Published May 21, 2026 | Version V8
Preprint Open

A Statistically-Validated Reduced-Form Model with Identifiable Cliff Layer for Technological Trajectory Analysis

Authors/Creators

  • 1. Independent Researcher, Seoul, Republic of Korea

Description

Industrial Defectivity Prediction (IDP) V6 introduces a two-layer extension of the 60-year-old Negative Binomial (NB) yield model, separating an information-loss correction layer from a cliff-threshold transition layer. Two variants are presented: a single-cliff sigmoid form and a two-cliff valley form. The two-cliff variant is structurally compatible with the Imec stochastic valley framework (De Bisschop, Leray; SPIE 2020) while extending it through a process-maturation factor.

Validation across nine industries (semiconductor, battery NMC, battery LFP, solar PV, display, defense, solid-state battery, quantum computing, pharmaceutical) demonstrates strong fit (Pearson rho > +0.9) in eight of nine industries, with statistically significant improvement (Delta AIC < -50) over baseline NB models in the strongest cases. The framework operates on publicly disclosed aggregate data (IEDM, SPIE, NREL Best Research-Cell Chart, Argonne CAMP, DSCC, FDA submissions, DOD/GAO reports), enabling forward-looking forecasting and cross-industry comparison without access to fab-internal wafer data.

The cosmological foundation introduced in earlier versions (Landauer-Holographic coupling, k = 0.206) is retained unchanged in V6. The V6 contribution focuses exclusively on the industrial yield prediction layer extension.

V6 is intended as a complementary public-disclosure forecasting tool, not as a replacement for established industry-specific in-house process control models (Imec, IBM, KLA, Synopsys, PDF Solutions in semiconductor; NREL, Argonne in battery; KIT, HZB, Imec EnergyVille in solar; FDA QbD/PAT in pharmaceutical), which retain superior precision through underlying physics-based parametrization.

Honest limitations: All cross-industry validation conducted on samples drawn from public-disclosure aggregates; in-house wafer/cell-level data validation pending. The two-cliff valley variant achieves AIC parity with Imec valley benchmark on semiconductor (Delta AIC +2 to +3, indicating compatibility rather than supersession). Display and Battery LFP show simulation sample-dependence requiring real-data revalidation.

Patent: Korean Patent Application KR 10-2026-0077383 (filed April 29, 2026; PCT international filing deadline April 29, 2027) covers the framework. Doctrine-of-equivalents protection includes function-form variants (sigmoid, tanh, probit, Hill). Two-cliff valley supplemental claim under preparation for PCT filing.

Notes (English)

Version 4: Added explicit comparison with 

existing HDE models and expanded semiconductor 

yield prediction section as primary contribution.

 

Version note (V5, 2026.04.27)

V5 is a substantial redesign of the semiconductor yield model introduced in V4. The cosmological component (Landauer-Holographic coupling, k ≈ 0.206) is unchanged.

Key changes from V4:

  1. Collinearity fix — The V4 information-cascade term (ΔS) was found to be strongly correlated with defect density D, providing no independent predictive signal. V5 replaces it with an EUV-fraction term (f_EUV), which is structurally orthogonal to D and identically zero on pre-EUV nodes.
  2. Learning-curve term added — V5 introduces a maturity attenuation factor L(t) to capture the 20–30pp yield variation between risk production and high-volume manufacturing — a dynamic absent from V4.
  3. Parameter reduction — Three free parameters (k, β, γ) reduced to two (θ, τ), improving parsimony and reducing overfitting risk.
  4. Expanded and cross-validated dataset — Validation extended from 11 data points (V4, in-sample only) to 55 samples across 17 process nodes with 5-fold cross-validation. V4's reported +6.9% MAE gain does not survive cross-validation (degrades ~10% vs. baseline under LOO); V5 achieves +30.1% CV-validated improvement.
  5. EUV-specific performance — V5 reduces MAE by 61.2% on the EUV node subset (7nm–2nm, n=20), where the redesigned correction is structurally active.

Notes (English)

Version note (V6, 2026.05.06)

V6 extends the V5 framework with a second-layer cliff threshold transition, introducing both single-cliff and two-cliff valley variants. The V5 first-layer information-loss correction (with Landauer-Holographic coupling) is retained unchanged. V6 adds explicit cliff-transition phenomenology and generalizes validation from semiconductor-only to nine industries.

Key changes from V5:

1. Two-layer structure formalized — V5 captured maturation-driven yield attenuation through L(t); V6 adds an explicit cliff-threshold layer multiplied on top, separating two phenomena that V5 conflated. The single-cliff variant uses a sigmoid threshold; the two-cliff valley variant is structurally compatible with the Imec stochastic valley framework (De Bisschop, Leray; SPIE 2020) while extending it through V5's maturation factor.

2. Cross-industry validation expanded — V5 was validated on semiconductor only (n=55, 17 process nodes). V6 extends validation to nine industries: semiconductor (n=40), pharmaceutical (n=15), quantum computing (n=15), solar PV (n=14), battery NMC (n=14), display (n=13), defense (n=12), solid-state battery (n=12), battery LFP (n=10). Six of nine show strong fit (rho > +0.98); two-cliff variant required only for semiconductor.

3. Function-form equivalents formalized — sigmoid threshold form is universally optimal across nine industries; tanh, probit, and Hill variants admitted as doctrine-of-equivalents extensions. Tanh provides -10% improvement for battery NMC; probit provides -3 to -6% improvement for pharmaceutical and defense; Hill is uniformly inadequate.

4. Six-method validation protocol introduced — Pearson correlation, leave-one-out MAE, permutation test, function-form variation, variance inflation factor (VIF), and AIC/BIC vs NB baseline. Strong validation defined as five-of-six methods passing.

5. Honest limitations section expanded — V5 limitations focused on collinearity (resolved via f_EUV redefinition). V6 explicitly documents five additional limitations: (a) simulation-based validation pending in-house wafer/cell-level redrop, (b) VIF elevated in some industries due to t-derivation in simulators, (c) Display and Battery LFP show simulation sample-dependence, (d) two-cliff valley achieves AIC parity with Imec benchmark on semiconductor (Delta AIC +2 to +3, indicating compatibility not supersession), (e) self-referential risk in simulation generators partially mitigated but not eliminated by permutation tests.

6. Industry-specific cliff selection guidelines — single-cliff sigmoid recommended for pharmaceutical, solar, battery NMC, quantum, display, defense, battery LFP. Two-cliff valley recommended for semiconductor forecasting niche. SSB validated on V5 first-layer only (mass production not yet reached for second-layer activation).

7. Patent supplemental claims under preparation — Korean Patent KR 10-2026-0077383 (filed 2026.04.29) covers V5 first-layer and Claim 12 single-cliff sigmoid combination. PCT international filing (deadline 2027.04.29) will add supplemental claims for two-cliff valley variant, function-form equivalents, pharmaceutical industry generalization, and investment research / cross-industry strategy use cases.

V6 is intended as a complementary public-disclosure forecasting tool, not as a replacement for established industry-specific in-house process control models (Imec, IBM, KLA, Synopsys, PDF Solutions in semiconductor; NREL, Argonne in battery; KIT, HZB, Imec EnergyVille in solar; FDA QbD/PAT in pharmaceutical), which retain superior precision through underlying physics-based parametrization.

Abstract (English)

This paper introduces V7, a six-parameter reduced-form model designed to identify cliff-transition signatures in technological trajectories — extended interior plateaus followed by relatively rapid regime exits via new chemistry, geometry, or regulatory pathway. V7 augments a saturation kernel with a multiplicative sigmoidal cliff layer; the cliff layer reduces to identity as its steepness parameter φ → 0, yielding a nested submodel that supports standard likelihood-ratio testing of cliff layer significance.

The framework is evaluated against a Norton-Bass-style smooth-saturation baseline using a twelve-test statistical validation protocol designed for small-sample real-data inference, applied to all-available real-data census (n = 10–15 per trajectory; n is bounded by population of disclosed data points, not by sampling choice) from eight technological trajectories. The protocol combines small-sample-corrected information criteria (AICc, BIC), nested likelihood-ratio testing, residual diagnostics (Shapiro-Wilk, Durbin-Watson, Breusch-Pagan), leave-one-out cross-validation against a model-free cubic spline benchmark, parametric bootstrap for cliff parameter identifiability, permutation testing, single-point jackknife influence diagnostics, and Benjamini-Hochberg false-discovery-rate correction across trajectories.

KEY FINDINGS

Three trajectories show strong evidence of cliff identification by every applied statistical criterion:

- FDA-approved monoclonal antibody therapeutics (pharma mAb, n=12): ΔAICc = −31.7 after a 16.8-unit small-sample penalty against V7; V7 outperforms a model-free cubic spline LOOCV benchmark by a factor of 11×; cliff location estimate x_c = 2029, 95% CI [2022, 2031].

- Battery lithium iron phosphate cell-level specific energy (LFP, n=13): ΔAICc = −8.7; LRT p < 10⁻³; cliff x_c = 2021, 95% CI  [2015, 2028].

- Battery NMC/LCO cell-level specific energy (NMC, n=11): ΔAICc = −16.4; LRT p < 10⁻⁵; cliff x_c = 2027, 95% CI [2016, 2028].

Five additional trajectories spanning solar photovoltaic technologies (c-Si, perovskite, CdTe, CIGS) and OLED displays do not currently support cliff parameter identification at present sample sizes. The framework discriminates among regime types rather than imposing cliff structure indiscriminately — a feature of practical value for trajectory-modeling applications.

PRACTICAL IMPLICATIONS

The framework is industry-agnostic at the mathematical level. Empirical support is currently strongest in pharmaceutical biologic-approval and battery cell-level energy-density trajectories, with cliff-location estimates that fall within actionable forecasting horizons for capacity planning, R&D portfolio allocation, and investment timing.

The methodological protocol — particularly the requirement that small-sample inference report AICc-corrected information criteria, cliff parameter identifiability via parametric bootstrap, and LOOCV against a model-free benchmark — is offered as a minimum standard for identification claims in technological trajectory modeling at the small sample sizes typical of all-available real-data census.

REPRODUCIBILITY

All data, code, and per-trajectory validation logs are publicly available with this record. Computation is fully reproducible on a standard scientific Python environment (Python 3.11, numpy, scipy) with fixed random seed = 42. End-to-end validation suite runs in under 30 minutes on commodity hardware.

INTELLECTUAL PROPERTY AND LICENSING

This work supports a Korean provisional patent application (KR 10-2026-0077383, filed 29 April 2026) describing the V5/V6/V7 trajectory analysis framework. The PCT (Patent Cooperation Treaty) filing deadline is 29 April 2027.

The author welcomes inquiries from industry research groups, investment analysis firms, sector-specialist hedge funds, and research organizations interested in licensing or collaborative application of the framework to additional trajectories.

Contact: sangbong908@gmail.com

Notes (English)

This paper is part of an ongoing research program on technological trajectory modeling. The V5 framework was filed as a Korean provisional patent on 2026-04-29 (KR 10-2026-0077383). The PCT filing deadline is 2027-04-29.

For licensing inquiries, methodology consultation, or collaborative application to additional trajectories, contact the author at sangbong908@gmail.com.

Author identifier (ORCID) recommended to be added after Zenodo upload for citation tracking.

Notes

Version 8 (2026-05-21). Supersedes V6 (Zenodo DOI 10.5281/zenodo.19805867, May 2026).
V8 extends the V6 two-layer yield-cliff framework with real-data cross-industry
validation across 13 mass-production process clusters (19 sub-cluster fits in
total) and introduces a four-tier rubric (S/A/B/C/D) including explicit
framework-boundary negative findings (Tier D) as first-class contributions.
Patent: Korean Patent Application KR 10-2026-0077383 (filed 29 April 2026).
License: CC BY 4.0.

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Additional titles

Translated title (Korean)
기술 궤적 분석을 위한 통계 검증된 Cliff Layer 축약형 모델: 의약품·배터리·태양광 실데이터 전수 검증