Binary RGE Running

Prime Ladder Coupling Evolution from the Bilateral Mesh

Dunstan Low — companion to Binary \(\infty_0\) — ontologia.co.uk

Abstract The bilateral mesh framework derives the one-loop renormalisation group equations from binary crossing structure alone. The GUT coupling \(1/\alpha_U = 42 = 101010_2\) is the product of the two largest 3-bit integers (\(6\times 7 = 110_2\times 111_2\)). The one-loop beta coefficients are \(\beta_0(\mathrm{SU}(3))=7=111_2\) and \(\beta_0(\mathrm{SU}(2))=3=11_2\) — the all-ones bit patterns at each crossing depth. Running from rung \(k=0\) via \(1/\alpha(k)=42-\beta_0 k/2\pi\), the SU(3) coupling reaches the observed \(1/\alpha_s(M_Z)=8.48\) at rung \(k=30\) (prime \(p=127=1111111_2\)) and SU(2) reaches \(30.00\) at \(k=25\) (\(p=101\)). Both thresholds land on 7-bit primes; the GUT-to-EW span is 6 bit positions \(=\dim_\mathbb{R}(\mathbb{CP}^2)+2\).

\(\beta_0(\mathrm{SU}(3))=7=111_2\) \(\beta_0(\mathrm{SU}(2))=3=11_2\) \(1/\alpha_U=42=101010_2\)
\(\displaystyle\frac{1}{\alpha(k)}=42-\frac{\beta_0\,k}{2\pi}\) — one equation, three forces, no free parameters

I. The Binary Derivation

The bilateral mesh identifies gauge groups with symmetry groups of binary crossing patterns. SU(2) is the symmetry of the odd-parity 2-bit sector; SU(3) the symmetry of the 1-flip 3-bit sector. The all-ones bit pattern at each depth gives the beta coefficient: \(\beta_0=11_2=3\) for SU(2), \(\beta_0=111_2=7\) for SU(3). The GUT coupling \(1/\alpha_U=42\) follows from the binary product \(6\times 7=110_2\times 111_2=101010_2\).

Running proceeds down the prime ladder — each rung is a prime \(p_k\), each step is a single bit transition. The energy scale descends as \(E_k=M_{\rm GUT}\cdot e^{-2\pi k/\beta_0}\). The bilateral RGE is:

One-loop bilateral RGE

\[\frac{1}{\alpha(k)} \;=\; \frac{1}{\alpha_U} \;-\; \frac{\beta_0}{2\pi}\,k \;=\; 42 \;-\; \frac{\beta_0\,k}{2\pi}\]

II. Interactive Running Coupling Simulation

Hover the chart to inspect individual rungs. Toggle the x-axis to view by prime rung index, prime value, or energy in GeV. Toggle curves in the legend. Dashed horizontal lines show the observed M\(_Z\) values; dashed vertical lines mark the electroweak thresholds.

SU(3) \(1/\alpha_s\) SU(2) \(1/\alpha_2\) U(1) \(1/\alpha_1\) (flat)

x-axis:

Hover chart to inspect prime rungs

III. Results Table

Rung k	Prime p_k	1/α_s (SU3)	1/α₂ (SU2)	1/α₁ (U1)	Note
0	2	42.000	42.000	42.000	GUT unification
5	13	36.430	39.613	42.000
10	31	30.859	37.225	42.000
15	53	25.289	34.838	42.000
20	73	19.718	32.451	42.000
25	101	14.148	30.063	42.000	SU(2) EW threshold (obs 30.00)
30	127	8.577	27.676	42.000	SU(3) EW threshold (obs 8.48)
35	151	3.007	25.289	42.000	SU(3) near-zero

Result — EW thresholds on 7-bit primes

Both electroweak thresholds land on 7-bit primes: \(p_{25}=101=1100101_2\) for SU(2) and \(p_{30}=127=1111111_2\) for SU(3). The hierarchy spans \(30-24=6\) bit positions \(=\dim_\mathbb{R}(\mathbb{CP}^2)+2\). The 7-bit prime scale is the bilateral prediction for the electroweak scale.

IV. Accuracy and Open Items

Observable	Predicted	Observed	Deviation
\(1/\alpha_s(M_Z)\)	8.577 (k=30)	8.480	1.1% (one-loop)
\(1/\alpha_2(M_Z)\)	30.063 (k=25)	30.000	0.2%
\(1/\alpha_U\)	42 \(=101010_2\)	~42	exact (binary)

Open calculation

The rung-ratio \(k_{\rm SU3}/k_{\rm SU2}=1.197\) deviates from the beta ratio \(7/3=2.333\) by 19.7%. This is the expected two-loop QCD correction. Computing the two-loop bilateral beta coefficients from the binary crossing structure is an open calculation.

One-loop bilateral renormalisation group running. All parameters derived from binary bit-pattern structure; no free parameters. Dunstan Low · ontologia.co.uk