8 — Claim Summary · 9 — Conditionality Footer

8 — Claim Summary

1 (Three regimes): STRUCTURAL. Unitary evolution (no records), measurement (one definite record), decoherence (many environmental records). Framework for the derivation.

2 (Environment as record): STRUCTURAL IDENTIFICATION. The environment identified with the accumulated record — the sum of all completed breaks. Definitional framing, not a deduction. Environmental coupling = record-writing by non-target degrees of freedom.

Monoid grows monotonically (Axiom R).

3 (Loss of coherence): DERIVATION (from AP09 formalism + factorisation assumptions) + STRUCTURAL INTERPRETATION (Axiom R). Coherence = off-diagonal terms of density matrix (AP09 formalism). Environmental entanglement multiplies coherence by overlap factor ⟨e_B|e_A⟩ per interaction.

Under stated factorisation assumptions (D1), product of N factors’ magnitudes decays exponentially. Specific contribution: Axiom R identifies the decay as structurally irreversible (formal no-go outstanding; D6).

Decoherence timescale τ_D derived from product form via per-interaction decoherence weight λ; quantitative parameters adopted from standard open quantum systems theory (D2).

4 (Classical world): STRUCTURAL. Classicality = saturated decoherence. Spectrum, not boundary. Quantum eraser = basis rotation, not record destruction (architectural prediction; interference recovered only in postselected sub-ensembles, not unconditionally; testable via KS-23).

5 (Structural meaning): STRUCTURAL. Three faces of Axiom R: arrow of time (cosmological), measurement irreversibility (single-event), decoherence (intermediate scale).

Quantum sector summary — foundational features of QM derived or structurally identified from {S, B, R, C}. Debts acknowledged: einselection, quantum Darwinism.

6 (Kill switches): STRUCTURAL. KS-22 (anomalous coherence persistence), KS-23 (quantum eraser record destruction), KS-24 (quantum sector completeness). All live. Six debts owed (D1–D6).

9 — Conditionality Footer

Dependencies: AP20 4 and 6 (the axioms {S, B, R, C}: completeness and minimality fenced KS-P.1 / KS-P.2; joint consistency from the undeniable premise; Axiom R’s record monoid). AP09 3–4 (superposition, measurement, Hilbert space). AP09 7.2 (unitary evolution). AP09 6 (Born rule). AP20 (EH and QRA proven).

Dependents: Any downstream result requiring the quantum-to-classical transition. The classical limit. The emergence of definite outcomes at macroscopic scale.

Open problems: None introduced beyond kill switches below.

Kill switches live: KS-22 (anomalous coherence persistence — if a system maintains coherence where the derivation predicts decay, under conditions where 3.3 assumptions hold).

KS-23 (quantum eraser record destruction — if unconditional interference is restored after which-path information has been redundantly encoded into uncontrolled environmental degrees of freedom; reversible unmarking of a controlled marker does not trigger).

KS-24 (quantum sector completeness — if a foundational quantum feature is identified that cannot be derived from or structurally identified with {S, B, R, C}).

Six debts owed: D1 (factorisation assumption), D2 (decoherence timescale parameters), D3 (quantum eraser formal characterisation), D4 (einselection), D5 (quantum Darwinism), D6 (scope of structural irreversibility / no-go theorem).

Inherited switches: AP20 kill switches (KS-P.1 through KS-P.3) propagate via the axiom and EH dependency. AP10’s KS-D.4 propagates via the Lorentzian-signature dependency. AP09 kill switches propagate via dependency on quantum foundations.

What is derived or structurally identified: Decoherence mechanism from standard QM formalism (AP09) under stated factorisation assumptions (D1). Irreversibility interpretation from Axiom R (monoid has no inverse; formal no-go outstanding, D6).

The environment structurally identified with the accumulated record. Exponential decay of coherence magnitude via environmental record-writing under stated assumptions. The classical world as saturated decoherence.

The quantum eraser as basis rotation, not record destruction (architectural prediction, testable via KS-23). Three faces of Axiom R (arrow of time, measurement irreversibility, decoherence) as one axiom at three scales.

Foundational features of the quantum sector derived or structurally identified; einselection and quantum Darwinism remain as debts.

References

Artist G (2026). AP06: The Leakage Constant. Artist’s Proof.

Artist G (2026). AP08: The Identity. Artist’s Proof.

Artist G (2026). AP09: The Break. Artist’s Proof.

Artist G (2026). AP11: The Spin. Artist’s Proof.

Artist G (2026). AP12: The Limit. Artist’s Proof.

Artist G (2026). AP10: The Dimension. Artist’s Proof.

Artist G (2026). AP20: The Proof. Artist’s Proof.

Joos, E. et al. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, 2nd edition.

Joos, E. & Zeh, H. D. (1985). The emergence of classical properties through interaction with the environment. Zeitschrift für Physik B, 59(2), 223–243.

Kim, Y.-H. et al. (2000). A delayed choice quantum eraser. Physical Review Letters, 84, 1–5.

Scully, M. O., Englert, B.-G., & Walther, H. (1991). Quantum optical tests of complementarity. Nature, 351, 111–116.

Walborn, S. P. et al. (2002). Double-slit quantum eraser. Physical Review A, 65, 033818.

Zurek, W. H. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75(3), 715–775.

Zurek, W. H. (2009). Quantum Darwinism. Nature Physics, 5(3), 181–188.

Cross-Reference Index

Environment = accumulated record: This paper 2.1

Environmental coupling = record-writing: This paper 2.2

Coherence loss from Axiom R: This paper 3

Exponential decay of coherence: This paper 3.3

Classical world = saturated decoherence: This paper 4.1

Quantum eraser = basis rotation: This paper 4.3

Three faces of Axiom R: This paper 5.1

Full quantum sector summary: This paper 5.2

Axiom R (record monotonicity): AP20 4

Measurement as the break: AP09 4

Born rule: AP09 6

Hilbert space: AP09 3.2

Unitary evolution: AP09 7.2

Entanglement: AP09 5

EH and QRA proven: AP20

N = 3 spatial dimensions: AP10

Spin (fermion/boson): AP11

Uncertainty principle / commutation relations: AP12

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the420code.org

Series The 420 Code

Catalogue Ø Artist’s Proofs

Proof Number AP13

Title The Grain

Subtitle Decoherence

Medium Physics

Artist G

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