M4 — Phase, Path, and Coherence Structure

M4 — Phase, Path, and Coherence Structure

Purpose and Scope

This paper formalizes the emergence of phase, path, interference, coherence, and decoherence within Cohesion Dynamics (CD), building on the ontological and structural results of M1–M3.

M4 does not introduce quantum mechanics, Hilbert space, amplitudes, probabilities, or linear dynamics. Its role is to identify and formalize the pre-quantum relational structures that make quantum-like behavior possible in later stages of the research programme.

Specifically, M4 establishes:

• phase as a relational property of constraint resolution cycles,
• paths as admissible realization histories of unresolved constraints,
• interference as recombination of compatible realizations,
• coherence as tolerance-limited compatibility,
• decoherence as tolerance violation with provenance partition.

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1. Unresolved Constraints and Realizations

In Cohesion Dynamics, a constraint graph evolves via local mismatch-reducing updates. In many situations, a constraint admits multiple equally admissible mismatch-reducing realizations.

Let a constraint $C$ admit a set of realizations $\{ r_i \}$ such that each $r_i$ reduces mismatch equally and satisfies all local admissibility conditions.

By the divergence axiom:

If multiple realizations are equally mismatch-reducing and admissible, no selection occurs.

All realizations are committed.

These realizations are not histories of particles or objects. They are candidate resolution trajectories of the same unresolved interaction.

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2. Paths as Realization Histories

A path is defined as a temporally ordered sequence of local commitments realizing a particular admissible resolution of an unresolved constraint.

Paths are:

• not spatial trajectories of objects,
• not worldlines,
• not primitives.

They are realization histories within the constraint graph.

Multiple paths may coexist so long as they remain mutually admissible under the tolerance structure defined below.

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3. Phase as Closure-Cycle Alignment

Constraint resolution in CD is not continuous. Cohesive Informational Units (CIUs) resolve constraints through discrete closure cycles, in which internal and external constraints are jointly satisfied.

We define phase as the relative alignment of these closure cycles between interacting CIUs and the surrounding constraint graph.

Let $\theta_i(t)$ denote the closure-cycle orientation of realization $r_i$ at time $t$.

The relative phase difference between two realizations is:

$$\Delta \phi_{ij}(t) = \theta_i(t) - \theta_j(t)$$

Phase accumulation occurs when realizations traverse different constraint structures or encounter different interaction sequences, causing their closure cycles to drift relative to one another.

Phase is therefore:

• relational,
• emergent,
• substrate-internal,
• not an angle in physical space.

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4. The Tolerance Vector $W$

Admissibility of coexistence is governed by a tolerance vector:

$$W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$$

with the following interpretations:

• $W_{\text{shape}}$: tolerance for spatial or structural strain,
• $W_{\text{clock}}$: tolerance for closure-rate mismatch,
• $W_{\text{spin}}$: tolerance for phase shear between closure cycles.

Two realizations $r_i$ and $r_j$ are compatible at time $t$ iff:

$$|\Delta \phi_{ij}(t)| \le W_{\text{spin}}$$

and corresponding shape and clock mismatches lie within $W_{\text{shape}}$ and $W_{\text{clock}}$.

Clarification on “compatibility”: In this context, “compatible” denotes admissibility for reconciliation attempts (per A-OPS operational semantics), not direct encoding of compatibility criteria in $W$. The tolerance window $W$ serves as a persistence window for exploratory reconciliation, gating whether divergent realization paths may be reconciled into a shared closure. This is consistent with the substrate-level interpretation established in A-OPS.

$W$ does not drive dynamics. It gates admissibility.

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5. Interference as Recombination of Compatible Realizations

Interference occurs when two or more realizations:

1. remain mutually admissible under $W$,
2. recombine at a later interaction,
3. jointly resolve a downstream constraint.

Constructive interference corresponds to recombination in which phase alignment reduces mismatch further.

Destructive interference corresponds to recombination in which phase misalignment suppresses resolution.

Interference is therefore not a superposition of amplitudes, but a joint constraint resolution outcome of still-compatible realizations.

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6. Coherence as a Compatibility Regime

Coherence is defined as the regime in which multiple realizations remain mutually admissible over extended interaction sequences.

Empirically, coherence is finite and characterized by a coherence timescale $\tau_{\text{coh}}$.

Within CD, coherence persists so long as accumulated phase drift satisfies:

$$|\Delta \phi_{ij}(t)| \le W_{\text{spin}} \quad \text{for } t \le \tau_{\text{coh}}$$

Beyond this regime, compatibility fails.

Coherence is therefore:

• conditional,
• bounded,
• parameter-dependent,
• not an intrinsic property of objects.

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7. Decoherence as Tolerance Violation

Decoherence occurs when accumulated mismatch exceeds tolerance:

$$|\Delta \phi_{ij}(t)| > W_{\text{spin}}$$

or when shape or clock mismatches exceed their tolerances.

At this point:

• realizations become mutually inadmissible,
• recombination is forbidden,
• the constraint graph partitions into distinct provenance domains.

Decoherence is not collapse. It is loss of joint admissibility.

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8. Provenance Partition and Branching

When compatibility fails, realizations inherit distinct provenance identifiers.

Let $p_i$ label the provenance of realization $r_i$ after partition.

Subsequent commitments are constrained to remain consistent with their provenance domain. Cross-provenance merges are disallowed by admissibility rules.

Branching is therefore:

• local,
• non-teleological,
• enforced by constraint compatibility,
• not selected or random.

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9. Entanglement as Shared Provenance Constraint

Entanglement arises when multiple CIUs participate in commitments that depend on the same unresolved constraint and subsequently share provenance partitioning.

Formally:

Two CIUs are entangled if their future admissible resolutions are constrained by shared provenance such that their joint state is not factorizable.

Entanglement propagates through ordinary interactions as provenance-conditioned commitments spread through the constraint graph.

Observers are not special; they are CIUs subject to the same rules.

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10. Limits of M4

M4 deliberately stops short of introducing:

• amplitudes,
• linear state spaces,
• probabilistic rules,
• Hilbert space structure.

These belong to later stages of the Quantum Emergence programme, where coherence-regime regularities may justify such representations as effective calculi.

M4 establishes the structural preconditions only.

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Summary

M4 formalizes phase, path, interference, coherence, decoherence, and entanglement as emergent relational structures arising from constraint resolution under tolerance.

The key result is that quantum-like phenomena are not primitive, but arise when multiple admissible realizations remain compatible long enough to recombine.

This completes the metaphysical foundation required for the Quantum Emergence programme to proceed from empirical classification to formal derivation.