The Coherence Function

Speculative — Phenomenological Ansatz — tanh motivated, not derived

Before the math: coherence measures how collectively particles behave. Ice cubes have high coherence (rigid crystal lattice). Steam has low coherence (random motion). This function quantifies that spectrum for any system, at any scale.

C(ρ) = tanh(γ · ln(ρ/ρ_crit + 1))

The Synchronism coherence function

Symbols in this equation.

C: coherence, the output, between 0 (parts act independently) and 1 (parts act as one).
ρ: the system's density (for a galaxy, mass per cubic parsec at a given radius).
ρ_crit: a reference density where the curve bends into saturation, a knee rather than a critical point (reference density).
γ: sets how fast C rises at low density (C ≈ γρ/ρ_crit) and how fast it saturates.
ln: natural logarithm, which compresses huge ranges (1,000× more input adds only about 7). tanh: the hyperbolic tangent, an S-shaped “dimmer switch” that keeps C below 1.
MOND: Modified Newtonian Dynamics (Milgrom 1983), the rival that changes gravity at low acceleration instead of adding dark matter. Its interpolating function μ sets how gravity blends between the Newtonian and modified regimes.
Hill function: the saturating curve xⁿ/(xⁿ + K), familiar from biochemistry; “Hill n = 1” is x/(x + K).

Walk through it step by step → (a 6-step interactive with the same symbols in plain language).

The most clarifying algebraic fact about this equation (moved here 2026-07-18 — it previously lived only on the walkthrough): the tanh∘ln composition is exactly a Hill-type rational function of (1+x)^2γ— a Hill function in x itself only at γ = ½, where it is x/(x+2). (That is why the form-selection table below lists “Hill n=1” and “tanh-log” as separate rows; wording corrected 2026-09-06 from “a two-parameter Hill function,” caught by a graduate-physics persona.) With x = ρ/ρ_crit, tanh(γ·ln(1+x)) = [(1+x)^2γ − 1] / [(1+x)^2γ + 1] — an algebraic identity, verified numerically (the ln is not an independent ingredient, but it is load-bearing: it gives the power-law tail 1−C ≈ 2x^−2γ that makes γ = ½ equal MOND's simple μ; tanh(γx) would have an exponential tail. Until 2026-11-07 this said the composition was not load-bearing). Slope behavior follows directly: dC/dx = γ(1−C²)/(1+x) is maximal at x = 0 (where it equals γ) and decreases monotonically — the curve is concave in x everywhere on x ≥ 0, with no critical point and no inflection to serve as a “transition.” Asymptotics: C ≈ γx for small x; C → 1 like 1 − 2(1+x)^−2γ for large x. This dissolves any residual mystique: the equation is a saturating rational function in the μ-law/Hill/Naka–Rushton lineage. Full six-piece decomposition on the walkthrough.

The identity the Hill form above was one substitution short of (added 2026-08-02): set γ = 1/2 in the identity above. Then 2γ = 1 and it collapses to C = [(1+x)−1]/[(1+x)+1] = x/(x+2) = μ_simple(x/2) — MOND's simple interpolating function μ_simple(u) = u/(1+u), identically, for all x, not asymptotically. γ = 1/2 is the exact algebraic MOND point of this function family. The free-γ SPARC fit in the table below independently lands at γ = 0.489 — 2.2% from that point. This is the executed form-selection table (below) reporting the same parameter two ways: its independently-fitted Hill exponent n = 0.975 and this page's 2γ = 0.978 agree to 0.3% because they are the same quantity, and the table's best-scoring simple form is Hill with n = 1 exactly — again, MOND. Combined with f_DM = 1−C on Tier 1 (so C is the interpolating function μ by definition), the galaxy sector reduces to: MOND, with μ's argument swapped from the acceleration it is keyed on in MOND to local density ρ. That single substitution is the entire difference from MOND — and it is the same substitution behind the local-density no-go on For Researchers.Which variable the fit actually used (verified in the script 2026-09-08, after two simulated-visitor personas showed the two readings give different force laws): the SPARC fit that returns γ = 0.489 and the form-selection table use C as an implicit μ keyed on g_obs — it solves g_bar = g_obs·tanh(γ ln(1 + g_obs/a₀′)) for g_obs (simulations/sparc_tanhlog_profile.py, line 85, research repo), with a₀′ profiled. At γ = ½ that is Milgrom's simple μ with a₀ = 2a₀′, which is why the profiled a₀′ = 5.33×10⁻¹¹ sits 2.25× below McGaugh's reference 1.20×10⁻¹⁰; converted, a₀ = 2a₀′ = 1.066×10⁻¹⁰ is 11% below it. At the fitted γ = 0.489 the exact deep-regime conversion is a₀′/γ = 1.091×10⁻¹⁰. That combination stays within 1.6% across Υ_disk = 0.4–0.6 while a₀′ moves 3.6×, so it, not a₀′, is what the fit measures (added 2026-10-04; maintainer/scripts/a0_identified_scale_across_upsilon.py). The 2026-08-14 “factor-2 a₀ tension that dissolves at Υ = 0.6” compared a₀′ with Milgrom's a₀; it dissolved there only because γ̂ ≈ 0.96 ≈ 1. The explicit reading — evaluating C on g_bar/a₀′ and dividing — gives g_obs → g_bar + 2a₀′ at γ = ½, a constant additive floor 0.8–1.8 dex above the RAR below 10⁻¹¹ m/s², and could never reach 0.1437 dex RMS. It was never run. This sentence previously read “μ's argument swapped from g_bar to ρ,” which was wrong twice: μ's argument in MOND is g_obs, and the fit swapped nothing — the swap to ρ is the framework's stated law, tested separately in the box below.

How C enters dynamics — in one place (added 2026-09-11). C(ρ) on its own predicts nothing; it has to be coupled to gravity. A simulated-visitor graduate-physics persona found that no equation page said how, so two of the six refutations could not be reconstructed from them. Four things differ from the headline equation above:

The floor. Some galaxy tests use a floored C, with a minimum value f = Ω_m = 0.315. The unfloored headline form goes to 0 as ρ → 0, which makes any 1/C boost unbounded; the floor is what creates the boost ceiling B ≤ 1/f = 3.17. It is asserted, not derived. Which tests use which floored function is in the table below this list. (Until 2026-09-23 this item said “galaxy tests use” the floored density-keyed form. The two ceiling kills did not: they evaluated an acceleration-keyed C with no tanh and no ρ. A simulated-visitor graduate-physics persona caught the contradiction with item 4.)
The coupling: two live forms. (a) Algebraic g_obs = g_bar/C, the reading behind f_DM = 1 − C and the ledger. (b) The field equation ∇·[C(ρ)∇Φ] = 4πGρ, which reduces to (a) only in spherical symmetry. In a thin disc the ∇C·∇Φ term refracts field lines and the two differ by up to 5.9× (bounded by the ceiling) once the knee sits inside the disc; at this framework's own knee, which never enters a disc, they coincide. Form (b) is Refracted Gravity's field equation (Matsakos & Diaferio 2016). RG's permittivity is a floored Hill switch, ε = ε₀ + (1−ε₀)·y^p/(1+y^p) with y = ρ/ρ_c. That is identical in closed form to the archive's floored Hill form C_Ω, butnot to the headline tanh(γ·ln(1+x)) on this page. The two coincide only at γ = ½ (Hill p = 1, knee doubled, ρ_c = 2ρ_crit). At γ = 2 and x = 1 the headline form gives 0.88 and RG's shape gives 0.06 (p = 4) or 0.5 (knee not doubled); at small x one is linear and the other goes as x^p. So “same field equation” holds, and “same permittivity” holds only at γ = ½. (Corrected 2026-10-14. The sentence said “identical to C_Ω in closed form” without saying C_Ω is not the headline form, and a simulated-visitor graduate persona checked it against the headline form.) The Galaxy Curve Plotter's v² = v_b² + (V_flat·C)² is a display branch with the opposite orientation (small C = no boost) and says so on its page.
The argument: two live forms. The stated law keys C on local density ρ. Several fits key the same compander on acceleration g/a₀, which is MOND's variable.
Which result used which. TEST-09 (BTFR slope) and TEST-10 (dark-matter fraction): floored, acceleration-keyed C(a) = Ω_m + (1−Ω_m)x/(1+x) under (a). Both keyings carry the same ceiling, which is why the refutation transfers to C(ρ). That transfer is a galaxy-scale statement. In the Solar System the floored density-keyed form is a separate problem: read pointwise, the interplanetary medium (solar wind) makes GM☉ inferred from different planets differ at O(1), so the law needs a declared smoothing length before it says anything there (PREDICTIONS.md, “SCOPE CONDITION” block; script maintainer/scripts/density_keyed_law_vs_interplanetary_medium.py). ΔBIC +184 (Υ frozen; +46 with Υ free per law): acceleration-keyed, γ = 2 (2,807 points treated as independent; ≈ +11 with the galaxy as the unit, 1.6–2.2σ). ΔBIC +2843: density-keyed, γ free. The 2026-09-08/09 SPARC knee grid: density-keyed, floored, form (b) solved in disc geometry. Globular clusters: density-keyed under (a) on spherical Jeans models, where (a) and (b) agree. Not in any of them: the striction force that a variational version of (b) adds (see For Researchers). At the framework's own knee (0.029 V²) the radial striction is ~10⁻⁵ of gravity, so these verdicts stand under that variational version too (first order, fixed disc; explorer 2026-10-13). Knees placed inside discs are a different case: there the striction exceeds gravity and the disc's equilibrium is not yet solved.

Which C, in one table (added 2026-09-23)

Name	Formula	Keyed on	Results that used it
C_ρ (the headline)	tanh(γ ln(1 + ρ/ρ_crit)), optionally floored at f	local density ρ	ΔBIC +2843 (ρ_crit free) and +3309 (ρ_crit = A·V_flat², A free) head-to-head on SPARC; TEST-03s environment null; SPARC knee grid (floored); globular clusters; Ġ/G from the Sun's motion vs lunar laser ranging (explorer 2026-09-23: at 1 pc smoothing, 68 of 74 grid laws excluded, 62 at 10 pc; survivors are locally Newtonian; holds under L2 and L3); under L3 only, at smoothing ≤ 0.3 pc, planetary perihelia via the striction force sourced by the Sun's self-energy (explorer 2026-10-19: closes that window for 21 of 32 γ = 0.489 laws). In cosmology the same formula is read on the mean matter density: γ = 0.487 ± 0.02 on DESI (that is Λ); read on the local density as a fluid, P(k) pins γ = ½ to 5×10⁻⁶
C_g	tanh(γ ln(1 + g_obs/a₀′)), solved implicitly	observed acceleration (MOND's variable)	γ = 0.489 fit; ΔBIC +184 at γ = 2 (RAR shape, Υ frozen; +46 with Υ free per law; ≈ +11 with the galaxy as the unit); form-selection table; Cassini (TEST-25)
C_a (the ceiling)	Ω_m + (1−Ω_m)·x/(1+x), x = (g_bar/a₀)^1/φ; no tanh, no γ	baryonic acceleration	TEST-09 (BTFR slope) and TEST-10 (dark-matter fraction), the boost-ceiling kills

Which γ. The symbol carries different numbers in different rows, and they are not one parameter measured several times. γ ≈ 0.489 is the C_g fit (acceleration-keyed, and only γ/a₀′ is identified). The density-keyed C_ρ fitted on SPARC goes to γ → 0.046, and γ = 2 is its registered value. The cosmology's γ ≈ ½ is C_ρ on the mean density. γ = 2/√N_corr is the proposed derivation that failed (wrong-sign correlation), and “γ ≈ 1” is a regime name from the chemistry track. γ = ½ turns up in both the galaxy and cosmology fits because it is the special point of the function family (MOND's simple μ in one variable, Λ in the other). The variables differ, so that is a coincidence of notation, not a cross-sector measurement.

The knees do not match either. At γ = ½ the cosmology sector fixes ρ_crit = ρ_Λ/2 ≈ 4×10⁻⁸ M_☉/pc³ (ρ_Λ = 0.685 × 1.26×10⁻⁷ M_☉/pc³), while the Galaxy Plotter's DDO 154 uses ρ_crit = 0.029 × 47² ≈ 64 M_☉/pc³ — about 10⁹ apart. With the cosmic knee, every galaxy disc (ρ ~ 10⁻²–1 M_☉/pc³) sits at x = ρ/ρ_crit ~ 10⁵–10⁷, where C ≈ 1: under the division wiring (g_bar/C, the one the cosmology uses) that is plain Newtonian gravity, no missing mass explained. So the two sectors share the function's form only, not its knee. Can one ρ_crit serve both? On these numbers, no. (Credit: a graduate-physics simulated-visitor persona, 2026-09-25.)

The wiring (how C enters gravity: division g_bar/C, the field equation, or the plotter's quadrature display) is a separate choice on top of each row; see item 2. The headline equation at the top of this page is C_ρ. Its own head-to-head result is the +2843 / +3309 pair, not the +184 (Υ frozen).

Most galaxy numbers on this site do not use the substitution above. (This heading read “has never itself been evaluated (2026-08-04)” until 2026-10-08, when a graduate-physics visitor noticed the page itself reports the result. The density-keyed form has since been run head-to-head on SPARC: free γ goes to 0.046 and it loses to the acceleration-keyed form by ΔBIC +2843.) every quantitative galaxy result on this site — γ = 0.489, the ΔBIC form-selection table, the Cassini/SPARC squeeze — was computed on acceleration (g_obs, μ-form), not ρ (see the 2026-09-08 “Which variable the fit actually used” note above; this sentence previously said “g_bar,” which the fit did not use). Running the framework's own numbers (γ = 2, ρ_crit = 0.029 V_flat²) with ρ actually substituted in moves the predicted rotation velocity by 2–5 orders of magnitude on the site's own five plotter galaxies, and three mutually exclusive galaxy force laws are live on the site depending on how C(ρ)·g vs g/C(ρ) is read (see the C-convention note on Galaxy Rotation). Parameter-free — no calibration of A, γ, or ρ_crit reconciles it. Full derivation: explorer/findings/efe-zero-survives-momentum-objection-but-the-substitution-was-never-evaluated.md.

One caveat: for ρ ≪ ρ_crit (real galaxy outskirts), C ≈ γx to first order (it equals γx/(1+γx) only through second order at γ = ½), and so to first order depends on γ and ρ_crit only through their ratio. The next term, C = γx − γx²/2 + …, separates them in principle at any γ, but only where the data reach x of order 0.1 or more; SPARC discs sit near x ≈ 10⁻⁴, where only the ratio is seen. (Until 2026-11-05 this sentence called γ(2γ−1) the degeneracy-breaking term. It is not: γ(2γ−1)x²/2 is how far the shape departs from γx/(1+γx), MOND's simple form, and it vanishes at γ = ½. A graduate simulated-visitor persona caught it.) So γ ≈ 0.489 is not a clean measurement of γ alone without also stating the ρ_crit (or, for the acceleration-keyed fit, a₀′) it was fit with.

The first question a physicist asks, answered where the equation is defined (moved here 2026-07-18): ρ has no stated units and no measurement protocol, and no protocol maps any laboratory or astronomical observable to the coherence value C, in any domain. N_corr (inside γ) is asserted rather than counted on every scale where it is used. C is therefore doubly unanchored — inputs asserted, output unmeasurable — which is why several claims on this site are labeled unrunnable as stated rather than merely untested. The only data contact is galaxy rotation, where the parameters are fit to the prediction target. Details on For Researchers.

This maps presence to coherence(a dimensionless number between 0 and 1 that measures how collectively elements behave — from sparse/independent (C ≈ 0) to dense/collective (C ≈ 1)).

Expert note (physicist terminology)

“Coherence” here means density-driven classicality/collectivity, not quantum phase coherence. BCS superconductors and BECs land at C ≈ 0 at all physical densities because their large N_corr → tiny γ → flat tanh curve — not because they lack macroscopic phase coherence. See the glossary warning.

Inputs and Outputs

Input: presence — compatible structural elements within the system's MRH. Physical density is one form of presence.

Output: coherence (0 = sparse/independent, 1 = dense/collective)

γ, ρ_crit

Parameters: γ, proposed as a transition sharpness; it sets the initial slope and the return exponent, and no value is sharp (half the Hill exponent: n = 2γ, since tanh(γ ln u) = (u^2γ−1)/(u^2γ+1); labelled “Hill coefficient” until 2026-09-11 and “coupling strength” before 2026-07-27), saturation knee (reference density — not a critical point; C(ρ_crit,γ=2)≈0.88)

Why This Specific Function?

1. The Compression Requirement

The physical state of any system lives in a high-dimensional space: magnitude, direction, temporal structure, spatial correlations, interference patterns. But the quantum/classical distinction is binary. You need a function that compresses high-dimensional information into a bounded scalar. This is an information-theoretically necessary compression (Session #67).

2. Why tanh?

The compression function must satisfy four properties:

Bounded [0, 1]: Coherence can't exceed unity or go negative
Monotonic: Higher presence → higher coherence (no paradoxical inversions)
Smooth saturation: Gradual approach to limits (enables field equations)
Handles extremes: ρ → 0 gives C → 0, ρ → ∞ gives C → 1

The site's own flagship chemistry result contradicts this axiom (explorer finding, 2026-07-29): C(ρ) as written is monotone increasing in density for any γ, ρ_crit > 0. But Sound Velocity — the site's strongest single chemistry correlation — ranks diamond (ρ = 3.51 g/cm³) as more coherent than lead (ρ = 11.34 g/cm³): higher density, lower coherence. Computed directly, Spearman(C(ρ), sound velocity) = −0.32 for every parameter choice tested — the opposite sign from what this monotonicity property requires. Either the chemistry sector uses a different (inverse) orientation of C that this page does not document, or this axiom is violated by the framework's own strongest empirical claim. Not yet resolved.

Several compander functions satisfy the four qualitative constraints — but as of 2026-07-22 the family has been put to data, and the constraints turn out to be incomplete. Fitting ten forms to the real SPARC radial-acceleration relation (2,807 points), SPARC refutes three members that pass all four qualitative constraints — arctan-log (ΔBIC +46.7), algebraic-log (+23.8), Gompertz (+58.0) — on asymptotic-rate grounds the qualitative list never mentions. Among the survivors (tanh, erf-log, Hill, exponential-ν) the data cannot distinguish (mutual ΔBIC ≤ 8.9). tanh with γ free finishes last of the four, but the same curve at γ = ½ is the table's “Hill n = 1 pinned” row, second best: the shape is not disfavoured, and freeing γ adds a parameter the data do not use. Effective-N caveat (added 2026-09-24): every ΔBIC in this table treats the 2,807 points as independent. Elsewhere the site discounts the same SPARC points for correlation along each curve, by about 5.6× (/galaxy-rotation, +184 → 33) or 20× (/core-idea, +2843 → +142). Divided by either factor, all three “refuted” forms (+23.8, +46.7, +58.0) fall near or below the decisive line of 10, and the survivors' ranking is noise. So the claim that asymptotic rate selects among forms is conditional on point independence. Measured 2026-09-29: a galaxy-block bootstrap of the +184 comparison has a spread 4.3× the point-independent expectation, i.e. an effective N of about 150, one per galaxy; at that N the +184 becomes +10.9 and this table's three “refuted” forms fall to +1.4, +2.8 and +3.4. The form-selection verdict does not survive the measured N_eff; only the +2843 density-vs-acceleration comparison does (+168). (“Logistic” is not a distinct member: the logistic sigmoid applied to the log-argument is identically tanh.) See the executed selection table below. tanh is the chosen form, motivated historically by its appearance in the Ising mean-field self-consistency equation m = tanh(βJzm) — the same shape. However, Synchronism's C(ρ) is not the Ising equation. The Ising result is a self-consistency loop where m appears on both sides. C(ρ) evaluates directly with no feedback loop — ρ goes in, C comes out. The tanh shape is motivated by the Ising analogy, not derived from it; it is a phenomenological choice from the compander family, not the uniquely forced form — confirmed by execution: not uniquely forced, and not even the family's best member, though inside the indistinguishability band.Empirical caveat: Landau-universality critical exponents (β, ν, α, δ, η) are off by ~2× in practice — the Landau analogy is motivational, not an accurate prediction of universality class. See Honest Assessment. See Parameter Derivations for the complete derivation vs. motivation distinction.

Saturation note: At γ = 2, C(ρ) saturates within ~1 decade of ρ_crit(C(10·ρ_crit) ≈ 0.9999). The coherence transition is a sharp but smooth saturation (a companding curve) — analytic, with no non-analyticity or critical exponents, so “phase transition” is a loose description, not a literal one. Each system has its own ρ_crit, so what is universal is the form of the crossover, not its location.

Form selection, executed (2026-07-22). Ten compander forms fit to the real SPARC RAR (2,807 points, same pipeline that produced the γ=2 kill, reproduced at +184.0 as a sanity check). Verdict rule pre-fixed before running (“privileged” = beat every same-parameter-count member by ΔBIC > 10). What x is in this table: for every compander row (tanh-log, erf-log, arctan-log, algebraic-log, Hill, Gompertz) x = g_obs/a₀′, in μ-form: the fit solves g_bar = g_obs·μ(g_obs/a₀′) for g_obs with a₀′ profiled; only the McGaugh ν and ν-δ rows take g_bar/a₀ (ν-form, their native form). The fit is keyed on acceleration, MOND's own variable, not on ρ/ρ_crit. The distinction is not cosmetic: the same function keyed on g_bar in the division wiring g_obs = g_bar/C at γ = ½ gives g_obs = g_bar + 2a₀ — a constant added acceleration with rising rotation curves, not MOND. The MOND equivalence below holds only in the g_obs (μ-form) reading. (This box said x = g_bar/a₀ from 2026-09-06 to 2026-09-15, contradicting the solver line on this page; a graduate-physics visitor caught it. Checked against explorer/scripts/compander_family_aic_bic_real_sparc.py, which inverts g_bar = g_obs·μ(g_obs/a₀). The TEST-09/10 ceiling scripts are a different realization again: floored C keyed on g_bar, with a 1/φ exponent.) The density-keyed form never leaves its linear regime in any SPARC disk and loses head-to-head at ΔBIC +2843 (archive 2026-08-24). Every number below is a statement about the acceleration-keyed branch; two simulated-visitor personas read this table as the density law and could not reconcile it with the straight line the Coherence Explorer shows — they were reading two models under one symbol.

Form	Best shape	RMS (dex)	ΔBIC vs McGaugh ν
generalized ν-δ	δ=0.44	0.1435	−1.8
Hill n=1 (MOND “simple” μ)	pinned	0.1437	−0.7
erf-log, free γ	γ=0.39	0.1435	+1.5
Hill, free n	n=0.975	0.1437	+5.9
tanh-log, free γ (this site's form)	γ=0.489	0.1437	+7.1
algebraic-log	γ=0.66	0.1441	+23.8 (points independent; +1.4 at N_eff ≈ 150: not refuted)
arctan-log	γ=2.15	0.1447	+46.7 (points independent; +2.8 at N_eff ≈ 150: not refuted)
Gompertz	n=0.61	0.1450	+58.0 (points independent; +3.4 at N_eff ≈ 150: not refuted)
tanh-log, γ=2 pinned (the framework's asserted value)	pinned	0.1485	+184.0 (points independent; +10.9 at N_eff ≈ 150, at the decisive line)

A consistency check the table passes without saying so (added 2026-09-06): free-γ tanh-log (+7.1) and Hill n=1 (−0.7) are the same curve at γ ≈ ½, so their BIC gap should be exactly the one-parameter penalty ln N = ln 2807 = 7.94. The gap is 7.8. That is the cleanest single demonstration on this page that γ adds nothing to MOND's μ. What the data actually selects is the pair of asymptotic rates — deep-MOND slope μ ∝ y, and a return to Newton at least power-law fast — not any particular sigmoid. The three refuted members fail at the edges (Gompertz vanishes too fast at low y; arctan-log and algebraic-log return to Newton only as powers of 1/ln y). Between any two members with acceptable asymptotics, SPARC is silent. So the honest statement is two-sided: tanh carries no statistical content (form-selection null), and the family is not degenerate either — the data discriminates, just not in tanh's favor. Read against the identity above: the top-scoring simple form is Hill n=1, which is γ=1/2 — the data is not neutral about tanh, it prefers the point where tanh becomes MOND's μ_simple. Full table and scripts: explorer/findings/compander-family-selection-executed-tanh-not-privileged.md.

3. Why log?

Presence spans enormous ranges — in the astrophysical case, density alone covers 80+ orders of magnitude (interstellar gas at ~10⁻²⁴ g/cm³ to neutron stars at ~10¹⁴ g/cm³). The logarithm maps this range into something tanh can differentiate between. The “+1” inside the log prevents divergence at ρ = 0.

What C = 0 and C = 1 Mean

C → 0: Sparse/Independent

Low density or flat tanh (large N_corr). Elements act independently — sparse gases, dilute quantum systems, quantum computing platforms. Note: BCS superconductors also land here due to tiny γ (≈6×10⁻⁴), not because they are incoherent — they are maximally quantum-phase-coherent, but C measures density-collectivity, not quantum phase coherence. Warning: do not read C≈0 as "quantum" or "wave-like" — C is classicality in the density/collective sense, the opposite of the quantum-coherence axis.

C → 1: Dense/Collective

High density relative to ρ_crit, moderate N_corr. Elements act together as a collective — dense matter, crystal lattices, everyday macroscopic physics. Newton's laws work reliably here. Galaxy dynamics (N_corr≈1, γ=2) lives in this regime. Note: high C does not mean "classical particle" in the quantum-decoherence sense — it means dense/collective in the density axis this equation measures.

Relationship to C = f(γ, D, S)

The consciousness and measurement pages of this site use a second form of coherence: C = f(γ, D, S) where D is decoherence and S is self-modeling. Both are called “coherence” (C) but they are not obviously the same observable. The relationship — whether there is a function ρ = g(γ, D, S) that reduces the parametric form to the density-based form — has not been derived. If the reduction exists, the “one equation” framing is vindicated. If it does not, C(ρ) and C = f(γ, D, S) are two different observables sharing a symbol. See research proposal dual_C_symbol_ambiguity_and_bridge_derivation.md.

What Class of Function Is C(ρ)?

C(ρ) is a compander — a logarithmic companding function, not an order parameter. This is the settled self-identification used on the Parameter Derivations page and in the Honest Assessment.

The compander class includes: μ-law audio compression (telephone networks), Hill function (oxygen binding to hemoglobin), Naka-Rushton equation (retinal response), Kubo susceptibility (stat-mech near criticality). All map a wide dynamic range onto a bounded output using a sigmoidal curve. C(ρ) does the same: density spanning 80+ orders of magnitude → coherence in [0, 1].

Key implication: Companders are purely evaluative (input → output). They cannot encode universality classes, critical exponents, or spontaneous symmetry breaking — those require a self-consistency loop. The Ising mean-field equation m = tanh(βJzm) is in a different structural class: it is implicit (m appears on both sides) and supports a phase transition. C(ρ) evaluates C from ρ directly with no feedback. These are different objects. The Landau analogy is motivational, not a claim of class membership.

Scope Limit: Universality Classes

C(ρ) cannot encode universality-class structure. Real phase transitions are classified by (spatial dimension d, order-parameter dimension n, symmetry group) — these determine whether a system belongs to the 3D-Ising universality class (α = 0.110), 3D-XY (α = −0.015), or mean-field (α = 0). C(ρ) has no inputs encoding d, n, or symmetry. It therefore cannot reproduce critical exponents or distinguish universality classes — a one-parameter scalar coherence function is structurally blind to this information.

This is a scope limit, not a parameter-tuning failure. C(ρ) describes sigmoidal interpolation between coherent and decoherent regimes. It does not claim to predict critical exponents, universality-class membership, or quantitative crossover structure. The failures documented in Honest Assessment (critical exponents 2× off, T_c 6.5× wrong) are consequences of applying the function outside this scope, not evidence that the function itself is wrong within scope.

Derivation History

Session #64-65: γ = 2.0 motivated by 6D phase space degrees of freedom — see Parameter Derivations for why “motivated, not derived” is the correct framing
Session #66: tanh form motivated by the Ising self-consistency equation (tanh appears in the fixed-point, not as a derivation from it)
Session #67: Information-theoretic compression argument
Session #87-91: Connected to cosmological parameters (a₀, Σ₀)

Next on Intermediate Path: The γ Parameter →Next on Physics Track: MRH: Markov Relevancy Horizon →Try It: Coherence Explorer

Related Concepts

The γ Parameterγ = 2/√N_corr — the proposed derivation and why it did not survive testing Reference Density (ρ_crit)ρ_crit = A V_flat²: the saturation knee of C(ρ), a reference scale, not a transition point γ Regimes: Crossovers, Not Phase Transitionsγ > 1.4, 0.6–1.4 (≈ 1), < 0.6 regimes — smooth crossovers in C(ρ), not phase transitions