C.4 Conjecture and obligation ledger
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C.4.1 How to read the ledger
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The ledger separates established statements, complete computations, partial checks, and open conjectures. “Open” means no proof is claimed in this monograph. “Computed” refers only to the stated finite model.
| ID | Statement | Status | Decisive next step |
|---|---|---|---|
| L1 | Canonical six-row transvection target has lengths \(7\) and \(6\) over nonzero commutative rings. | Proved | Independent external proof audit. |
| L2 | Dimensions below six are rigid over every coefficient field or ring. | Open | Symbolic bounded-skeleton elimination for \(n\le5\). |
| L3 | Maximum unitriangular defect is \(\Theta(n\log n)\). | Open | Matching upper bound or larger interacting-catalyst family. |
| L4 | Multiplicative unitriangular distortion is bounded by \(3/2\). | Open | Normal form beyond one clean carrier. |
| L5 | Canonical atom has lower border length seven. | Open | Eliminate six-gate word-map components algebraically. |
| L6 | A one-copy rigid target exhibits tensor activation. | Open | Exact tensor-power search or asymptotic construction. |
| C1 | Two-qubit parabolic Clifford entangling model has four \(3\to2\) defects in one free-phase orbit. | Analytic costs and orbit; finite classification | Complete: Theorem 10.5 plus the exhaustive census. |
| C2 | Three-qubit swap-free total Clifford model has twenty-four defects. | Computed | Analytic motif classification. |
| C3 | Unit-cost SWAP eliminates all three-qubit total defects. | Computed | Determine whether rigidity persists for larger \(n\). |
| C4 | Parabolic-versus-full Clifford defect is unbounded. | Open | Construct and certify an amplifiable atom. |
| N1 | Three-bit affine targets receive no Toffoli-assisted saving in the two tested metrics. | Computed | Independent implementation and formal certificate. |
| N2 | Some finite affine target is shortened by temporary nonlinearity. | Open | Complete four-bit normal-form search. |
| P1 | A diagonal target has a Pareto improvement from non-diagonal Clifford escape. | Open | Exact small-width multiobjective atlas. |
| P2 | Phase-polynomial \(T\)-count displays tensor activation. | Open | Joint synthesis of repeated phase targets. |
| G1 | A number-conserving Gaussian target is shortened by Bogoliubov escape. | Open | Exact low-mode orthogonal/symplectic search. |
| G2 | A nearest-neighbor matchgate target is shortened by restored mode permutation. | Open | Architecture-weighted routing theorem. |
| S1 | A constant-size symmetry-breaking excursion can save unbounded cost. | Open | Explicit sector-coupled family and lower bound. |
| A1 | A fixed number of dirty ancillas can yield unbounded saving in a non-linear quantum gate model. | Open | Family with all-input restoration proof. |
| R1 | A nonsimply laced root system has catalytic ratio exceeding \(3/2\). | Open | Low-rank Chevalley word search and restricted lower bound. |
| H1 | Typed Green kernels yield non-enumerative distance lower bounds. | Open | Representation-theoretic coefficient inequalities. |
| H2 | Finite-degree transition moments suffice to distinguish all Latin squares of fixed order. | Open | Moment hierarchy and counterexample search. |
| Q1 | The order-six quantum-Latin product realization is empty while the entangled realization is nonempty. | Established externally | See Ball–Simoens and cited entangled constructions. |
C.4.2 Negative results worth preserving
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The following failed or negative conclusions are part of the research record.
Comparisons involving the separate hard Latin line-spread computation remain pending until its input representatives and verifier are supplied.
Abstract row-column-symbol eigenvalues do not distinguish order-six Latin main classes.
No classification of the two hard classes by a projective spread operator is certified by the present archive.
Three-bit Toffoli assistance does not shorten affine targets in the tested costs.
Adding unit-cost SWAP removes the three-qubit total-count Clifford defects in the declared quotient model.
These results prevent overgeneralization and sharpen the next search spaces.
C.4.3 Publication obligations
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Before any skeleton chapter is separated into a theorem paper, complete:
targeted literature review,
independent verification of all exact computations,
a formal statement of equivalence conventions,
proof or certificate for every lower bound,
archive of source, expected outputs, and manifests,
explicit discussion of what the model omits.