryon equivalence as prediction: inherits 28's family. No new parameter; one old one eliminated.**