Introduction

Classical complexity theory is built on the idea that the class \(\mathbf{P}\) captures all problems solvable in “feasible time” on any reasonable model of computation. Cobham’s thesis expresses two linked intuitions:

1. \(\mathbf{P}\) is the class of efficiently solvable problems.

2. Any two “reasonable” computational models can simulate each other with at most polynomial overhead (invariance principle).

Standard models (multi-tape Turing machines, random-access machines, boolean circuits) are mutually polynomially equivalent, so the invariance principle has appeared empirically justified.

In this paper we describe a computational model derived from Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL), where coherence constraints are explicitly enforced at every step of a computation. The model is:

• geometric (hexagonal lattice in the complex plane),

• prime-indexed (each vertex carries a distinct prime),

• phase-based (each active vertex carries a phase angle),

• coherence-constrained by PAL (Pattern Alignment Lock).

We define deterministic and nondeterministic coherence classes \(\mathbf{PAOL}\) and \(\mathbf{NPAOL}\), show \(\mathbf{PAOL} \subsetneq \mathbf{NPAOL}\), and demonstrate that \(\mathbf{PAOL}\) is strictly contained in classical \(\mathbf{P}\), breaking Cobham’s invariance in this setting.

Companion observation-theory paper.

The present work develops Coherence-Constrained Computation Theory (CCCT) on the Allen Orbital Lattice and analyses coherence as a third axis of complexity beside time and space. A companion paper by the author, Non-Invasive Field Observation on the Allen Orbital Lattice: The NFO Operator and Fractal Observation Windows , extends this framework with a formal observation layer. That paper introduces fractal observation windows and a Non-Invasive Field Observation (NFO) operator that extracts structural information from PAL-coherent field configurations without altering the underlying dynamics or identity map. Together, CCCT and the NFO framework describe both the computational use of coherence and the ways an external observer-pattern can monitor coherent cascades on the AOL without disturbing them.

The Allen Orbital Lattice Machine

We model computation on a hexagonal lattice with prime-indexed vertices and phase labels.

We say that the computation accepts if the final active set \(S_T\) is grounded in the sense that a coherence functional \(C(S_T)\) (for example a sum of cosine terms over edges inside \(S_T\)) exceeds \(G^*\).

Coherence-Constrained Complexity Classes

We now define deterministic and nondeterministic coherence classes induced by AOL-Machines.

Classical \(\mathbf{P}\) is defined as usual in terms of deterministic Turing machines.

Separation of \(\mathbf{PAOL}\) and \(\mathbf{NPAOL}\)

We now show that deterministic and nondeterministic coherence classes are strictly separated.

\(\mathbf{PAOL}\) Compared to Classical \(\mathbf{P}\)

We now compare \(\mathbf{PAOL}\) to the classical class \(\mathbf{P}\).

Coherence Collapse and Cobham’s Invariance

We summarise the geometric obstruction underlying the previous section.

Cobham’s invariance principle asserts that any two “reasonable” computational models can simulate each other with polynomial overhead. In our setting we obtain:

The CCCT Hierarchy

The results above define a new hierarchy of complexity classes under coherence constraints. At the level of deterministic computation we have \[\mathbf{P} \;\supsetneq\; \mathbf{PAOL} \;\supseteq\; \mathbf{PAOL} \cap \mathbf{L},\] where \(\mathbf{L}\) denotes logarithmic space in the classical model, and at the level of coherence-constrained nondeterminism we have \[\mathbf{NPAOL} \;\supsetneq\; \mathbf{PAOL}.\]

Time and space remain classical resources, but coherence appears as a third independent axis. Informally:

• Time: number of symbolic steps (\(T(n)\)).

• Space: number of symbols or cells used.

• Coherence: structure of admissible phase relations under PAL.

In this picture, \(\mathbf{PAOL}\) represents efficiently computable problems under strict coherence constraints, while \(\mathbf{P}\) represents efficiently computable problems without coherence restrictions.

Non-Invasive Field Observation and Its Impact on CCCT

The introduction of Non-Invasive Field Observation (NFO) on the Allen Orbital Lattice fundamentally refines the interpretation of coherence as a computational resource. In CCCT, coherence is treated as an intrinsic structural limitation on the evolution of an AOL-Machine, independent of time and space. The NFO framework adds a complementary dimension: the ability to extract structural information from a PAL-coherent computation without disturbing it.

Observation without coherence expenditure.

Classical models treat observation as a state-altering operation, and quantum models treat measurement as a projection that destroys superposition. In contrast, NFO introduces observation protocols that preserve:

• the field configuration \(\Phi\),

• PAL-coherence of the active set,

• the identity map associated with the computation.

This yields a new mode of “transparent monitoring” not present in standard complexity theory.

Observation budget as a new external resource.

Although NFO does not increase the internal computational power of an AOL-Machine, it changes the information profile available to an external verifier or controller. This motivates an extended resource description: \[\text{Time},\quad \text{Space},\quad \text{Coherence},\quad \text{Observation}.\] Let \(f(n)\) bound the number of fractal window evaluations an observer-pattern may apply during a computation on inputs of size \(n\). We define: \[\mathbf{PAOL}^{\mathrm{NFO}}[f]\] to be the class of languages decidable by PAL-coherent AOL-Machines running in polynomial time and using at most \(f(n)\) non-invasive observations.

Monitoring coherent cascades.

Because NFO-operators commute with PAL-coherent dynamics, an observer-pattern can inspect the structural evolution of a coherent cascade at arbitrary time steps without changing its outcome or reachable configurations. This is in contrast with classical and quantum models, where inspection modifies computation. The existence of NFO-operators therefore strengthens CCCT by showing that:

1. coherence constraints limit computation but do not limit non-invasive structural access;

2. computational trajectories in \(\mathbf{PAOL}\) and \(\mathbf{NPAOL}\) can be externally monitored without altering complexity-theoretic class membership;

3. destructive measurement and non-invasive observation are cleanly separable layers.

Consequences for the CCCT hierarchy.

The extended hierarchy becomes: \[P \;\supsetneq\; \mathbf{PAOL} \;\subsetneq\; \mathbf{NPAOL}\] together with an orthogonal observation axis: \[\mathbf{PAOL}^{\mathrm{NFO}}[0] = \mathbf{PAOL}, \qquad \mathbf{PAOL}^{\mathrm{NFO}}[\mathrm{poly}] \text{ allows polynomially many non-invasive inspections.}\] This does not collapse any existing separations but enriches them by clarifying which algorithms permit continuous monitoring. In particular, computations that are coherence-feasible but not classically simulatable under PAL constraints can still be observed non-invasively in ways unavailable to classical models.

Summary.

NFO strengthens CCCT by decoupling observation from coherence expenditure. It establishes a principled framework for observing coherent computation without interference, clarifying that PAL-coherent processes may be transparent to external analysis while still possessing structural limitations not present in classical computation models.

Provenance

All concepts introduced in this paper originate in the Pattern Field Theory programme developed by the author in 2025. For clarity, we list the main definitions and results with their origin:

• Allen Orbital Lattice and AOL-Machine model (Allen, 2025).

• Pattern Alignment Lock (PAL) coherence condition (Allen, 2025).

• Coherence-constrained classes \(\mathbf{PAOL}\) and \(\mathbf{NPAOL}\) (Allen, 2025).

• Coherence Collapse Theorem (Allen, 2025).

• Violation of Cobham invariance in PAL-coherent models (Allen, 2025).

This document is intended as the foundational reference for Coherence-Constrained Computation Theory (CCCT).

Conclusion

We have defined a coherence-constrained model of computation on the Allen Orbital Lattice, introduced the classes \(\mathbf{PAOL}\) and \(\mathbf{NPAOL}\), and proved the strict separations \[\mathbf{PAOL} \subsetneq \mathbf{NPAOL} \qquad\text{and}\qquad \mathbf{PAOL} \subsetneq \mathbf{P}.\] These results show that coherence constraints impose structural limitations on computation that are not captured by classical models and that Cobham’s invariance principle does not extend to PAL-coherent computation.

In this framework, computation is not only about symbolic transitions and resource bounds, but also about the geometry of coherence in prime-indexed phase space.