SIGNAL FORMATION THEORY (SFT)

Formal Structure of Operative Input Formation
Aligned with the Algebra of Becoming
By: Jordan Vallejo

Introduction

Signal Formation Theory (SFT) specifies how systems resolve signal into operative input.

It formalizes the upstream structural relations by which a system moves from signal availability to a resolved input capable of entering downstream continuation processes.

SFT does not govern continuation. It governs the formation of the input into continuation.

Its terminal output is:

x

This operative input is then consumed by the Algebra of Becoming.

The Algebra of Becoming begins only once operative input is available:

x, σ, E → Q

where candidate meanings are generated relative to realized state and environment. SFT terminates prior to that boundary.

Scope

SFT governs:

  • signal exposure

  • signal structuring

  • admissible configuration space

  • engagement of an admissible configuration

  • resolution into operative input

SFT does not govern:

  • candidate meaning generation

  • candidate admissibility over meanings

  • evaluation among candidate meanings

  • continuation admissibility

  • trajectory selection

  • realization of successor state

Those are specified by the Algebra of Becoming. In AoB, continuation admissibility is generated specifically from K(σ) through Ω(σ ; K(σ)), while the governing bundle at state σ is ⊠(σ) = ⟨K(σ), B(σ), F(σ)⟩.

Domain

Let:

D denote the signal field
Ψ denote exposure mapping
A denote the exposed signal set
J denote the admissible configuration family
H denote configuration constraints
I denote structural invariants
W denote the engaged configuration
◇ denote the resolution mapping
Y denote the resolution output space
x denote operative input

All SFT relations terminate at x.

Composite Mapping

The composite structure of signal formation is:

D → A → J → W → Y ∋ x

with internal mappings:

A = Ψ(D)
J = { j ⊆ A ∣ j satisfies H and I }
W ∈ J
Y = ◇(W)
x ∈ Y

This is not a temporal sequence. It is a structural dependency relation.

Part I. Exposure Mapping (EM)

1. Scope

Exposure Mapping specifies the structural relations governing signal exposure and admissible configuration space.

It defines:

  • signal field

  • exposure mapping

  • exposed signal set

  • admissible configuration family

It does not define engagement, candidate meanings, interpretation, or continuation.

2. Core Variables

D — Signal field
The set of signals present under exposure conditions.

Ψ — Exposure mapping
A mapping from signal field to exposed signal set.

Ψ : D → A

A — Exposed signal set
Signals accessible to the system after exposure.

J — Admissible configuration family
The family of configurations admissible over A under structural constraints.

H — Configuration constraints
Rules governing admissibility of configurations.

I — Structural invariants
Properties preserved across admissible configurations.

3. Structural Relations

Signal exposure is defined by:

A = Ψ(D)

Configuration admissibility is defined by:

J = { j ⊆ A ∣ j satisfies H and I }

Thus:

J ⊆ 𝒫(A)

4. Interpretation of EM

EM defines what signal material is available and which configurations of that material are structurally admissible.

These are not candidate meanings. They are admissible signal configurations.

EM therefore remains strictly pre-interpretive and upstream of AoB candidate generation.

Part II. Resolution Mapping (RM)

5. Scope

Resolution Mapping specifies the structural relations governing engagement and resolution into operative input.

It defines:

  • engagement of an admissible configuration

  • resolution into output space

  • production of operative input

It does not define candidate meaning generation, admissibility over meanings, evaluation, or continuation.

6. Core Variables

W — Engaged configuration
An admissible configuration instantiated for resolution.

W ∈ J

◇ — Resolution mapping
A mapping from engaged configuration to resolution output space.

◇ : W → Y

Y — Resolution output space
The set of admissible resolved outputs generated from engagement.

x — Operative input
The operative input extracted from resolved output.

x ∈ Y

7. Structural Relations

Engagement:

W ∈ J

Resolution:

Y = ◇(W)

Operative input:

x ∈ Y

8. Interpretation of RM

RM specifies how an admissible signal configuration is resolved into operative input.

It defines the structural relation between engagement and resolved output without invoking semantics, candidate meanings, or continuation.

Part III. System Placement

9. Layered Architecture

The layered dependency is:

EM → RM → AoB

equivalently:

SFT → AoB

with terminal interface:

x → Q

expanded within AoB as:

x, σ, E → Q

EM defines exposure and admissible configuration space.
RM defines engagement and resolution into operative input.
AoB defines candidate generation, admissibility over meanings, evaluation, selection, and continuation.

10. Boundary Condition

SFT terminates at:

x

AoB begins at:

x, σ, E → Q

Therefore SFT does not:

  • generate Q

  • filter Q

  • evaluate Q

  • select among Q

  • define candidate trajectories

  • define AGM

  • define successor realization

Those belong entirely to AoB.

11. Interpretive Activation Boundary

Interpretation is not a function of SFT.

Interpretation occurs only downstream within AoB when:

ADL(σ) = 1 ∧ |Q^⊠| ≥ 2 ∧ N ≥ 2

SFT supplies input into this condition through x, but does not determine whether the condition holds.

SFT therefore terminates prior to all downstream multiplicity conditions over candidate meanings, including N.

12. Environmental Role

The environment may participate upstream in:

  • signal field composition

  • exposure conditions

  • admissibility of configurations

But within AoB the environment participates specifically in candidate generation:

x, σ, E → Q

The environment does not, within SFT, define continuation admissibility or realization.

Part IV. Formal Relations

13. Composite Law of Signal Formation

Signal formation is given by the composite relation:

D → A → J → W → Y ∋ x

with:

A = Ψ(D)
J = { j ⊆ A ∣ j satisfies H and I }
W ∈ J
Y = ◇(W)
x ∈ Y

This composite law terminates at operative input.

14. Upstream Irreducibility Theorem

SFT cannot be derived from AoB.

AoB takes x, σ, and E as given in candidate generation, but does not specify how x is formed.

Therefore the operative input layer is upstream-irreducible relative to AoB.

15. Downstream Irreducibility Theorem

AoB cannot be reduced to SFT.

SFT terminates at x.

AoB additionally requires:

  • realized state σ

  • environment E

  • continuation admissibility Ω(σ ; K(σ))

  • governing structure ⊠(σ) = ⟨K(σ), B(σ), F(σ)⟩

  • candidate evaluation and successor realization

Therefore SFT does not substitute for AoB.

16. Layer Distinction Theorem

SFT and AoB define distinct structural layers.

SFT governs operative input formation.
AoB governs continuation resolution.

Their relation is not symmetric independence. It is layer distinction with directional dependency:

SFT → AoB

AoB depends on SFT output at the input boundary.
SFT does not depend on AoB for its own formal definition.

17. Non-Temporal Clarification

The signal formation structure:

D → A → J → W → Y → x

is not a primitive time sequence.

It is a dependency structure specifying how operative input is structurally formed from signal.

Likewise, its coordination with AoB is not a claim about clock time, but about formal order of dependence.

Part V. System Generality

SFT applies to:

  • physical systems

  • biological systems

  • cognitive systems

  • artificial systems

It does not require:

  • semantics

  • interpretation

  • consciousness

  • language

These may occur in downstream systems, but they are not presupposed by signal formation as such.

Symbol Dictionary

D — Signal field
Ψ — Exposure mapping
A — Exposed signal set
J — Admissible configuration family
H — Configuration constraints
I — Structural invariants
W — Engaged configuration
◇ — Resolution mapping
Y — Resolution output space
x — Operative input

Closure

Signal Formation Theory specifies how systems move from signal field to operative input through exposure, admissible configuration, engagement, and resolution.

It ends at:

x

The Algebra of Becoming begins at:

x, σ, E → Q

SFT therefore governs the upstream formation of operative input.
AoB governs everything that follows.

Final Statement

Signal Formation Theory formalizes the structural layer by which signal becomes operative input.

Exposure Mapping defines signal availability and admissible configuration space.
Resolution Mapping defines engagement and transformation into resolved output.
Together they terminate in operative input x.

This output is then consumed by the Algebra of Becoming, where candidate meanings are generated, admissibility is evaluated relative to Ω(σ ; K(σ)), and continuation is resolved under the governing structure ⊠(σ)