Sunday, June 29, 2025

Grok Generated LaTex for Fine-structure Constant Equations

### Given Equations

We start with the following system of equations:

1. **$\alpha^2 = \pi \cdot r_p \cdot R_\infty \cdot \frac{R_H}{R_\infty - R_H}$**

- This equation relates $\alpha^2$ to the variables $r_p$, $R_\infty$, and $R_H$.

2. **$r_p = \frac{2}{\pi R_H} \cdot \left( \frac{v_u}{c} \right)^2$**

- This equation expresses $r_p$ in terms of $R_H$, $v_u$, and $c$.

The goal is to solve for $\alpha$ in terms of $R_\infty$, $R_H$, $v_u$, and $c$ by eliminating $r_p$.

---

### Solution Process

To find $\alpha$, we substitute the expression for $r_p$ from the second equation into the first equation and simplify step-by-step.

#### Step 1: Substitution

- Substitute $r_p = \frac{2}{\pi R_H} \cdot \left( \frac{v_u}{c} \right)^2$ into the first equation:

$\alpha^2 = \pi \cdot \left( \frac{2}{\pi R_H} \cdot \left( \frac{v_u}{c} \right)^2 \right) \cdot R_\infty \cdot \frac{R_H}{R_\infty - R_H}$

#### Step 2: Simplification

- **Cancel the $\pi$ terms:**

The $\pi$ in the numerator and denominator cancels out:

$\alpha^2 = \frac{2}{R_H} \cdot \left( \frac{v_u}{c} \right)^2 \cdot R_\infty \cdot \frac{R_H}{R_\infty - R_H}$

- **Simplify the $R_H$ terms:**

The $R_H$ in the numerator and denominator simplifies:

$\frac{2}{R_H} \cdot \frac{R_H}{R_\infty - R_H} = \frac{2}{R_\infty - R_H}$

Thus:

$\alpha^2 = \frac{2}{R_\infty - R_H} \cdot \left( \frac{v_u}{c} \right)^2 \cdot R_\infty$

- **Rearrange:**

Rewrite the expression for clarity:

$\alpha^2 = \frac{2 R_\infty}{R_\infty - R_H} \cdot \left( \frac{v_u}{c} \right)^2$

#### Step 3: Solve for $\alpha$

- Take the square root of both sides:

$\alpha = \sqrt{ \frac{2 R_\infty}{R_\infty - R_H} \cdot \left( \frac{v_u}{c} \right)^2 }$

- **Apply the square root property:**

Since $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ (for positive $a$ and $b$):

$\alpha = \sqrt{ \frac{2 R_\infty}{R_\infty - R_H} } \cdot \sqrt{ \left( \frac{v_u}{c} \right)^2 }$

- **Simplify:**

The square root of the squared term simplifies:

$\alpha = \sqrt{ \frac{2 R_\infty}{R_\infty - R_H} } \cdot \frac{v_u}{c}$

- **Alternative form:**

Rearrange for conciseness:

$\alpha = \frac{v_u}{c} \cdot \sqrt{ \frac{2 R_\infty}{R_\infty - R_H} }$

This form is clear and expresses $\alpha$ in terms of the required variables.

---

### Final Answer

The solution for $\alpha$ is:

$\alpha = \frac{v_u}{c} \sqrt{\frac{2 R_\infty}{R_\infty - R_H}}$

---

### Notes

- The expression assumes $R_\infty - R_H > 0$ to ensure the denominator is positive and the square root is defined.

- All variables ($R_\infty$, $R_H$, $v_u$, $c$) are treated as positive quantities for simplicity, consistent with typical mathematical or physical contexts.

---

This response provides the complete mathematical content with all dollar sign pre and postfixes removed, as requested, while maintaining readability through markdown formatting.

∞∞∞ Proton Radius Puzzle Solution ∞∞∞

Proton Radius Puzzle Solution

The Proton Radius Puzzle Solution

Introduction

The proton radius puzzle emerged from a discrepancy in the measured charge radius of the proton. Traditional measurements using electron scattering and hydrogen spectroscopy, such as the CODATA 2014 value of 0.8751 fm, conflicted with the smaller value of 0.84184 fm obtained from muonic hydrogen spectroscopy. This puzzle persisted until recent experiments and theoretical models reconciled the values, favoring the smaller radius.

This presentation introduces a novel solution based on a quantized superfluid model. By assuming a circulation condition with velocity equal to the speed of light ($ v = c $) and a quantization number $ n = 4 $, we derive a proton radius that matches the muonic hydrogen measurement, offering a theoretical resolution to the puzzle.

Key Equations

The derivation starts with the quantization condition for a superfluid:

\[ \oint \vec{v} \cdot d\vec{l} = \frac{n h}{m} \]

For a circular path with radius $ r_p $, velocity $ v = c $, and mass $ m = m_p $ (proton mass), this becomes:

\[ c \cdot 2 \pi r_p = \frac{n h}{m_p} \]

Solving for the proton radius $ r_p $:

\[ r_p = \frac{n h}{2 \pi c m_p} \]

With $ n = 4 $:

\[ r_p = \frac{4 h}{2 \pi c m_p} = \frac{2 h}{\pi c m_p} \]

Using constants ($ h = 6.62607015 \times 10^{-34} \, \text{J·s} $, $ c = 2.99792458 \times 10^8 \, \text{m/s} $, $ m_p = 1.67262192369 \times 10^{-27} \, \text{kg} $), the computed radius is approximately 0.84132 fm, closely matching the muonic hydrogen value.

Comparison Graph

Comparison Table

Source	Value (fm)	Uncertainty (fm)
CODATA 2014	0.8751	0.0061
Muonic Hydrogen (2010)	0.84184	0.00067
PRad Experiment (2019)	0.831	0.007
CODATA 2018	0.8414	0.0019
Computed Value ($ n = 4 $)	0.84132	-

Interactive Calculation

Adjust the quantization number $ n $ below to see how the proton radius $ r_p $ changes. Note that $ n = 4 $ yields a value matching the muonic hydrogen measurement.

Quantization number $ n $:

Computed Proton Radius: 0.84132 fm

If this model is correct, there will be signature in the proton mass spectrum data at 1.15MeV, 1.19MeV, and 2.11MeV offset from peak of 938MeV according to Grok:

Grok

Conclusion

This quantized superfluid model resolves the proton radius puzzle by deriving a radius of 0.84132 fm for $ n = 4 $, aligning closely with the muonic hydrogen measurement and the updated CODATA 2018 value. It suggests a novel internal structure for the proton, combining quantum mechanics and relativistic principles. The choice of $ n = 4 $, inspired by the fourfold symmetry of the Tetragrammaton, adds a unique perspective, though the model's strength lies in its empirical agreement with experimental data.

AI (Grok3) Derivation of the Proton Radius

https://t.co/gQ8sKvfAZw
— CornDog🌽🐶🍊🦢🚚🚛🚒𓆏🐸𓆏 (@Corndog98368908) June 29, 2025

https://x.com/i/grok/share/ZFjSaloVT2a5mRxL3l8QZSljk

Saturday, June 28, 2025

Proton Radius Puzzle 1

Proton Radius Puzzle Solution

The following equations relate the proton radius to fundamental constants, potentially addressing the proton radius puzzle:

$\mu = \frac{\alpha^2}{\pi r_p R_\infty}$
$r_p = \frac{2h}{\pi c m_p}$
$R_H = \frac{R_\infty}{1 + \frac{m_e}{m_p}}$
$R_\infty = \frac{m_e e^4}{8 \epsilon_0^2 h^3 c}$

Where:

$r_p$: proton radius
$\alpha$: fine-structure constant
$h$: Planck's constant
$c$: speed of light
$m_p$: proton mass
$m_e$: electron mass
$R_\infty$: Rydberg constant (infinite mass)
$R_H$: Rydberg constant for hydrogen

Calculating $r_p$ using the second equation:

This matches the muonic hydrogen measurement (~0.842 fm), suggesting $r_p = \frac{2h}{\pi c m_p}$ as the true radius. The first equation shows $\mu = \frac{m_p}{m_e}$, linking these constants consistently.

Google AI on the Proton Radius and Fundamental Constants


	Google AI Link - Proton Radius & Constants

https://x.com/i/grok/share/feexEtV1vCYPPrES64yya4MIk Grok analyzes trends of constants, however, did not dig into historical trends enough,

Friday, June 27, 2025

Solution for the Constants Including Proton Radius Dependence

https://x.com/i/grok/share/vDRWWB79lKRhtbFcetL9JBq8D

It'll take some reviewing to determine if these equations are valid...

The Surfer, OM-IV

Grok3: Corrections to Constants by Including Proton Radius Solution

#Correction to #Constants
considering #ProtonRadius #Puzzle #Solution https://t.co/9aHeVPqOVQ
— CornDog🌽🐶🍊🦢🚚🚛🚒𓆏🐸𓆏 (@Corndog98368908) June 28, 2025

#ProtonRadiusPuzzle #Solution #Update #WIP Grok3's Attempt

#ProtonRadiusPuzzle #Solution #Update #WIP future corrections may be coming as well https://x.com/i/grok/share/adFRO9Dkj6gGLbdt7Z65y0RLc

Embed:

#ProtonRadiusPuzzle #Solution #Update #WIP future corrections may be coming as wellhttps://t.co/F6XZdidPOG
— CornDog🌽🐶🍊🦢🚚🚛🚒𓆏🐸𓆏 (@Corndog98368908) June 27, 2025

Correction: n=1 (DRAFT#2): A Physics Letter A (Mathematical Physics): Proton Radius, the Rydberg Equation, and Fundamental Physics Constants

(This is an updated to an earlier draft)

Proton Radius, the Rydberg Equation, and Fundamental Physics Constants

n=1 Bohr_model

Highlights

Complete solution to proton radius puzzle

Correction to Rydberg Equation (extends with proton to electron mass ratio)

Roots of Full Rydberg Equation define fundamental physics constants

Abstract

A mathematical solution for the proton radius has been derived which adds a term to the Rydberg equation. By avoiding using the reduced mass approximation (as has been done in solid-state theory and the hydrogen atom's analytical solution to the Schrödinger Wave Equation) and including the proton to electron mass ratio which brings in the proton radius. A proton to electron mass ratio of 1836.15267 is calculated with a proton radius of 0.8412fm.

Keywords

Proton radius; proton to electron mass ratio; Rydberg equation

1.0 Introduction

This derivation makes use of, as a starting point,: $$R_M={R_\infty\over {1+{m_e\over M}}}[1]$$

[1] https://en.wikipedia.org/wiki/Rydberg_constant#Alternative_expressions

1.1 Governing model

Schrödinger Equation (and Full Wave Equation)

(At lowest energies, n=1, Bohr model and Schrödinger model are equivalent for 1H hydrogen)

2.0 Derivation:

$R_M={R_\infty\over {1+{m_e\over M}}}$

$R_M(1+{m_e\over M})=R_\infty$

$(1+{m_e\over M})={R_\infty\over R_M}$

$1={R_\infty\over R_M}-{m_e\over M}$

For hydrogen, 1H @ 0°K:, $M=m_p$

$1={R_\infty\over R_H}-{m_e\over m_p}$

For the electron to proton mass ratio (inverse of proton to electron mass ratio), the derivation of the proton to electron mass ratio:

$$\mu={m_p\over m_e}=1836.15267\dots$$

$$m_e={2R_{\infty}h\over c\alpha^2}$$

( $m_e$ for the electron mass: Derivation of Rydberg equation for electron mass, also see [1] above)

$$m_e{\alpha^2\over R_{\infty}}={2h\over c}$$

$$m_e{\alpha^2\over \pi R_{\infty}}={2h\over \pi c}$$

By substitution, let $r_e={\alpha^2\over\pi R_{\infty}}$, then:

$$m_er_e={2h\over\pi c}$$

Now, because for every action there is an equal and opposite reaction, for every force there is an equal and opposite force, for every torque, there is an equal and opposite torque, equate $m_pr_p$ to $m_er_e$ to balance torque/spin between proton and electron:

$$m_er_e={2h\over\pi c}=m_pr_p$$

$$\therefore {m_p\over m_e}={r_e\over r_p}={\alpha^2\over\pi r_pR_{\infty}}=1836.15267$$

Where:

$$m_pr_p={2h\over\pi c}=4\ell m_{\ell}$$

$$r_p=0.841235640(294664)\;fm$$(Google Calculator for r_p)

$\mu={m_p\over m_e}={\alpha^2\over\pi r_pR_ \infty}=1836.15267$ (Google Calculator for μ)

$1={R_\infty\over R_H}-{\pi r_pR_ \infty\over \alpha^2}$

$R_\infty={m_ee^4\over 8{\epsilon_0}^2h^3c}$

$$1={{m_ee^4\over 8{\epsilon_0}^2h^3cR_H}}-{\pi r_p{m_ee^4\over 8{\epsilon_0}^2h^3c}\over \alpha^2}$$

$m_p=$ proton mass

$m_e=$ electron mass

$e=$ elementary charge

$\epsilon_0=$ permittivity of free space

$r_p=$ proton radius

$r_e=$ effective torque arm radius for electron

$\alpha=$ fine-structure constant

$h=$ Planck constant

$c=$ speed of light

$R_{H}=$ Rydberg constant for 1H hydrogen atom

$R_{\infty}=$ Rydberg constant for massive nucleus atom

$R_{M}=$ Rydberg constant for Multi-proton atom

$\ell=$ Planck length

$m_{\ell}=$ Planck mass

3. Other Solutions for proton mass (ratio)

Boundary Value Problem method of determining coefficients use with wave equation for both electron and proton; Geometrical assuming vacuum density and information theory of black hole event horizon - Haramein et al

4. Conclusions

This derivation provides a polynomial that predicts future trends of fundamental physics constants coefficient values as the measurements are refined over the years to be more precise. The polynomial may also actually DEFINE the constants - ongoing work with the polynomial investigating stability and convergence continues.

There are various algorithms one can envision setting some constants as known and iterating to find the others, and likely all constants in the equation. A timebase reference is also needed and likely why c is defined in mainstream via decree rather than derivation.

Q.E.D.

Credit authorship contribution statement

Mark's derivation with Lyz's input to equate/balance the proton and electron

Declaration of Competing Interest

n/a

Data availability

n/a

References

[standard]

*Mµ - Master of the µniverse, Macro and micro, inner and outer-dimensional

The Surfer, OM-IV

Correction Coming! Found a mistake where R_H was used instead of R_∞

Found a mistake where $R_H$ was used instead of $R_\infty$, will be correcting after more review. I am re-reviewing with Grok3's assistance.... may be a few weeks or months....

THAT'S WHY IS WAS TITLED DRAFT:!

https://phxmarker.blogspot.com/2023/06/draft-physics-letter-mathematical.html?sc=1751058483083#c8638999812085907832

The Surfer, OM-IV

Latest Summary: Proton to Electron Mass Ratio and Proton Radius Related Equations

The proton to electron mass ratio:

$$\mu={\alpha^2\over {\pi r_p R_\infty}}$$

The proton radius:

$$r_p={2h\over {\pi c m_p}}$$

Rydberg Relationship between Hydrogen and infinite mass nucleus:

$$R_H={R_\infty\over 1 + {m_e\over m_p}}$$

Rydberg Constant:

$$R_\infty={m_ee^4\over{8\epsilon_0^2h^3c}}$$

$R_\infty$	$R_\infty=	Rydberg constant
	$m_e=	rest mass of the electron$
	$e==	elementary charge$
$\varepsilon_0$	$\epsilon_0=	permittivity of free space$
	$h=	Planck constant$
	$c=	speed of light in vacuum$

The Surfer, OM-IV

Monday, June 23, 2025

Claude Created Proton Radius Dashboard

Just testing, need to review:

https://claude.ai/public/artifacts/0aa07ab9-b9e6-41c0-a09b-bb440a362fa8

Sunday, June 22, 2025

claude.ai Attempt at Proton Radius Puzzle

https://claude.ai/public/artifacts/1f8bd120-0c79-4cf8-9ea1-6df662a5d484?fullscreen=true

https://claude.ai/chat/dddc5689-f4ad-48cf-9cbe-416768ffd0b2

Saturday, June 21, 2025

Master Dashboard for Constants of Physics - Proton Radius Related

Master Dashboard for Constants of Physics

Significant Digits (4-16):

Equations

Calculated Values

Calculated μ:

Measured μ:

Error δμ:

Defined r_p:

Select Constant to Vary:

Proton to Electron Mass Ratio Dashboard - WIP

Proton Radius Dashboard

Proton Radius and μ Calculation Dashboard

Equation: μ = (α)² / (π * r_p * R_∞), where r_p is in meters (r_p (m) = r_p (fm) * 10⁻¹⁵)

Current Values

Proton radius r_p (fm):

Fine-structure constant α:

Rydberg constant R_∞ (m⁻¹):

Calculated μ:

Accepted μ: 1836.15267343

Error (|μ_calculated - μ_accepted|):

Set Values

Set r_p (fm):
Set α:
Set R_∞ (m⁻¹):
Set δ_r_p (fm):
Set δ_α:
Set δ_R_∞ (m⁻¹):

Controls

Sensitivity Analysis

Friday, June 20, 2025

Draft: Non-functional Radar Chart Star Glyph for $\mu$

Proton-to-Electron Mass Ratio Sensitivity Analysis

Equation: μ = α² / (π r_p R_∞)

Adjust Constants

Adjustment Step Size:

Percentage Variation (%):

Sensitivity Analysis

Sunday, June 15, 2025

3D Visualization of Proton Radius, Mass Ratio, and Rydberg Constant

3D Interactive Physics Visualization

Proton Radius Theory vs. NIST CODATA Measurement

Proton Radius Calculation

Sunday, June 8, 2025

3D Scatter Plot Test

Interactive 3D Spiral

Number of turns: 5

Radius: 1

Saturday, May 31, 2025

Proton Decay Solved - Lifetime ∞Infinite∞ Due To Topological Protection

https://en.wikipedia.org/wiki/Proton_decay

https://x.com/i/grok/share/HS3o5fY8B57CHdUDBY8OBa7yf

Conclusion
If the proton is considered a vortex in a superfluid, its stability arises from the topological protection inherent to vortices, making it highly resistant to decay. This provides an alternative explanation for the proton’s extraordinary lifetime, potentially resolving the proton decay problem by suggesting that protons don’t decay spontaneously due to their vortex nature, rather than a conservation law. While this idea is speculative and requires much more development to become a viable theory, it highlights an intriguing connection between superfluid dynamics and particle physics, offering a novel lens through which to view the proton’s enduring stability.

The Surfer, OM-IV

Friday, May 30, 2025

Grok Assisted Theory of The Fundamental Physics Constants

#Fundamental #Physics #Constants #Theory https://t.co/840JQCqt47
— CornDog🌽🐶🍊🦢🚚🚛🚒𓆏🐸𓆏 (@Corndog98368908) May 30, 2025

Thursday, May 29, 2025

Proton Radius Solution and Deterministic Correlation of Constants

https://t.co/8PG7uLTz0x
— CornDog🌽🐶🍊🦢🚚🚛🚒𓆏🐸𓆏 (@Corndog98368908) May 29, 2025