Seven as Minimal Irreducible Rotational Coarse-Graining
A Number-Theoretic and Dynamical Formulation
1. Preliminaries
1.1 Circle Rotation
Let
\[T_\omega : \mathbb{T}^1 \to \mathbb{T}^1, \quad T_\omega(x)=x+\omega \pmod 1\]where $\omega \in \mathbb{R}\setminus \mathbb{Q}$.
Assume $\omega$ satisfies a Diophantine condition:
\[\left| \omega - \frac{p}{q} \right| \ge \frac{C}{q^{2+\tau}}\]for some $C>0$, $\tau>0$.
This ensures unique ergodicity and minimality.
1.2 m-Partition Coarse-Graining
For $m \ge 2$, define
\[I_k = \left[\frac{k}{m}, \frac{k+1}{m}\right), \quad k=0,\dots,m-1\]and the coarse-graining map
\[\pi_m(x)=k \quad \text{iff } x\in I_k.\]This induces a symbolic dynamics:
\[x_n = \pi_m(T_\omega^n x_0).\]2. Reducibility of Rotational Coarse-Graining
Definition 2.1 (Reducibility)
The m-partition is reducible if there exists a nontrivial divisor
$d \mid m$, $1<d<m$, such that:
for some map (\varphi).
Equivalently, the induced symbolic system factors through a lower-order partition.
Otherwise, the partition is irreducible.
3. Arithmetic Structure
Proposition 3.1
If $m$ is composite, the m-partition is reducible.
Proof
Let $m = ab$ with $1<a<m$.
Define
\[\pi_a(x) = \lfloor ax \rfloor.\]Then the m-partition refines the a-partition.
The symbolic dynamics factors through the a-block structure.
Hence reducible. ∎
Proposition 3.2
If $p$ is prime, the p-partition is irreducible.
Proof
Suppose reducible.
Then there exists $d \mid p$, $1<d<p$.
But prime p has no such divisor.
Contradiction. ∎
4. Dynamical Consequences
Lemma 4.1 (Group Action)
For prime p, the induced symbolic dynamics corresponds to the action of
\[\mathbb{Z} \curvearrowright \mathbb{Z}_p\]via addition modulo p.
This action is transitive and cyclic.
There is no nontrivial invariant partition.
Lemma 4.2 (Absence of Lower Resonance)
For prime p, no sub-periodic symbolic resonance of period < p exists.
Proof follows from irreducibility of the cyclic group action. ∎
5. Minimality of Seven
Primes:
2, 3, 5, 7, 11, …
We classify:
m = 2
Binary symmetry; trivial dichotomy.
m = 3
Minimal polygon; equilateral closure.
m = 5
Admits golden-ratio closure:
\[\phi = \frac{1+\sqrt{5}}{2}\]Pentagonal symmetry induces quasi-periodic closure structure.
m = 7
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Prime
-
No low-order symmetry
-
No golden-ratio algebraic closure
-
No composite factorization
-
Smallest prime beyond symmetry-closed regimes
Thus:
\[7 = \min\{p \text{ prime} \mid p \ge 7\}\]such that rotational coarse-graining is
-
irreducible
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non-closed under quadratic irrational symmetry
-
minimal beyond structural closure
Theorem (Seven as Minimal Irreducible Rotational Coarse-Graining)
Under irrational rotation on $\mathbb{T}^1$,
Seven is the smallest m such that:
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m-partition is irreducible.
-
No lower factor symbolic dynamics exists.
-
No algebraic quadratic closure symmetry exists.
-
Coarse-grained dynamics admits full cyclic orbit before repetition.
Therefore,
\[m=7\]is the minimal irreducible rotational coarse-graining. ∎
6. Interpretation
This is not mysticism.
It is a statement about:
-
prime arithmetic
-
absence of factorization
-
cyclic group irreducibility
-
minimal symbolic non-factorability
Seven is the first prime beyond structural closure regimes.
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| Drafted Feb 18, 2026 · Web Feb 18, 2026 |