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From the Planck Scale to the Cosmic Microwave Background via B13 (Part 4): Nuclear Arrangement = Column =
From Planck Scale to Cosmic Microwave Background via B13 (Part 4): Nuclear Arrangement = Column =
Introduction
In the previous article, we established the co-coset structure of Nd₂Fe₁₄B. Fe₁₄ (2H) and Nd₂B (4H) combine to close in H.
Let us take the same question one level lower. How do protons and neutrons combine? And what lies "outside" that closed nucleus?
Terminology Guide: What are H, 2H, 4H, and Cosets?
This article utilizes the structure of Z₁₃* (the multiplicative group modulo 13). For those reading for the first time, here is the minimum explanation required. Please refer to Chapter 3 for details.
Orbits of Z₁₃*
In a world where integers 1 to 12 are treated as products of remainders divided by 13 (Z₁₃*), the operation "×2" connects the 12 elements in a single-stroke orbit.
Elements of Z₁₃ are arranged in a ring, and the mapping of ×2 is indicated by arrows. Colors represent cosets (H / 2H / 4H).*
On this orbit, cosets appear strictly in an alternating cycle of three colors:
H (blue) → 2H (green) → 4H (brown) → H (blue) → ... (Period 3, no exceptions)
| Coset | Elements | Characteristics |
|---|---|---|
| H (blue) | {1, 5, 8, 12} | Stable region / Subgroup (closed within itself) |
| 2H (green) | {2, 3, 10, 11} | Metastable (2x of H) |
| 4H (brown) | {4, 6, 7, 9} | Maximum deviation (4x of H) |
Why 4H is the "Maximum Deviation"
Specific recurrence relations, such as the Fibonacci and Pell sequences, never appear in 4H under mod 13 (algebraically proven in Chapter 3). The additive structure inherent in these recurrence relations avoids 4H. Note that nuclear structure is a quantum many-body system governed by the strong interaction, which is a different dynamic from the properties of recurrence relations.
Co-coset Relationship
"Multiplying" cosets lands you in a different coset:
2H × 4H = H ← Green and brown combined return to blue (Co-coset relationship)
H × H = H
4H × 4H × 4H = H (Closes when the sum of distances is a multiple of 3)
Closure Condition via Sum of Distances
Assign "distance" to each coset: H=0, 2H=1, 4H=2. When combining multiple terms, if the sum of distances ≡ 0 (mod 3), then the product closes in H.
1. Cosets of Protons and Neutrons
Reiterating the structure established in the previous article:
u quark = 1 (H, blue)
d quark = 2 (2H, green)
Proton (uud): 1 × 1 × 2 = 2 → 2H (green) Distance 1
Neutron (udd): 1 × 2 × 2 = 4 → 4H (brown) Distance 2
Proton (2H) × Neutron (4H) = H ✓ Distance 1+2=3≡0 (Nuclear closure)
In terms of the diagram, it is "Green × Brown = Blue." Two adjacent colors on the ×2 orbit combine to return to the stable region.
Regarding N=Z Nuclei (Nuclei with equal number of protons and neutrons)
Distance sum = 1×Z + 2×Z = 3Z → mod 3 = 0. Algebraically, it must close.
This is a structural necessity that precedes calculation. Since 2^Z · 4^Z = 8^Z (mod 13) and 8 ∈ H, all N=Z nuclei fall into H. The fact that N=Z nuclei serve as the starting point for stability is inscribed in the distance structure of Z₁₃.
2. Arrangement of Major Stable Nuclei
Evaluation formula: Product of Z protons = 2^Z mod 13, Product of N neutrons = 4^N mod 13, Total product = Product of both mod 13
| Nucleus | Z | N | Total Product | Coset |
|---|---|---|---|---|
| ²H (Deuterium) | 1 | 1 | 8 | H ✓ |
| ⁴He (Helium-4/alpha particle) | 2 | 2 | 12 | H ✓ |
| ¹²C (Carbon-12) | 6 | 6 | 12 | H ✓ |
| ¹⁶O (Oxygen-16) | 8 | 8 | 1 | H ✓ |
| ⁵⁶Fe (Iron-56) | 26 | 30 | 4 | 4H |
From deuterium to helium, carbon, and oxygen, light stable nuclei all fall into H. This is because the distance sum is maintained as a multiple of 3 in light nuclei where N=Z is approximate.
⁵⁶Fe (Iron-56) falls into 4H. We record this as an observation. The reason nuclear fusion ends near iron is a problem of continuous quantities known as the binding energy curve, and the discrete classification of mod 13 lacks that resolution. We stop at juxtaposing these two facts.
3. The Question of ⁸Be
Evaluating ⁸Be (Beryllium-8, Z=4, N=4):
Proton product: 2⁴ mod 13 = 3
Neutron product: 4⁴ mod 13 = 9
Total product: 3 × 9 = 27 mod 13 = 1 → H
It falls into H. Yet, ⁸Beryllium is unstable and immediately decays into two ⁴He (Helium-4/alpha particles).
⁸Be → ⁴He + ⁴He (Half-life ≈ 10⁻¹⁶ seconds)
Here lies the core of this article:
Falling into H is a necessary condition for stability, but not a sufficient one.
In the case of ⁸Beryllium, the two ⁴Helium nuclei (both H) cannot function as co-coset partners for each other within the nucleus. Two H blocks are simply juxtaposed without a co-coset mediator present inside. While closure is achieved, the internal structure does not form a co-coset pair.
B13 does not observe binding energy, spatial arrangement, or energy levels. Landing in H is a confirmation of "topological consistency," not the entirety of physical stability. This question is reserved for now.
4. Patterns in the Boundary Node Elemental Sequence
Elements where Z mod 13 = 0 are called "Boundary Nodes." This implies they are connection points to lower scales. We track the pattern of the proton products (2^Z mod 13) of these elements:
| Z | Element | Proton Product mod 13 | Coset |
|---|---|---|---|
| 13 | Aluminum (Al) | 4 | 4H |
| 26 | Iron (Fe) | 4 | 4H |
| 39 | Yttrium (Y) | 3 | 2H |
| 52 | Tellurium (Te) | 3 | 2H |
| 65 | Terbium (Tb) | 8 | H |
| 78 | Platinum (Pt) | 8 | H |
| 91 | Protactinium (Pa) | 11 | 2H |
| 104 | Rutherfordium (Rf) | 9 | 4H |
| 117 | Tennessine (Ts) | 5 | H |
It changes as 4H → 2H → H → 2H → 4H → H... This is not monotonic convergence; the power structure of the period 12 of 2^Z mod 13 appears as is. This is a mathematical necessity, not a discovery. We record it as a visualization.
5. Magic Numbers in mod 13
Let us check the mod 13 of magic numbers considered to provide "closed shells" in nuclear physics:
| Magic Number | mod 13 | Coset |
|---|---|---|
| 2 | 2 | 2H |
| 8 | 8 | H |
| 20 | 7 | 4H |
| 28 | 2 | 2H |
| 50 | 11 | 2H |
| 82 | 4 | 4H |
| 126 | 9 | 4H |
It is not monochromatic H. A simple correspondence of Magic Number = H does not hold. H, 2H, and 4H are mixed.
Magic numbers are determined by spin-orbit interaction and the shell model. Since mod 13 does not observe those, it is natural that they are mixed. We record this as an observation.
6. The Question of What Lies Outside the Closure
After protons and neutrons close into H through a co-coset relationship, electrons exist outside that closure.
Are electrons "residues" excluded from the nuclear closure, or are they independent terms forming another co-coset structure?
This article ends with this question left open.
Inside the nucleus, 2H × 4H = H operates.
Electrons are something outside of H.
Their algebraic positioning will be questioned in the next article.
Preview of the Next Article: Electron Closed Shells and Nuclear Magic Numbers
Before we step outside the nucleus, let us record one observation.
Comparing nuclear magic numbers with electron configuration closed shell numbers (maximum occupancy for each principal quantum number n, 2n²):
Nuclear magic numbers: 2, 8, 20, 28, 50, 82, 126
Electron closed shells: 2, 8, 18, 32, 50, 72, 98, 128
There is a mixture of numbers that match (2, 8, 50) and those that do not in the two sequences. Evaluating this with mod 13 reveals a striking structure:
| Value | Nuclear Magic | Electron Closed | Coset |
|---|---|---|---|
| 2 | ✓ | ✓ | 2H |
| 8 | ✓ | ✓ | H |
| 18 | — | ✓ | H |
| 20 | ✓ | — | 4H |
| 28 | ✓ | — | 2H |
| 32 | — | ✓ | 4H |
| 50 | ✓ | ✓ | 2H |
| 72 | — | ✓ | 4H |
| 82 | ✓ | — | 4H |
| 98 | — | ✓ | 4H |
| 126 | ✓ | — | 4H |
| 128 | — | ✓ | 2H |
Cosets of closed shells shared by nuclei and electrons (2, 8, 50): 2H, H, 2H → No 4H.
Closed shells unique to nuclei or electrons: 4H dominates.
Only when nuclei and electrons "have the same closed shell number" does their coset avoid 4H. We leave this as an observation for now. Only after the algebraic positioning of electrons is established can a logical foothold be created for this observation.
Summary
B13 Positioning:
A "coarse classifier" that categorizes nuclear structure using mod 13 cosets.
Describes the shadow of symmetry that remains even after compression.
Does not handle the entirety of physical stability (energy, spin, arrangement).
Stability of N=Z nuclei is a structural necessity:
2^Z · 4^Z = 8^Z (mod 13), 8 ∈ H → Must go to H.
Major stable nuclei:
²H, ⁴He, ¹²C, ¹⁶O → H (near N=Z, distance sum = multiple of 3)
⁵⁶Fe → 4H (recorded as an observation; we do not claim correspondence with continuous quantities).
The question of ⁸Be (Beryllium-8):
Total product = H, but decays immediately.
Falling into H is a necessary condition but not a sufficient one (reserved).
Boundary node elemental sequence (Z mod 13 = 0):
Pattern following the power structure of period 12 (mathematical necessity).
Magic numbers in mod 13:
H, 2H, 4H mixed (spin-orbit interaction is outside of mod 13).
Next question:
Only closed shells shared by nuclei and electrons avoid 4H (observation).
Toward the algebraic positioning of electrons.
From B13 Framework / Handover v86
Toward the establishment of the Second Major Node (atomic/molecular scale)
Discussion