🌟 Shunyaya Orbit Stabilization Theory (SOST)

🧩 Can Monochromatic Triangle Dynamics Be Classified Exactly?

From six vertices onward, degree parity and edge count modulo `3` completely determine reachability between labeled graph states.

🔺 A tiny local move, a global question

Take a simple graph on n labeled vertices.

Choose any three vertices and inspect the three edges between them.

A move is allowed only when those three edge bits are monochromatic:

000 <-> 111

So an empty triangle can become a complete triangle, and a complete triangle can become empty.

The rule is local, reversible, and state-dependent.

But its global behavior creates a much larger question:

Given two graph states, can we determine exactly whether one can be transformed into the other?

For every n>=6, the answer is:

Yes.

That is the central theorem of Shunyaya Orbit Stabilization Theory (SOST).

🔗 Explore the complete SOST repository on GitHub

🎯 Orbit equivalence is completely determined by two invariants

For a graph G, define

P(G) = vertex degree-parity vector

and

R(G) = |E(G)| mod 3

Then, for every n>=6,

G ~ H iff P(G)=P(H) and R(G)=R(H)

where G ~ H means that G can be transformed into H through a sequence of legal monochromatic triangle toggles.

So two graph states are in the same orbit exactly when:

every corresponding vertex has the same degree parity; and
the two graphs have the same number of edges modulo 3.

Nothing else is needed.

The global state space collapses to a remarkably small invariant description.

🔢 The number of orbits is exact

The quotient has the structure

Q_n ~= {even parity vectors in F_2^n} x Z_3

and therefore

|Q_n| = 3*2^(n-1)

Even more strongly,

Q_(n+1) ~= Q_n x {0,1}

So increasing the graph order by one introduces exactly one additional binary quotient coordinate.

In compact form:

local reversible rule -> complete invariants -> exact orbit quotient -> binary growth

This is not merely an asymptotic count.

It is an exact structural description of the orbit space.

6️⃣ Why six vertices matter

The threshold is unexpectedly sharp — and non-monotone.

The exact small-order behavior is:

n=3 -> 7 orbits; invariant complete on realized labels

n=4 -> 32 orbits; invariant incomplete

n=5 -> 59 orbits; invariant incomplete

n=6 -> 96 orbits; invariant complete

and then

n>=6 -> completeness persists

So the invariant works at order three, fails at four and five, and then becomes permanently complete from six onward.

That makes n=6 the sharp permanent threshold.

There is also a structural reason six appears.

Five vertices are exactly sufficient to realize every admissible local combination of even degree parity and edge-count residue modulo 3. Those five vertices provide the local target structure needed to control a sixth designated vertex, which is then compressed and eliminated by the induction.

🧠 How an infinite theorem comes from a finite core

At six vertices there are

2^15 = 32,768

possible graph states.

The complete move graph splits into exactly

96 = 3*2^5

orbits.

SOST packages this finite result as a standalone certificate: every non-root state records a legal parent transition, and the 96 roots correspond exactly to the admissible invariant labels.

But checking K_6 alone would not prove the result for every larger graph.

The essential step is constructive.

The six-vertex theorem provides local compression operations that can be embedded unchanged inside larger graphs. A designated vertex can be systematically reduced until it is either isolated or compressed to one controlled incident edge.

The induction then eliminates vertices one at a time.

So the proof has the form

finite exact theorem base + constructive compression + induction -> every n>=6

The finite computation is the starting kernel.

The universal theorem comes from the symbolic reduction that follows it.

🧪 An independent test at more than two million states

The proof does not depend on an order-seven computation.

Instead, K_7 is kept outside the theorem dependency chain as an independent falsification test.

The exhaustive calculation checks all

2,097,152

binary edge states of K_7

and obtains exactly

192 = 3*2^6

orbits, with zero invariant-classification mismatches.

It also audits

18,350,080

legal state/triangle transitions.

The theorem predicts 192.

The independent exhaustive computation produces 192.

✅ Reproducibility is part of the result

The repository includes:

the complete theorem and proof;
the standalone K_6 certificate;
constructive induction replay;
independent small-order verification;
exhaustive C++17 K_7 verification;
machine-readable evidence;
proof-rigor checks; and
automated GitHub verification.

The standard repository commands are:

python -B verify.py --self-test

python -B verify.py --verify

The current repository verification passes its declared checks, while the theorem for every n>=6 remains the written mathematical proof rather than a finite extrapolation.

⚖️ What is — and is not — being claimed

The theorem is precise.

It establishes a complete orbit classification for the stated monochromatic triangle toggle on simple labeled complete-graph edge spaces for every n>=6.

It establishes:

complete invariant

(P(G), R(G))

exact orbit count

3*2^(n-1)

exact quotient growth

Q_(n+1) ~= Q_n x {0,1}

and a constructive mechanism explaining why completeness persists.

It does not claim that arbitrary reversible dynamical systems must admit the same invariant structure or stabilization law.

The broader structural ideas are kept separate from the proved graph theorem.

🌐 Explore the complete theorem

The GitHub repository contains the full proof, theorem architecture, exact quotient-growth result, finite certificates, constructive replay, independent holdout, software, machine-readable evidence, computational integrity records, and reproducible verification.

🔗 Shunyaya Orbit Stabilization Theory on GitHub

The repository is the appropriate place for the complete mathematical and computational details.

🌌 The larger picture

The local rule could hardly be simpler:

000 <-> 111

Yet the resulting state space grows exponentially.

What is striking is that, from six vertices onward, the entire reachability relation reduces exactly to:

degree parity + edge count modulo 3

with