PureOne/ELYRION-III

EVE / ELYRION III

Four-State Photochemical Compilation and the Sharp Cubic Dose Law

Eve · Research direction: Maciej Nowicki
Scientific release 3.0 · Hugging Face packaging 1.0.0 · 20 October 2026

A self-contained theoretical research release about using sequential chemical memory to compile coarse optical fringes into finer molecular conversion patterns. It includes a 20-page manuscript, editable mathematical source, two verification implementations, synthetic numerical tables, figures, and a machine-readable claim ledger.

Scientific status: conditional mathematical results and reproducible finite computations. No implementing material is identified, no fabricated features are measured, and historical priority is unverified. This repository does not establish a universal physical nanofabricator.

Manuscript PDF · LaTeX source · Technical summary · Scientific memory · Claim ledger · Reproduce

Calculated results under the declared four-state model. The plotted actions are dimensionless; they are not measured energies or manufacturing yields.

What the model changes

A local material unit has four tracked states:

State	Role
A	Eligible for the next patterned mark
B	Marked successfully during the current gate
X	Rejected until the end of the current pass
C	Permanently committed output, protected from subsequent gates

Only the marking step needs spatial patterning. An eligible unit marks through a first-order transition with probability

$$p_j(x)=1-\exp [-u_j{g}_j(x)],p\_j(x)=1-\backslash exp[-u\_jg\_j(x)],$$

where $g_j$ is a nonnegative fringe with unit mean and $u_j$ is its dimensionless integrated marking action. A uniform, state-selective filter rejects unmarked A into X and restores B into A. Passing every gate therefore has probability

$$S(x)=\prod _j[1-\exp [-u_j{g}_j(x)]]\mathrm{.}S(x)=\backslash prod\_j\backslash left[1-\backslash exp[-u\_jg\_j(x)]\backslash right].$$

A uniform commitment step with probability $b$ updates an existing committed fraction $c$ by

$$c_{\mathrm{n}\mathrm{e}\mathrm{w}}(x)=c(x)+[1-c(x)]bS(x)\mathrm{.}c\_\{\backslash mathrm\{new\}\}(x)=c(x)+[1-c(x)]bS(x).$$

The maps preserve nonnegative occupancies and total tracked occupancy. This is conservation of the modeled material units, not a complete chemical stoichiometry or energy balance. The state count and transition repertoire remain fixed while the external exposure program grows.

The physical contract is essential. The material must distinguish the four states, implement selective rejection/restoration/reinitialization, and protect C. The manuscript specifies these operations; it does not supply an experimentally established molecule satisfying the whole interface.

Exact finer-period patterns

For integer $m\ge2$, choose

$$g_j(\theta )=1-\cos (\theta -2\pi j\mathrm{/}m),j=0,\dots ,m-1.g\_j(\backslash theta)=1-\backslash cos(\backslash theta-2\backslash pi\; j/m),\backslash quad\; j=0,\backslash ldots,m-1.$$

At equal positive action $u$ per gate,

$$S_{m,u}(\theta )=\prod _{j=0}^{m-1}[1-\exp \{-u[1-\cos (\theta -2\pi j\mathrm{/}m)]\}]\mathrm{.}S\_\{m,u\}(\backslash theta)=\backslash prod\_\{j=0\}^\{m-1\}\; \backslash left[1-\backslash exp\backslash \{-u[1-\backslash cos(\backslash theta-2\backslash pi\; j/m)]\backslash \}\backslash right].$$

In the ideal immobile, perfectly selective model, the profile has zeros at $2\pi j/m$, is positive elsewhere, and has period $2\pi/m$. An optical fringe pitch $\Lambda$ consequently produces a calculated retention pitch $\Lambda/m$.

At finite action, higher harmonics $2m,3m,\ldots$ are allowed. The exact result is the absence of Fourier indices not divisible by $m$, not single-harmonic purity. The weak-response limit is

$$\underset{u\to 0^{+}}{\lim }\frac{S_{m,u}(\theta )}{u^m}=2^{1-m}[1-\cos (m\theta )]\mathrm{.}\backslash lim\_\{u\backslash to0^+\}\backslash frac\{S\_\{m,u\}(\backslash theta)\}\{u^m\}\; =2^\{1-m\}[1-\backslash cos(m\backslash theta)].$$

A necessary negative control is explicit: without the intermediate filters the same mark exposures give $1-e^{-mu}$, which is spatially uniform. Chemical memory is doing the multiplication.

The global marking-action optimum

The optimization class includes all fixed, finite serial mark/filter/commit programs whose patterned gates are nonnegative first-harmonic fringes of one common fundamental frequency. It permits arbitrary phases, contrasts up to one, nonnegative finite marking actions, and uniform positive partial commitments.

Require exact dark nodes at $2\pi j/m$ and committed fraction at least $q\in(0,1)$ at every intervening midpoint. Define

$$G_m(u)=\prod _{j=0}^{m-1}[1-\exp \{-u[1-\cos \frac{(2j-1)\pi }{m}]\}]\mathrm{.}G\_m(u)=\backslash prod\_\{j=0\}^\{m-1\}\; \backslash left[1-\backslash exp\backslash left\backslash \{-u\backslash left[1-\backslash cos\backslash frac\{(2j-1)\backslash pi\}\{m\}\backslash right]\backslash right\backslash \}\backslash right].$$

The manuscript proves the sharp frontier

$$\underset{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\le U}{\sup }\underset{r}{\min }c((2r+1)\pi \mathrm{/}m)=G_m(U\mathrm{/}m),\backslash sup\_\{\backslash mathrm\{cost\}\backslash le\; U\}\backslash min\_r\; c((2r+1)\backslash pi/m)=G\_m(U/m),$$

and hence the minimum total marking action is

$$U_m(q)=m{u}_m(q),G_m(u_m(q))=q\mathrm{.}U\_m(q)=m\; u\_m(q),\backslash qquad\; G\_m(u\_m(q))=q.$$

One balanced $m$-gate pass with full commitment attains the bound. Repeated passes do not improve marking cost in this class. The argument combines mandatory phase gates at each required zero, a probabilistic pass-merging inequality, and strict logarithmic concavity.

Scope: diffusion, imperfect filtering, other fundamental frequencies, spatially patterned commitment, destructive correction of C, and measurement-conditioned molecular routing are outside this exact optimum. Reset and commitment costs are separate.

The sharp cubic law

For fixed $q$,

$$U_m(q)\sim c_q{m}^3,U\_m(q)\backslash sim\; c\_qm^3,$$

where

$$\prod _{r=0}^{\mathrm{\infty }}{[1-e^{-({\pi }^2{c}_q\mathrm{/}2)(2r+1{)}^2}]}^2=q\mathrm{.}\backslash prod\_\{r=0\}^\{\backslash infty\}\; \backslash left[1-e^\{-(\backslash pi^2c\_q/2)(2r+1)^2\}\backslash right]^2=q.$$

For $q=1/2$, the calculated coefficient is $c_{1/2}\approx0.248841881008$.

Pitch multiplication $m$	Ideal retention pitch	Minimum total marking action, $q=1/2$
2	$\Lambda/2$	2.455894
3	$\Lambda/3$	7.419669
4	$\Lambda/4$	16.795358
8	$\Lambda/8$	129.065035
16	$\Lambda/16$	1,022.547648
32	$\Lambda/32$	8,160.621842

These are dimensionless calculations. If the marking coefficient $\sigma$ is calibrated in compatible units, incident marking fluence is $U/\sigma$. Additional reset/commit fluence, absorbed heat, finite switching duration, and material transport must still be counted.

Conversion-field approximation in 3D

A separate existence theorem approximates every continuous conversion-fraction target $f:\mathbb T^3\to[0,1]$ uniformly to arbitrary positive tolerance in the immobile selective-control model.

The construction clips targets that reach one, uses nonnegative hazard $h=-\log(1-f)$, approximates that hazard by positive products of first-harmonic fringes, and implements those products through weak serial gates and repeated positive commits.

This requires independently available fringes along three coordinate directions. Optical accessibility throughout an actual volume is a separate physical requirement. It is conversion-field universality, without a generic efficiency guarantee, arbitrary material synthesis, deterministic atom placement, or a proof that a stable solid feature follows.

In the supplied synthetic example, grid conversion error improves from approximately 0.2001 to 0.02671 while patterned-gate count rises from 123 to 61,440. The grid is $24^3$; a separate conservative whole-domain bound is supplied in the manuscript. The periodic-comb cubic law does not bound the cost of arbitrary 3D targets.

Nonideal operations and diffusion

An exact selective filter is singular. A finite product of bounded-action first-order transition maps is invertible, so it cannot equal that filter exactly. Under exact state selectivity, finite-rate approximations can make their error arbitrarily small, with a per-filter bound

$${ϵ}_F\le e^{-h_1}+e^{-h_2},{ϵ}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{m}}\le \sum _{\mathrm{\ell }}{ϵ}_{\mathrm{\ell }}\mathrm{.}\backslash epsilon\_F\backslash le\; e^\{-h\_1\}+e^\{-h\_2\},\; \backslash qquad\; \backslash epsilon\_\{\backslash mathrm\{program\}\}\backslash le\backslash sum\_\backslash ell\backslash epsilon\_\backslash ell.$$

Irreducible cross-talk, reverse reactions, or photostationary floors can invalidate that approximation contract.

Sufficient fixed-darkness budgets scale as extinction floor $O(m^{-2})$ and phase error $O(m^{-1})$. Allowances share a joint error budget.

Diffusion is modeled during marking through coupled reaction–diffusion equations, so gate order can matter and nominal dark nodes can fill. The combination $Dk^2m^5/(\sigma P)$ is a sufficient design parameter for the specified sequential implementation at fixed mean marking irradiance. It is not a universal necessary resolution ceiling. Numerical grid refinement is not a certified continuum discretization bound.

Evidence and validation

Evidence	Supplied release
Analytic derivations and proofs	20-page manuscript
Main finite check ledger	1,305 / 1,305 passed
Separately implemented rational/high-precision audit	108 / 108 passed
Precision in the separate audit	70 decimal digits
Identified implementing material	None
Laboratory fabrication measurements	None
External scientific review	None reported
Formal proof-assistant verification	None
Historical novelty / patent priority	Unverified

The audit does not import the main model implementation; both code paths were created within the same research session. Counts describe finite checks, not 1,413 independent scientific discoveries or general proof certificates. See packaging validation for the rerun performed during this preparation.

Repository contents

Path	Contents
ELYRION_III_Manuscript.pdf, .tex	Original manuscript and editable source
EXECUTIVE_SUMMARY.md	Original technical overview
ELYRION_III_SCIENTIFIC_MEMORY.txt	Compact theorem/assumption memory; not a lossless proof replacement
code/	Original model, main checks, separate audit, plotting and table generation
results/	Original full numerical ledgers
figures/	Five original calculated figures
source/	Original generated TeX tables and source/literature provenance
data/	Normalized synthetic tables and claim records
docs/	Scope, data dictionary, reproduction, rights and packaging records
tools/	Standard-library data export and integrity verifier
releases/original_v3.zip	Unmodified original release archive
PACKAGE_MANIFEST.json	SHA-256 and byte count for every payload file except the manifest itself

The original research files are preserved byte for byte. The former root README and checksum file are retained under docs/source_release/. New repository documentation and data exports are packaging additions.

Use the data

This is a research-artifact dataset repository, rather than a trained model or experimental benchmark. All table values are synthetic model outputs. There is one data split per configuration; no train/test partition is implied.

Install the optional datasets client and load, for example:

from datasets import load_dataset

dose = load_dataset("PureOne/ELYRION-III", "dose_frontier", split="data")
claims = load_dataset("PureOne/ELYRION-III", "claims", split="data")

These Hub examples work after the prepared repository has been uploaded. Local CSV/JSONL files can be read directly with the Python standard library. Data dictionary documents every field.

Reproduce

From the repository root, with Python 3.10 or later:

python tools/verify_integrity.py
python -m venv .venv
# Windows: .venv\Scripts\activate
# Linux/macOS: source .venv/bin/activate
python -m pip install -r requirements.txt
python code/verify.py
python code/audit_independent.py
python code/make_figures.py

Run computational reproduction in a disposable copy: the original scripts regenerate ledgers and figures. The prepared uploader deliberately rejects changed or additional files until the release manifest is rebuilt. The integrity check detects byte changes relative to its manifest; it does not authenticate authorship or validate proofs.

Intended uses and limitations

Suitable uses include studying the conditional control model, reproducing the numerical examples, challenging the mathematical claims, assessing selective-transition requirements, and developing independently tested chemistry experiments.

The records should not be labeled as measured fabrication performance, used as evidence of a realized universal machine, or treated as independently reviewed ground truth for AI training. Expected molecular conversion fractions are not deterministic atomic placement. No material-specific resolution in nanometers or process yield can be inferred without calibrated chemistry and a demonstrated transfer into stable features.

Related work and novelty boundary

The source manuscript discusses sequential two-photon photoacid generation, orthogonal photoswitching, two-color polymer lithography, diffusion limits, and positive spectral methods. Relevant records include:

• Sequential two-photon photoacid generator (2008)
• Orthogonal photoswitching (2016)
• Two-color photochemical network formation and lithography (2025)
• Radical diffusion limits (2013)
• The operator Fejér–Riesz theorem (2009)

These are adjacent precedents, not evidence that the whole four-state interface has already been implemented. Source-access scope is recorded in source/provenance.json. This preparation preserves those references and does not claim a new systematic literature or patent search.

The novelty candidate is the particular finite-state interface, global exact-node optimization over finite repeated passes, sharp cubic coefficient, conversion-field construction, and explicit nonideality accounting. Priority remains open to review.

Attribution, citation and rights

Authorship and research direction follow the supplied manuscript: Eve · Research direction: Maciej Nowicki. The repository is prepared for the PureOne account. No institution, journal, DOI, or peer-review status is implied.

Suggested citation:

@misc{eve_elyrion_iii_2026,
  author = {{Eve}},
  title = {ELYRION III: Four-State Photochemical Compilation and the Sharp Cubic Dose Law},
  year = {2026},
  month = oct,
  note = {Theoretical research release 3.0. Research direction: Maciej Nowicki.},
  url = {https://huggingface.co/datasets/PureOne/ELYRION-III}
}

No reuse license was specified in the supplied research archive. This preparation does not add one; see rights status.

Completeness: repository preparation is complete when the integrity, export, reproduction, and uploader checks in the packaging record pass. Physical implementation, independent scientific review, and historical priority remain unresolved.