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Purpose, assumptions and theory status

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# Theory and mechanism registry

Status is attached to a particular claim and assumptions. An undisproven conjecture
is not an established theorem; unrelated conjectures are not combined as premises.
Candidate benefits require their own preregistered comparison and retained failures.

| Ingredient | Status / assumptions | Pipeline role and testable benefit | Falsification / current gate |
|---|---|---|---|
| Finite permutation/group composition | Exact implemented control | Preserve order and inverse behavior under a coordinate change | Exhaustive transition and composition disagreements reject control |
| Complex roots of unity / Fourier coordinates | Exact finite-cycle construction; supplied features in REP-002–004 | Represent operations compactly while preserving decoded consequences | Commution alone insufficient; inspect collisions, decoder and OOD composition errors |
| Quotients / Markov aggregation | Exact operational sufficiency control under stated partition | Identify which distinctions can be removed for a named operation | Nonzero quotient residual or lost-context witness rejects sufficiency |
| Covariance / variational boundary terms | Mathematical controls under explicit regularity and boundary assumptions | Demonstrate expression changes with preserved specified observations | Boundary or transformation violations are counterexamples; no ML conclusion follows automatically |
| Softmax, sparsemax, entmax | Implemented selector comparisons | Learn selective memory routing under equal initialization/data/budget | Compare exact answers, export agreement, all seeds and cost; zeros alone imply no speedup |
| Persistent homology / Ripser | Bounded optional diagnostic | Observe selected latent-state structure | Synthetic geometry controls and sampling sensitivity; no certification of meaning or order |
| Graph heat / PDE-inspired diffusion | State-changing intervention, experimentally assessed | Candidate noise smoothing with explicit graph assumptions | Identity/permutation controls and binding damage; default remains zero |
| Prime-based phases / arithmetic encodings | Conditional research | Candidate phase dictionaries or routing periods | Compare matched non-prime controls for collisions, recall and cost; no RH premise |
| Riemann hypothesis | Unproved conjecture; no executable dependency | No accepted operational advantage | A concrete independent mechanism and test must precede an experiment |
| GPU/fiber/neuron analogies | Representational analogies only | Batched slots and value-channel dimensions clarify operations | Hardware measurements and tensor contracts; no biological equivalence claim |
| Learned language/context discovery | Planned research | Map unseen expressions to exact operations and expose needed distinctions | Frozen held-out paraphrases, negation/order/ambiguity and erroneous state changes |

Primary mathematical and implementation references remain with each existing
protocol. This registry adds no copied implementation, checkpoint or external data.