K-LENS Series — Kernel-Level Semantic Lenses

The K-LENS series establishes formal mappings between the Cohesion Dynamics kernel grammar (K-KERN) and established mathematical and physical frameworks.

These papers prove representation theorems that clarify what additional assumptions are required for each framework, and show how CD subsumes and unifies multiple semantics of concurrency, causality, and composition.

Purpose

Goal: Map the kernel grammar into familiar mathematical frameworks to:

Prove representation theorems
Clarify required additional assumptions
Show how CD unifies multiple semantics
Enable formalism-independent understanding

Not Goal: Privilege any framework as ontological, or claim physical inevitability

What is a Lens?

A lens is a semantic view that answers:

“Given the kernel grammar, when can it be represented as X?”

Not:

“The kernel is X.”

All K-LENS papers:

Start from K-KERN definitions only
Make extra assumptions explicit
Preserve carrier independence
Treat lenses as views, not commitments

Core Object: CD Resolution Structure

All K-LENS papers operate on the same kernel-derived object:

Events = resolutions (grammar acts)
Dependency = grammatical enabling
Conflict = irreconcilability
Histories = admissible event sets
Types/modes = reconciliation-stable equivalence classes

Each lens shows how this object can be represented under extra constraints.

Papers

Note: Order-theoretic structure (K-ORD) is classified as a kernel consequence, not a lens, because it requires no additional assumptions beyond K-KERN. See K-ORD paper for details.

K-LENS-ES — Event Structure Lens

Status: Planned

Maps K-KERN grammar to:

Prime event structures
Stable event structures
General event structures

Defines:

Causality = dependency
Conflict = irreconcilability
Enabling = admissibility

States required assumptions explicitly (e.g., conflict heredity).

Role: Primary concurrency semantics paper

K-LENS-PN — Petri Net / Occurrence Net Lens

Status: Planned

Maps kernel grammar to:

Safe Petri nets
Occurrence nets
Asynchronous transition systems

Requires assumptions about:

Resource interpretation
Conservation
Boundedness

Role: Underpins executable carriers and M-series simulations

K-LENS-CS — Causal Set Lens (Conditional)

Status: Optional/Deferred

Shows that under additional constraints:

Local finiteness
No Zeno accumulation
Antisymmetry

A CD history can be represented as a causal set.

Explicitly conditional — no claim that CD generally yields CST.

K-LENS-CT — Constructor Theory Lens (Optional, Advanced)

Status: Optional/Deferred

Clarifies correspondence between:

CD admissibility / irreconcilability
CT possible / impossible tasks

Key Result: CT tasks correspond to type-preserving grammatical substructures

Role: Semantic alignment only — not reduction

K-LENS-QM — Quantum Semantics Lens (Deferred)

Status: Deferred (future tranche)

Would address:

Branching histories
Phase as symmetry of reconciliation
Interference as admissible history overlap

Requires careful governance — not part of initial tranche.

Epistemic Status

K-LENS papers establish conditional representation theorems:

Conditional: Mappings hold under stated additional assumptions
Not sufficient: Lenses do not guarantee kernel realizability
Not ontological: Lenses are semantic views, not substrate commitments

Governance: All K-LENS papers must comply with K-GOV epistemic framework. See K-GOV for necessity language discipline.

Relationship to Other Series

K-LENS vs K-KERN

K-KERN: Canonical kernel specification (grammar-first)
K-LENS: Maps kernel into external frameworks (understanding)

K-LENS vs K-ADAPT

K-LENS: Maps to external frameworks for semantic clarity
K-ADAPT: Maps to carrier implementations for realization

K-LENS vs A-series

K-LENS: Kernel-level semantic mappings
A-series: Carrier implementations showing feasibility

K-LENS vs M-series

K-LENS: Structural mappings (no mechanisms)
M-series: May depend on K-LENS results (not vice versa)

Out of Scope

K-LENS papers do NOT:

Add axioms
Introduce dynamics, clocks, or metrics
Claim physical inevitability
Replace kernel definitions
Assert substrate realizability
Introduce carrier implementations
Recover physical constants

Integration Rules

All K-LENS Papers Must:

Cite K-KERN
Cite K-GOV
Make extra assumptions explicit
Preserve carrier independence

No K-LENS Paper May:

Introduce new primitives
Assert necessity without qualification
Privilege one lens as “the” representation

How to Read K-LENS Papers

Appropriate Criticism

Rigor of representation theorems
Completeness of stated assumptions
Correctness of mappings
Whether extra assumptions are minimal
Clarity of scope and non-claims

Not Appropriate

Demanding physics derivations (that’s B/G-series work)
Criticizing for not being ontological (lenses are semantic)
Expecting carrier implementations (that’s A-series work)
Treating one lens as privileged
Demanding sufficiency proofs

Recommendation on A-ORD and Order Theory

A-ORD (Paper A-ORD — Order Structure as Necessary Consequence) contains order-theoretic derivation work from Axioms v2.

Status Change:

✔ Order-theoretic result extracted and reclassified as kernel consequence (K-ORD)
✔ K-ORD correctly classified as necessary structure, not optional lens
✔ Keep A-ORD as historical/legacy evidence

Rationale: The partial order is a necessary consequence of K-KERN grammar (dependency + admissibility) without requiring additional assumptions. Under K-GOV, such results must be classified as kernel consequences, not lenses.

Series Evolution

Current State:

Series outline established
Papers planned but not yet written
A-ORD identified for extraction

Next Steps:

K-ORD created (kernel consequence, not lens)
Create K-LENS-ES
Create K-LENS-PN
Optional: K-LENS-CS, K-LENS-CT
Defer: K-LENS-QM

Rationale

The K-LENS series:

Replaces ad-hoc “lens” discussions with disciplined programme
Resolves confusion between derivation, compatibility, and ontology
Gives reviewers a clear semantic story
Positions CD as unifying semantics of constrained composition

K-LENS makes precise what CD means — without claiming what reality must be.