# n187 — The dark inversion: what a dark-matter measurement would teach the theory's parameters

**Date:** 2026-07-11 · **Question:** can DM observations *inform* parameters, not just kill? **Answer:** yes — the dark sector is the framework's most invertible sector: its three inputs (capacity band, KZ relic law, born-cold distribution) each map back to a parameter nothing else reaches. Five channels, each with its arithmetic.

## Channel 1 — the DM–knee cross-lock (the sharpest)

The capacity band (n137) ties the dark quantum to the anchor: m_DM ∈ [1.9, 13.5] keV = **0.51–3.62 × ħγ/c²** under the electron anchor. Inverted: **a measured m_DM pins the knee** — the anchor window collapses from ×1836 (3.26 decades) to the band's internal factor ×7.1 (0.85 decade). A keV DM mass from structure data (or a line-free spectral cutoff) would tell the Auditometer where to look *before the audit experiment exists*. And the two measurements then check each other: knee and m_DM must land at a ratio in [0.51, 3.62], or the capacity selection dies. **This is a registered cross-lock: two future measurements, one consistency band, no free parameter between them.**

## Channel 2 — the quench clock (the cooling debt's first number)

The relic law (n136, in-house): n ∝ τ_Q^(−0.615). Today's mean density is known (ρ_DM ≈ 2.2×10⁻²⁷ kg/m³), so a measured mass fixes the number density — n = 6.5×10⁵ /m³ (1.9 keV) to 9.2×10⁴ /m³ (13.5 keV) — and *that* inverts through the KZ exponent to **τ_Q, the crystallization quench time**: the first number the unbuilt cooling history would ever receive from data. Precision propagates as δτ_Q/τ_Q = 1.63 × δm/m; across the band τ_Q spans ×24. Honest limit: the KZ prefactor (substrate correlation scale at the transition) is unpriced — the inversion delivers τ_Q *in substrate units*, a relation, until the cooling sector prices the prefactor.

## Channel 3 — the crystallization horizon (the second number)

KZ formation leaves order-unity density granularity at the defect correlation length — a **minicluster population** (the axion-minicluster analogue). A measured minicluster mass scale (microlensing surveys, pulsar-timing dips, stellar-stream gaps) measures the **comoving correlation length at crystallization** — the second cooling-history number, independent of channel 2. The two together over-determine the quench: a cross-check inside the debt.

## Channel 4 — the statistics of the dark quantum

Phase-space data discriminates spin-statistics: the Tremaine–Gunn bound applies only to fermionic DM (dwarf-spheroidal TG floor ≈ 0.4–1 keV — below the band's 1.9 keV edge, so the band survives *either* assignment today). But a future measured phase-space density **above** the fermionic limit at measured m would force the dark quantum bosonic — directly informing the topological branch's spin assignment (windings = orientable cycles vs the non-orientable fermion handles of n121). One astrophysical measurement adjudicating a topology question the lattice cannot.

## Channel 5 — the 3.5 keV lesson (already banked)

The disputed 3.5 keV X-ray line sits exactly where a ~7 keV decaying candidate would put it — mid-band. The framework *forbids* the decay interpretation (28c: topological stability, no line at any depth): it was structurally committed to that line being atomic/instrumental. Deep follow-ups (Hitomi-class, blank-sky stacking) have moved against the decay reading — a retrodictive pass the sector never claimed credit for. [Honesty flag: observational status from training knowledge; verify against primary sources before public use as a pass.]

## Summary table

| Measurement | Parameter informed | Precision map |
|---|---|---|
| m_DM | the knee γ (cross-lock) | window ×1836 → ×7.1 |
| m_DM + Ω_DM | τ_Q, the quench clock | δτ/τ = 1.63·δm/m (substrate units) |
| minicluster scale | crystallization correlation length | independent of τ_Q channel |
| phase-space density | dark quantum statistics (topology) | binary: TG-violating ⇒ bosonic |
| line searches | stability (kill-only) | no parameter, kill 28c |

**Classes: channel 1 [B — band is anchor-extended]; channel 2 [B/C — prefactor unpriced]; channels 3–4 [C — mechanisms real, scales unpriced]; channel 5 [flagged retrodiction]. Nothing here adds a parameter; every channel consumes a future measurement.**