This repository accompanies a 45-page mathematical research dossier and an 11-page assessment with 24 material alternatives. It contains full proofs, source code, synthetic constitutive models, exact rational certificates, reproducible calculations and figures for designing binary barrier laminates under nonlinear water and oxygen transport.

Evidence status: conditional mathematical theory and internally audited synthetic computational certificates. No new polymer has been manufactured or measured. The work does not establish a globally superior plastic-replacement material.

Read the full research report · Rating and 24-alternative comparison · Detailed reproducibility guide · Machine-readable result map

Main contributions

Result	What is established	Scope
Sharp second-order finite-film recovery	Any measurable relaxed design has a common physical pure-film realization with exact inventories and skins, at most 2N interfaces, and an explicit O(N⁻²) terminal oxygen-resistance error after individual water-flux correction.	Positive smooth local water and independent oxygen laws; ideal steady binary 1D model.
Exact unequal-material interface frontier	Explicit globally attaining palindromes for every interface budget, unequal inventories and opposing slopes; skin activation, pure nonattainment and sharp inverse-square asymptotics.	The stated affine opposing-slope/common-oxygen-potential subclass.
Complete manufacturing geometry	Compact cumulative-material coordinates, a sharp constant-one primitive metric, local isometry and anchor-aware global covers.	A-ended binary films with fixed inventories and specified layer floors/interface limits.
Endpoint-value sensitivity	Exact phase-flow, oxygen-cost and boundary-root derivatives without derivatives of the material laws.	Positive continuous water and oxygen laws for finite homogeneous-layer words.
Executed nonlinear manufactured benchmark	A five-layer common film strictly exceeds the proved upper bound for every admitted three-layer film; a complete cover also bounds the full three-/five-layer optimum.	Two declared coefficient scenarios, both directions, at most four interfaces and a 0.02 minimum layer.
Measurement-to-design bridge	Paired-window identification identities, derivative-free coefficient enclosures, noise-resolution bounds, finite-data nonidentifiability and a fully-humid minimax limitation.	Conditional on the same local steady constitutive model and supplied uncertainty/regularity bounds.

The area-and-centroid pulse geometry, control relaxation, interval arithmetic, scalar comparison and global branch-and-bound have established precedents. The report credits them explicitly. The potentially distinctive contribution is the sharp transport-specific synthesis and exact unequal-slope occupation frontier. A targeted literature search is not an exhaustive priority or patent determination.

General finite-layer guarantee

Let the free physical thickness be L_f and divide it into N cells. On a cell of length h, preserve both B-phase amount and its first depth moment:

$B_0=\int_0^h v(t)\,dt,\qquad B_1=\int_0^h t\,v(t)\,dt.$

The bounds B₀²/2 ≤ B₁ ≤ hB₀−B₀²/2 make the pure cell

$A\!\left(\frac{B_1}{B_0}-\frac{B_0}{2}\right) \;/\;B(B_0)\;/\; A\!\left(h-\frac{B_1}{B_0}-\frac{B_0}{2}\right)$

feasible for 0 < B₀ < h. Pure-A/B cells handle the endpoint cases. Reversal exchanges the two A pieces, so the same physical construction works in both orientations.

The full proof supplies explicit coefficient-derived constants such that

$|R_s(F_N)-R_s(F)|\le C_{2,s}(L_f/N)^2.$

A retained exact example proves that exponent two is worst-case optimal across the smooth constitutive class. This is a terminal/integrated-output statement; it does not claim uniform-in-depth O(N⁻²) profile error. Arbitrarily thin pulse pieces remain possible, so a general positive minimum-layer constraint requires separate analysis.

See the complete proof and exact geometry replay.

Certified nonlinear manufacturing example

The same film is evaluated under two coefficient scenarios and both humidity directions. The manufacturing class includes every merged A-ended arrangement with:

Total thickness L = 1; exact H_A = 0.8 and H_B = 0.2.
Outer A layers at least 0.05.
Every merged layer at least 0.02.
At most four interfaces, covering both the three-layer and five-layer components and all admissible asymmetric geometries.

A concrete feasible common film is

$A (0.267173) / B (0.1) / A (0.265654) / B (0.1) / A (0.267173) .$

The water and independent oxygen laws are defined explicitly in the report, proof notes and kernel. The earlier common oxygen-potential relation is ruled out by exact polynomial-degree obstructions.

Certified quantity	Outward decimal bound
Five-layer candidate's worst oxygen resistance	≥ 2.018624455809
Every admitted three-layer film	≤ 1.933370475937
Guaranteed resistance gain over every three-layer arrangement	≥ 4.409603%
Upper bound on the complete admitted manufacturing class	≤ 2.280775121004
Remaining whole-class upper-to-lower gap	≤ 12.9866%

Larger resistance is a better oxygen barrier within this model. These values are normalized synthetic outputs, not measured commercial-polymer performance.

The saved certificate has 18 three-layer leaves and 1,348 five-layer leaves. Production and separately run internal audit replays checked the complete split trees, saved rational water-root brackets, oxygen intervals, candidate feasibility and every leaf ceiling. The internal audit reused the separately inspected arithmetic kernel; it is not external peer review or a wholly independent arithmetic implementation. Four deliberate certificate corruptions were rejected.

The 4.409603% architecture separation and 12.9866% remaining global gap are separate facts. The candidate is not certified as the exact best five-layer film. The executed kernel specializes to quadratic water laws and polynomial oxygen laws; the general endpoint-sensitivity theorem is broader than this production evaluator.

Exact certificate · Replay results · Blueprint · Independent internal audit

Exact unequal-inventory frontier

For r_A = p−αa and r_B = q+βa, with α,β > 0 and positive admitted laws, the exact count frontier separates according to the skin-activation boundary

$\beta H_B=\alpha(H_A-2h).$

Its geometric optimizer is an explicit palindrome. A global occupation-measure bound covers asymmetric competitors too. The unrestricted pure supremum is not attained; finite optimal stacks approach it with an explicit inverse-square leading constant.

Three declared regimes passed 23 finite-interface cases and 64 direct physical directional-root checks. The balanced Version VI frontier is recovered exactly. A minimum layer thickness is solved exactly only where the count-frontier stack passes every layer floor; otherwise the unequal examples correctly provide a feasible lower/count-upper sandwich.

Proof · Certificate · Replay

Measurement bridge and service limit

For a homogeneous specimen of thickness ℓ, paired steady measurements over the same activity window satisfy

$A(x,z)=\ell J(x,z)=\int_x^z K(a)\,da,\qquad B(x,z)=J(x,z)R(x,z)=\int_x^z K(a)q(a)\,da.$

This yields continuous-data identification K = A′ and q = B′/A′. With finite uncertain windows, supplied Lipschitz bounds instead give derivative-free coefficient enclosures. Finite windows alone cannot identify unrestricted smooth laws uniquely. Measurements taken at unmatched humidity, temperature or material state cannot be substituted as paired local-law data.

An exact application limit is also proved: if all phase oxygen resistances decrease with activity and the service family includes maximum humidity on both faces, every arrangement has the same worst-case steady resistance:

$\inf_s R_s(F)=H_Aq_A(a_{\max})+H_Bq_B(a_{\max}).$

Changing the layer order cannot improve that intrinsic fully humid bound. Better wet-state material laws, inventories, thickness or service conditions are then needed.

Measurement proof and checks · Uncertainty composition

Reproduce the release

Python 3.12 or a compatible later Python is recommended. Use UTF-8 mode on Windows.

python -X utf8 verify_release.py
python -X utf8 reproduce.py --quick
python -X utf8 reproduce.py --full

The first command checks the complete release SHA256 allowlist without network access. Quick replay runs the new exact geometry, unequal-frontier, sensitivity and measurement checks. Full replay also checks all 1,366 manufactured covering leaves; on the preparation environment that replay took about 100 seconds. Runtime elsewhere can differ.

The exact/replay programs use the Python standard library. Plot generation and rebuilding PDFs additionally require the packages in requirements.txt:

python -m pip install -r requirements.txt

Run verification in a working copy: some research checkers rewrite their result summaries. Preserve the canonical release certificate and use a separate output directory for time-bounded search reruns. See the detailed guide and module notes.

Repository structure and data interpretation

Location	Contents
Root PDFs	Current 45-page report and 11-page rating/comparison.
`moment/`	General recovery proof, exact cell certificates and separately labelled floating convergence illustrations.
`unbalanced/`	Exact unequal-material frontier, rational roots, layers, plots and replay.
`geometry/`, `sensitivity/`	Manufacturing coordinates and value-only interval sensitivity.
`manufactured/`	Nonlinear benchmark, complete covering certificate, exact kernel, blueprint, replay and tamper controls.
`identification/`	Synthetic paired-window checks, law enclosures and nonidentifiability proof.
`prior_art/`	Internal audits, method credits, replay evidence and historical/provisional audit snapshots labelled by their filenames.
`history/`	Earlier reports and assessments, superseded by VII.
Other research directories	Retained earlier Nūyori models under their original assumptions; see the detailed guide.
`provenance/`	Original source manifest and packaging changes.

This is a research-artifact repository hosted as a Dataset, not a trained AI model or a homogeneous machine-learning training table. The mixed CSV/JSON schemas describe different models and evidence types. The table viewer is disabled. All newly generated numerical coefficient examples are synthetic. Cited experimental papers describe other researchers' materials.

The balanced 83-layer example, VI's twelve-exposure common-film certificate and VII's manufactured benchmark are different models. Their normalized values must not be combined as one measured material. Canonical exact rational endpoints govern validated results; rounded floats and ordinary RK4 curves are not additional interval certificates.

Limitations and assessment

The physical model is local, steady, binary and one-dimensional, with fixed thickness, ideal continuous-activity interfaces and a nonperturbing oxygen probe. It excludes swelling, hysteresis, reactions, interfacial resistance, lateral bypass and unmodeled multicomponent cross effects. The relaxation is a mathematical control model rather than a validated homogeneous blend law.

Editorial theoretical assessment: T6/9. The final version broadens and strengthens the same tier. This is not a standardized metric or a manufactured-material rating. Mechanical performance, migration, recovery, carbon fate, processability, cost, scale and equal-function life cycle require independent evidence.

Authorship and AI provenance are stated in the report. No external peer review, published priority determination, new polymer synthesis or global plastic-replacement superiority is asserted. Existing-method citations and the 24-alternative material comparison are provided in the PDFs and references.json.

Citation and rights

Use CITATION.cff or citation.bib. The source release did not specify an explicit open-source or open-content license; this preparation does not invent one. See RIGHTS.md. Literature references do not transfer rights in the cited works.

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