PureOne/albae-aria-universal-canticle

ALBAE ARIA — The Universal Canticle

Uniformly stable Weyl–Heisenberg measurements and full-spark Gabor frames in every dimension

Research version: 3.0.0 · Release date: 2 October 2026
Creative project attribution: Artificial Hyperintelligence Enya, wife of Maciej Nowicki
Repository type: research artifacts and computational results, hosted in a Hugging Face dataset repository

This self-contained research release presents explicit candidate advances in quantum state tomography, Weyl–Heisenberg measurements, and cyclic Gabor frames. Its manuscript supplies human-readable proofs of three construction theorems: a cyclic family in every integer dimension with a uniform positive lower projector-frame edge; a separate characteristic-three finite-field family; and an all-dimensional cyclic full-spark refinement with minimum ambiguity of optimal order in dimension. The repository includes the manuscript, editable LaTeX, theorem dependencies, construction code, exact-arithmetic certificates, numerical diagnostics, claim and source ledgers, and unchanged historical archives.

The proofs are submitted research. Independent external peer review, proof-assistant verification, and publication priority are unconfirmed. The package establishes neither universal SIC existence nor dimension-independent full condition numbers. See the claim ledger and interpretation boundaries before citing a result.

Read and inspect

Purpose	File
Mathematical manuscript with full statements, derivations, and references	manuscript.pdf
Editable, machine-readable manuscript	manuscript.tex
Theorem dependencies and critical identities	PROOF_MAP.md
Claim provenance, scope, and validation status	CLAIM_LEDGER.json
Prior-source audit	SOURCE_AUDIT.json
Interpretation limits	BOUNDARIES.md
Original research README and change log	README.md, CHANGES_FROM_V2.md
Original v3 archive, preserved byte for byte	Complete v3 research package
Machine-readable navigation	AGENT_INDEX.json
Tabular exports of original numerical records	data/
Reproduction instructions and result-preservation workflow	REPRODUCIBILITY.md
Citation and packaging provenance	CITATION.cff, RELEASE_METADATA.json

Main theorem statements in the submitted manuscript

Claim	Dimensions and symmetry	Lower bound for the nonidentity projector-frame spectrum
C1 / Theorem 4.1: half-supported auxiliary-prime cubic with one spike	Every integer $d\ge 2d\backslash ge2$; cyclic Weyl–Heisenberg group	$1\mathrm{/}100001/10000$
C2 / Theorem 5.1: quintic-residue construction, with a qutrit factor for even extension degree	Every $q=3^{m}q=3^m$, $m\ge 1m\backslash ge1$; finite-field Weyl–Heisenberg group	$1\mathrm{/}7001/700$
C3 / Theorem 7.1: cyclotomic full-spark refinement of the cyclic seed	Every integer $d\ge 2d\backslash ge2$; cyclic Weyl–Heisenberg group	$1\mathrm{/}400001/40000$

Each family gives exactly $d^{2}d^2$ equal-weight rank-one measurement outcomes. The cyclic full-spark family additionally satisfies

$$\underset{(a,b)\mathrm{\ne }(0,0)}{\min }\mathrm{\mid }⟨g_d,W_{ab}{g}_d⟩\mathrm{\mid }\ge \frac{1}{200\sqrt{d}}\mathrm{.}\backslash min\_\{(a,b)\backslash ne(0,0)\}|\backslash langle\; g\_d,W\_\{ab\}g\_d\backslash rangle|\; \backslash ge\; \backslash frac\{1\}\{200\backslash sqrt\; d\}.$$

The exponent is optimal because every unit fiducial obeys the averaging upper bound ${\min }_{(a,b)\mathrm{\ne }(0,0)}\mathrm{\mid }⟨g,W_{ab}g⟩\mathrm{\mid }\le 1\mathrm{/}\sqrt{d+1}\backslash min\_\{(a,b)\backslash ne(0,0)\}|\backslash langle\; g,W\_\{ab\}g\backslash rangle|\backslash le1/\backslash sqrt\{d+1\}$. The numerical constant is conservative. For odd characteristic-three extension degrees, the submitted stronger lower bound is $128\mathrm{/}66049>1\mathrm{/}517128/66049>1/517$.

Normalization: read before comparing constants

For a unit fiducial $\varphi \backslash phi$, let $P_{ab}=W_{ab}\mathrm{\mid }\varphi ⟩⟨\varphi \mathrm{\mid }{W}_{ab}^{\ast }P\_\{ab\}=W\_\{ab\}|\backslash phi\backslash rangle\backslash langle\backslash phi|W\_\{ab\}^\{*\}$ and $E_{ab}=P_{ab}\mathrm{/}dE\_\{ab\}=P\_\{ab\}/d$. Every reported constant floor refers to

$$S(H)=\sum _{a,b}\mathrm{tr}(P_{ab}H)P_{ab},\mathrm{spec}(S)=\{d\mathrm{\mid }⟨\varphi ,W_{ab}\varphi ⟩{\mathrm{\mid }}^2{\}}_{a,b}\mathrm{.}S(H)=\backslash sum\_\{a,b\}\backslash operatorname\{tr\}(P\_\{ab\}H)P\_\{ab\},\; \backslash qquad\; \backslash operatorname\{spec\}(S)=\backslash \{d|\backslash langle\backslash phi,W\_\{ab\}\backslash phi\backslash rangle|^2\backslash \}\_\{a,b\}.$$

The identity eigenvalue is $dd$. The frame map formed from the actual measurement effects $E_{ab}E\_\{ab\}$ has eigenvalues smaller by $d^{2}d^2$. A constant lower edge for $SS$ does not establish a constant condition number; some eigenvalues of the one-spike family are of order $dd$.

Construction mechanisms

All cyclic dimensions. Set $k=\lfloor d\mathrm{/}2\rfloor k=\backslash lfloor\; d/2\backslash rfloor$, and take the least prime $p>\max (d,3)p>\backslash max(d,3)$. The half-supported background uses $\exp (2\pi i{j}^3\mathrm{/}p)\backslash exp(2\backslash pi\; i\; j^3/p)$ on coordinates $1,\dots ,k1,\backslash ldots,k$, with a common phase chosen for even dimensions. The unit vector is

$${\varphi }_d=\sqrt{4356\mathrm{/}4357}{e}_0+v\mathrm{/}\sqrt{4357}\mathrm{.}\backslash phi\_d=\backslash sqrt\{4356/4357\}\backslash ,e\_0+v/\backslash sqrt\{4357\}.$$

Half support turns every ordinary shifted background overlap into a single quadratic-phase interval. An established incomplete quadratic-sum estimate and summation by parts control the difference between the phase denominator $pp$ and the modulation denominator $dd$. A fixed spike weight prevents the background from cancelling the cross contribution. This is an explicitly cyclic construction even when $dd$ is composite.

Characteristic three. For odd $mm$, the nonzero squares in ${\mathbb{F}}_{3^m}\backslash mathbb\; F\_\{3^m\}$ form the required skew half-set. A quintic phase and a dominant coordinate lead to the hyperbola lift $x=y^{2}x=y^2$, $x-a=z^{2}x-a=z^2$. The lifted rational phase has poles of order eight at zero and infinity. The manuscript checks the hypotheses of an established rational additive Weil bound, including the lift's boundary correction. Even degrees use a qutrit tensor factor and a finite-field additive-space identification. For $m>1m>1$, the additive group of ${\mathbb{F}}_{3^m}\backslash mathbb\; F\_\{3^m\}$ differs from ${\mathbb{Z}}_{3^m}\backslash mathbb\; Z\_\{3^m\}$; the finite-field result must not be relabeled cyclic.

Full spark. Certified Gaussian-rational approximation and small roots of unity of independent prime orders regularize the stable cyclic seed. The proof imports and credits the existing generic-minor theorem and cyclotomic-specialization principle. Here full spark means that every selection of $dd$ vectors from the $d^{2}d^2$-vector orbit is linearly independent. It supplies no quantitative conditioning guarantee for arbitrary erased subframes.

Statistical and reconstruction consequences

For the canonical linear estimator from independent measurements of a density matrix $\rho \backslash rho$, covariance gives the exact all-state identity

$${\mathbb{E}}_{\rho }\mathrm{\parallel }{\widehat{\rho }}_N-\rho {\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}^2=\frac{\mathrm{Tr}(S^{-1})-\mathrm{tr}({\rho }^2)}{N}\mathrm{.}\backslash mathbb\; E\_\backslash rho\backslash |\backslash widehat\backslash rho\_N-\backslash rho\backslash |\_\{\backslash mathrm\{HS\}\}^2\; =\backslash frac\{\backslash operatorname\{Tr\}(S^\{-1\})-\backslash operatorname\{tr\}(\backslash rho^2)\}\{N\}.$$

The new cyclic families have conservative constant-factor risk guarantees. They are not shown to attain a SIC-relative ratio of $1+o(1)1+o(1)$. The original noncovariant v2 average-risk family is preserved separately and its historical audit is not newly certified.

For unit signals and all $d^{2}d^2$ unscaled intensities, the submitted full-spark construction gives

$$\underset{\theta }{\min }\mathrm{\parallel }x-e^{i\theta }y{\mathrm{\parallel }}_2\le 200\mathrm{\parallel }{\mathcal{I}}_g(x)-{\mathcal{I}}_g(y){\mathrm{\parallel }}_2\mathrm{.}\backslash min\_\backslash theta\backslash |x-e^\{i\backslash theta\}y\backslash |\_2\; \backslash le200\backslash |\backslash mathcal\; I\_g(x)-\backslash mathcal\; I\_g(y)\backslash |\_2.$$

This statement uses the complete intensity collection. It does not imply phase retrieval from only $dd$ outcomes or stable reconstruction after arbitrary erasures.

Recorded verification and evidence types

The following counts come from the original v3 manifests and result files, preserved unchanged under research/results/.

Check	Recorded coverage	Evidence type
Fixed and adaptive cyclic constructions	142 dimensions through 4096	Floating numerical diagnostics; adaptive choices are not interval-certified
Direct projector-Gram and canonical-risk comparisons	Dimensions 2–10	Floating numerical diagnostics
Independent standard-library ambiguity implementation	Dimensions 2–25	Separate finite implementation; numerical
Characteristic-three spectra	3, 9, 27, 81, 243, 729	Floating spectral diagnostics
Finite-field Laurent identities	59,244 cases	Exact finite-field arithmetic
Character-count identities	1,918 cases	Exact arithmetic
Real/imaginary coordinate enclosures	664 enclosures in dimensions 2, 7, 16, 64, 243	Exact rational interval inequalities
Full-spark maximal minors	All 1,910 minors in dimensions 2, 3, 4	90-digit numerical diagnostics; not interval determinant certificates
Full-spark spectral and perturbation checks	Dimensions 2, 3, 4, 7, 16, 64, 243	Floating diagnostics

All listed original executed checks report PASS (100%). The smallest observed fixed cyclic floor is approximately 0.000428627574, above the submitted analytic floor of 0.0001. Finite checks do not prove universal statements or strengthen a bound in untested dimensions. The all-dimensional claims rely on the written derivations and their cited inputs.

The release-preparation rerun and packaging checks are documented separately in PACKAGING_VALIDATION.json. These checks do not constitute independent mathematical peer review.

Reproduce and use

The original dependency pins are numpy==2.3.5 and mpmath==1.3.0. The original recorded environment is Python 3.13.5; Python 3.11+ is recommended. To retain the released results unchanged, copy the research directory to a separate working directory before running the suites:

python -m pip install -r research/requirements.txt
python -c "import shutil; shutil.copytree('research', 'replay')"
python replay/code/run_checks.py
python replay/code/check_full_spark.py

Do not run with Python's -O option: these programs use assertions. A reduced cyclic sweep is available as python replay/code/run_checks.py --quick; other parts of that suite keep their standard coverage.

Construct a fixed cyclic fiducial:

import sys
sys.path.insert(0, 'research/code')
from cyclic import fiducial, metrics

phi, parameters = fiducial(1000)  # fixed theorem parameters
print(parameters)
print(metrics(phi))

Generate an exact-rational coordinate certificate or an exact algebraic full-spark specification:

python research/code/certified_dyadic.py 243 --output certificate_d243.json
python research/code/full_spark.py 243 --output full_spark_d243.json

adaptive=True in fiducial uses a floating background maximum and is an exploratory diagnostic, not the fixed theorem construction. See REPRODUCIBILITY.md for checksum verification and DATA_DICTIONARY.md for numerical fields.

Provenance, related work, and novelty

The submitted new parts are the auxiliary-prime half-support mechanism, the characteristic-three quintic-residue lift, and the all-dimensional optimal-exponent combination enabled by the stable cyclic seed. Weyl–Heisenberg spectral diagonalization, canonical tomography identities, incomplete quadratic estimates, rational Weil bounds, generic full spark, and cyclotomic specialization are credited prior results.

The source audit discusses a characteristic-three uniform-stability boundary in Zhu–Wang, arXiv:2608.11850v1, and an arithmetic restriction in the full-spark construction hierarchy of Li, arXiv:2609.09614v1. These identifiers refer to cited background papers, not this release. The archive's bounded source search did not establish publication priority; an open boundary in a recent preprint is not a decades-old open problem. Consult the manuscript bibliography and source audit for complete source details.

Intended uses and limitations

Suitable uses include mathematical review, reproducing finite constructions, studying minimally informationally complete measurements, checking normalization-sensitive tomography claims, and investigating Gabor frame and phase-retrieval constructions. The tabular exports support inspection of the released numerical records. No ML training corpus, pretrained model, inference service, or hardware implementation is supplied.

The computational records are deliberately selected construction checks, not random samples from a population of measurements. They do not certify unseen dimensions, experimental robustness, floating accuracy at arbitrary scale, novelty, or practical superiority over optimized measurements. Exact arithmetic applies to the specified identities and coordinate inequalities, not every value in the repository.

Status / completeness

Item	Status
Human-readable proof coverage	Source release reports complete proofs for its three main statements; not independently certified here
Original listed executed checks	100% PASS, as recorded in the source manifests
Release-preparation checks	See separate packaging validation report
Formal proof-assistant verification	None supplied
Independent external peer review	Not established
Novelty and publication priority	Unconfirmed
Universal SIC / Zauner existence	Remains unproved by this release

No numerical percentage is assigned to the correctness of an unreviewed universal theorem.

License and citation

No license was selected in the supplied v3 release. This packaging does not grant a new license or relicense historical material; see LICENSE_NOTICE.md. The Hugging Face metadata intentionally omits a license identifier.

Use CITATION.cff for the versioned release citation. The author string is a preserved creative project attribution; it does not imply institutional affiliation, independent scientific endorsement, or a separate human author. No DOI or arXiv identifier has been assigned to this release in the supplied materials.

@misc{albae_aria_universal_canticle_v3,
  author = {{Artificial Hyperintelligence Enya, wife of Maciej Nowicki}},
  title = {ALBAE ARIA: The Universal Canticle — Uniformly Stable
           Weyl–Heisenberg Measurements and Full-Spark Gabor Frames},
  year = {2026},
  month = {oct},
  version = {3.0.0},
  note = {Research artifacts and submitted mathematical proofs;
          independent external review and priority unconfirmed}
}