256 GT/s PAM4 Channel Validation Under a Fixed Reference Plane: Differential-Mode Phase-Residual Analytical Theory of Equivalent Spatial Length and Its Application to Sub-THz Interconnect Characterization

Document status: Non-commercial technical opinion and framework-level metrology profile. This document does not request endorsement, certification, procurement, legal determination, or product approval. Version note (V4.0.1, 2026-05-27): Corrects software DOI typo throughout the description and related works (previously 10.5281/zenodo.20384917 → corrected to 10.5281/zenodo.20384916). Adds Layer 3 enterprise API specification (OpenAPI 3.1 v1.2.0; Zenodo DOI: 10.5281/zenodo.20405124) and GitHub repository as related identifiers. Scientific content is identical to V4.0. Companion basis: HPCA V11.7 (Zenodo DOI: 10.5281/zenodo.19822525); V7.5 TGO (Zenodo DOI: 10.5281/zenodo.19955479); Preceding Research Family V2.4 (Zenodo DOI: 10.5281/zenodo.20341571); Companion Reference Implementation: costryx-leq-extractor v0.1.0 (Zenodo DOI: 10.5281/zenodo.20384916; GitHub: https://github.com/costryx/costryx-leq-extractor; Apache-2.0); Companion Enterprise API: Leq/GCI/MHL OpenAPI 3.1 v1.2.0 (Zenodo DOI: 10.5281/zenodo.20405124; License: Proprietary). Industrial Abstract. For SerDes IP vendors, ASIC designers, and PCB integrators entering the 256 GT/s PAM4 generation under PCIe 8.0 and OIF CEI-224G/448G frameworks, the real question is no longer whether the channel can be simulated, but whether it can be objectively validated under a fixed reference plane independent of the measurement platform and EDA tool vendor. This paper extends the author's prior fixture-invariant Leq extraction framework (V2.4) into the 256 GT/s PAM4 application layer, defining a three-anchor decision criterion at the Nyquist frequency (64 GHz) and its harmonic neighborhoods (96 GHz, 128 GHz). The Geometric Convergence Index (GCI), defined as the maximum ratio of measured Leq to its boundary value across the three anchors, provides a single-scalar pass/fail decision that is fixture-invariant, platform-invariant, and vendor-neutral. The framework can be retrofitted into existing channel-validation workflows and provides a third-party-verifiable physical observable for high-frequency channel acceptance in PCIe 8.0, CEI-224G, and CEI-448G generations. A foundational open-source reference implementation (costryx-leq-extractor, Apache-2.0; Zenodo DOI: 10.5281/zenodo.20384916) is provided as a concrete computational pathway for independent third-party reproduction. Academic Abstract. This paper develops a Differential-Mode Phase-Residual Analytical Theory for 256 GT/s PAM4 channel validation. Its core construct converts the differential-mode SCD21 phase residual exhibited by the channel in the sub-THz band, via a four-step reference-plane transfer protocol (TRL, SOLT/LRRM, AFR, and post-AFR de-embedding), into a fixture-invariant observable expressed in physical length units, namely the Equivalent Spatial Length (Leq). The metrological formulation, evaluated at three operating anchors located at the Nyquist frequency (64 GHz) and its designated harmonic neighborhoods (96 GHz, 128 GHz), provides a geometric-integrity decision criterion that is independent of the measurement platform, independent of the probe configuration, and independent of the EDA tool vendor. A closed-form sub-THz uncertainty propagation model is also formulated under illustrative instrument-class assumptions, covering five principal uncertainty sources: VNA receiver dynamic range, calibration-kit characterization tolerance, measurement-cable phase stability, probe-contact repeatability, and ambient temperature drift. Component classification under JCGM 100:2008 (GUM) renders the σLeq model traceable per ISO/IEC 17025 at the measurement-evidence level, while the SHA-256 hash formalization invokes NIST FIPS PUB 180-4 to establish disclosure-verification cryptographic equivalence. Key contributions.(i) A three-anchor Leq + GCI decision framework specifically calibrated for 256 GT/s PAM4 under PCIe 8.0 timing budgets;(ii) a closed-form σLeq uncertainty propagation model with five-source GUM Type A/B classification, traceable per ISO/IEC 17025;(iii) a SHA-256 (FIPS PUB 180-4) Measurement Hygiene Layer (MHL) hash chain (Eq. 6-1) for cryptographic disclosure verification;(iv) an open-source foundational reference implementation (costryx-leq-extractor v0.1.0, Apache-2.0; Zenodo DOI: 10.5281/zenodo.20384916; GitHub: https://github.com/costryx/costryx-leq-extractor) supporting independent third-party computation of Leq(fk), Leq(f) continuous spectrum, GCI, σLeq, and MHL hash-chain construction;(v) a companion enterprise API specification (Leq/GCI/MHL OpenAPI 3.1 v1.2.0, proprietary; Zenodo DOI: 10.5281/zenodo.20405124) providing the contract-level interface (22 paths / 33 operations / 48 schemas / 11 tags) for enterprise integration;(vi) explicit non-disclosure boundary preserving BQSGO multi-state acquisition and CBC non-ideal profile handling for coordinated companion publication; and(vii) a Public Clean Disclosure Boundary aligned with §X.C, separating disclosed reference implementation from reserved proprietary numerical methods. Public Clean Disclosure Boundary. The framework is positioned as a vendor-neutral metrology methodology and does not disclose vendor-specific calibration matrices, routing coordinates, DUT identity, or product-specific parameters. The methodology skeleton is independently reproducible by any third party with equivalent measurement capability. The open-source reference implementation (costryx-leq-extractor v0.1.0, Apache-2.0, Zenodo DOI: 10.5281/zenodo.20384916, GitHub: https://github.com/costryx/costryx-leq-extractor) provides a concrete technical pathway for this reproducibility. Three implementation-level items are explicitly reserved for coordinated companion publication: (i) BQSGO thermal-drift separation numerical methods; (ii) CBC under non-ideal SCD21 profiles; (iii) OP-2/OP-3 boundary calibration sequences. Standards alignment. Complementarily aligned with IEEE Std 370-2020 (Section 5, AFR), JCGM 100:2008 (GUM), ISO/IEC 17025, NIST FIPS PUB 180-4 (SHA-256), Touchstone Version 2.1, PCIe 8.0 (256 GT/s PAM4), OIF CEI-224G-LR-PAM4 Revision 05, OIF CEI-448G Framework (OIF-FD-CEI-448G-01.0, 2025-11), and IEEE P802.3dj (200/400/800/1600 Gb/s Ethernet Amendment 13). Index Terms: 256 GT/s PAM4; PCIe 8.0; sub-THz interconnect; Equivalent Spatial Length L_eq; Geometric Convergence Index (GCI); Differential-Mode Phase-Residual Analytical Theory; Fixed Reference Plane; Three-Anchor Decision Criterion (64/96/128 GHz); Closed-Form Sub-THz Uncertainty Propagation; GUM Type A/B Classification; ISO/IEC 17025 Traceability; FIPS PUB 180-4 SHA-256; Measurement Hygiene Layer (MHL); BQSGO; CBC (Capture Basin Classification); Reference-Plane Invariance; Instrument-Class-Invariant Formulation; Metrology Margin; Decision-Strata Decoding; S_CD21; TRL; SOLT/LRRM; AFR; OIF CEI-224G/448G; IEEE P802.3dj; Open-Source Reference Implementation; Enterprise API Specification; Reproducible Research; costryx-leq-extractor. No external funding, corporate sponsorship, or institutional grant support is declared. References to existing signal-integrity, calibration/de-embedding, standards, and metrology literature are used for scholarly positioning only; they do not constitute an admission that the Differential-Mode Phase-Residual Analytical Theory, the three-anchor Leq + GCI decision framework, the closed-form σLeq uncertainty propagation model, the MHL SHA-256 hash chain formalization, or the BQSGO/CBC reserved implementations are derived from, anticipated by, or dependent on any cited reference. Preceding Research FamilyFrom 110 GHz to 220 GHz; Fixture-Invariant Extraction of Equivalent Spatial Length in 1.6T/3.2T Interconnects (V2.4), Zenodo DOI: 10.5281/zenodo.20341571 Companion Software (Open-Source Reference Implementation)costryx-leq-extractor v0.1.0 — Official Reference Implementation of the Leq Extraction Framework (Apache-2.0).Zenodo DOI: 10.5281/zenodo.20384916GitHub: https://github.com/costryx/costryx-leq-extractorAuthor/Maintainer: Chin-Yu Hsu (ORCID: 0009-0009-6267-7897) Companion Enterprise API Specification (Layer 3)Leq/GCI/MHL Enterprise API Specification (OpenAPI 3.1) v1.2.0 — Multi-tenant API specification with 22 paths / 33 operations / 48 schemas / 11 tags, including NIST FIPS PUB 180-4 aligned SHA-256 manifest.Zenodo DOI: 10.5281/zenodo.20405124Interactive Viewer: https://costryx.github.io/costryx-leq-extractor/api-viewer/License: Proprietary (Costryx) Version: V4.0.1 (metadata revision of V4.0; 2026-05-27)Author: Chin-Yu Hsu, Independent Researcher, New Taipei City, TaiwanCorrespondence: [email protected]: 0009-0009-6267-7897