PMS Theory

Composition & Semantics

Purpose of this Page

This page specifies how PMS operators compose, constrain one another, and acquire meaning purely through structure.

PMS does not assign semantics by interpretation or narrative. Meaning arises from operator position, order, and constraint.

Sequential Composition

PMS expressions are defined as ordered operator chains.

Formally, a PMS composition has the shape:

E = o_n ∘ o_{n-1} ∘ ... ∘ o_1(S_0)

where:

• each o_i is one of: Δ, ∇, □, Λ, Α, Ω, Θ, Φ, Χ, Σ, Ψ
• the order is non-commutative
• every adjacent pair must satisfy dependency rules

Key properties:

• operator order is semantically binding
• reordering operators changes the structure
• some operators are invalid without prior structure

There is no implicit symmetry or automatic normalization.

Frames (□) and Reframing (Φ)

Frames (□)

A frame defines the active structural domain in which operators apply.

Formally:

• all operators act within a frame
• frames restrict scope and admissibility
• multiple frames may exist, but only one is active at a time

Unframed operators are structurally invalid.

Reframing (Φ)

Φ transforms an existing frame without erasing prior structure.

Properties:

• Φ requires an active □
• Φ changes interpretive context, not raw structure
• Φ does not reset dependencies
• Φ is not allowed past certain boundaries (see Σ)

Reframing is structural reinterpretation, not semantic reinterpretation.

Σ as Commit Boundary

Σ marks a structural integration point.

Once Σ is applied:

• preceding operators are treated as committed
• prior structure becomes non-reversible
• reinterpretation is prohibited unless explicitly stabilised

Formally:

Σ(E) ⇒ E becomes invariant with respect to Φ

Σ serves as:

• a consolidation operator
• a boundary between construction and use
• a point of structural finality

Σ does not imply evaluation, execution, or measurement. It implies commitment.

Ψ as Invariant

Ψ introduces self-binding constraints.

Properties:

• Ψ can only act on stabilised structures
• Ψ restricts admissible future compositions
• Ψ does not modify structure — it constrains it

Formally:

• Ψ defines a set of forbidden successor chains
• violations are structurally detectable

Ψ establishes:

• invariants
• limits
• long-term constraints

Ψ is not control flow.
Ψ is not feedback.
Ψ is structural binding.

Semantics without Interpretation

PMS semantics are positional, not referential.

Meaning is determined by:

• operator identity
• relative position
• dependency satisfaction
• boundary placement (Σ)
• invariant constraints (Ψ)

There is no hidden semantic layer.

Two identical operator chains are semantically equivalent, regardless of domain.

Structural Consequences

From the above, several consequences follow:

• PMS expressions are auditable
• invalid structures are detectable without context
• interpretation can be layered on top, but is not required
• PMS remains substrate-independent

This makes PMS suitable as a meta-grammar, not an application language.

What Comes Next

Up to this point, PMS has been treated in isolation: as a closed operator grammar with internal rules.

The next step is to clarify how PMS relates to existing systems without replacing them, competing with them, or subsuming them.

→ Continue to:
Relation to Other Systems
(Formal comparison, no reduction)