# Auditable Drift Removal for Sub–2 nm EUV Metrology (v1. 1b) **Finite-Rank Neutralization in a Hilbert Observation Model** Lee Byoungwoo ## OverviewSub–2 nm EUV metrology increasingly fails in a regime where stochastic noise is already reduced and **small but coherent systematic drift** (tool + environment + servo imprint) dominates the uncertainty budget. This preprint proposes a **deterministic, registry-driven, rank-limited drift-removal primitive** designed to be auditable by a third party. The method treats each metrology stream on a declared operational window as an element of a (real) Hilbert space \ (H\), optionally whitened by a frozen procedure. A finite dictionary of drift atoms \ (V=\v₁, , vᵣ\ H\) is declared in a registry. Drift is removed by a weighted orthogonal projection onto \ (span (V) ^\). The core guarantee is theorem-level: **if the systematic component lies in \ (span (V) \), it is removed exactly on the operational window**, and the remaining decisions reduce to paper-fixed audit gates and frozen registry artifacts. This document is explicitly **application-facing**: it is intended to accompany the author’s “neutralization theory” papers, but it **does not assume number-theoretic structure** and makes no spectral-rate claims beyond finite-rank removal + auditable empirical gates. ## Technical summary (what is fixed) ### Hilbert observation modelA channel-restricted observation \ (y H\) is modeled as = s + d +, \ (s\) is the process-relevant component, \ (d\) is systematic drift, and \ (\) is stochastic noise. Whitening is represented by a positive self-adjoint weight \ (W 0\) defining\ y₁, y₂W: = Wy₁, y₂. \ ### Finite-rank neutralization operatorLet \ (V: Rʳ H\) be the dictionary synthesis map and \ (G: =V^\*WV^r r\) the weighted Gram matrix. The weighted projection onto \ (span (V) \) is^ (W) V: = V\, (V^\*WV) ^-1V^\*W, the neutralizer is^ (W) V: = I - P^ (W) V. \**Exact removal**: for any \ (d (V) \), \ (N^ (W) V d = 0\). ### Audit invariants and diagnosticsA practical invariant is the post-orthogonality residual: (y): = \|V^\*W y\|₂\|V^\*Wy\|₂ + 10^{-12}, y: = N^ (W) V y. is enforced via \ ( (G) _\) (registry-fixed). ### Metrics (including the \ (_\) baseline rule) A primary example metric is the log–log PSD slope on a declared band \ (I=fₐ, fb\): fit \ (Sᵧ (f) \) vs \ (f\) on \ (I\), denote slope by \ ( (y) \). A baseline slope \ (_\) is declared *ex ante* either by: - **spec-based** rule (from a channel physics/controls model), or- **reference-based** rule (from a “golden” tool/dataset). Bias-reduction factor (with an audit floor \ (₅₋₎₎ₑ>0\) ): : = M (yₑ₄) \{M (y₎ₒₓ), ₅₋₎₎ₑ\}. \ ## Paper-fixed PASS/FAIL policy (AC1–AC8) To eliminate ambiguity, the PASS/FAIL semantics are fixed at the sentence level: 1) Declare window, whitening, dictionary type, rank \ (r\), and fit bands in `pbconfig. json`. 2) Compute \ (G=V^\*WV\) and require \ ( (G) _\) (else FAIL). 3) Produce \ (y = N^ (W) V y\) and require \ (OrthRes (y) _\) (else FAIL). 4) Run a negative control (no projection / \ (r=0\) ) and require drift persists beyond noise (else FAIL). 5) Require improvement: \ (BRF BRF_\) (else FAIL). 6) Require perturbation sensitivity: a mismatched dictionary degrades performance by \ (₃₄₆\) (else FAIL). 7) Require injection/leakage loss below \ (₈₍₉\) for declared probes (else FAIL). 8) Require minimum bins/points \ (n_\) on all reported fits (else FAIL; otherwise PASS). These are recorded as booleans AC1–AC8 in `acceptance. json`. ## Output artifacts (certificate-ready) Each run is designed to be auditable from frozen artifacts, without per-run tuning: - `datasetᵣegistry. json`: immutable dataset identity + raw-file hashes (per channel) - `pbconfig. json`: window, whitening, dictionary, rank, fit bands, thresholds - `metrics. csv`: metrics for pre/post/control/perturbed/injection variants - `acceptance. json`: PASS/FAIL + supporting fields - `neutralizationcertificate. json`: canonical payload + SHA256 digest - `reportcard. pdf` (or `. md`): one-page dossier per channel - `SHA256SUMS. txt` (or equivalent): digests of the above If raw traces are confidential, the dossier remains auditable via **content hashes** and frozen registry identifiers. ## Industrial deployment guidance (production) This preprint upgrades the operational layer into a deployment-ready audit loop: - **Shadow-mode lifecycle**: run in parallel with an incumbent baseline \ (R₀\) without affecting tool decisions; require sustained PASS and stable non-regression. - **Reference baseline \ (R₀\) **: explicitly declare the legacy/vendor detrending/filtering pipeline and report a competitiveness gap\ₑ䃐 (M): = M (y₎ₒₓ) \{M (R₀ (yₑ₄) ), ₅₋₎₎ₑ\}. \- **Change control**: any update to whitening \ (W\) or dictionary \ (V\) increments a registry version and requires a new shadow-mode validation window. ## Validation templates and exemplars- **Synthetic worked example**: standard figures (pre/post, negative control, bias vs drift strength) and a Monte Carlo table. - **Scenario templates (Appendix) **: - S1: CD–SEM scan-path drift (with injection/leakage check and degradation vs rank) - S2: Overlay wafer-map drift (pre/post maps, extracted drift component, residual map) - **KPI bridge (optional) **: a conservative, pre-registered association analysis linking PASS/FAIL to downstream fab KPIs (yield/rework/alarm proxies) without overriding technical gates. ## Scope it is not a claim of global stationarity, nor an inference of \ (_\) from the same audited run. - Success requires that drift be well-approximated by the declared \ (span (V) \), and that preservation of process signatures is certified by injection/leakage probes. - The “KPI bridge” is explicitly **non-causal** and reported only as a value-assessment layer. ## Related neutralization theory papers (background) This work is designed to be compatible with the author’s neutralization theory stack: - Lee, Byoungwoo (2026). *Mock Theta Neutralization v2. 0d: Full-Stack Neutralization Theorem with Calibrated Exponents, Pipeline, and a Worked Application on. . . * Zenodo. - Lee, Byoungwoo (2026). *Neutralizing the Eisenstein Continuum (v1. 8): A First-Principles Tool for High-Precision Arithmetic Quantum Chaos. * Zenodo. - Lee, Byoungwoo (2025). *Neutralizing Shadow Terms in BPS and Black Hole Microstate Counts: An Automorphic Projection and L-Value Transduction. * Zenodo. ## Suggested keywordsEUV metrology; High-NA EUV; CD-SEM; overlay; drift removal; orthogonal projection; finite-rank model; auditability; reproducibility; registry-driven pipeline; acceptance criteria; wafer map; process control.
Byoungwoo Lee (Sun,) studied this question.