docs/PROBLEMS.md

Problem Status & Solved Instances

This document tracks the status of all combinatorial problems, algebraic proofs, and competition benchmarks handled by the Short Exact Sequence (SES) Framework.

1. Decompositions of $\mathbb{Z}_m^k$ (Claude's Cycles)

The core task is to find a set of $k$ permutations $\sigma_c$ that decompose the Cayley graph of $\mathbb{Z}_m^k$ into $k$ disjoint Hamiltonian cycles.

Problem	Parameters	Method	Iterations	Best Score	Status
P1	$k=4, m=4$	Fiber-Structured SA	50M	0	Solved
P2	$k=3, m=6$	Multi-Fiber Basin Escape	500k	0 (via Repair)	Solved
P3	$k=3, m=8$	Sovereign Solver (Obstruction)	$O(1)$	--	Proven Impossible
Odd $m$	$k=3, m \in {3, 5, \dots}$	Sovereign Spike	$O(m)$	0	Analytically Proven

2. Multi-Modal Manifolds

Domain	Sizing	Metric	Status
Vision	$G_{256}^5$	Cohomological Gradient	Stable (v2.0)
Neural	$G_{255}^3$	Topological Entropy	Stable
Knowledge	$G_{256}^4$	Closure Hash Density	Stable (v16.0)
Frontier	$G_{256}^{128}$	Hilbert Spectrum	Stable (v1.0)

3. Proven Impossibilities ($H^2$ Parity Obstructions)

Configurations are strictly PROVED IMPOSSIBLE if $m$ is even and $k$ is odd.

Configuration	Parameters	Group	Reason
Even $m$, $k=3$	$m \in {4, 6, 8, \dots}$	$\mathbb{Z}_m^3$	$H^2$ Parity Obstruction
Heisenberg	$m=6, k=3$	$Heis(\mathbb{Z}_6)$	Non-Abelian $H^2$ Block
Icosahedral	$k=3$	$2I$ (Binary)	$H^2$ Parity Obstruction

4. The Non-Canonical Obstruction

Even when the $H^2$ parity obstruction vanishes (Odd $m$), certain r-triples may be blocked by the joint-sum constraint.

• Thm 14.1: For $m=9$, the triple $r=(2, 2, 5)$ is OBSTRUCTED despite having $\gcd(r_i, m)=1$.
• Golden Path Immunity: The canonical Spike $r=(1, m-2, 1)$ is analytically proven to be immune to this obstruction for all odd $m$.

5. Verified Theorems

• Thm 11.1: Analytic Proof of Spike Construction (Golden Path) for all odd $m$.
• Thm 14.1: Non-Canonical Obstruction for composite $m$.
• Thm 6.1: Finalized Parity Obstruction Law (Even $m$ + Odd $k$).

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Last Updated: March 2026