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/ρcrit + 1))
The Synchronism coherence function
New to the notation? Walk through it step by step → — a 6-step interactive that defines every symbol (C, ρ, ρcrit, γ, ln, tanh) 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 two-parameter Hill function. With x = ρ/ρcrit, tanh(γ·ln(1+x)) = [(1+x)2γ − 1] / [(1+x)2γ + 1] — an algebraic identity, verified numerically (the ln is cosmetic; nothing about the composition is 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) = μsimple(x/2) — MOND's simple interpolating function μsimple(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 fDM = 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 enclosed-mass acceleration gbar 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.
The substitution above has never itself been evaluated (2026-08-04): every quantitative galaxy result on this site — γ = 0.489, the ΔBIC form-selection table, the Cassini/SPARC squeeze — was computed with the argument gbar, not ρ. Running the framework's own numbers (γ = 2, ρcrit = 0.029 Vflat²) 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 ρcrit reconciles it. Full derivation: explorer/findings/efe-zero-survives-momentum-objection-but-the-substitution-was-never-evaluated.md.
One caveat: for ρ ≪ ρcrit (real galaxy outskirts), C ≈ γx/(1+γx) depends on γ and ρcrit only through their ratio — the degeneracy-breaking term is γ(2γ−1), which vanishes at exactly γ=1/2. So near the fitted value, γ ≈ 0.489 is not a clean measurement of γ alone without also stating the ρcrit prior it was fit under.
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. Ncorr (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 Ncorr → tiny γ → flat tanh curve — not because they lack macroscopic phase coherence. See the glossary warning.

Inputs and Outputs

ρ
Input: presence — compatible structural elements within the system's MRH. Physical density is one form of presence.
C
Output: coherence (0 = sparse/independent, 1 = dense/collective)
γ, ρcrit
Parameters: transition sharpness (Hill coefficient — corrected 2026-07-27 from “coupling strength,” the wrong kind of quantity), saturation knee (reference density — not a critical point; C(ρcrit,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:

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, ρcrit > 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), and tanh finishes last of the four. (“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 ρcrit(C(10·ρcrit) ≈ 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 ρcrit, 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):
FormBest shapeRMS (dex)ΔBIC vs McGaugh ν
generalized ν-δδ=0.440.1435−1.8
Hill n=1 (MOND “simple” μ)pinned0.1437−0.7
erf-log, free γγ=0.390.1435+1.5
Hill, free nn=0.9750.1437+5.9
tanh-log, free γ (this site's form)γ=0.4890.1437+7.1
algebraic-logγ=0.660.1441+23.8 (refuted)
arctan-logγ=2.150.1447+46.7 (refuted)
Gompertzn=0.610.1450+58.0 (refuted)
tanh-log, γ=2 pinned (the framework's asserted value)pinned0.1485+184.0 (refuted)
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 μsimple. 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 Ncorr). 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 ρcrit, moderate Ncorr. Elements act together as a collective — dense matter, crystal lattices, everyday macroscopic physics. Newton's laws work reliably here. Galaxy dynamics (Ncorr≈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, Tc 6.5× wrong) are consequences of applying the function outside this scope, not evidence that the function itself is wrong within scope.

Derivation History

Next: The γ Parameter →Try It: Coherence Explorer

Related Concepts

The γ Parameterγ = 2/√N_corr: why 2, why √NCritical Densityρ_crit = A V_flat²: the transition pointPhase Transitionsγ < 1, γ ≈ 1, γ > 1 regimes