Critical Density
Audited-Negative — A-from-Jeans | scale-dependent, not derivedThe critical density ρcrit is the density at which the coherence function transitions from quantum (C → 0) to classical (C → 1). It is per-system, not a universal constant: Vflat is the observed flat rotation velocity of each specific system, so each galaxy (or other gravitationally bound object) has its own ρcrit. A is the universal proportionality constant; Vflat enters as an input. The rotation-curve fit is not zero-parameter — it requires an observed Vflat per galaxy.
The A Parameter
A is calibrated against the Jeans criterion (Session 53), not derived from it: βJ = λJeans / Rhalf is the dimensionless Jeans-length-to-galaxy-size ratio, empirically βJ ≈ 1.1 ± 0.2 across SPARC galaxies. A carries density units: since ρcrit = A·Vflat² with ρ in M☉ pc−3, A is in M☉ pc−3 (km/s)−2 — not (km/s)−2. Earlier versions of this page dropped the density dimensions, which is exactly where a large numerical discrepancy can hide unnoticed.
Correction (2026-08-05) — this page previously claimed “5% agreement” under a “Validated” badge. It asserted that the formula above yields A ≈ 0.029 against an empirical 0.028. It does not. With βJ = 1 and R0 = 8 kpc the stated formula yields A ≈ 4.6×10−5 — the value parameter derivations has marked audited-negative since 2026-06-07, and 635× smaller than the 0.029 used in every galaxy computation on this site. This page was a fossil contradicting its own linked prerequisite; it was caught by an outside reader recomputing the integral, not by the badge system.
What the 644× actually is (2026-08-05, revised same day). The archive bridges the two numbers with an unexplained “644× unit conversion.” It is not a unit conversion, and it is not a new quantity: the 644× was already decomposed on 2026-06-07. A depends only on the product βJ·R0, and Session687_A_From_Jeans_Arithmetic_Audit.md §1.2 records Session 66's own factorization: βJ = 4.5, R0 = 0.07 kpc — product 0.315 kpc, within 0.8% of the “317 pc” that setting βJ = 1 produces. Forcing βJ = 1 simply relabels that product as a length; matching it to the galaxy plotter's pinned h = 300 pc is a 5% coincidence.
Retracted (2026-08-05, same day): “A is a proxy for a coarse-graining length ℓ.” That reading was published here for a few hours and is withdrawn. A quantity that appears in a formula only as a product cannot be inverted into one of its factors without an independent measurement of the other — and the inversion contradicted this page's own calibration. The two residual questions are not about an unspecified length:(1) why βJ = 4.5 when the calibration above gives 1.1 ± 0.2 (a 17σ gap), and (2) why this site's rendering of the formula carries a 4π that Session 53's does not — 12.57 of the 635.
Consequence — the knee is unreachable, and no length choice rescues it. If ℓ were a coarse-graining length it would have to smooth ρ and ρcritalike — that is what a coarse-graining length is. Doing both, ℓ cancels:
x is a virial ratio. For any bound system Vc ≲ Vflat, so x ≲ 0.019βJ² ≈ 0.02 in every sector at every ℓ — the knee is out of reach by roughly 40× for free, with no fitted parameter. Verified numerically on all five plotter disks (max over ℓ: 1.7×10−3 to 1.1×10−2) and to four digits at Cassini and wide-binary scales. It is kernel-robust: a Gaussian kernel gives coefficient 0.00505, moving the ceiling down a further 3.8×. This is the only galaxy-sector statement on this site that requires no estimator choice, no velocity definition, and no contested external measurement.The single escape found is βJ = 4.5, which lifts the self-consistent x to 0.385 (C = 0.57) — at 17σ from this page's own calibration. No refutation is added to the ledger from this (it remains at 6): it closes a question rather than opening one.
At the site's working A = 0.029, NGC 3198 gives ρ/ρcrit = 9.45×10−4 — recomputed from the plotter's own midplaneDensity() (ρ(0) = 0.6164 M☉ pc−3), so it can be re-run. Values of 0.934, 1.43×10−3 and 0.91 briefly shown here on 2026-08-05 were high by a factor of 1.515 and are corrected.
Symbol note: this page previously wrote the Jeans ratio as α, which forced a paragraph disclaiming confusion with the fine-structure constant. It is now βJ throughout, matching parameter derivations; the disclaimer is no longer needed.
Physical Meaning
Every gravitationally bound system has a characteristic rotation velocity Vflat — the velocity at which its rotation curve flattens. This velocity encodes the total mass and size of the system. The critical density is where internal gravitational binding energy equals the coherence threshold.
Below ρcrit: the system is under-dense, loosely bound, quantum effects persist. Above ρcrit: the system is gravitationally coherent, classical behavior dominates.
Connection to MOND
From ρcrit = A × Vflat² and the coherence function, two cosmological results emerge:
MOND's a₀
13% error vs Milgrom's value (at site-standard H₀ = 67.4)
Reparametrization — dimensional bookkeeping — prior artFreeman's Σ₀
≈4% error vs Freeman's value (corrected 2026-07-09; not independent of a₀'s ≈13% gap — Σ₀ = a₀/2πG is the same number, not a second success)
Reparametrization — same number as a₀