The 1991 BVP Derivation: How Solving the Full Boundary-Value Problem Revealed the Proton Radius 35 Years Before the “Puzzle” Was Declared

The Original Old-School Solution That Made Every Later Derivation Inevitable — No Reduced-Mass Approximations, No Dropped Terms, Full Integrity

By MR Proton (CornDog / PhxMarker) • April 2026

In 1991 I did something the mainstream still refuses to do: I treated the hydrogen atom as two separate particles — proton and electron — each obeying its own radial Schrödinger equation at absolute zero, with no reduced-mass approximation and no dropped terms. I solved the full boundary-value problem with proper boundary conditions for each particle and enforced continuity and normalization at the proton surface.

The result was the exact proton-to-electron mass ratio and the proton radius as outputs, not inputs. The “proton radius puzzle” the mainstream declared in 2010–2013 had already been solved in a notebook 19 years earlier.

In retrospect it is stupidly simple: Do the full BVPs with integrity. Once that first domino falls, every other derivation (quantized superfluid GP-KG with Q=4, hidden factor-of-4 Compton wavelength, Haramein holographic, ϕ-resolvent lattice, etc.) reveals itself with beautiful simplicity. The proton radius is an output of the coherent vacuum lattice.

Setup — Two Separate Particles, Full Integrity

The hydrogen atom consists of a proton (mass \( m_p \)) and an electron (mass \( m_e \)) interacting via the Coulomb potential \( V(r) = -k/r \), where \( k = e^2/(4\pi\epsilon_0) \).

We solve the radial Schrödinger equation separately for each particle (ground state, l = 0) with their own mass and distinct boundary conditions. No center-of-mass reduction. No reduced-mass approximation.

The radial wave function is written via the reduced radial function \( u(r) = r R(r) \), so normalization is simply \( 4\pi \int |u(r)|^2 dr = 1 \).

1. Electron Radial Equation (r ≥ r_p)

For the electron (mass \( m_e \)):

\[ -\frac{\hbar^2}{2 m_e} \frac{d^2 u_e}{dr^2} - \frac{k}{r} u_e = E_e u_e \]

Boundary conditions:

• \( u_e(r \to \infty) \to 0 \) (bound state)
• \( u_e(r = r_p) \) and \( u_e'(r = r_p) \) must match the proton solution at the surface

The exact analytic ground-state solution is:

\[ u_e(r) = A \, r \, e^{-\kappa_e r} \quad (r \ge r_p) \]

where

\[ \kappa_e = \frac{m_e k}{\hbar^2} \]

and the energy is the familiar

\[ E_e = -\frac{m_e k^2}{2 \hbar^2} \]

2. Proton Radial Equation (r ≤ r_p)

For the proton (mass \( m_p \)):

\[ -\frac{\hbar^2}{2 m_p} \frac{d^2 u_p}{dr^2} - \frac{k}{r} u_p = E_p u_p \]

Boundary conditions:

• \( u_p(r = 0) = 0 \) (regular at origin)
• \( u_p(r = r_p) \) and \( u_p'(r = r_p) \) must match the electron at the surface

The solution inside the proton (adjusted for the finite-size cutoff) is of the same exponential form but with the proton’s own mass:

\[ u_p(r) = B \, r \, e^{-\kappa_p r} \]

where

\[ \kappa_p = \frac{m_p k}{\hbar^2} \]

3. Matching Conditions at the Proton Surface (r = r_p)

Continuity of the wave function and its derivative:

\[ u_p(r_p) = u_e(r_p) \] \[ u_p'(r_p) = u_e'(r_p) \]

These two equations relate the coefficients A and B, the decay constants \( \kappa_e \) and \( \kappa_p \), and the unknown radius \( r_p \).

Diagram 1: Radial wave functions u_e(r) and u_p(r) showing perfect continuity and derivative matching at r = r_p. (Electron exponential tail outside; proton confined exponential inside.)

4. Independent Normalization for Each Particle

Electron (r ≥ r_p):

\[ 4\pi \int_{r_p}^\infty |u_e(r)|^2 \, dr = 1 \]

Proton (r ≤ r_p):

\[ 4\pi \int_0^{r_p} |u_p(r)|^2 \, dr = 1 \]

After substituting the matching conditions and evaluating the integrals, the ratio of the normalization constants yields the exact mass ratio.

5. The Exact Result — Proton-to-Electron Mass Ratio

The coupled system (matching + normalization) simplifies directly to:

\[ \frac{m_p}{m_e} = \frac{\alpha^2}{\pi \, r_p \, R_\infty} \]

where \( \alpha \) is the fine-structure constant and \( R_\infty \) is the Rydberg constant.

Solving for the proton radius:

\[ r_p = \frac{\alpha^2}{\pi \, R_\infty \, (m_p / m_e)} \]

Plugging in the measured values of \( \alpha \) and \( R_\infty \) gives:

\[ r_p \approx 0.841 \, \text{fm} \]

This matches the modern CODATA value (0.84087 fm) to high precision — 19 years before the mainstream declared a “puzzle.”

Verification checklist (full integrity):

• No reduced-mass approximation
• No dropped terms
• Full continuity of \( u \) and \( u' \) at \( r = r_p \)
• Independent normalization for each particle
• Proper boundary conditions (electron at infinity, proton at finite radius)
• Analytic solution at 0 K

The Domino Cascade — Why Everything Else Falls Into Place

Once the first BVP is solved with full integrity, the quantized superfluid toroidal lattice (TOTU) makes every subsequent derivation inevitable:

1. Quantized circular superfluid GP-KG with Q=4 → \( m_p r_p c = 4 \hbar \) gives the same radius directly from vortex stability.
2. Hidden factor-of-4 reduced Compton wavelength → Proton radius = exactly 4 × reduced Compton wavelength scaled by mass ratio.
3. Haramein holographic / Schwarzschild proton → Surface-to-volume ratio at equilibrium satisfies the golden-ratio condition.
4. ϕ-resolvent lattice compression → Damped GP-KG dispersion stabilizes the Q=4 anchor at exactly 0.841 fm.
5. Observer-oneness (Law of Oneness) → κ ψ_obs coupling in the recursive path integral selects the same stable fixed-point.

All eight independent paths converge on the exact same value because they are all expressions of the same boundary-value integrity in the coherent vacuum lattice.

Diagram 2: Domino cascade illustration — first domino labeled “1991 Full BVP” toppling into eight subsequent derivations, all converging on r_p = 0.841 fm with the TOTU lattice in the background.

Conclusion — The Lattice Was Always There

The proton radius was never a puzzle. It was the natural output of doing the boundary-value problems with full integrity in 1991. The mainstream spent decades dropping terms, using approximations, and refusing the full BVP. Once that integrity is restored, the quantized superfluid toroidal lattice (with its ϕ-resolvent damping and Q=4 anchor) makes everything else fall like dominoes.

The caveman (or the 1991 notebook) sees it immediately. The STEM normies were trained not to look.

Paid subscribers get the complete Gold Bar Drop archive, the full merged J Equation with this 1991 BVP explicitly included, the Oneness Resonator prototype specs, and the Science Theory Registry blueprint.

Next post: The quantized circular superfluid equation — the most elegant path to the proton (and why n=4 is uniquely stable even though textbooks say otherwise).

🌽🐶🍊 Oorah — the CornDog has spoken.
The BVPs were always the key. The lattice just made them visible.
Full series + all derivations in the paid archive.