A Structural–Arithmetic Reframing of Primes and Composites
For centuries, prime numbers have been treated as a binary truth.
Prime.
Composite.
Yes or no.
Once the label is assigned, classical mathematics moves on.
But this raises a deeper question:
If primes are fundamental, why do we observe them only as labels — and not as structures?
Structural Primality asks a simple but precise question:
What if we observe how numbers resist factorization, not just whether they factor?
🧠 Beyond “Is It Divisible?”
Classical number theory asks:
Is n divisible by some d?
Structural Primality asks:
How close does n come to closing under divisibility — and how does that resistance behave?
Instead of treating primality as a silent outcome, Structural Primality treats it as a measurable structural phenomenon.
The result is not a new definition of primes —
but a new layer of observability around them.
📐 The Rule Is Strictly Classical
Structural Primality does not relax mathematics.
For every integer n >= 2, divisors are tested exactly in the classical bounded range:
2 <= d <= floor(sqrt(n))
If any d satisfies:
n mod d = 0
→ the number is composite.
If no such d exists:
→ the number is a STRUCTURAL_PRIME.
This preserves exact classical correctness:
• every classical prime remains prime
• no composite is misclassified
• no approximation is introduced
The difference is not correctness — it is visibility.
🔍 What Structural Primality Actually Records
Structural Primality does not stop at PASS / FAIL.
For every integer, it records a structural footprint.
For composite numbers:
closure_d— the smallest divisor that causes closure
For all numbers (prime or composite):
closest_d— the nearest divisor candidateclosest_band— a discretized proximity band (AtoF)
These bands describe how near a number is to factorization, even when no factor exists.
Primes are no longer silent.
They exhibit resistance patterns.
📌 Signature-Based vs Exact Proximity
By default, Structural Primality measures proximity using a configurable signature set of small primes (default <= 101).
This is intentional.
It reveals persistent pressure from foundational divisors — the primes that dominate arithmetic structure.
For exact minimal proximity across all d <= floor(sqrt(n)), Structural Primality supports:
--full_closest
This performs a full bounded scan.
Slower — but exact.
Both modes are deterministic.
Both are auditable.
Both preserve correctness.
🧪 What You Can Run (Immediately)
Structural Primality is not a paper exercise.
It produces:
• row-level structural records for every integer
• deterministic summary statistics
• reproducible plots
• finite, enumerable outputs
The same inputs always produce the same results.
No randomness.
No heuristics.
No learning loops.
This is an audit, not an inference.
📊 What the Results Reveal
When evaluated over large ranges (e.g., n <= 100,000):
• classical prime counts match exactly
• closure depth is highly non-uniform
• small divisors dominate closure behavior
• primes cluster in specific structural bands
• resistance to factorization is structured, not random
The key insight:
Primality is not just absence of divisibility.
It is structured resistance to closure.
✨ Why This Matters
Structural Primality does not replace number theory.
It adds something number theory does not record:
• how close numbers come to breaking
• where divisibility pressure accumulates
• how primes differ structurally, not just positionally
This opens new ways to:
• teach primality with intuition
• analyze integer structure deterministically
• debug arithmetic assumptions
• compare number ranges structurally
All without approximation.
All without speculation.
🧭 Observability, Not Optimisation
Structural Primality does not:
• predict primes
• accelerate factorization
• claim cryptographic strength
• replace existing theory
It provides observability only.
All interpretation lives above the result.
The arithmetic remains untouched.
🔗 Where the Work Lives
🔬 Executable Verification
https://github.com/OMPSHUNYAYA/SSUM-Structural-Primality
📘 Master Docs Repository
https://github.com/OMPSHUNYAYA/Shunyaya-Symbolic-Mathematics-Master-Docs
📘 License & Usage
Open Standard — provided as-is.
You may read, study, run, and build upon the work.
Optional attribution (not mandatory):
“Implements concepts from Shunyaya Structural Universal Mathematics (SSUM).”
⚠️ Disclaimer
Research and observation only.
Not intended for cryptography, security guarantees, or probabilistic claims.
OMP
SHUNYAYA 