From Witness-Space Sharpness To Family-Pointwise Exactness For The Solvability Complexity Index

Abstract: We study how exact Solvability Complexity Index (SCI) statements should be formulated for families of computational problems rather than for single problems. While the equality (\operatorname{SCI}G (\mathcal P)=k) is unambiguous for an individual computational problem (\mathcal P), the family setting requires one to distinguish family-pointwise exactness, witness-space sharpness, and worst-case exactness. We formalize this trichotomy, prove that witness-space sharpness coincides with worst-case exactness but is, in general, strictly weaker than family-pointwise exactness, and show that certain Koopman-operator classification results are sharp only in this worst-case sense. We then establish two positive upgrade theorems: an abstract pullback principle and a concrete finite-query criterion guaranteeing that witness-space sharpness upgrades to family-pointwise exactness. Next, we introduce a decoder-regular finite-query transport preorder on SCI computational problems, prove that it is a preorder, derive a transport-saturation sufficient criterion extending the principal-source package, and show that the associated transport degrees need not form a lattice in full generality. We analyze the natural decoder classes (\mathscr R{\mathrm{cont}}) and (\mathscr R_{\mathrm{Bor}}): on the full class the corresponding quotients are not upper semilattices, while on the nondegenerate subclass the preorder is upward and downward directed. Finally, we exhibit two natural positive families realizing the principal transport mechanism: exact integration on compact intervals and a fixed-window spectral decision family obtained by block-diagonal stabilization.

• The paper establishes an upgrade principle that guarantees family-pointwise exactness by leveraging a source problem and uniform finite-query reductions.
• It clarifies that while witness-space sharpness and worst-case exactness are equivalent, they do not inherently ensure exactness across all family members.
• Explicit case studies, such as exact interval integration and block-diagonal stabilization, illustrate the practical implementation of decoder-regular finite-query transport.

Solvability Complexity Index for Problem Families: From Witness-Based Sharpness to Family-Pointwise Exactness

Introduction and Motivation

This work investigates the formulation and transfer of Solvability Complexity Index (SCI) exactness statements from singular computational problems to families of problems. Traditionally, SCI assigns to a problem $\mathcal{P}$ the minimal height $k$ such that a type-$G$ algorithm tower computes the target map. While the condition $SCI(\mathcal{P}) = k$ is transparent for a single problem, nontrivial ambiguities arise when extending this notion to families $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$. Three logically distinct sharpness criteria emerge: family-pointwise exactness, witness-space sharpness, and worst-case exactness.

The central question addressed is: Under what conditions does the existence of an extremal witness with $SCI = k$ in the family, along with a uniform upper bound, guarantee that all family members achieve exactness at level $k$? The paper establishes a precise formalism for these notions, demonstrates inherent differences between them using Koopman operator examples, and devises transfer and obstruction principles for upgrading witness-space sharpness to family-pointwise exactness.

Exactness Notions for Families and their Logical Relations

Given a family $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$ and fixed $k \in \mathbb{N}_0$, three sharpness regimes are formalized:

1. Family-Pointwise Exactness at $k$: $k$0, $k$1.
2. Witness-Space Sharpness at $k$2: $k$3, $k$4 and $k$5 s.t. $k$6.
3. Worst-Case Exactness at $k$7: $k$8.

Witness-space sharpness and worst-case exactness are proven equivalent, but both are generally weaker than family-pointwise exactness. This fine distinction is significant in applied analysis: for instance, in spectral problems, one can typically construct explicit "hard" instances attaining the upper bound, but this does not ensure that all family members exhibit critical complexity.

Using classes of Koopman operator spectral approximation problems, the paper provides explicit cases where worst-case exactness holds, but family-pointwise exactness fails due to the presence of trivial subproblems (e.g., degenerate finite spaces, or spaces with explicitly solvable subclasses).

Abstract and Concrete Upgrade Theorems

The work presents a general upgrade principle: If a family $k$9 satisfies:

• There is a "source" problem $G$0 with $G$1,
• Every hypothetical type-$G$2 solution to any family member $G$3 can be transformed (pulled back) into a type-$G$4 solution for $G$5,
• Uniform type-$G$6 upper bounds across $G$7,

then it follows that all family members must have $G$8, and in combination with the upper bound, $G$9 holds family-pointwise. This logic rules out the possibility that a lower-complexity algorithm exists for any member without contradicting the hardness of the source problem.

Finite-query evaluation reductions provide a concrete and verifiable transfer mechanism: If every $SCI(\mathcal{P}) = k$0 admits an encoding and continuous decoder mapping from $SCI(\mathcal{P}) = k$1, and all queries are simulable using finitely many evaluations from $SCI(\mathcal{P}) = k$2, then $SCI(\mathcal{P}) = k$3. If this structure is present throughout the family, it yields family-pointwise exactness.

These results are not only sufficient but are shown (by explicit counterexample, e.g., for full Koopman families) to be necessary in the sense that, without such a global transport structure, witness-based sharpness does not generically upgrade to family-pointwise exactness.

Structural Theory: Decoder-Regular Transport Preorders and Degrees

A general theory of decoder-regular finite-query transport is developed, introducing preorders $SCI(\mathcal{P}) = k$4 on SCI computational problems for arbitrary decoder classes $SCI(\mathcal{P}) = k$5 (e.g., continuous or Borel measurable decoders). Key properties include:

• The transport relation is a preorder, yielding partially ordered transport degrees via suitable quotients.
• Ambient classes, transport bases, and cones: For an ambient class $SCI(\mathcal{P}) = k$6 and layer of problems with $SCI(\mathcal{P}) = k$7, one can ask whether a (principal or non-principal) basis exists such that all such problems are reducible from the basis.
• The degree structures for natural decoder classes (continuous, Borel) are shown not to be lattices in general, and various open problems are posed regarding the algebraic structure of the transport-degree posets, particularly on nondegenerate subclasses.

Positive Examples: Families Satisfying Family-Pointwise Exactness

Two case studies demonstrate natural infinite families for which the upgrade theorems apply:

Exact Interval Integration Family

• The family of exact integration problems on compact real intervals $SCI(\mathcal{P}) = k$8 (with $SCI(\mathcal{P}) = k$9) under pointwise evaluation admits a uniform finite-query reduction from the canonical unit interval $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$0 problem.
• Explicit encodings (affine transformation) and decoders (identity) exist, and all evaluation queries are simulable.
• It is shown that $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$1 is precisely $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$2 for every such interval, and only $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$3 for degenerate intervals ($\mathcal{F} = (\mathcal{P}_i)_{i \in I}$4).
• The class admits a principal ambient structure: every nontrivial interval is in the transport cone of the unit interval problem.

Block-Diagonal Stabilization in Spectral Decision

• Consider the problem of deciding spectral emptiness for singleton windows within a fixed interval for a class of diagonal operators.
• By block-diagonal stabilization (adding a block with spectrum outside the window), a family-pointwise structure with $\mathcal{F} = (\mathcal{P}_i)_{i \in I}$5 is constructed, uniformly validating the transfer principle.
• Two-sided finite-query reductions exist between the canonical singleton-window decision problem and each stabilized member; thus all family members form a single transport-degree.

Obstruction Theory and Non-Principality

The analysis also treats obstructions:

• Explicit non-principal ambient classes are exhibited (e.g., for discrete spectral problems from the SCI literature), where the exact SCI layer decomposes into multiple incomparable transport degrees.
• Full Koopman families do not admit universal transport bases, prohibiting upgrade from witness-based to pointwise statements.

Theoretical and Practical Implications

The paper's results and constructions refine the logical landscape of algorithmic complexity for computational problem classes in numerical analysis, operator theory, and information-based complexity:

• Practical: For method designers, these results precisely delimit when class-wide lower bounds are robust (i.e., immune to trivial or degenerate subcases), and when complexity analyses must be stratified or localized.
• Theoretical: The abstract framework generalizes beyond classical settings and feeds into ongoing efforts to understand the structure of algorithmic reductions, completeness, and degree-theoretic phenomena in continuous mathematics and scientific computation.

The search for necessary and sufficient conditions for upgradeability of worst-case sharpness to pointwise exactness, and the full algebraic characterization of transport-degree posets for major decoder classes, remain open directions with significant import for algorithmic complexity theory in analysis.

Conclusion

This work provides a rigorous and granular taxonomy for SCI exactness statements over families of computational problems, precisely distinguishes between family-pointwise exactness and witness-based sharpness, and constructs both upgrade mechanisms and counterexamples. Through the development of decoder-regular finite-query reductions and associated preorder/degree structures, it offers a general transport-obstruction framework that captures the essence of complexity propagation across computational problem families. Key open problems are posed concerning the universal structure of these degrees and transfer mechanisms, inviting further investigation into the landscape of algorithmic complexity in nonlinear analysis and beyond.