Functorial Depth Criterion

Status and purpose

This document is a criterion for formulating and auditing claims of recurrent physical computation. It is not itself an empirical conjecture and does not predict that any particular material can satisfy it.

Its purpose is to prevent physical motion, decoder computation, hidden control, unpriced resources, or inefficient repetition from being mistaken for increasing computational depth.

Logical process theory

A candidate claim must declare a finitely presented logical process theory. At minimum it must specify:

  • logical objects or information types;
  • admissible primitive transformations;
  • sequential and parallel composition;
  • equations or semantic equivalences between transformations;
  • the task family and its input domain;
  • the resources relative to which depth is measured.

An ordinary extensional category is insufficient when multiple implementations of the same function have different costs. The preferred structure separates implementation from semantics, for example with:

  • a category or 2-category of algorithms, circuits, or process implementations;
  • a category of extensional task semantics;
  • a semantics functor between them;
  • 2-cells representing simulation, refinement, or proofs of semantic equivalence.

Presented and model-relative depth

Computational depth is not an invariant of a bare category. It depends on a declared presentation and generator set (\Gamma). Adding a complex transformation as a primitive can reduce its apparent depth to one.

For a logical transformation (g), define depth relative to the declared implementation theory and generator library. If ([g^t]) denotes implementations with the same declared semantics, the no-fast-forward quantity is

[ \operatorname{Depth}_{\Gamma}([g^t])

\inf_{h\in[g^t]} \operatorname{depth}_{\Gamma}(h). ]

A claimed depth family must satisfy a predeclared lower-bound hypothesis such as

[ \operatorname{Depth}_{\Gamma}([g^t]) \geq c t ]

over its claimed domain. This is necessarily relative to the selected comparison model; no model-independent no-fast-forward claim is assumed.

Short cycles, identity maps, relabelled trajectories, and transformations with cheap direct implementations do not demonstrate intrinsic increasing depth merely because they are physically iterated.

Material-specific physical process theory

For a material (M), device regime (C), and primitive set (\Pi), define a physical process theory

[ \mathcal P_{M,C,\Pi}. ]

Its objects are physical state spaces and interfaces. Its morphisms are physically realizable processes under the declared fabrication, control, boundary, and environmental constraints. Sequential composition represents cascading or recurrence, while the monoidal product represents parallel physical composition.

A material alone does not determine this process theory. Geometry, temperature, wavelength, drive regime, fabrication tolerance, measurement apparatus, and allowed controllers are part of its definition.

Physical primitives such as propagation, interference, phase accumulation, refraction, scattering, resonance, nonlinear mixing, measurement, and feedback should appear as generators or derived factorizations. They should not be assumed independent merely because natural language assigns them different names.

Explicit physical resources

Every resource participating in a computation must be represented either as an explicit object or by a compositional resource grading. The account includes:

  • elapsed time and sequential uses;
  • energy and fresh signal carriers;
  • spatial volume and component count;
  • precision and calibration;
  • external memory and stored programs;
  • clocks and recurrence-dependent control;
  • measurement and feedback;
  • fresh ancillas, reset reservoirs, and entropy sinks;
  • initialization, encoding, decoding, and communication;
  • waste products and environmental consumption.

An apparent endomorphism

[ f:X\rightarrow X ]

may conceal fresh resources at every use. Where appropriate it must instead be represented as

[ f:X\otimes R\rightarrow X\otimes W, ]

where (R) is consumed and (W) records waste. Repeating the process then visibly requires (R^{\otimes t}).

Semantics and realization

The logical semantics, encoding, and decoding must be declared independently of the observed physical trajectory.

A valid realization must connect a logical task family to physical processes so that observed physical composition agrees with logical composition within a declared tolerance. Schematically,

[ \operatorname{Observe}(f^t) \simeq g^t. ]

This correspondence is evidence that a device implements the declared process. The mere existence of a functor from the free category generated by (g) is not evidence: any chosen physical endomorphism (f) induces such a functor automatically.

The substantive tests are semantic adequacy, counterfactual correctness, uniformity, resource accounting, and the absence of cheaper equivalent implementations.

Interface discipline

Encoders and decoders must:

  • remain fixed across recurrence counts;
  • be specified before evaluation;
  • have their physical implementation and resource costs included;
  • not calculate (g^t), select a precomputed answer, or use recurrence-indexed lookup tables;
  • be generated uniformly across problem sizes;
  • remain asymptotically subordinate to the computation attributed to the substrate.

An interface that is constant only with respect to (t) can still conceal an arbitrarily large lookup table when the input domain is fixed. Claims of scalable depth therefore require a uniform family indexed by problem size, not a collection of separately synthesized finite devices.

Robustness as compositional structure

Approximate equality must be compatible with composition. A claim should use an error-enriched category, bicategory, or other structure in which error witnesses compose according to a declared law.

If physical implementations (\tilde f_i) lie within an allowed disturbance class around (f), the relevant condition is of the form

[ d!\left( \operatorname{Observe}(\tilde f_t\circ\cdots\circ\tilde f_1), g^t \right) \leq \epsilon(t). ]

The disturbance class, metric, composition law, and acceptance threshold must be declared before testing. Local error measurements are useful only if their compositional law predicts the observed usable horizon.

Fixed rule versus fixed total state

A finite noisy state space contains only finitely many reliably distinguishable states. It cannot support an unbounded nonperiodic computation without fresh resources, growing precision, external memory, or an expanding physical system.

Scalable claims should therefore distinguish:

[ \text{fixed total physical state} ]

from

[ \text{fixed local physical rule applied to a uniform family }X_n. ]

The first permits only a finite-horizon claim. The second may support asymptotic claims if all resource growth is explicit.

Audit requirements

A future recurrent-computation claim must:

  1. declare the logical process theory, presentation, generator library, and semantic equivalence;
  2. specify a uniform task family rather than one fixed finite lookup domain;
  3. state a no-fast-forward hypothesis relative to a named comparison model;
  4. define the material, device regime, and allowed physical primitives;
  5. expose every controller, clock, measurement, memory, fresh carrier, reset, and entropy sink;
  6. declare encoders, decoders, perturbations, metrics, and tolerances before evaluation;
  7. test unseen inputs, compositions, perturbations, devices, and configuration horizons;
  8. verify that measured local errors predict composed behavior;
  9. search actively for logical and physical shortcut implementations;
  10. report total resources rather than recurrence count alone.

Changing any of these foundations after observing failure defines a new hypothesis.

Role in the categorical siege

Theoretical computational models determine candidate logical process theories. Materials and operating regimes determine constrained physical process theories. Physical mechanisms supply generators and relations.

The research target is not the bare existence of a functor. It is a resource-explicit, semantically adequate realization whose availability or impossibility follows from a material-specific compositional property.

The central open problem is to identify an invariant or monotone (\mu) that produces a nontrivial bound between the logical and physical theories.

A positive future conjecture should have the form

[ \text{primitive }\Pi\text{ in material }M \Rightarrow \mu(f^t)\text{ grows at a specified rate}, ]

thereby enabling a declared logical family.

An obstruction conjecture should have the form

[ \mu(f)\leq B \quad \text{for every allowed physical construction}, ]

while the target logical family requires

[ \mu(g^t)>B. ]

Candidate quantities may include distinguishability, accessible mode rank, polynomial degree, non-Gaussianity, memory capacity, or information-flow structure. None is accepted merely because it correlates with rich dynamics; it must compose predictively and yield a quantitative forbidden outcome.

Decision rule for Conjecture 5

Do not formulate Conjecture 5 by adding another universal abstraction.

It may be formulated only after selecting a material-and-primitive regime and identifying a compositional invariant or obstruction capable of making a quantitative prediction. The preferred first conjecture is an obstruction claim, because it can eliminate a broad physical design class and identify exactly which additional resource would be required to escape the bound.