Perf-Consts: Tunable Performance Constants

• Perf-Consts are fixed constants embedded in code or mathematical frameworks that control performance-critical parameters without affecting overall correctness.
• In operating systems, perf-consts determine key settings like batching thresholds and scaling factors, directly impacting metrics such as latency and throughput.
• In algebraic geometry, 'Perf–constants' quantify the minimal categorical complexity required to generate the perfect derived category, aiding in precise categorical analysis.

A perf-const is a fixed constant value in software or mathematical frameworks that governs the magnitude of a performance-critical parameter while remaining neutral with respect to correctness. The term arises in modern operating systems, where perf-consts directly encode system trade-offs (such as batching thresholds, time intervals, or scaling factors) into kernel binaries and subsystems. The concept also appears in algebraic geometry and category theory as “Perf–constants,” denoting the generation time required for the perfect derived category $\operatorname{Perf}(X)$. This article synthesizes the technical and empirical context of perf-consts, emphasizing their formal definition, role, tunability, and measurement across operating systems and algebraic geometry.

1. Formal Definition and Significance

A perf-const is defined as a numeric constant $p \in \mathbb{R}_{>0}$ embedded in source or binary code, influencing the value or scale of a behavior impacting system or categorical performance without affecting global correctness (Chen et al., 14 Dec 2025). In operating systems, these include batch sizes, softirq rerun counts, shrink batch sizes, migration thresholds, and congestion window scaling factors. For a batching threshold, the typical parametric impact is:

$$L(p) \simeq \frac{\alpha}{p} + \beta \cdot p, \qquad T(p) \simeq \gamma(1 - e^{-kp}),$$

where $L$ is latency, $T$ is throughput, and parameters $(\alpha, \beta, \gamma, k)$ describe hardware or algorithmic costs (Chen et al., 14 Dec 2025).

Within derived category theory, the Perf–constant of a scheme $X$ is the minimal integer $n$ such that $D^{\mathrm{perf}}(X) = \langle G \rangle_n$ for some perfect complex $G$ and inductively-generated subcategories $\langle G \rangle_k$; this quantifies the minimum categorical complexity to generate $\operatorname{Perf}(X)$ (Neeman, 2017).

2. Perf-Consts in Operating System Kernels

Modern OS kernels, especially Linux, contain hundreds of perf-consts—hardcoded values (“magic numbers”) controlling performance-critical logic such as I/O scheduling (e.g., BLK_MAX_REQUEST_COUNT), memory reclamation (SHRINK_BATCH), CPU softirq handling (SOFTIRQ_RESTART), migration control (NR_MAX_MIGRATION), and TCP algorithms (CUBIC_Hystart_factor). These values encode workload and hardware assumptions. Empirical studies show static selection of $p$ leads to suboptimal performance as hardware and application characteristics evolve: for instance, raising BLK_MAX_REQUEST_COUNT dramatically increases achievable throughput on HDD I/O.

The majority of perf-consts are not dynamically tunable; only $\sim$145 are exposed by sysctl, leaving hundreds baked into kernel images. Changing them historically requires source edits and full kernel rebuilds (Chen et al., 14 Dec 2025).

3. Limitations of Classic Tuning Approaches

Available mechanisms for modifying perf-consts are inadequate for real-time or context-sensitive optimization:

• Static Compilation: Changing $p$ necessitates editing code followed by a rebuild and reboot cycle—prohibitive for rapid or frequent adaptation.
• Sysctl/Sysfs Interfaces: A small, statically-enumerated subset is exposed, requiring considerable developer effort for each new parameter. Lack of setter/getter atomicity and bound-checking leads to race conditions and safety issues.
• Kernel Live Patching (KLP): Existing tools (e.g., Kpatch, Ksplice) patch at function scope, requiring bytecode regeneration and slow, whole-function rollouts. They guarantee only per-thread “version atomicity” and do not track or prevent side-effect races when perf-consts are inlined in multiple sites (Chen et al., 14 Dec 2025).

4. Principled In-situ Perf-Const Tuning: KernelX and Scoped Indirect Execution

KernelX is an architecture for principled in-situ tuning of arbitrary perf-consts at millisecond timescale on deployed, unmodified kernels (Chen et al., 14 Dec 2025). Its core mechanism, Scoped Indirect Execution (SIE), enables:

• Offline Scope Extraction: For a given perf-const $p$, KernelX rebuilds the kernel with alternate values, discovers the minimal instruction spans (Critical Spans, CS) in machine code where $p$ is materialized, and symbolically recovers the functional relation $R/M \leftarrow f(R/M, p)$.
• Runtime Indirections: At runtime, indirection snippets are injected at CS boundaries to update architectural state as if a new $p$ had been compiled in. For transition safety:
  • Version atomicity ensures threads see either old or new semantics per CS entry.
  • Side-effect safety tracks all PCs within the extended Safe Span (SS) post-CS; global transitions wait for all threads to exit the SS, enforced via guard/un-guard kprobes and atomic refcounts.
• Programmable Tuning Policies: An in-kernel, type-verifiable interface exposes API functions such as x_set() (schedule new value) and x_transition_done() (detect safe switchover), allowing eBPF-based “X-tune” programs to implement adaptive logic utilizing application hints and hardware signals. Example policy: set $p=64$ for queue depths above 100.

This permits millisecond-scale, race-free adaptation of perf-consts, previously infeasible with static compilation or coarse-grained patching.

5. Empirical Outcomes and System Overheads

Representative case studies demonstrate the substantial impact of perf-const tuning:

Subsystem	Perf-Const	Default $p$	Tuned Range $p'$	Measured Improvement
Block I/O	BLK_MAX_REQUEST_COUNT	32	[1,128]	$54\times$ write throughput
CPU softirq	SOFTIRQ_RESTART	10	[1,100]	Latency-CPU trade-off (149–560µs)
Memory (shr)	SHRINK_BATCH	128	$\leq$24	$70\%$ latency reduction
Memory (TLB)	NR_MAX_MIGRATION	512	[64,256]	$40\%$ FCT tail reduction, $2\times$ TLB
Network (CUBIC)	Hystart scaling factor	3	1 or 3	$81\%$ FCT$_{99.99}$ reduction

Transition overheads are competitive: SIE probe triggers cost 243 cycles (jump-optimized) to 1858 cycles (INT3-fallback), policy application (X-tune) loads per CS in $\sim$30 ms, and global transitions (with side-effect safety) take $<$150 ms under heavy load. The offline cost to prepare for a given perf-const is dominated by two parallel kernel builds and symbolic span analysis ($<$20 minutes), amortized over future adaptations (Chen et al., 14 Dec 2025).

6. Perf–Constants in Derived Categories and Algebraic Geometry

In the context of triangulated and derived categories, particularly $\operatorname{Perf}(X)$, the Perf–constant is the minimal number of cone and direct summand extensions needed to generate the category from a single object $G$:

• On affine schemes $\operatorname{Spec} R$ with $\operatorname{gldim} R = r$, $D^{\mathrm{perf}}(R) = \langle R \rangle_{r+1}$.
• For projective $\mathbb{P}^d_k$, Beilinson’s result gives $$\operatorname{Perf}(\mathbb{P}^d) = \langle \oplus_{i=0}^{d} \mathcal{O}(i) \rangle_{d+1}$$.
• In general, for separated schemes of finite type and Krull dimension $D$: $\operatorname{Perf\textrm{–}constant} \leq D+1$.

Explicitly, for quasi-projective $X$ embedded via $f: X \to \mathbb{P}^n_R$, $D^{\mathrm{perf}}(X) = \langle G \rangle_{2(n+1)}$ for $G = \bigoplus_{i=-n}^{0} f^*\mathcal{O}(i)$. These estimates allow precise bounding of categorical complexity for generation and have consequences in geometric representation theory, K-theory, and Fourier–Mukai equivalence (Neeman, 2017).

7. Related Notions and Cross-Domain Synonyms

The notion of perf-const is context-dependent but shares features across disciplines:

• Operating Systems: Perf-consts are tightly-coupled constants encoding trade-offs in system performance. Their tunability and discoverability are crucial for system adaptive behavior and productivity (Chen et al., 14 Dec 2025).
• Derived Algebraic Geometry: The Perf–constant quantifies categorical generation, with bounds driving the analysis of generators, exceptional collections, and quiver-tensor structures (Neeman, 2017).
• Performance Analysis Tools: In low-level system monitoring (e.g., RISC-V perf_events), “constants” also appear as event codes, masks, mapping fields—integral for flexible profiling and hardware-specific analysis (Domingos et al., 2021).

A plausible implication is that, in software engineering and algebraic geometry alike, formalizing and exposing perf-consts as tunable or measurable quantities is essential for advanced control, optimization, and understanding of system or categorical dynamics.