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The organizing picture: a three-certificate loop
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Closure operators, reflections, and idempotents
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Coarse-graining of Markov dynamics and lumpability
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Graph cycles, affinities, and nonequilibrium network structure
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Protocol geometry and stochastic pumps
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Finite state spaces, distributions, and kernels
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Paths, time reversal, and relative entropy
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A unified theory package viewpoint
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Support graphs and discrete 1-forms
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Assumption bundles
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Order-theoretic closure and fixed points
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Closure ladders and saturation
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Idempotent endomaps and induced closures
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Idempotent endomaps
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Existence Requires Choosing a Scale
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AUT + REV + ACC regime and graph 1-forms
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Cycle integrals, exactness, and the null regime
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Accounting as coordinates on cycle space
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Drive Is Coordinate-Free
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Data processing: coarse-graining cannot create asymmetry
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No Fake Arrows
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P3 Loves P6 Law
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Generic extension and the finite forcing lemma
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Counting lemma: definable predicates are rare
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Almost Nothing Is Definable
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The "Nothing Stays Constant" Lemma
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Primitives P1—P6 as closure-changing operations
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Definitions of P1—P6
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Mapping to the spine
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Downward influence across theories
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What Six Birds does *not* claim
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Constraints Kill Engines
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Spotting the Six Birds in the wild (examples)
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Protocol trap: external schedule vs autonomous lifted model
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Finite forcing count: definability is exponentially rare
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What the theory does and does not claim
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Outlook: forthcoming instantiations
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A.1 Repository integrity checks (from repo root)
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A.2 Python evidence harness (deterministic tests)
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Appendix B: Lean formalization map
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B.2 File map and key declarations
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Reproduce it: how to build the project
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Reproduce it: how to run the experiments
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C.2 Evidence by theme (tests and scripts)
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Protocol trap and the "P3 needs P6 drive" correction under autonomy
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Graph topology effects of P2 (edge deletion) and P1 (rewrites)
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Finite forcing / definability rarity
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Balanced-atom route (definitions + kernel-mass hinge)
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ECT compression and capacity witnesses
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Appendix D: Zeno cascades and depth
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Setup: frontier and Zeno criterion
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Storage-based activity and the WORK quantum (Option B)
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Integrated throughput (ICAP) and feasibility
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Hard lemma slots (WORK/CAP/route)
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HL-CAP-X1 (bounded dissipation density)
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HL-CAP-X2 (finite memory / kernel mass)
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HL-ROUTE (route mismatch controls gain growth)
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Checkable divergence criteria
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Zeno by fast capacity growth (DIV fails)
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Zeno by vanishing work quantum (WORK fails)
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Appendix E: Toolkit theory—defects
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Defects as quantitative relaxations of exact laws
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Route mismatch defect (abstract)
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Abstract packaging maps
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Linear-operator specialization
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Emergent coercivity template via sector compression
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Summary: slots and divergence consequence
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Coercivity from feasibility gating (P2)
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Mode compression from sectorization (P4) and minimality (P5)
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Defect propagation rules (toolkit)
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What this paper adds (Notch)
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Six Birds Theory recap: primitives and closures
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Bird 2 — Gate: Constraints (feasibility carving)
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Bird 3 — Protocol: Protocol holonomy
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Bird 5 — Package: Packaging
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Bird 6 — Audit: Accounting
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The three ingredients: order, measure, and arrow
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Ordering: time without clocks
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Arrow (irreversible bookkeeping)
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Two arrows: causation-time vs enablement-time
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Enablement-time (between layers)
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A Diagnostic: Does the Variable Set Change?
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Claim 1: Arrow-of-Time from Packaging and Accounting
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Claim 2: No Fake Arrows Under Coarse-Graining
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Claim 4: No Global Time Under Protocol Holonomy
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Methods: A Finite-State Laboratory and Audit Suite
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Audit 1: Entropy Production as an Arrow Proxy
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Audit 2: Path-Reversal KL and No Fake Arrows Under Coarse-Graining
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Audit 4: Enablement as Forced Theory Extension
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Audit 5: Constraints and Reachability Cones
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Audit 7: Constraint Boxes vs Signalling Channels
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Reproducibility and Auto-Generated Paper Tables
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Results I: Arrows and Clocks
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Arrow Audit II: Path-Reversal KL and 'No Fake Arrows'
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Clock Viability Is Paid: Budgeted Stabilization and Anti-Stall Progress Metrics
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Results II: Enablement and Constraints
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Enablement births time: forced theory extension with a no-birth control
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Tick disappearance and undefined tick failure
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No global time from protocol holonomy
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The holonomy obstruction (informal theorem)
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