Testable Predictions

WILL Relational Geometry derives quantitative, falsifiable predictions across cosmology, gravitation, and quantum mechanics—all from the single closure condition \(\kappa^{2}=2\beta^{2}\) and without free parameters. Every result below is linked to its derivation, an interactive lab, or a runnable notebook.

1. Cosmological Predictions

6 predictions

a. Derivation of the Hubble Parameter \(H_0\)

WILL RG derives \(H_0\) solely from the CMB temperature \(T_0\) and the fine-structure constant \(\alpha\), bridging microphysical constants with cosmic expansion.

Radiation energy density:

$$\rho_\gamma \;=\; \frac{4\,\sigma_{\mathrm{SB}}\,T_0^{\,4}}{c^{3}}$$

Maximum geometric density, set by the EM coupling limit:

$$\rho_{\max} \;=\; \frac{\rho_\gamma}{3\,\alpha^{2}}$$

Hubble parameter from first principles:

$$H_0 \;=\; \sqrt{8\pi\,G\,\rho_{\max}}$$

\(H_0 \approx 68.15\;\mathrm{km\,s^{-1}\,Mpc^{-1}}\) — within 1% of Planck 2018 (\(67.4\pm0.5\)).

Part II — “Deriving \(H_0\) from CMB Temperature and \(\alpha\)”

📄 Open derivation in Part II

b. Distant Supernova Flux Levels

WILL RG predicts the cosmic expansion history with fixed geometric density parameters derived from topological energy partitioning:

$$\Omega_m = \tfrac{1}{3},\qquad \Omega_\Lambda = \tfrac{2}{3}$$

Theoretical distance modulus:

$$\mu_{\mathrm{WILL}}(z) \;=\; 5\log_{10}\!\left( \frac{c\,(1+z)}{H_0} \int_{0}^{z} \frac{dz'}{\sqrt{\tfrac{1}{3}(1+z')^{3}+\tfrac{2}{3}}} \right)+25$$

Residuals < 0.015 mag across the full Pantheon+ redshift range.

Part II — “Geometric Expansion Law: Distant Supernova Flux Levels Test”

📄 Open derivation in Part II

c. CMB Acoustic Spectrum

The CMB acoustic spectrum emerges as resonant harmonics of an \(S^{2}\) topology loaded by baryons. The first acoustic peak position follows from vacuum stiffness and baryonic mass loading alone:

$$\ell_{\mathrm{pred}}(\mathrm{WILL}) \;=\; \ell_{\mathrm{vac}}\, \frac{\Omega_\Lambda}{\Omega_\Lambda + \Omega_{\mathrm{bary}}} \;\approx\; 220.59$$

\(\ell_{\mathrm{obs}}\approx 220.60\) — agreement within ~0.01%, offers an alternative explanation “missing mass” without invoking dark matter.

Part II — “Geometric Origin of the CMB Acoustic Spectrum”

📄 Open derivation in Part II

d. Resolution of the Low Quadrupole Anomaly

Vacuum tension acts as a geometric high-pass filter, naturally suppressing the largest angular scales. The predicted quadrupole power:

$$P(\ell\!=\!2) \;\propto\; \left(\frac{1}{1 + R_{\mathrm{eff}}\,\lambda_2}\right)^{2}, \qquad \lambda_2 = 6$$

Predicted corridor: 0.132 – 0.285  |  Planck observation: \(D_{\ell=2}\approx 0.20\) — falls within the WILL RG corridor.

Part II — “Resolution of the Low Quadrupole Anomaly”

📄 Open derivation in Part II

e. Galactic Rotation Curves & Radial Acceleration Relation

Rotation curves and the RAR follow from a single, theoretically fixed acceleration scale linked to the cosmic horizon:

$$V_{\mathrm{RG}}(r) \;=\; \sqrt{V_b^{2}(r) \;+\; \sqrt{V_b^{2}(r)\;\cdot\; a_\kappa\, r}}, \qquad a_\kappa = \frac{c\,H_0}{3\pi}\approx 0.70\times10^{-10}\;\mathrm{m\,s^{-2}}$$

Tested on the SPARC database (175 galaxies) with zero fitting parameters. Median RMSE ≈ 12.64 km/s — comparable to or exceeding MOND, with near-zero systematic bias.

Part II — “Galactic Dynamics: The Law of Resonant Interference” and “The Universal RAR”

📄 Open derivation in Part II

f. Wide Binary Stars

Wide binary systems couple to the horizon via a distinct kinematic resonance channel, predicting a boost factor different from the galactic (structural) one:

$$\gamma_{\mathrm{WILL}} \;=\; 1 + \frac{a_\beta}{g_N}, \qquad a_\beta = \frac{c\,H_0}{6\pi}\approx 0.35\times10^{-10}\;\mathrm{m\,s^{-2}}$$

Matches Gaia DR3 observational data, offering an explanation to the wide-binary anomaly that challenges standard MOND.

Part II — “The Kinetic Resonance: Potential explanation of the Wide Binary Anomaly”

📄 Open derivation in Part II

2. Relativistic & Gravitational Predictions

5 predictions

a. Suggesting resolution of GR Singularities

WILL RG imposes a natural, geometry-derived upper bound on energy density, desolving black-hole singularities:

$$\rho_{\max} \;=\; \frac{c^{2}}{8\pi\,G\,r^{2}}$$

Black holes become energetically saturated, non-singular regions.

Part I — “No Singularities, No Hidden Regions”

📄 Open derivation in Part I

b. Derivation of the Equivalence Principle

The equality of gravitational and inertial mass is not postulated but derived as a structural identity from relational compositional closure:

$$E_{\mathrm{loc}} \;=\; \frac{E_0}{\beta_Y\,\kappa_X} \;=\; \frac{E_0}{\sqrt{1-\beta^{2}}\;\sqrt{1-\kappa^{2}}}$$

\(m_g \equiv m_i \equiv m_{\mathrm{eff}}\) follows necessarily from the unified scaling of invariant rest energy \(E_0\).

Part I — Theorem “Equivalence of Inertial and Gravitational Response”

📄 Open derivation in Part I

c. Perihelion Precession

The precession formula emerges algebraically from the ratio of gravitational and kinematic projections at periapsis—no mass no G no differential equations or metric tensors required:

$$\Delta\varphi \;=\; \frac{3\pi}{2}\,\frac{\kappa_p^{\,4}}{\beta_p^{\,2}}$$

Accurately predicts Mercury’s 43″/century observed precession for star S2.

Part I — “The Universal Precession Law: Derivation via \(Q_a\)”

📄 Open derivation in Part I

d. Factor of 2 in Gravitational Lensing & Shapiro Delay

Light (\(\beta=1\), \(\beta_Y=0\)) concentrates its entire transformation resource on a single geometric axis, removing the 1/2 partitioning factor that applies to massive bodies:

$$\Phi_\gamma = \kappa^{2}\,c^{2}, \qquad\qquad \Phi_{\mathrm{mass}} = \tfrac{1}{2}\,\kappa^{2}\,c^{2}$$

Factor of 2 explained geometrically — no ad-hoc doubling needed.

Part I — “Single-Axis Energy Transfer and the Nature of Light”

📄 Open derivation in Part I

e. Algebraic Light Deflection

The total deflection of light by a gravitational lens is derived as a strict, exact algebraic difference between measurable relational phase states at periapsis—without differential equations, background manifolds, or Taylor series:

Photonic closure defect (shape parameter) for light (\(\beta=1\)):

$$\delta_\gamma \;=\; \kappa_p^{\,2}, \qquad e_\gamma \;=\; \frac{1}{\kappa_p^{\,2}} - 1$$

Phase state equation solved for the orbital angle \(o\):

$$\cos\,o \;=\; \frac{\kappa_o^{\,2} - \kappa_p^{\,4}} {\kappa_p^{\,2}\,\kappa_{Xp}^{\,2}}$$

Evaluating at spatial infinity (\(\kappa_o \to 0\)) and applying \(\cos(\tfrac{\pi}{2}+x)=-\sin x\), the total deflection angle is:

$$\boxed{\;\Delta\varphi \;=\; 2\,\arcsin\!\left(\frac{\kappa_p^{\,2}}{\kappa_{Xp}^{\,2}}\right)\;}$$

where \(\kappa_p^{2} = R_s/r_p\) and \(\kappa_{Xp}^{2} = 1 - \kappa_p^{2}\). For weak fields (\(\kappa_p^{2}\ll 1\)), \(\arcsin(x)\approx x\) and \(\kappa_{Xp}^{2}\to 1\), recovering the empirical value \(\Delta\varphi\approx 2\kappa_p^{2}=4GM/rc^{2}\).

\(\Delta\varphi_\odot \approx 1.7500\;\mathrm{arcsec}\) — matches the observed solar deflection (\(1.7512\pm 0.001\;\mathrm{arcsec}\), VLBI) to within measurement uncertainty. Difference from exact GR: \(\sim 2\times 10^{-7}\;\mathrm{arcsec}\)—far below any foreseeable detection threshold.

Part I — “Algebraic Derivation of Light Deflection”

📄 Open derivation in Part I

3. Quantum Mechanical Predictions

5 predictions

a. Quantized Atomic Radii & Energy Levels

From topological closure (\(n\lambda_n = 2\pi r_n\)), the Universal Scale Principle, and the Geometric Closure Condition (\(\kappa_q^{2}=2\beta_q^{2}\)), the Bohr radius and quantized energy levels are derived without force analogues or wavefunctions:

$$r_n \;=\; \frac{4\pi\varepsilon_0\,n^{2}\hbar^{2}}{m_e\,e^{2}}$$
$$E_n \;=\; -\,\frac{\alpha^{2}\,m_e\,c^{2}}{2\,n^{2}}$$

Reproduces standard QM results from purely geometric principles.

Part III — “Derivation of Atomic Structure from First Principles”

📄 Open derivation in Part III

b. The Fine Structure Constant \(\alpha\)

The fine-structure constant is derived as the kinetic projection of the electron in the hydrogen ground state—not an empirical input but the unique kinematic amplitude compatible with geometric closure:

$$\alpha \;\equiv\; \beta_1$$

\(\alpha\) derived, not assumed — establishes it as the ground-state kinematic projection of a charged fermion.

Part III — “The Geometric Origin of the Fine Structure Constant”

📄 Open derivation in Part III

c. Unified Relativistic Energy Formula (Fine Structure)

The total relativistic energy formula, numerically identical to the Sommerfeld–Dirac result, arises from circular geometric closure in phase space:

$$E_{n,j} \;=\; m_e\,c^{2}\, \sqrt{1 + \left(\frac{Z\alpha}{n_{\mathrm{eff}}}\right)^{\!2}}, \qquad n_{\mathrm{eff}} = (n-k)+\sqrt{k^{2}-(Z\alpha)^{2}}$$

Matches Sommerfeld–Dirac exactly — derived from geometry, not abstract spinor equations.

Part III — Theorem “The Unified Relativistic Energy Formula”

📄 Open derivation in Part III

d. Geometric Origin of Spin & Pauli Exclusion

Spin is the topological chirality of the standing wave’s phase integration: right-handed (\(+\tfrac{1}{2}\)) and left-handed (\(-\tfrac{1}{2}\)) winding on \(S^{1}\).

The Pauli Exclusion Principle follows from the Principle of Unique Address: co-located energy flows must differ in at least one geometric invariant. With \((n,k,m)\) fixed, chirality is the only remaining degree of freedom.

Two electrons per orbital—derived, not postulated.

Part III — “The Geometric Origin of Spin and Exclusion”

📄 Open derivation in Part III

e. Geometric Uncertainty Principle

The uncertainty principle is a geometric necessity from the closure of energy projection on \(S^{1}\). For orthogonal projections \(\Delta\beta_X\) and \(\Delta\beta_Y\):

$$\Delta\beta_X\;\Delta\beta_Y \;\ge\; \frac{1}{2}\,\lvert\sin 2\theta\rvert\,(2\pi/n)^{2}$$

Calibrated to physical observables this yields the Heisenberg relation, with \(\Delta x\cdot\Delta p\approx\hbar\) — reinterpreting \(\hbar\) as a conversion factor for topological winding.

Part III — “Geometric Origin of the Uncertainty Principle”

📄 Open derivation in Part III

Interactive Labs

Run the predictions yourself. Each lab links to its derivation above and includes a clear falsifiability clause.

1) Galactic Dynamics Lab — SPARC rotation curves

Headline metric: median RMSE ≈ 12.64 km/s over ~175 galaxies.

Open the Galactic Dynamics tool

Falsification Clause

If, with fixed data-selection rules on the SPARC dataset, the median RMSE exceeds 50 km/s for ≥ 25% of the sample, the prediction is considered falsified.

2) Relational Orbital Mechanics Lab

Headline metric: all observable orbital structure follows from directly measurable light-frequency projections—no mass or \(G\) required.

Open the R.O.M. tool

Falsification Clause

If, with accurate input data, disagreement with observations > 5%, the prediction is considered falsified.

3) Relativistic Time Offset Lab

Primary calculation: the daily relativistic time offset between a surface observer and an orbiting body. Secondary check: classical energy consistency. Object-agnostic—applies to any circular orbit.

Presets:

Δt per day (μs)

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Geometric Energy (ΔE)

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Normalized Physical Energy

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Energy Ratio

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Show Calculation Breakdown

1) Define projections

Gravitational projection at the surface \(A\): \(\kappa_A^2 = \frac{2GM}{R_A c^2}\).
Gravitational projection at the orbit \(B\) (radius \(r\)): \(\kappa_B^2 = \frac{2GM}{r\,c^2}\).
Kinematic projection for a circular orbit at \(B\): from \(v^2 = GM/r\) we get \(\beta_B^2 = \frac{v^2}{c^2} = \frac{GM}{r\,c^2}\). On the surface we take \(\beta_A^2 = 0\).

2) Combine into \(Q^2\) and \(Q_t\)

\(Q^2 = \kappa^2 + \beta^2\). Thus \(Q_A^2 = \kappa_A^2 + \beta_A^2\) and \(Q_B^2 = \kappa_B^2 + \beta_B^2\).
The time-axis complement is \(Q_t = \sqrt{1 - Q^2}\).

3) Time offset (core result)

Daily clock difference in microseconds per day: \(\Delta t_{B\to A}[\mu s/\text{day}] = \bigl(1 - \frac{Q_{tA}}{Q_{tB}}\bigr)\times 86400\times 10^6\).

4) Classical energy consistency

Fix the potential zero at the surface \(R_A\). For a circular orbit at radius \(r\):
Potential: \(E_p = \bigl(-\frac{GMm}{r}\bigr) - \bigl(-\frac{GMm}{R_A}\bigr)\).
Kinetic: \(E_k = \frac{1}{2}mv^2\) with \(v^2 = GM/r\).
Total: \(E_{tot} = E_p + E_k\). Normalize by \(mc^2\) to get \(E_{tot}/(mc^2)\).
Geometric energy (mass-independent): \(\Delta E_{A\to B} = \frac{1}{2}\bigl[(\kappa_A^2-\kappa_B^2)+(\beta_B^2-\beta_A^2)\bigr]\).
Consistency: \(\frac{E_{tot}/(mc^2)}{\Delta E_{A\to B}} = 1\).

Reproducible Notebooks

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