The Core Idea
One function, proposed for every scale from Planck to cosmic — what it says, where it was tested, and where it broke.
Failed contradicted by data · Reparametrization existing physics in new notation (“Null-Class”: a simpler model without the framework does as well) · Audited-Negative an internal check found the claim wrong; kept on record · Speculative no test exists · Untested a test exists but hasn't been run
Full definitions: badge definitions on Honest Assessment; terms are defined in the Glossary.
Think of it as a dimmer switch from sparse/independent to dense/collective. Feed it the local density of a system, and it outputs a number between 0 (sparse/independent) and 1 (dense/collective). The proposal was that the same switch, on the same scale, works for a single atom and a galaxy cluster — that is what was tested, and where it broke is below.
What does coherence do to gravity? (added 2026-09-14 — a reader finished the beginner path without learning this.) In galaxies, the proposal was that gravity looks stronger than the visible matter explains by a factor set by C. Two wirings were tried, and they point opposite ways. In the one the site's galaxy refutations used, the boost is 1/C: gravity looks strongest where coherence is low(the sparse outskirts), capped at about 3.2×. In the other (“quadrature”), the extra speed grows with C, so it switches on in dense centres — where it isn't needed — and fails outright. The 1/C wiring fits only after its input is switched from density to acceleration and its dial is fitted, at which point it is MOND; with the cap it fails the most dark-matter-dominated galaxies. Both are on the Galaxy Curve Plotter.
tanh, in one picture. tanh is just an S-shaped curve — flat, then steep, then flat again. Nothing happens at low density, a small change flips the behavior in the middle, and past the knee more density changes almost nothing. That's the whole “dimmer switch.”
Drawn with γ = 2 (the galaxy-scale pin — refuted on SPARC data — SPARC is a set of 175 galaxies with carefully measured rotation speeds; see the caveat below) on a logarithmic density axis. Note the two honest markers: the curve's true midpoint (C = 0.5) sits near 0.32·ρcrit, and at ρcrit itself C ≈ 0.88 — ρcrit is a saturation knee, not a critical point. Both markers move with γ, so neither number means anything without it: C(ρcrit) = tanh(γ·ln 2), which is 0.88 at γ = 2 but only 0.33 at the SPARC-fitted γ ≈ 0.49 — below the halfway point rather than near saturation. The Coherence Explorer defaults to γ = 0.49 and reads 0.327; this figure is drawn at the refuted pin. (Flagged 2026-09-18 by a visitor who found the two figures 2.7× apart with neither stating its γ.)
Why These Specific Choices?
Why tanh?
tanh is an S-shaped curve that smoothly transitions from 0 to 1 — think of it as a dimmer switch from sparse/independent to dense/collective. The function must be bounded [0, 1], monotonic, and smooth. tanh's form recalls the Ising model (the textbook model of a magnet, where each spin lines up with its neighbours) self-consistency equation m = tanh(βJzm), but the resemblance is only visual: C(ρ) has no feedback loop (ρ goes in, C comes out), so the Ising analogy carries no motivating force. Other sigmoids (logistic, error function, Hill) satisfy the same four constraints. tanh is a phenomenological choice, not a derived result. Note: C(ρ) is real-analytic everywhere on its domain — no singularity, no free-energy functional, and no length scale anywhere in the equation — so it has no critical point and therefore no critical exponents (β, ν) to compare. Asking for them is a category error, not a near-miss with Landau theory (the standard theory of how phase transitions behave near their critical point); the analogy is motivational only.
Need an analogy first? · Why tanh? (motivation, not derivation) →
Why γ = 2/√Ncorr?
The 1/√Ncorr dependence resembles central-limit-theorem scaling (fluctuations ~ 1/√N), which is generic statistics for correlated ensembles. Ncorr (how many particles move together as one unit) was meant to be the physically measurable input. The factor of 2 is motivated by phase-space arguments (6D contracted to 3 effective) but should be understood as a motivated ansatz rather than a rigorous derivation.
So what sets γ physically? Honestly: nothing yet. No protocol exists to measure Ncorr in any system independently of fitting γ to the same data — Ncorr is back-solved from the fitted γ, not counted — so the relation predicts nothing on its own, and where it is applied it points the wrong way for collective systems. It is badged audited-negative; see the γ Calculator. (Until 2026-09-15 this paragraph called Ncorr “the physically measurable quantity,” contradicting the calculator.)
Why 2/√Ncorr? (motivation, not derivation) →
Why log? (Honest answer: it's cosmetic)
The traditional motivation was compression: density spans 80+ orders of magnitude (from interstellar gas at 10−24 g/cm³ to neutron stars at 1014 g/cm³), and the logarithm squeezes that range into something the tanh can work with. But an exact identity retires that story. With x = ρ/ρcrit:
tanh(γ·ln(1+x)) ≡ [(1+x)2γ − 1] / [(1+x)2γ + 1] — exactly.
The equation is an ordinary rational (Hill-type) saturation function of (1+x)2γ; the tanh∘ln decomposition is notation, not mechanism. Any saturating rational function “spans 80 orders of magnitude” for free — the ln does no work that the exponent 2γ doesn't undo. So the Hill functions and Naka–Rushton curves named below (standard saturation curves from biochemistry and vision science; see compander, a curve that squeezes a huge input range into a fixed output range) are not merely similar alternatives; the flagship equation is one of them, exactly. (Identity stated 2026-07-10. Provenance: derived independently by two successive internal review passes, 2026-07-09 and 2026-07-10 — the audit machinery caught it; the identity then took a day to reach this page. No external derivation is on record.)
Precision (2026-09-14): “Hill-type” is exact in the variable (1+x). In x = ρ/ρcrititself, C is exactly a Hill function only at γ = ½, where C = x/(x+2); at other γ the Hill index n = 2γ is the large-x slope, and near x = 0 the curve is simply linear (C ≈ γx).
What It Predicts
Two dials, one vocabulary — don't mix them up: C near 1 means collective; γ runs the other way (small γ = many correlated particles = a gentle switch).
γ « 1: Collective Regime
Many correlated particles (large Ncorr → small γ = 2/√Ncorr). BECs, superconductors. Despite being quantum systems in the real-world sense, these score C ≈ 0 here — the S-curve is so flat that typical densities leave C near zero. This is why “coherence” in Synchronism is not quantum coherence.
γ ≈ 1: The Boundary
Phase transitions, chemistry, catalysis, biology. Where the S-curve has intermediate steepness. 1,703 phenomena cluster here at 89% boundary-consistency rate.
Null model run (2026-05-10): a plain 2-parameter polynomial in atomic number matches these correlations (|Δr| ≤ 0.07) and sometimes beats them, so they reflect density-monotonic chemistry, not anything specific to this framework (null-class). (This card said “pending null model” until 2026-09-15 — stale; Honest Assessment has carried the result since May.)
γ » 1: Independent Regime
Few correlated particles (Ncorr ≈ 1 → large γ). Ideal gas, independent stars in a galaxy. Steepest S-curve — C rises quickly with density. Galaxy dynamics is here (stars treated as independent, Ncorr=1, γ=2).
The data say the same thing, independently. The framework's whole distinctive claim is that density drives a collective transition. Tested in the density variable itself, the data reject that: fit head-to-head on the same SPARC galaxy points with γ free, the density-keyed law loses to the acceleration-keyed one at ΔBIC +2843 (+142 after allowing for neighbouring points on a rotation curve not being independent), and its best-fit γ runs down to 0.046 — the fit switches its own density dependence off. In plain words: when the density version is allowed to choose, it chooses to barely respond to density at all — the best fit says the parts don't switch to acting together, the opposite of the idea. That is the criticality retraction above arriving a second time, through the data rather than through the algebra.
The better-known galaxy fit, γ ≈ 0.489, is a different law: there the same tanh-log shape is keyed on acceleration and used as an implicit interpolating function (μ-form) on the observed acceleration — MOND's own variable, not ρ. Written as a Hill function its index is n = 2γ ≈ 0.98 ≈ 1, the non-cooperative limit, so that switch is not a collective one either; but on its own it says nothing about density. At exactly γ = 1/2 the identity closes: C(x) = x/(x+2) = μsimple(x/2), MOND's simple interpolating function — so “the acceleration-keyed fit converges to MOND” is not an empirical coincidence but algebra (see the RAR entry, Galaxy Rotation for how C entered that fit, and For Researchers).
Revision note
This passage said the SPARC fit γ ≈ 0.489 (Hill index ≈ 1) meant “its own best fit sets cooperativity to zero” for the claim that density drives a collective transition. That fit was run in the acceleration-keyed, implicit μ-form law, so it cannot speak for the density-keyed law; the density-keyed law's own head-to-head result (ΔBIC +2843, best-fit γ → 0.046) now carries the conclusion.Choose Your Path
The Math
Dive into the equation: derivations, proofs, parameter origins
The Evidence
See it tested against 14,610 galaxies (175 SPARC + 14,435 ALFALFA–SDSS, a large radio-plus-optical galaxy survey, after quality cuts)
The Chemistry
Explore 1,703 phenomena at the γ ≈ 1 boundary
The Failures
Where the theory falls short and what that teaches us
The Foundations
Four axioms everything else flows from — including what Intent actually is