Neutralizing EUV Drift (v1.1b) — Auditable Finite-Rank Neutralization for Sub–2 nm Metrology

# 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 \(V=\{v_1,\dots,v_r\}\subset H\) is declared in a registry. Drift is removed by a weighted orthogonal projection onto \(\mathrm{span}(V)^\perp\). The core guarantee is theorem-level: **if the systematic component lies in \(\mathrm{span}(V)\), 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 \(y\in H\) is modeled as\[y = s + d + \varepsilon,\]where \(s\) is the process-relevant component, \(d\) is systematic drift, and \(\varepsilon\) is stochastic noise. Whitening is represented by a positive self-adjoint weight \(W\succ 0\) defining\[\langle y_1,y_2\rangle_W := \langle Wy_1, y_2\rangle .\] ### Finite-rank neutralization operatorLet \(V:\mathbb{R}^r\to H\) be the dictionary synthesis map and \(G:=V^\*WV\in\mathbb{R}^{r\times r}\) the weighted Gram matrix. The weighted projection onto \(\mathrm{span}(V)\) is\[P^{(W)}_V := V\,(V^\*WV)^{-1}V^\*W,\]and the neutralizer is\[N^{(W)}_V := I - P^{(W)}_V.\]**Exact removal**: for any \(d\in\mathrm{span}(V)\), \(N^{(W)}_V d = 0\). ### Audit invariants and diagnosticsA practical invariant is the post-orthogonality residual:\[\mathrm{OrthRes}(y) := \frac{\|V^\*W\widehat y\|_2}{\|V^\*Wy\|_2 + 10^{-12}},\quad \widehat y := N^{(W)}_V y.\]Conditioning is enforced via \(\kappa(G)\le \kappa_{\max}\) (registry-fixed). ### Metrics (including the \(\beta_\star\) baseline rule)A primary example metric is the log–log PSD slope on a declared band \(I=[f_a,f_b]\):fit \(\log \widehat S_y(f)\) vs \(\log f\) on \(I\), denote slope by \(\widehat\beta(y)\).A baseline slope \(\beta_\star\) 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 \(\varepsilon_{\mathrm{floor}}>0\)):\[\mathrm{BRF} := \frac{M(y_{\mathrm{pre}})}{\max\{M(y_{\mathrm{post}}),\varepsilon_{\mathrm{floor}}\}}.\] ## 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 `pb_config.json`. 2) Compute \(G=V^\*WV\) and require \(\kappa(G)\le\kappa_{\max}\) (else FAIL). 3) Produce \(\widehat y = N^{(W)}_V y\) and require \(\mathrm{OrthRes}(y)\le\tau_\perp\) (else FAIL). 4) Run a negative control (no projection / \(r=0\)) and require drift persists beyond noise (else FAIL). 5) Require improvement: \(\mathrm{BRF}\ge \mathrm{BRF}_{\min}\) (else FAIL). 6) Require perturbation sensitivity: a mismatched dictionary degrades performance by \(\rho_{\mathrm{deg}}\) (else FAIL). 7) Require injection/leakage loss below \(\lambda_{\mathrm{inj}}\) for declared probes (else FAIL). 8) Require minimum bins/points \(n_{\min}\) 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:- `dataset_registry.json`: immutable dataset identity + raw-file hashes (per channel) - `pb_config.json`: window, whitening, dictionary, rank, fit bands, thresholds - `metrics.csv`: metrics for pre/post/control/perturbed/injection variants - `acceptance.json`: PASS/FAIL + supporting fields - `neutralization_certificate.json`: canonical payload + SHA256 digest - `report_card.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 \(R_0\) without affecting tool decisions; require sustained PASS and stable non-regression. - **Reference baseline \(R_0\)**: explicitly declare the legacy/vendor detrending/filtering pipeline and report a competitiveness gap\[\Delta_{R_0}(M) := \frac{M(y_{\mathrm{post}})}{\max\{M(R_0(y_{\mathrm{pre}})),\varepsilon_{\mathrm{floor}}\}}.\]- **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 & limitations- The mathematical claim is **finite-rank removal on a declared window**; it is not a claim of global stationarity, nor an inference of \(\beta_\star\) from the same audited run.- Success requires that drift be well-approximated by the declared \(\mathrm{span}(V)\), 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.