🧮 Shunyaya Residual Capacity Theory (SRCT)

🧩 Can Modulus 9 Appear Exactly Twice in an Odd Covering System?

A computer-assisted mathematical proof rules out an entire finite covering-system class.

🔢 A surprisingly specific question

In number theory, a covering system is a finite collection of congruences whose residue classes together cover every integer.

Here, the modulus is the repeating step size of a congruence class.

For example, each congruence has the form

x == a_i (mod m_i)

Now impose three conditions:

every modulus is odd and greater than 1;
modulus 9 appears exactly twice;
every other modulus appears at most once.

A natural question is:

Can such a covering system exist?

The answer is:

No.

That is the central result of Shunyaya Residual Capacity Theory (SRCT), presented as a computer-assisted mathematical proof.

The theorem is class-wide: it does not eliminate a few examples or a bounded search region. It rules out every finite covering system satisfying those exact conditions.

🔗 Explore the complete SRCT repository on GitHub

🧩 Why modulus 9 matters

A standard quantity in this area is t_9: the least number of times modulus 9 must occur in a covering system whose remaining moduli are odd, distinct, and greater than 1.

Published constructions give

t_9 <= 3

SRCT proves

t_9 != 2

So the remaining possibilities are

t_9 in {0,1,3}

That distinction matters.

SRCT does not prove t_9 = 3.

The cases where modulus 9 occurs zero times or once remain outside this theorem.

What has been eliminated is the entire exact-two case.

🌐 The difficult part: going from finite computation to a universal statement

A computer can exhaust a finite state space.

But the theorem concerns arbitrary finite covering systems, whose least common multiple may contain additional primes or higher prime powers.

So a finite search alone is not enough.

SRCT uses the finite modulus

K = 11486475 = 3^3 * 5^2 * 7 * 11 * 13 * 17

as an obstruction core.

Importantly,

K is not an assumed upper bound on the LCM.

Instead, the proof establishes a mathematical bridge showing that additional prime support and higher prime-power depth cannot escape the obstruction detected at the finite core.

In simplified form:

finite exact obstruction + universal extension argument -> class-wide nonexistence

That bridge is what turns the finite computation into a theorem about every finite covering system satisfying the three conditions above.

🧮 What the computer actually does

The proof separates two responsibilities.

The mathematical layer establishes the reductions, projection principles, lifting arguments, residual-deficit transport, p-adic layer behavior, non-erosion inequalities, and the final contradiction.

The computational layer verifies the finite obstruction using exact integer and rational arithmetic.

Among the frozen finite landmarks are:

45 unordered placements of the two modulus-9 residues
896 balanced q45 states
19712 direct q63 continuations
3030 hard late-parent groups
138448 hard engaged edges
0 nonpositive q99 terminal bundles

No floating-point approximation is needed for the theorem-critical finite calculations.

✅ Reproducibility is part of the result

The repository was built so that the theorem-supporting computation can be checked rather than simply trusted.

Current verification endpoints include:

556/556 PASS — repository self-test

565/565 PASS — standard theorem binding

335105/335105 PASS — strengthened adversarial proof audit

The repository also contains 114 scientific files whose exact identities are cryptographically bound, including the theorem manuscripts, theorem-binding objects, and a byte-preserved 102-object predecessor chain.

The verification system checks issues such as:

quantifier_gap:NONE

domain_escape:NONE

inequality_direction_failure:NONE

finite_interface_mismatch:NONE

circular_dependency:NONE

scope_overclaim:NONE

These checks do not replace the mathematics.

They help verify that the mathematical argument, finite evidence, and computational interfaces remain consistently bound together.

⚖️ What is — and is not — being claimed

The result is intentionally narrow.

SRCT establishes:

No finite odd covering system exists with modulus 9 occurring exactly twice while every other modulus occurs at most once.

It does not establish:

impossibility when modulus 9 occurs zero times;
impossibility when modulus 9 occurs once;
t_9 = 3;
formal verification in a proof assistant;
absolute bibliographic priority.

The universal mathematical lemmas have not been formally verified in a proof assistant. The result is therefore presented as a computer-assisted mathematical proof, supported by exact computation, reproducible verification, and strengthened consistency audits.

🔍 Why This Result Is Interesting

The most interesting part is not simply that a large computation returned “no.”

The deeper structure is:

a finite obstruction can survive arbitrary finite enlargement when the proof establishes the right transport and non-erosion principles.

That changes the role of computation.

Instead of asking the machine to somehow inspect an unbounded family of finite cases, the proof identifies a finite obstruction and then proves mathematically that additional prime support or higher prime-power depth cannot repair it.

In compact form:

universal proof + exact finite obstruction + reproducible theorem binding

-> class-wide modulus-9 exact-two nonexistence

🌐 Explore the full proof

The GitHub repository contains the normative theorem manuscript, exact computational evidence, frozen scientific chain, theorem dependency map, reproducibility records, verification tools, SHA-256 scientific identities, proof-audit certificates, and automated verification workflow.

🔗 Shunyaya Residual Capacity Theory on GitHub

The repository is the right place for the complete mathematical argument and technical details.

🌌 The larger question

A simple-looking question started the journey:

Can modulus 9 occur exactly twice?

For this entire class of finite odd covering systems, the answer is now:

No.

What remains is just as interesting:

t_9 in {0,1,3}

And that leaves the deeper unresolved cases still open.

One exact multiplicity is gone. The surrounding number-theoretic landscape remains.

OMP