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Cohesion Dynamics (CD)

K-series — Kernel Grammar & Invariants

The K-series defines the kernel grammar of Cohesion Dynamics: the minimal structural invariants and compositional relations that any substrate must support to be CD-compatible.

The K-series separates what must be structurally possible (kernel invariants) from how it is realized (carrier implementations in A-series).

Series Purpose

The K-series serves to:

  1. Define the kernel grammar: Specify the formal language of admissible composition over configurations and resolutions
  2. State structural invariants: Identify capabilities substrates must support (conditionally necessary, not sufficient)
  3. Separate kernel from carrier: Make explicit that kernel defines capabilities; carriers realize them
  4. Enable formalism independence: Allow multiple mathematical representations (event structures, Petri nets, categories) without privileging any
  5. Support carrier diversity: Allow alternative implementations (graph-based, order-theoretic, category-theoretic, field-theoretic)

Key Papers

K-GOV — Governance of Epistemic Scope and Necessity Claims

Status: Normative/governance
Paper ID: K-GOV
Location: publishing/staging/K-GOV.md

K-GOV is the authoritative governance framework for interpreting claims in the CD programme. It governs:

  • The epistemic status of kernel capabilities (K-CAP)
  • Interpretation of “necessary”, “permissive”, and “required” capabilities
  • Carrier independence vs. substrate independence
  • Conditional necessity vs. logical necessity
  • Hardware/implementation/run distinctions
  • Correct interpretation of M-series and E-series results

All papers making scope-sensitive, necessity, or independence claims must cite K-GOV.

K-GOV introduces no new ontology or mechanisms; it governs how existing results are to be read. See also the plain-language How to Read Cohesion Dynamics page.

K-KERN — The Cohesion Dynamics Kernel as a Grammar of Composition

Status: Normative/formal specification
Paper ID: K-KERN
Location: publishing/staging/K-KERN.md

K-KERN is the canonical kernel specification. It defines:

  • Primitive objects: Configurations (immutable informational structures) and Resolutions (atomic compositional acts)
  • Grammar components:
    • Consistency predicate: Consistent(S)\mathrm{Consistent}(S)
    • Admissibility relation: Admissible(r,C,C′)\mathrm{Admissible}(r, \mathcal{C}, \mathcal{C}') (structure-relative, ternary)
    • Dependency relation: c≺c′c \prec c' (partial order, acyclic by construction)
  • Derived structures: Histories (downward-closed consistent sets), continuation space, reconciliation, modes (interaction-induced equivalence classes), structural resilience, constructors
  • Non-commitments: No spacetime, locality, simultaneity, metric structure, background manifold, clocks, schedulers, or privileged formalism

Key clarifications:

  • Grammar is kernel-defining, not an optional lens
  • Resolutions are compositional facts in the grammar, not spacetime events
  • Admissibility is structure-relative; failure induces structural divergence
  • Circular justification forbidden by well-founded dependency structure
  • Kernel specifies which histories are well-formed; carriers enforce/compute grounding

K-ORD — Order Structure Induced by the Kernel Grammar

Status: Staging
Paper ID: K-ORD
Location: publishing/staging/K-ORD.md

K-ORD establishes a necessary structural consequence of K-KERN: admissible histories are necessarily partially ordered by dependency. No representational formalism, carrier dynamics, or additional assumptions are introduced.

Key Result: Given K-KERN grammar (resolution persistence, admissible reachability), a partial order structure on configuration equivalence classes follows directly.

Classification: This is a kernel consequence, not a lens. The partial order is derived necessarily from K-KERN without additional assumptions, distinguishing it from optional representational lenses.

Why not a lens?

  • Requires no additional assumptions beyond K-KERN
  • Not optional or representational
  • Follows necessarily from kernel grammar

Contrast with lenses: K-LENS-ES requires conflict heredity; K-LENS-PN requires resource interpretation. K-ORD requires nothing beyond K-KERN.

K-LENS Series — Kernel-Level Semantic Lenses

Status: Active
Purpose: Formal semantic mappings between K-KERN grammar and established mathematical frameworks

The K-LENS subseries provides representation theorems showing how K-KERN grammar maps into familiar mathematical frameworks. These are semantic lenses that clarify what additional assumptions are required for each framework representation.

Key Papers:

  • K-LENS-ES — Event Structure Lens: Maps K-KERN to prime, stable, and general event structures
  • K-LENS-PN — Petri Net Lens: Maps K-KERN to safe Petri nets, occurrence nets, and asynchronous transition systems

Future K-LENS Papers:

  • K-LENS-CS (Causal Set lens, conditional)
  • K-LENS-CT (Constructor Theory lens, optional)
  • K-LENS-QM (Quantum Semantics lens, deferred)

Role: K-LENS papers answer “Given K-KERN, when can it be represented as X?” — not “The kernel is X.” They make explicit what additional assumptions are needed for each framework and preserve carrier independence.

Read full K-LENS Series overview →

Future K-ADAPT Series

Planned papers mapping K-KERN grammar into specific carrier implementations for execution and simulation.

Role: These papers introduce no new ontology. They state additional assumptions required for carrier implementation and demonstrate how K-KERN grammar can be realized computationally. Dependency direction: K-KERN → K-ADAPT-*.

Epistemic Status

K-series papers state conditionally necessary structural capabilities:

  • Conditional: Required in explored regimes (M/E-series results), not claimed as logically inevitable
  • Necessary: If certain operational regimes are to be realizable, these capabilities must be supported
  • Not sufficient: Kernel invariants do not guarantee emergence of constructors, physics, or specific phenomena

Governance: The interpretation of necessity claims, kernel capabilities, and scope is governed by K-GOV. All K-series papers must align with K-GOV’s epistemic framework. See also How to Read Cohesion Dynamics for plain-language guidance.

What K-series does NOT do:

  • Prove sufficiency of any capability
  • Guarantee emergence of constructors or quantum structure
  • Fix parameters or recover SM/GR
  • Make ontological commitments beyond Axioms v2
  • Privilege any mathematical formalism
  • Specify enforcement mechanisms (carrier responsibility)

Relationship to Other Series

K-series vs. F-series (Foundational Postulates)

  • F-series: Ontological commitments (what exists: CIUs, constraints, closure)
  • K-series: Structural capabilities (what must be possible for certain regimes)

K-series vs. A-series (Carrier Architectures)

  • K-series: Kernel invariants (capabilities substrates must support)
  • A-series: Carrier implementations (explicit realizations showing feasibility)
  • Key distinction: Kernel specifies “what”; carriers demonstrate “how”
  • Citation discipline: Papers needing only structural capabilities cite K-KERN; papers needing specific mechanisms (e.g., height functionals, scalar mismatch) cite carrier papers (e.g., Paper A)

K-series vs. M-series (Formal Mechanisms)

  • K-series: Structural invariants enabling mechanisms
  • M-series: Mechanisms themselves (cohesion, modes, tolerance, constructors)
  • M-series results inform kernel capability classification (conditional necessity established via M/E evidence)

K-series vs. B-series (Representational Consequences)

  • K-series: Grammar of well-formed histories
  • B-series: Structural consequences within histories (branching, irreconcilability, superposition)

How to Read K-series Papers

Appropriate Criticism

  • Formal coherence and internal consistency
  • Minimality of primitives and relations
  • Evidence grounding (are invariants justified by M/E results?)
  • Clarity of kernel-carrier separation
  • Whether grammar truly defines kernel (not just one representation)
  • Whether formalism independence is maintained

Not Appropriate

  • Demanding sufficiency proofs (K-series states conditional necessity only)
  • Expecting physics derivations (that’s B/G-series scope)
  • Criticizing lack of enforcement mechanisms (carrier responsibility)
  • Treating K-KERN as one option among many formalisms (grammar is kernel-defining)
  • Demanding metric, spacetime, or locality (explicitly excluded)

Series Evolution

Current State:

  • K-KERN established as canonical kernel specification
  • K1 (transitional/pedagogical paper with capability layers) superseded by K-KERN
  • Grammar-first formalism locked in as kernel definition

Future Work:

  • K-ADAPT-* series for formalism mappings
  • Possible refinements to kernel grammar based on M/E evidence
  • Universe-specific regimes forward-declared but not yet formalized

Series Map Context

After Kernel v3 refactor, the CD programme structure is:

  • K-series — Kernel grammar & invariants (what must be possible)
  • A-series — Carrier architectures (how it can be realized)
  • M-series — Emergent mechanisms (what can arise in regimes)
  • E-series — Empirical narrowing & elimination
  • T-series — Toy models
  • (future) K-ADAPT — Mappings from kernel grammar to formalisms

This structure cleanly separates kernel-level (K), realization-level (A), and emergence-level (M) concerns.