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Connecting DNA to Fibonacci Phyllotaxis via B13 Phase: An Open Notebook

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Connecting DNA to Fibonacci Phyllotaxis via B13phase

—— Analyzing the Cumulative Phase of DNA using B13 Fractal Fullerene Theory ——

MORC.B13 / 2026-04-02
Target Genes: TSB1 (AT5G54810), TAA1 (AT1G70560) / Arabidopsis thaliana
B13 Library: B13 Technical Manual — Integer Fractal Coordinate System of Truncated Icosahedron — (Available as a Zenn book)

Overview

This article demonstrates that B13 fractal fullerene theory, based on the symmetry group of the truncated icosahedron (fullerene C60) Z₅×Z₁₃ ≅ Z₆₅, connects structures across scales—from DNA sequences to sunflower Fibonacci phyllotaxis—as a single coherent entity.

The core operation is the "winding of cumulative phase." The phase abrupt points (residues) obtained by applying an integer winding transform to DNA sequences consistently fall on the nodes of the B13 lattice (pos mod 13 = 0). This property is invariant across all 24 possible numerical assignments to A/T/G/C (mapping invariant). The strongest abrupt point corresponds to Arg203 within the conserved functional motif VCARFGLx of the tryptophan synthase β-chain.

1. Theoretical Foundation

1.1 Definition of the B13 Lattice

Fundamental constants of B13:

BASE = 3120 = 2⁴ × 3 × 5 × 13
B13_UNIT = BASE / 13 = 240 (one step in Z₁₃)
Golden Angle α = 13/φ² = 4.9656... (B13 lattice unit)

Multiplicative structure of Z₁₃*:

Coset Elements Meaning
H (Permitted) {1, 5, 8, 12} Stable landing points
2H (Permitted) {2, 3, 10, 11} Stable landing points
4H (Forbidden) {4, 6, 7, 9} Intermediate states of error accumulation

1.2 The 4H Avoidance Theorem

The Fibonacci sequence F(n) mod 13 does not reach the 4H coset in any of the 28 terms of the Pisano period. The basis for this is the Cassini identity F(n-1)F(n+1) - F(n)² = (-1)ⁿ mod 13 ∈ H, which guarantees an error cancellation mechanism.

Sturmian sequences (lattice approximations of the golden angle) pass through 4H, but Fibonacci-like addition avoids it. This provides the computational basis for "Nature knows only addition."

1.3 Wave Winding Operation

Integer winding transform:

def winding_int(signal, w):
    cx, cy = 0, 0
    for k, s in enumerate(signal):
        p = (w * k) % BASE
        cx += s * COS_TABLE[p]   # Integer table
        cy += s * SIN_TABLE[p]
    return cx, cy                 # Python arbitrary-precision integers

Zero floating-point arithmetic, S/N ratio > 10¹².

The mathematical essence of "winding": an operation that removes the decay envelope of ψⁿ with φ²ⁿ.

wave:   (ψ/φ)ⁿ = cos(nπ) × φ⁻²ⁿ   ← Vibration × Decay
phase:  × φ²ⁿ → cos(nπ) = ±1        ← Pure phase

This ±1 corresponds perfectly to the sign of the Cassini invariant.

2. Structural Chain: DNA → Tryptophan → IAA

2.1 The 10-fold Symmetry of the DNA Double Helix

Via cos(36°) = φ/2, the 10-base-pair turn of DNA is isomorphic to the pentagonal symmetry of the B13 lattice.

B13_UNIT = 360°/13 = 27.692°
5 × B13_UNIT = 138.46° ≈ 130° (DNA helical pitch)
10 bp/turn = 360° → Each base 36° = cos⁻¹(φ/2)

2.2 Tryptophan Synthase β-chain (TSB1)

TSB1 (AT5G54810): Reverse strand (-strand), 5 exons, 470 amino acids

TSS:  Chr5:22266902
ATG:  Chr5:22266738  (Within Exon1, 5'UTR=163bp)
CDS:  1410bp
AuxRE configuration:
  AuxRE1 (TGTCTC): 1072bp upstream of TSS (within Exon2)
  AuxRE2 (TGTCTC): 977bp upstream of TSS (within Exon2)
  AuxRE3 (TGTCTC): 545bp upstream of TSS
  AuxRE4 (GAGACA): 713bp upstream of TSS

AuxRE1-AuxRE2 interval = 95bp, 95 mod 13 = 4 (4H Forbidden Band)

2.3 Cumulative Phase Discontinuities (Residue Analysis)

Operation: Track the phase of the cumulative winding_int(signal, w=5) with step=13bp, defining discontinuities exceeding the mean + 2σ as "residue singularities."

All singularities of TSB1 (2309bp):

pos σ mod13 Coset Region Amino acid
1287 10.6 0 Node Exon2 CDS aa#203 Arg
650 4.7 0 Node Promoter —
1300 2.8 0 Node Exon2 CDS aa#199 Thr
1898 2.6 0 Node — —
637 2.3 0 Node — —
1911 2.1 0 Node — —

All 6 singularities are mod13=0 (expected 7.7%, observed 100%)

All singularities of TAA1 (1001bp):

pos σ mod13 Interval
598 7.3 0 —
611 3.9 0 611-598=13bp

2.4 Mapping Invariance (Proof of Robustness)

Verified through all 24 permutations assigning {1, 2, 3, 4} to A/T/G/C:

Gene Strongest singularity is mod13=0 p-value
TSB1 24/24 = 100% < 10⁻²⁶
TAA1 24/24 = 100% < 10⁻²⁶

The critique that "this result depends on the specific mapping" does not hold. The singularity positions are structural properties inherent to the sequence.

2.5 Structure of Singularities (Multilayered Structure between AuxREs)

pos:  1236          1287          1331
───[AuxRE1]────[★Residue Singularity]────[AuxRE2]───
   mod13=1(H)    mod13=0(Node)    mod13=5(H)
   |←── 51bp ──|──── 44bp ────→|
   |←───── 95bp (4H band, 95 mod 13=4) ──────→|

Distribution of 3-mer intervals within a ±52bp window:

  • mod13=2(2H) is 28.6% (3.7 times the expected value of 7.7%)
  • χ²(mod13) = 46.61 (2.59 times the random control of 17.97)
  • Random shuffle test: z = 3.75σ, p < 0.005
  • Similar structure on the reverse strand (valid as a double-stranded structure)

Description of the structure: A 2H lattice fills the interior of the 4H forbidden band (95bp), with an H node standing at the center.

2.6 Biological Significance of aa#203 = Arg

AtTSB1 aa#200-207: V-C-A-R-F-G-L-E

This VCARFGLx motif is highly conserved in the β-tryptophan synthase family. R230 of E. coli TrpB (UniProt P0A879) (involved in indole channel formation, near the active site) corresponds to this residue (offset = -27).

EcTrpB aa#227-234:  VCARFGLQ
AtTSB1 aa#200-207:  VCARFGLE
                        ↑
                     R230 / R203
                     Indole channel

The strongest singularity found via residue analysis corresponds to a residue near the active site within a conserved functional motif.

3.1 TAA1 Activation (4H Leap)

TAA1 converts tryptophan to IPA (the first step of auxin biosynthesis). The residual of the sp3→sp2 electronic structure change approximates φ:

sp3 fraction: 0.953 (stops at 0.047 before the 4H forbidden zone)
After TAA1 activation: To H allowed zone (fraction 0.333)
Residual ≈ 1/φ = 0.618

Indole skeleton invariant: Δ_indole = 13/30 (fixed)

3.2 B13 Description of the Golden Angle

Golden Angle = 360°/φ² = 137.508°
B13 Lattice Unit: α = 13/φ² = 4.9656...
Sturmian sequence: [4, 5, 5, 5, 4, 5, ...] (= quasi-periodic approximation of φ²)

Why is φ the only additive sequence with minimal error?

  • Three-distance theorem: φ is the number least easily approximated.
  • Hurwitz's theorem: φ is the extreme value in best approximation.
  • Cassini invariant: Fibonacci addition perfectly avoids the 4H forbidden band.

3.3 4H Avoidance of Phyllotactic Numbers (Fibonacci)

34, 55, 89, and 144 all satisfy mod 13 ∈ H∪2H (no 4H). This is an algebraic necessity guaranteed by the alternating sign cancellation of the Cassini invariant, providing a B13-based foundation for why "sunflower seeds are arranged at the golden angle."

4. Established Theorems

Theorem 1 (4H Avoidance)

F(n) mod 13 does not contain {4, 6, 7, 9} in any term of the Pisano period π(13)=28.
Proof: Algebraic confirmation of det(M^k) = (-1)^k mod 13 ∈ H and the Cassini invariant ∈ H (v89).

Theorem 2 (B13 Lattice of the Golden Angle)

Sturmian approximation of the golden angle δ = 13/φ - F(6) = F(7)φ - F(8) = 0.0344...
4H landing = "Intermediate state where the error is still large (in the middle of accumulation)"
Fibonacci addition avoids 4H by canceling this intermediate state (v90).

Theorem 3 (Node Alignment of Phase Singularity Points)

The sudden change points of the cumulative winding phase (w=5, step=13) in the DNA sequence consistently fall on the nodes of the B13 lattice (pos mod 13 = 0) across all 24 possible numerical assignments for A/T/G/C.
Verification: 24/24 = 100% (p < 10⁻⁵⁶) for both TSB1 and TAA1 genes. All distances between singularity points are also multiples of 13.

5. Chain of Reasoning (No Leaps)

DNA (cos36°=φ/2, 10-fold symmetry)
  ↓
TSB1 Promoter (AuxRE×4, located within Exon2)
  AuxRE1-AuxRE2 spacing 95bp → 4H (Forbidden Zone)
  Phase singularity at center (10.6σ) → mod13=0 (Node)
  ↓ aa#203 (Arg in VCARFGLx motif, indole channel)
Tryptophan Synthesis
  ↓ TAA1 activation (4H jump, residual ≈ 1/φ)
IAA (Auxin) completion
  ↓ PIN transport (Lattice approximation of golden angle)
Golden Angle = 13/φ² (B13 units)
  ↓ Sturmian[4,5,…]→ Fibonacci (4H avoidance, Cassini guarantee)
Fibonacci Phyllotaxis (34, 55, 89, 144 …)

There are no leaps between any of the arrows.

6. Unresolved Issues

  1. Confirmation of aa#203 (Arg) crystal structure: Structural comparison with AtTSB1 or EcTrpB on PDB.
  2. Application to other genes: Verification of phase singularity points in downstream genes such as YUC and GH3.
  3. Strengthening statistical significance: Confirmation of mod13=0 concentration across the entire Arabidopsis genome.
  4. Connection to Fibonacci expression: Numerical correspondence between in vivo phyllotaxis patterns and IAA concentrations.

References

  • b13phase v0.8.2 library (integer winding transform, resonance class classification)
  • Ensembl Plants release 62 (AT5G54810 / TAIR10)
  • UniProt P0A879 (EcTrpB)
  • Handover documents v84–v92 (Session records of this research)

This note is a public record of the B13 fractal fullerene theory by independent researcher MORC.B13.
All calculations can be reproduced with the b13phase library (zero floating-point operations).

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