Technical Note XXXIX
Where the theory stands, what remains open, and what to test next
This is a roadmap note, not a result note. It records the consolidated state of BeeTheory, the questions that are genuinely open (including one idea tested today that did not yield a law), and the concrete objectives that would move the theory from a working phenomenology to a derivation.
1. Consolidated state — what is established
1.1 The point-mass regime is derived, not fitted
A regularized exponential wave function \( \psi(r) = (1/N)\exp[-\sqrt{r^2+a^2}/a] \), with \(a=a_0\), produces three Laplacian terms. Only \(T_2 \to -2/(a\,r)\) survives at macroscopic distance, reproducing the Newtonian \(-1/r\) potential. With \(K = G M_1 M_2\, a/2\), the interaction energy is \(U = -GM_1M_2/r\) exactly. This is validated — parameter-free — on the eight planets to twelve decimal places, on the Cavendish scale (\(G_{eff}\) matches the measured \(G\)), and on the Earth’s PREM profile (\(g = 9.81\,\text{m/s}^2\)).
1.2 The galactic model is calibrated and blind-tested
Extended to a continuous density via superposition, the three-parameter model reproduces 101 disk-galaxy rotation curves:
| Parameter | Value | Calibration (20) | Blind (81) |
|---|---|---|---|
| λ | 12.696 | median |err| 16.0% | median |err| 17.3% |
| c | 0.163 | mean signed −4.3% | mean signed −0.9% |
| ℓ_floor | 3.00 kpc | — | no bias vs R_d / M_vis |
1.3 New result: the wave-mass profile is not an ansatz
The postulated enclosed wave mass \( M_{wave}(
2. Open questions
2.1 The quantum-to-force bridge open
The Schrödinger equation governs a probability amplitude, not a force-bearing field. The step from interfering wave peaks to a real attractive force is interpreted, not yet derived from a dynamical equation. Until this is explicit, \(T_2 \propto 1/r\) is a match of form rather than a derivation of gravity.
2.2 The solar / galactic discontinuity open
The solar regime is parameter-free; the galactic regime needs three fitted parameters, with \(\lambda\) jumping from ≈2 (Milky Way, with bulge) to ≈13 (bulgeless disks). The continuity between the two regimes is not yet demonstrated from first principles.
2.3 The origin of ℓ_floor open
A constant 3 kpc coherence length, independent of mass, separated from the Bohr radius \(a_0\) by some thirty orders of magnitude, is currently a parameter rather than a derived quantity.
2.4 A hierarchy of coherence levels — tested, no law found negative result
An attractive hypothesis: matter organizes into nested coherence levels — electron, nucleus, molecule, body, planet, solar system, galaxy, cluster — each with its own wave-coherence length \(\ell_n\), the \(\ell_n\) linked by a recurrence that would connect \(a_0\) to \(\ell_{floor}\) and remove its arbitrariness.
A single power law \( \ell \propto M^{p} \) across the levels does not hold: residuals reach 3.6 dex (factor ~4000), with a 2.3 dex scatter, and the extrapolation to clusters fails by three orders of magnitude. The likely reason: the levels are bound by different forces (Coulomb for atoms/molecules, gravity for planets and above), so no single equation should span them. The idea is not refuted in its restricted form — a recurrence within the gravitational regime only (solar system → galaxy → cluster) remains untested and is the honest way forward.
3. Objectives — what to test next
| Objective | What it would settle | Status |
|---|---|---|
| Disk convolution to 2 parameters | Whether a universal kernel \(e^{-s/\ell_{floor}}\) convolved on a Freeman disk reproduces rotation curves with c=0, collapsing the model to (λ, ℓ_floor). | to do |
| Cluster coherence length | Apply the wave-mass formalism to Coma (σ≈1000 km/s, R_vir≈2.6 Mpc, M~10¹⁵ M☉), extract the required ℓ_cluster, and check whether it sits in a sensible sequence with ℓ_floor. | to do |
| Radial Acceleration Relation | Reproduce McGaugh (2016) \(g_{obs}(g_{bar})\) point by point — the strictest constraint on any modified-gravity proposal. | to do |
| Bulged galaxies (T ≤ 3) | Extend the 3-parameter model to NGC2841, the Milky Way, and the 13 bulged SPARC blind galaxies. | to do |
| Upper bound on ℓ_floor | Resolve the +25–45% over-prediction on massive Sc/Sbc galaxies (NGC3198) — does ℓ_floor saturate when the galaxy dominates? | to do |
| Dwarf spheroidals | Test the formalism on pressure-supported (dispersion, not rotation) systems. | to do |
Two objectives are decisive and independent. The RAR test determines whether BeeTheory reproduces the true underlying law, not just the flat velocities. The cluster coherence length probes the regime where MOND is weakest — if a single sensible ℓ_cluster reproduces Coma, it is a strong, independent signal. Neither has been done; both are the next steps.
BeeTheory.com — State, open questions and objectives · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026