Minimal cycle length in a restricted class of cyclic structures

• To: The mathematical research community
• Purpose: Academic assessment and possible further analysis
• Project: The Horizon Equation
• Status: Working note (not a proof)

1. Problem Statement
We investigate the following question:

Can a given set of structural requirements generate a necessary class of cyclic structures with a minimal closed identity cycle?

In particular, we examine:

• whether such a structure exists,
• what the smallest valid cycle length is,
• and whether this structure necessarily leads to the geometry expressed in the Horizon Equation (see section 6).

The goal is not to derive the number 104 directly from a counting process, but to examine whether a specific set of structural requirements necessarily implies a minimal cyclic geometry.

2. Background and Motivation
This work springs from a larger ontological model, but the problem formulation below may be assessed independently of that context.

The point of departure is a set of structural requirements related to:

• identity,
• relation,
• cycle,
• phase,
• periodicity,
• closure,
• coordinated dynamics,
• and relational stability.

The hypothesis is that these requirements jointly define a restricted class of valid cyclic structures.

The mathematical task therefore becomes:

What is the smallest structure that can simultaneously realise all the requirements?

3. Basic Structure
Let:

$S = {s_0, s_1, \ldots, s_{N-1}}$

Define a cyclic transition:

$s_i \mapsto s_{i+1 \bmod N}$

This yields a directed cycle graph with period $N$.

4. Requirements for a Valid Structure

(A) Distinctness
All states shall be distinct:

$s_i \neq s_j \quad \text{for } i \neq j$

(B) Minimal period
The cycle shall not be reducible to a shorter period:

$s_k \neq s_0 \quad \text{for } 0 < k < N$

(C) Continuity
Only local transitions are permitted:

$s_i \to s_{i+1}$

(D) Observation requirement
There exists an injective mapping between internal states and observable structural states:

$f : V \to X$

(E) Periodicity
There exists an operator $T$ such that:

$R(t + T) = R(t)$

This establishes phase and coordinated periodic structure.

(F) Internal operative structure ($\Pi_8$)
It is assumed that each complete transition requires a minimal internal operative sequence consisting of eight distinct transformations.

$\Pi_8$ is provisionally defined as a threshold operator:

$\Pi_8(B) = \mathcal{L}(B)$

with information cost:

$C(\Pi_8) = \ln 2$

where $\mathcal{L}$ is a binary decision operator of the form:

$\mathcal{L}(B) = \begin{cases} E_1 & \text{if } B \geq B_c \ E_0 & \text{otherwise} \end{cases}$

The number 8 is motivated from decision theory:

Three necessary operations:

• accumulate signal,
• compare with threshold,
• lock result,

each requiring a verification, giving:

$3 \times 2 = 6$

plus start and end phase:

$6 + 2 = 8$

$\Pi_8$ is thus understood as a candidate for the minimal operative closure sequence.

A complete formalisation of $\Pi_8$ as an explicit mathematical object remains to be done, but the basic structure is established.

This is the point that provisionally motivates the factor:

$104 = 8 \times 13$

(G) Global coherence requirement
The structure must be globally consistent throughout the entire period.

Intuitively this means:

• no collapse to a shorter cycle,
• no degeneration of states,
• stable phase coordination,
• preserved identity throughout the entire cycle,
• consistent relational homecoming.

This requirement is not yet fully formalised and represents the main open problem.

5. Candidate Structure
Based on the structural requirements set out above, the following is proposed:

$N = 104 = 8 \times 13$

where:

• $8$ represents minimal internal operative structure ($\Pi_8$),
• $13$ represents minimal closed identity cycle.

The hypothesis is that this is the smallest structure that simultaneously satisfies all the requirements above.

Through non-formal analysis, $N < 104$ (tested for $N = 2$–$96$) appears to violate at least one requirement, whilst $N = 104$ emerges as the first consistent candidate.

This is not an exhaustive classification.

6. Horizon Geometry
The central hypothesis is that requirements (A)–(G) do not merely generate a cycle, but a specific geometric/topological structure — and that the candidate value $N = 104$ is the smallest structure that realises this geometry.

As contextual motivation for the fact that $N = 104$ is not arbitrarily chosen, the following observation is noted:

The candidate value appears independently in a geometric construction — the Horizon Equation — of the form:

$\alpha = \frac{1}{104} \cos!\left[\left(12 + \eta_{\text{port}} + \eta_{\text{EM}} + \eta_G - \frac{1}{4} - \frac{\ln 2}{56}\right) \frac{2\pi}{104}\right]$

where all terms are structurally motivated without using $\alpha$ as input data. This equation matches the experimentally established value of the fine-structure constant (CODATA) with a relative error of $8.4 \times 10^{-13}$.

This is not presented as part of the mathematical proof basis, and no part of the mathematical task depends on it. It is mentioned solely because it indicates that the candidate value $N = 104$ has independent geometric grounding — and because a mathematician assessing the interest of the problem should have access to this information.

The mathematical task is to examine:

• whether requirements (A)–(G) necessarily generate this geometry,
• and whether that geometry necessarily implies a minimal closed structure with $N = 104$.

7. Open Mathematical Problems
Given requirements (A)–(G):

1. Can it be shown that a valid structure exists?
2. Can it be shown that no structure with $N < 104$ satisfies all the requirements simultaneously?
3. Is $N = 104$ unique, or unique up to symmetry/isomorphism?
4. How must the coherence requirement (G) be formalised for the problem to become well-defined?
5. Can $\Pi_8$ be formalised as an explicit mathematical object — operator, automaton or graph structure?
6. Can the horizon geometry be derived as a necessary consequence of the requirements?

8. Clarification
This note does not claim:

• that $N = 104$ is proved,
• that the requirements are fully formalised,
• that the model is mathematically closed,
• or that the horizon geometry has been derived.

It is claimed only:

• that a concrete structural problem has been identified,
• that a specific candidate structure exists with partially established internal structure,
• and that the problem appears to be mathematically analysable.

9. Request to the Research Community
We request:

• assessment of the precision of the problem,
• proposals for formalisation of the coherence requirement,
• analysis of the lower bound for $N$,
• assessment of minimality and uniqueness,
• analysis of $\Pi_8$ as a possible operator or transition structure,
• and assessment of whether the requirements generate a necessary horizon geometry.

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