Remainders, Ghosts, and the Hidden Structure of Physical Constants
By John Gavel
1. The Mystery We Want to Explain
The fine-structure constant α⁻¹ = 137.035999084…
This single number controls how light interacts with matter. It determines:
- The exact frequencies of light atoms emit (atomic spectra)
- The strength of chemical bonds (molecular structure)
- Whether atoms are stable or collapse (the edge of existence)
- Even why gold is yellow instead of silver
We've measured it to ten decimal places.
We use it every day in quantum mechanics, particle physics, and engineering.
Yet after a century of modern physics, we have no idea why it has this value.
Why not 136?
If α⁻¹ were 136, electrons would spiral into nuclei. Atoms would collapse. Chemistry wouldn't exist.
Why not 138?
If α⁻¹ were 138, electrons would barely bind to nuclei. No molecules. No life.
Why exactly 137.036…?
Not 137.0. Not 137.04. But 137.035999084… measured to breathtaking precision—and we have absolutely no theoretical explanation.
For decades, physicists puzzled over it. Feynman called it 'a magic number' and said, 'It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding.' He half-joked about putting it on his tombstone.
What if this number isn't arbitrary?
What if 137.036 is a remainder—the visible gap between what a discrete universe can build recursively and what continuous reality requires?
What if α⁻¹, along with other mysterious constants (tiny neutrino masses, the absurdly small cosmological constant), follows a deeper pattern we've been missing?
This essay develops that idea.
2. The Core Principle: Constructible + Remainder
Let's start with something simple—division always leaves a remainder:
\[ \frac{N}{N-1} = 1 + \frac{1}{N-1} \]
Examples:
- 12/11 = 1 + 1/11 = 1.0909…
- 132/131 = 1 + 1/131 = 1.0076…
The pattern:
- Whole part (1): What you can build directly with your tools
- Remainder (1/(N-1)): The gap your tools can't reach
This isn't deep—it's arithmetic. But what if physical observables work the same way?
\[ Q_{\text{measured}} = Q_{\text{constructible}} + R_{\text{remainder}} \]
Where:
- Q_constructible = what a discrete substrate can build through recursive operations
- R_remainder = structured gaps (ghosts) from operations the substrate cannot perform
The claim is simple but powerful:
Physical constants aren't fundamental numbers dropped from the sky. They're sums of:
- A constructible bulk (from combinatorial rules, discrete geometry)
- Irreducible remainders (ghosts from circular closure, irrational packing, transcendental limits)
Let's see why this makes sense.
3. Angle Trisection: When Your Tools Fall Short
The impossibility of angle trisection is the perfect classical analogy.
The problem: Given angle θ, construct θ/3 using only compass and straightedge.
Your tools:
- Bisect angles (\(\theta \to \theta/2\))
- Add and subtract angles
- Take square roots (geometric mean)
What you can build:
- \(\theta/2, \theta/4, \theta/8, \theta/16, …\) (powers of 2)
- \(\theta/2 \pm \theta/4, …\) (rational combinations)
What you CANNOT build:
- \(\theta/3\) (in general)
Why? Because \(\theta/3\) requires solving:
\[ \cos(\theta/3) = \text{root of a cubic equation} \]
Compass and straightedge give quadratic extensions (square roots). Trisection needs cubic extensions. Your toolkit is one degree short.
Mathematically:
\[ \frac{\theta}{3} \notin \text{field generated by square roots alone} \]
But \(\theta/3\) still exists! It's well-defined, can be measured or computed numerically. You just can't construct it with allowed operations.
The decomposition:
\[ \frac{\theta}{3} = \underbrace{\text{best binary approximation}}_{\text{constructible}} + \underbrace{\text{gap}}_{\text{remainder}} \]
Key insight: The gap isn't an error—it's information about:
- Your toolkit is incomplete
- The size of what you're missing
- What operation would close the gap (cube roots)
This is exactly the structure we'll apply to physical constants.
4. Ghosts: Numbers That Exist But Can't Be Built
Definition: A ghost is a mathematically valid quantity that:
- Exists (well-defined, computable, measurable)
- Cannot be constructed with your current allowed operations
- Still constrains the system by appearing as a remainder
Example 1: Primes as Ghosts
Suppose your toolkit is addition and multiplication. You can construct:
- 4 = 2×2
- 6 = 2×3
- 8 = 2×4
- 9 = 3×3
- 10 = 2×5
You CANNOT construct: 2, 3, 5, 7, 11, 13… (primes)
Primes constrain everything: 15 = 3×5, 30 = 2×3×5. They exist but cannot be built directly; they structure the number line.
Example 2: π as a Ghost
Toolkit: rational arithmetic (fractions)
- 22/7 ≈ 3.142857
- 355/113 ≈ 3.1415929
You cannot construct π exactly. It governs circular geometry:
\[ C = 2 \pi r, \quad A = \pi r^2 \]
Example 3: Golden Ratio Φ as a Ghost
Toolkit: rational arithmetic (Fibonacci ratios)
- 1/1, 2/1, 3/2, 5/3, 8/5, 13/8 … → approach Φ ≈ 1.6180339…
Φ governs pentagonal packing, icosahedral ratios, and Fibonacci spirals. It is maximally un-constructible from rationals.
5. Why Ghosts Create Problems: Three Geometric Tensions
Problem 1: Closure Problem (Sphericity Ghost)
Icosahedron: 20 triangular faces, 30 edges, 12 vertices
Surface areas:
\[ A_{\text{ico}} = 5 \sqrt{3} a^2 \approx 8.6603 a^2, \quad V_{\text{ico}} = \frac{5}{12}(3+\sqrt{5}) a^3 \approx 2.1817 a^3 \]
Equivalent sphere radius:
\[ r = \left(\frac{3V}{4\pi}\right)^{1/3} \approx 0.8051 a, \quad A_{\text{sphere}} = 4 \pi r^2 \approx 8.1398 a^2 \]
Sphericity ratio:
\[ \Psi = \frac{A_{\text{sphere}}}{A_{\text{ico}}} \approx 0.9399 \]
Problem 2: Holonomy Problem (Circular Ghost)
Vector transported around a closed loop:
\[ \sum_i \theta_i = 2 \pi \]
When you transport a vector around a closed loop using discrete angular steps, the phase accumulates:
θ₁ + θ₂ + θ₃ + ... = total phase
For perfect closure, this sum must equal exactly 2π. But 2π is transcendental—it cannot be expressed as a finite sum of rational angles. No matter how cleverly you choose discrete steps, there's always a remainder.
This is the circular ghost: the irreducible gap between discrete steps and continuous closure.
Problem 3: Packing Problem (Golden Ghost)
Pentagons + triangles cannot tile perfectly (3-fold vs 5-fold symmetry).
Pentagon interior angle:
\[ \cos(108^\circ) = -\frac{1}{2\Phi} = -\frac{\Phi-1}{\Phi} \]
Φ is the maximal irrationality, producing unavoidable remainder.
6. Reflection Principle
Forward recursion fails → boundary reflection enforces unreachable ghosts.
Example: soap bubbles (pressure vs tension enforces volume, creating perfect sphere).
7. Application to α⁻¹
Discrete substrate parameters: H = 132, K = 12, Ψ ≈ 0.9399
Constructible bulk:
\[ Q_{\text{bulk}} = \frac{H(K-1)}{K \Psi} = \frac{132 \times 11}{12 \times 0.9399} \approx 128.753 \]
Ghost remainder decomposition:
\[ R = 2 \pi + \Phi + \Phi^{-2} \approx 6.283 + 1.618 + 0.382 = 8.283 \]
Final α⁻¹:
\[
\alpha^{-1} = Q_{\text{bulk}} + R = 128.753 + 8.283 = 137.036
\]
8. N/(N-1) Pattern and Hierarchy
To see how remainders naturally suppress with capacity, consider the simple division pattern N/(N-1):
As N (capacity) grows, the whole part stays constant at 1, but the remainder shrinks exponentially. This is the mathematical engine behind hierarchical smallness—not fine-tuning, but geometric necessity.
\[ \frac{N}{N-1} = 1 + \frac{1}{N-1} \]
| N | N/(N-1) | Whole | Remainder |
|---|---|---|---|
| 2 | 2/1 | 1 | 1.000 |
| 3 | 3/2 | 1 | 0.500 |
| 4 | 4/3 | 1 | 0.333 |
| 12 | 12/11 | 1 | 0.091 |
| 132 | 132/131 | 1 | 0.0076 |
| 1000 | 1000/999 | 1 | 0.001 |
9. Testable Predictions
Higher-order terms in α⁻¹, universal patterns in other constants, hierarchy scaling as 1/Hⁿ.
10. Summary
- Division leaves a remainder \(\frac{N}{N-1} = 1 + \frac{1}{N-1}\)
- Physical constants = Constructible + Remainder \(\, Q = Q_{\text{constructible}} + R_{\text{gap}} \,\)
- Remainders have structure: \(R = 2\pi + \Phi + \Phi^{-2} + \text{higher-order}\)
11. Appendix: Geometric Model of Ψ
Here is a proposed decomposition that matches α⁻¹ and arises naturally from geometric closure constraints.
\[ V_{\text{ico}} = \frac{5}{12}(3+\sqrt{5}) a^3 \approx 2.1817 a^3, \quad A_{\text{ico}} = 5 \sqrt{3} a^2 \approx 8.6603 a^2 \]
\[ r = \left(\frac{3 \times 2.1817}{4 \pi}\right)^{1/3} a \approx 0.8051 a, \quad A_{\text{sphere}} = 4 \pi r^2 \approx 8.1398 a^2, \quad \Psi = \frac{A_{\text{sphere}}}{A_{\text{ico}}} \approx 0.9399 \]
Final Note — The Dynamics Behind Ghosts
When geometric closures are incompatible, the system accumulates tension. A discrete substrate tries to “fit” itself into a continuous geometry, but when the required closure cannot be achieved with the available operations, the mismatch builds pressure. That tension must be relieved for the system to remain stable.
The relief appears as a remainder — a ghost — which becomes a physical constant.
In this view, constants like , , and are not arbitrary numbers. They are the equilibrium points where geometric tension is minimized. They are the structured residues of closure attempts that cannot be completed with the system’s constructive toolkit.
These ghosts are not unreal. They are incomplete in the sense that they cannot be expressed directly by the substrate’s operations — they are the parts the system cannot build, but must still account for. They are the “missing moves” that geometry demands but the substrate cannot perform.
The constant emerges as the sum of:
what the system can construct
plus the tension-relief terms it must include to avoid tearing
This is why physical constants look like remainders: they are remainders — the visible scars of incompatible closures inside a discrete universe.
