Section 1 — Primitive Foundations of Temporal Flow Physics (TFP) (v11.1)

Section 1 — Primitive Foundations of Temporal Flow Physics (TFP) (v11.1)

(Relational Formulation)
By John Gavel

1.0 Purpose of Section 1

This section defines the irreducible foundations of Temporal Flow Physics (TFP).
Only logically necessary assumptions appear here.
No geometric, vectorial, dynamical, or interpretive structures are assumed.
All higher-level concepts — including time, direction, space, force, mass, symmetry, and observables — emerge from relational closure of primitive differences.

1.1 Primitive Ontological Entities

Axiom 1 — Discrete Relational Sites

Reality consists of a finite or countable set of discrete sites.
Sites possess no intrinsic properties beyond the capacity to participate in relational comparisons.

Axiom 2 — Binary Relational State

Each site carries a binary relational state:

\[ \text{State} \in \{+1, -1\} \]

This state has no intrinsic meaning.
It does not represent direction, motion, charge, spin, or orientation.
All interpretations of this state are emergent and contextual.

Axiom 3 — Relational Difference

For any two sites \(i\) and \(j\), a difference is defined as:

\[ \text{Difference}(i, j) = \begin{cases} -1 & \text{if their states differ} \\ +1 & \text{if their states agree} \end{cases} \]

Differences are primitive.
They are not vectors, do not possess direction, and are not embedded in a coordinate system.

1.2 Locality and Comparison

Axiom 4 — Primitive Adjacency

Sites are related by primitive adjacency relations.
Comparisons and interactions occur only between adjacent sites.
No global comparisons, nonlocal interactions, or absolute reference frames exist at the fundamental level.

Axiom 5 — Local Update Exclusivity

A site may update its state only as a function of:
• Its current state
• The states of adjacent sites

All updates are local.
All information propagation is relational.

1.3 Difference, Memory, and Closure

Axiom 6 — Difference Without Closure Is Non-Determinative

A single difference, or any finite set of non-overlapping differences, carries no locational, directional, or temporal meaning.
Differences become informative only when they participate in overlapping relational constraints.

Axiom 7 — Relational Closure (Determinacy Axiom)

A system of sites is said to be determinate if and only if the set of local relational differences is sufficient to uniquely determine all site states up to global symmetry.
This requires that differences overlap with sufficient independence to close all degrees of freedom.
No global structure may be assumed unless this condition is met.

1.4 Discreteness, Capacity, and Update Structure

Axiom 8 — Discrete Updates

All state changes occur in discrete update steps.
There is no underlying continuous time parameter at the fundamental level.
Sequential ordering is emergent.

Axiom 9 — Finite Relational Capacity

Each site can participate in only a finite number of relational updates per update step.
When relational demand exceeds local capacity, unresolved differences accumulate.
This accumulation is the primitive origin of persistence, delay, and effective inertia.

Axiom 10 — Local Autonomy

There is no universal clock, global synchronization, or global conservation law at the fundamental level.
Sites update independently and in parallel, constrained only by local relations.

1.5 Interpretive Non-Commitment

Axiom 11 — Non-Primitive Interpretation

No primitive concept of:
• Direction
• Flow
• Vector
• Force
• Mass
• Time
• Symmetry
• Geometry
exists at the foundational level.
Such concepts arise only as compressed descriptions of stable, closed patterns of relational difference.

1.6 Minimal Summary of First Principles

TFP is grounded on the following irreducible pillars:

• Discrete relational sites
• Binary relational states
• Local adjacency and comparison
• Difference as the sole primitive observable
• Determinacy arising from overlapping relational closure
• Finite local capacity and discrete updates
• All higher-level physical structure emerges from these principles