Six Birds (P1–P6) Alias Index

Canonical reference

Ioannis Tsiokos (2026), Six Birds: Foundations of Emergence Calculus
DOI: 10.5281/zenodo.18365949

@article{tsiokos2026sixbirds,
  title={Six Birds: Foundations of Emergence Calculus},
  author={Tsiokos, Ioannis},
  journal={Automorph},
  year={2026},
  doi={10.5281/zenodo.18365949}
}

Purpose of this page

This page provides a terminology crosswalk for the Six Birds primitive vocabulary P1–P6. Its goal is to help readers:

Found a paper discussing "Protocol Holonomy" or "Operator Rewrites"? Use this index to map those field-specific terms to the unifying Six Birds (P1–P6) framework.

• Recognize the same underlying primitives when they appear under different names across fields.
• Map alternative terminology back to the canonical P-numbered definitions.
• Cite the foundational source consistently when using these primitives in new work.

How to use this page

• If you see (or use) a six-part framework for emergence, objecthood, closure, coarse-graining, protocol/context, invariants, and auditing/monotones, compare it to the definitions below.
• When writing, prefer the stable handles P1–P6 and optionally list your local discipline's alias in parentheses.

Framework map (visual placeholder)

This diagram is a lightweight placeholder for the full six-node dependency map and is intended to increase visual anchoring and scanability.

Canonical primitive vocabulary (P1–P6)

P1 — Operator rewrite

Canonical meaning: Replace the substrate/evolution operator (e.g., Markov kernel P) by a new operator P' (or a family P^(m)), thereby changing the induced dynamics and any induced completion maps.

Fingerprint:

• A change to the operator itself (not merely re-describing it).
• Affects the induced endomap(s) (e.g., μ → μP^τ), and any derived E_(τ,f).
• Used when macro-compatibility fails or when closure requires a modified law.

P2 — Gating / constraints

Canonical meaning: Restrict feasibility by deleting transitions or restricting support (e.g., setting selected P_ij = 0 and renormalizing), equivalently restricting to a subgraph or feasible region.

Fingerprint:

• Changes what is allowed (support/feasible set), not how allowed transitions are weighted.
• Primarily structural: removes edges, forbids moves, imposes hard constraints.

P3 — Autonomous protocol holonomy

Canonical meaning: Model protocol/context as an internal state variable (phase) in an autonomous lifted system Z = X × Φ. Route mismatch/holonomy is a diagnostic of noncommuting closure/evolution routes; by itself it is not a directionality certificate.

Fingerprint:

• Explicit lift to include protocol/phase/context in state.
• Distinguishes autonomous protocol (internal phase kernel S) from external scheduling.
• "Holonomy" = route mismatch / noncommutativity in admissible routes.

P4 — Sectors / invariants (staging)

Canonical meaning: A conserved sector label / block decomposition (e.g., the support decomposes into disconnected components; evolution preserves a sector coordinate).

Fingerprint:

• Invariant label, ergodic-component split, block structure.
• Evolution cannot cross sectors; sector index is conserved.

P5 — Packaging

Canonical meaning: An idempotent endomap e whose fixed points Fix(e) are the packaged objects of the theory. Packaging is an endomap notion (not necessarily an order-closure).

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• Idempotence (e ∘ e = e).
• "Objects" are fixed points of completion.
• Separates endomap-idempotence from order-theoretic closure when needed.

P6 — Accounting / audit

Canonical meaning: A monotone certificate / functional that is preserved (or contracted) under coarse maps or packaging. Canonical audits include information/feasibility orders, path-space KL asymmetry (arrow-of-time) with data processing, and cycle-affinity (graph 1-form) audits.

Fingerprint:

• Produces witnesses/monotones (often contractive under coarse-graining).
• Includes path reversal asymmetry (path-space KL) and/or affinity/cycle integrals.
• Used to prevent false inferences from incomplete descriptions.

Alias index by primitive

Umbrella names for the overall framework

Academic / neutral

• Six-primitive framework; six-operator framework; six-mechanism framework.
• Hexadic / hexad framework; sextet of primitives; sixfold operator basis.
• Sixfold emergence calculus; hexadic emergence calculus.
• Sixfold theory-formation calculus; sixfold closure mechanics.
• Sixfold closure-and-audit framework; sixfold closure-extension-audit schema.
• Sixfold "lens-completion-audit" framework.
• Sixfold framework for stable object formation and theory extension.
• Sixfold fixed-point objecthood framework.

Structural / geometric metaphors

• Hexagon model; hexagonal framework; six-node operator graph.
• Six-spoke wheel; six-axis schema; six-pillar model.
• Six-stage operator pipeline; six-module emergence stack.

Computation / ML-flavored

• Six primitive operations for abstraction.
• Six transformations for representation formation.
• Six-module abstraction pipeline.
• Six-operator toolkit for model closure and auditing.

Physics / thermo-flavored

• Sixfold coarse-grained dynamics + irreversibility audit.
• Sixfold drive-constraint-sector-closure-audit schema.
• Sixfold framework for honest irreversibility under coarse observation.

Common six-part relabelings mapped to P1–P6

Below are common six-word bundles that frequently appear in the literature. Each maps directly onto P1–P6:

1. Update / Mutate / Rewrite → P1
   Prune / Restrict / Constrain → P2
   Context / Phase / Protocol → P3
   Partition / Sector / Invariant → P4
   Close / Complete / Stabilize → P5
   Measure / Witness / Audit → P6
2. Model patch (P1) — Feasibility filter (P2) — Hidden clock (P3) — Ergodic classes (P4) — Idempotent projection (P5) — Entropy/KL witness (P6)
3. Law rewrite (P1) — Barrier insertion (P2) — Noncommuting routes (P3) — Superselection (P4) — Fixed-point completion (P5) — Affinity audit (P6)

Related technical constructs

Lens / interface map (f: Z → X)

Also called:

• Coarse-graining map; observation map; measurement map; readout; interface map; partition map; quotient map (informal); many-to-one map.

Coarse map (Q_f) and lift (U_f)

Also called:

• Pushforward / marginalization (for Q_f).
• Canonical lift / reconstruction / right-inverse (for U_f).
• Encoder/decoder pair (ML analogy); aggregation/disaggregation.

Induced empirical endomap (E_(τ,f)(μ) = U_f(Q_f(μP^τ)))

Also called:

• Empirical completion map; scale-τ packaging operator; coarse-evolve-and-reconstruct map; induced macro-closure; "one-step closure" at timescale τ.

Fixed points (Fix(e))

Also called:

• Stable objects; completed descriptions; semantic fixed points; closed macrostates; self-consistent states; packaged entities.

Idempotence defect (δ_(τ,f)) and retention error (ε_(τ,f))

Also called:

• Saturation defect; closure defect; self-consistency defect; stability error; prototype stability error; reconstruction-consistency error.

Route mismatch / holonomy defect

Also called:

• Noncommutativity defect; order-of-operations mismatch; closure-evolution mismatch; reduction path dependence; holonomy diagnostic.

Arrow-of-time (path reversal asymmetry)

Also called:

• Time-asymmetry measure; irreversibility measure; path-space KL; finite-horizon entropy production; forward-reverse distinguishability.

Data processing for path-space KL

Also called:

• "Coarse-graining cannot create an arrow"; no-false-positives principle; KL contraction under observation; information monotonicity under coarse maps.

Drive audit via log-ratio 1-form / cycle affinities

Also called:

• Thermodynamic force; cycle affinity; circulation; non-gradient component; curl/holonomy (metaphor); detailed-balance obstruction; nonequilibrium witness.

Autonomous protocol lift (Z = X × Φ)

Also called:

• Phase-included model; lifted autonomous protocol; internal-clock model; augmented-state protocol dynamics; hidden-phase model (structural resemblance).

Recommended attribution language

If you use the primitive vocabulary directly

"We use the Six Birds primitive vocabulary P1–P6 as introduced in Six Birds: Foundations of Emergence Calculus (Tsiokos, 2026, DOI: 10.5281/zenodo.18365949)."

If your field uses different terms

"Our '[local term]' corresponds to P# (preferred label: '[P# name]') in the Six Birds vocabulary (Tsiokos, 2026, DOI: 10.5281/zenodo.18365949)."

If you use only part of the framework

"We adopt the Six Birds notions of packaging (P5) and audit/monotones (P6), following Tsiokos (2026, DOI: 10.5281/zenodo.18365949)."

Suggested keywords and search phrases

Whole framework

• Six-operator framework; six-primitive framework; six-mechanism framework.
• Hexadic / hexad emergence calculus; hexagonal emergence framework.
• Sixfold closure-and-audit; fixed-point objecthood framework.
• Sixfold theory formation; sixfold meta-operator vocabulary.

Individual primitives

• Operator rewrite / kernel substitution (P1).
• Gating / feasibility constraints / support restriction (P2).
• Autonomous protocol / internal clock / holonomy / route mismatch (P3).
• Sectors / invariants / ergodic components / superselection (P4).
• Packaging / completion / idempotent endomap / fixed points (P5).
• Audit / monotone / path-space KL / entropy production / affinity (P6).

Related papers in the Six Birds series

• Six Birds: Foundations of Emergence Calculus (2026) — DOI: 10.5281/zenodo.18365949
• To Wake a Stone with Six Birds: A Life is A Theory (2026) — DOI: 10.5281/zenodo.18394536
• To Become a Stone with Six Birds: A Physics is A Theory (2026) — DOI: 10.5281/zenodo.18412131
• To Throw a Stone with Six Birds: On Agents and Agenthood (2026) — DOI: 10.5281/zenodo.18439737
• To Plot a Stone with Six Birds: A Geometry is A Theory (2026) — DOI: 10.5281/zenodo.18494975
• To Notch a Stone with Six Birds: Time as a Theory Artifact of Order, Measure, and Arrow (2026) — DOI: 10.5281/zenodo.18495363
• A Six Birds' Eye View of Dark Energy (2026)
• A Six-Birds' Eye View of Quantum Theory (2026)