Research program and paper architecture

For a nested pair of gate models, determine the distortion function

where \(\mathcal T_n\) is the target class. The basic questions are:

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1. Does any defect exist?

2. What is the first size threshold?

3. What is the minimal catalytic motif?

4. Is the effect generic, sparse, or confined to special strata?

5. How does it scale under direct sum, recursion, and tensor products?

6. Which resource vector reveals the first defect?

7. Can the effect survive realistic architecture and fault-tolerant costs?

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The unitriangular theory provides a complete first example:

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• a universal six-row atom over nonzero commutative rings;

• exact binary and ternary first-threshold evidence;

• exact classification of twenty binary minimal defects;

• connected families with two reverse gates and linear saving;

• microscopic rank-one, bandwidth-one escape;

• generic rigidity and defect codimension;

• recursive and tensor amplification to \(\Omega(n\log n)\).

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The small Clifford atlas adds a second gate logic with genuine exact defects and shows that changing the primitive library by adding SWAP can remove them.

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The highest-priority quantum project is a paper on parabolic-versus-ambient Clifford synthesis.

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1. Extend the analytic four-target free-phase orbit certificate to other cost-preserving Clifford equivalences and larger widths.

2. Classify the twenty-four three-qubit total-count defects under parabolic conjugation and qubit permutations.

3. Derive analytic normal forms for the witness circuits.

4. Prove lower bounds independent of full group enumeration.

5. Test depth, weighted, and architecture-aware metrics.

6. Determine whether a finite defective atom tensor-amplifies.

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The three-bit affine-versus-Toffoli census is rigid. The next target is a complete four-bit normal-form search without enumerating \(16!\) permutations.

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1. Compute exact affine distances for all \(|\operatorname{AGL}(4,2)|=322{,}560\) targets.

2. Enumerate Toffoli sandwiches modulo affine left and right actions.

3. Use ANF equations to retain only affine endpoints.

4. Compare total, Toffoli, CNOT, and fault-tolerant costs.

5. Produce SAT certificates below every claimed optimum.

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The phase setting is likely to produce a defect first in a resource other than total primitive count.

Priority metrics:

A first paper should choose one metric and avoid a diffuse comparison across all resources.

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Two concrete first searches are recommended.

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1. Number-conserving versus general fermionic Gaussian synthesis on a small number of modes.

2. Nearest-neighbor versus all-to-all matchgate circuits with weighted fermionic SWAP cost.

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Both admit exact orthogonal-matrix representations and have clear architecture relevance.

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This program should begin with a narrowly declared resource theory, such as:

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• excitation-preserving two-qubit gates versus arbitrary two-qubit gates;

• unitary Clifford+\(T\) versus measurement-assisted Clifford+\(T\);

• clean versus dirty ancillas for a fixed arithmetic family;

• local architecture versus one reusable nonlocal bus gate.

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The first objective is not maximal generality but one theorem with exact restoration and cost accounting.

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For a split reductive group with allowed roots \(\Psi\subset\Phi\), classify configurations in which a forbidden root acts as a reusable catalyst through Chevalley commutator relations.

Suggested sequence:

Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.

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For each finite-dimensional gate hierarchy, define bounded word loci and defect strata. Determine:

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• irreducible components,

• exact dimension and codimension,

• singularities and degenerations,

• border length,

• finite-field point counts,

• behavior under tensor products.

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The packet hierarchy should be developed as a separate mathematical framework rather than appended to every circuit paper.

Core objectives:

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1. define spherical, type, and holonomy functors for finite transition groupoids;

2. prove information-loss theorems for commutativization;

3. classify when typed Green kernels are sufficient for distance;

4. relate imprimitivity to combinatorial decomposition;

5. formalize realization fibers and descent obstructions.

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Version \(0.1\) of the atlas supports exact small finite groups. A mature package should add:

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1. generic multiobjective Dijkstra search;

2. disk-backed and parallel BFS;

3. SAT and SMT export;

4. group action and orbit classification;

5. symbolic word-map elimination;

6. circuit diagrams in Quantikz and SVG;

7. architecture graphs and routing costs;

8. certificate verification independent of the search engine;

9. a machine-readable theorem-status ledger.

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A result should be labeled proved only when every lower bound and restoration claim has a proof. A complete finite census should identify the state quotient, enumerate its full size, and make the traversal reproducible. A symbolic skeleton should be valued as a research object but not described as an established shortcut.

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If developed, the theory may contribute to:

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• exact and heuristic quantum circuit compilers,

• architecture-aware routing,

• parity and phase-network synthesis,

• stabilizer and Clifford optimization,

• reversible arithmetic,

• Gaussian and fermionic simulation,

• lower-bound methodology,

• geometric group theory of gate sets,

• algebraic geometry of circuit word maps.

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The near-term value is methodological and algorithmic. Hardware benefit requires a device-specific cost and noise model.

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1. Prove or refute a universal six-dimensional linear threshold over all coefficient rings.

2. Determine the true asymptotic order of the maximum unitriangular defect.

3. Decide whether multiplicative distortion is bounded.

4. Determine the lower border length of the canonical six-row atom.

5. Classify tree-tight catalytic matroids.

6. Find a one-copy rigid target with tensor activation.

7. Extend the two-qubit analytic orbit certificate to larger Clifford systems.

8. Analytically classify the three-qubit total-count Clifford defects.

9. Determine whether SWAP restores rigidity for all \(n\) in the chosen primitive model.

10. Find the first affine target shortened by temporary nonlinearity, or prove none exists.

11. Find the first diagonal target improved by non-diagonal Clifford catalysis.

12. Determine a phase-polynomial Pareto defect.

13. Find a number-conserving Gaussian target shortened by Bogoliubov escape.

14. Find a matchgate routing catalyst with a certified architecture advantage.

15. Establish a symmetry-sector microscopic-escape theorem.

16. Classify when a fixed number of dirty ancillas gives unbounded saving.

17. Develop typed Green-kernel lower bounds that avoid full enumeration.

18. Determine finite moment bounds sufficient for transition-holonomy classification.

19. Classify catalytic motifs in \(B_2\), \(C_2\), and \(G_2\).

20. Prove a general theorem connecting algebraic codimension and catalytic defect.

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The common phenomenon is not that every restriction can be beaten. It is that the path space matters. A target may obey a symmetry, flag, degree bound, locality rule, or factorization constraint while the cheapest route to that target does not.

The disciplined research question is therefore:

Which temporary resources can be opened, reused, and exactly removed at a net computational gain?

The established linear theory answers this question completely for one important model. The monograph supplies the symbolic and computational machinery to ask it across quantum gate logic without conflating identities, experiments, and theorems.

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