Functional Metrology of Complex Digital Systems

< 10 nm - tunneling, quantization

L3 caches, interconnects - emergent modes

tau = tau_0 * Nways^alpha * Nsets^beta

ILLUSTRATIVE MODEL - Deterministic demonstration based on values reported in the publication. Not a live hardware measurement.

Paper result: 98.2 +/- 0.8% at full compensation

Metrology · Hardware · Complex systems

Functional Metrology of Complex Digital Systems

Unified theory for in vivo characterization of emerging dynamic properties

Introduction: metrology of complex digital systems

Contemporary microprocessors feature transistor densities above 100 million per mm2 and interconnection complexity that generates non-trivial collective behaviors escaping discrete logic gate models. Functional metrology aims to characterize these emergent dynamic properties in situ, quantify hardware reliability via non-destructive signatures, predict long-term behavior, and extract certified entropy for cryptographic applications.

This work presents the first experimental demonstration of stable hardware signatures (98.2 +/- 0.8%) extracted from metastable collective modes in L3 cache networks across 300 CPUs spanning four architectures (Intel RaptorLake, AMD Zen4, AMD Zen3, Intel Skylake). The methodology combines strict real-time temporal isolation, multi-scale thermal compensation, and a statistical validation protocol compliant with NIST standards.

The three coexisting physical regimes in a modern processor: quantum (< 10 nm), analog (jitter/skew) and collective (L3 caches, interconnects).

The hybrid nature of digital systems

At short transistor channel levels (< 10 nm), quantum effects become significant: tunneling, energy quantization, charge fluctuations. These phenomena impose fundamental limits to miniaturization and create intrinsic noise. In timing paths and clock circuits, signals exhibit continuous value ranges with variable propagation times - jitter and inter-signal skews create analog behaviors.

In shared structures (L1/L2/L3 caches, interconnects, memory controllers), collective modes emerge from the interaction of millions of transistors. These modes exhibit nonlinear dynamics and metastable states. This third collective regime is the central subject of this work.

Theorem 1.1: Existence of metastable collective modes

In an L3 cache network of capacity C shared between N cores, there exist metastable states characterized by relaxation times tau satisfying:

where tau_0 is the characteristic relaxation time, and alpha, beta are micro-architecture dependent coefficients.

Proof: The cache structure forms a network of coupled oscillators with frozen disorder. Complex systems theory predicts the existence of localized modes (Anderson) with long relaxation times when N_ways * N_sets gg 1.

The following table presents the cache geometries of the four studied architectures. Each micro-architecture presents a unique cache geometry that must be adapted to excitation sequences.

The L3 cache modeled as a network of coupled oscillators. The highlighted cluster represents a localized metastable collective mode (Anderson localization).

The measurement challenge: ergonicity and thermal compensation

The operating system scheduler continuously restores ergonicity through several mechanisms that corrupt hardware signature measurements: context switches (~10 us), periodic interrupts (timer tick ~1 ms: HRTIMER, SCHED), task migrations between physical cores, and preemption of priority tasks.

A hardware signature measurement is reproducible if the relative standard deviation sigma/mu satisfies:

where varepsilon_measure = 0.01 for high-precision characterization. This imposes strict temporal isolation.

The implemented solution relies on strict real-time isolation: (1) switching to FIFO real-time mode with priority 99, (2) pinning to a specific core via sched_setaffinity, (3) disabling local interrupts in the kernel environment. This isolation guarantees sigma/mu < 0.01 as required by Theorem 2.1.

Thermal variation induces non-linear changes in several physical parameters: carrier mobility (~1.5%/degC), threshold voltage (Vth, -2 mV/degC), RC constants (+0.3%/degC). The thermal compensation model (Theorem 2.2) relates the measured latency lambda_mes(T) to the reference latency lambda_0(T_0) via the Caughey-Thomas mobility ratio (exponent m = -1.5) and threshold voltage corrections.

Raw signal (dashed) corrupted by OS ergonicity artifacts vs compensated signal (solid cyan) stable to 98.2 +/- 0.8%.

Extracting the stable signature: experimental results

The standardized experimental protocol comprises three defined excitation sequences: (1) 1D Frustration Pattern - alternating reads and writes on adjacent lines to maximize contention in the L3 cache, optimized for Intel Ring Bus; (2) 2D Resonance Pattern - constructive interference between cache slices, revealing the system's eigenmodes; (3) Aging Probe - sequences specifically designed to excite degradation mechanisms (BTI, HCI, EM) with maximum sensitivity.

The inter-architecture reproducibility results on N=50 CPUs per architecture confirm the validity of Theorem 1.1. The 98.2 +/- 0.8% stability on Intel RaptorLake and comparable results on AMD Zen4 (97.8 +/- 1.1%), AMD Zen3 (97.9 +/- 0.9%) and Intel Skylake (97.5 +/- 1.2%) demonstrate the generality of the approach.

The interactive simulator below deterministically illustrates the relationship between compensation level and signature stability, based on values reported in the publication. This is not a live hardware measurement.

Left: relaxation exponent beta increasing with thermal stress (aging detection). Right: min-entropy gauge H_min = 4.2 bits exceeds the NIST SP800-90B threshold of 3.5 bits.

Applications: silicon diagnostics and certified entropy

The relaxation exponent beta increases with silicon aging, offering an early detection metric before functional failures appear. Experimental results after 1000 hours of stress confirm significant Deltabeta values with high sensitivity and very low false positive rates.

The hardware signatures extracted from metastable collective modes satisfy the NIST SP800-90B test suite requirements. The measured min-entropy H_min = 4.2 pm 0.3 bits significantly exceeds the certification threshold of 3.5 bits. This property opens the way to using L3 caches as certifiable physical entropy sources for cryptographic applications.

Theorem 5.1 establishes an accelerated degradation model based on the modified Arrhenius law for estimating the remaining useful life (t_RUL). Compared to existing methods (ECC tests, burn-in, our method, scanning acoustic microscopy), our approach offers the best cost/precision/test-time trade-off (4 hours, 92% precision) and is the only one combining early detection with moderate cost.

The publication defines five falsifiable hypotheses with precise experimental protocols. These hypotheses are formulated so they can be confirmed or refuted by independent third parties.

H1 (Stability): hardware signatures extracted from L3 cache networks exhibit > 95\% stability over 30-day periods after thermal and software compensation. Decision threshold: CV < 5\%.

H2 (Aging): the relaxation exponent beta increases significantly (delta > 0.05) before functional failures appear, with sensitivity > 3. Protocol: 30 CPUs under accelerated stress (1000 h at 85 degC), paired Student t-test.

H3 (Reproducibility): CPUs of the same architecture exhibit correlated signatures with r > 0.85, enabling family classification. 100 CPUs per architecture, Pearson correlation matrix.

H4 (NIST Entropy): the extracted entropy satisfies H_min > 3.5 bits, meeting NIST SP800-90B requirements. 1000 batches of 1 Mb each, full NIST SP800-90B test suite.

H5 (Remaining useful life): the predictive reliability model based on modified Arrhenius law predicts remaining time before failure with < 20\% error. 50 CPUs, 40/10 train/validation split.

The current methodology covers four x86 architectures (RaptorLake, Zen4, Zen3, Skylake). Extension to new architectures (ARM Neoverse, RISC-V) and non-volatile memories is underway. Integration with physical reliability models (TDDB, EM, BTI) and open standardization of hardware signatures constitute the next steps. Tier 1 validation across 3 independent laboratories, 300 CPUs and 6 months of accelerated testing is the threshold required for industrial certification (inter-laboratory reproducibility > 90%, correlation with standard reliability tests > 0.85, false positives < 1%).

If you use this work in your research, please cite:

The full publication (22 pages) including mathematical derivations, C and Python implementations, and the complete set of statistical validation tests is available on Zenodo.

Caughey & Thomas (1967) - carrier mobility in silicon

DESVAUX G.J.Y. · 2026 · contact@hopenmind.com