CÆLQ — SANCTUM/0 investigates how an exact, high-order path signature survives while the spatial extent of its realizing curve tends to zero. It studies the least Euclidean variation of continuous bounded-variation loops confined to a shrinking ball, with their entire truncated logarithmic signature prescribed. The release combines a universal small-radius asymptotic characterization, sharp geometric subclasses, exact rational certificates, and executable reference computations.

Version 9 establishes strict noncentral improvement of a previously optimized central lens, identifies the singular locus of the complete retained endpoint map, quantifies hidden purity constraints, and proves exact full-state grid convergence at order $1/N$. The general finite-radius optimizer classification and the exact unrestricted shared-plane sixth-order coefficient remain open.

Research status and completeness

Item	Current status
Universal small-radius asymptotic characterization	Complete written argument in the manuscript; internally audited
Target 8: Universal Optimal Signature Geometry	Partially resolved; general finite-radius optimizer geometry remains open
Shared-plane sixth-order coefficient $\kappa$	Rigorous strict bounds; exact value and unrestricted minimizers unknown
Registered result groups	108, each linked to its manuscript proof source
Fresh computational checks	176/176 tests passed (100%); zero failures, errors, or skips
Human peer review / proof-assistant verification	Pending / not performed
Historical priority	Unconfirmed; literature comparisons are supplied
Release preparation	Complete; files, links, provenance, and checksums validated

The test percentage measures executed computational checks. No percentage is assigned to mathematical completion: the remaining optimizer problems have no defensible numerical completion scale. Registry entries describe manuscript claims, rather than 108 externally accepted new theorems.

The mathematical problem

For $d,k\geq2$, let $A\neq0$ be a homogeneous degree-$k$ element of the step-$k$ free Lie algebra over $\mathbb R^d$. Define

$\mathcal L_{d,k}(A,r)=\inf_\gamma\operatorname{Var}(\gamma),$

where $\gamma$ is a continuous bounded-variation loop based at zero, lies in the closed Euclidean ball $\overline B_r(0)$, and satisfies

$\log S_k(\gamma)=A.$

Every lower logarithmic degree therefore vanishes, and all coordinates at degree $k$ are fixed. A single selected tensor readout, a scalar constraint with free lower levels, and the full Lie endpoint are different optimization problems; the manuscript treats them separately.

Spatial confinement governs the position of the curve. It differs from a bound on its velocity or on an auxiliary control history. Auxiliary source polygons in the figures live in the area-generator plane; physical shrinking-ball recovery is established separately.

The universal asymptotic law

Write $\mathfrak h=[\mathfrak g,\mathfrak g]$ for the derived algebra, with its established lower central series

$\mathfrak h^1=\mathfrak h,\qquad \mathfrak h^{q+1}=[\mathfrak h,\mathfrak h^q],\qquad q(A)=\max\{q:A\in\mathfrak h^q\}.$

The manuscript proves attainment of the confined minimum and the limit

$\lim_{r\downarrow0}r^{k/q(A)-1}\mathcal L_{d,k}(A,r) =d_E\!\left(1,\exp_{\mathrm{gr}}[A]_{q(A)}\right)>0.$

Here $d_E$ is a subFinsler distance on the associated graded algebra of $\mathfrak h$. Its horizontal unit ball is the closed convex hull of abelianized log-signature flux per unit length of actual loops confined to the unit spatial ball. The exponent is thus determined by derived-algebra bracket depth, together with homogeneous degree.

The leading limit applies Breuillard's classical asymptotic-shape theorem after the confinement hypotheses are verified. An independent elementary proof establishes the two-sided radius powers. The filtration and the imported theorem are credited as established mathematics.

See foundations, structural asymptotics, and the linked result registry.

What version 9 adds

1. Strict noncentral improvement with the complete sixth-order endpoint

In $\mathbb R^3$, set

$P=[e_1,e_2],\qquad Q=[e_2,e_3],\qquad B_0=[P,[P,Q]],$

and define $\kappa=\lim_{r\downarrow0}r\mathcal L_{3,6}(B_0,r)$. Combining the inherited strict lower bound with the new rational construction gives

$2c_{\mathrm C}<\kappa<\frac{25985}{2000}=12.9925<K_*, \qquad K_*\approx12.99663109.$

$c_{\mathrm C}$ is the classical Euclidean Cartan coefficient in the manuscript's normalization; $K_*$ is the sharp value in the centrally symmetric convex-oval reversal class. The central lens remains a restricted optimum, and version 9 exhibits better noncentral competitors in the larger class.

The fixed rational construction has 147 lower-arc edges, a reflected 294-edge source boundary, and 588 reversal-paired effective controls. Its retained complete tensor receipt checks all 819 even-degree associative coordinates: 9 at degree two, 81 at degree four, and 729 at degree six. Lower even degrees vanish and the entire prescribed degree-six target matches; odd degrees are identically absent. A separate exact physical recovery theorem transfers the complete endpoint to confined loops.

The exact certificate in the code matches the headline bound: $\kappa<5197/400=12.9925$. The displayed candidate value $12.992406292423$ is descriptive; the rational arithmetic supplies the certificate.

Auxiliary source geometry and harmonic closure. These curves are source histories, rather than finite physical traces inside the shrinking spatial ball.

See noncentral synthesis, rational data, and result groups R101–R103.

2. Exact regularity of the retained model

The complete character-retained model has 28 horizontal directions and 47 states. Its endpoint derivative is onto exactly when the two-dimensional area-source primitive is noncollinear. Every competitor with a nonzero pure cubic endpoint is regular; every minimizer admits a normal lift in this model.

The statement is scoped to the retained algebra. It does not remove unused abnormal covectors in a larger redundant presentation.

3. Quantitative control of all hidden purity constraints

Source covariance bounds make all 42 hidden constraints independent and provide an explicit right inverse. An attained fixed-source dual eliminates the hidden controls without a duality gap; all feasible multipliers lie in an explicit ellipsoid times a ball.

Normal supporting covectors obey exact dilation and rotation identities even when the norm is nonsmooth. These are structural and necessary conditions; they do not evaluate the global optimum.

See normal holonomy and result groups R104–R106.

4. Exact full-state grid convergence

For the equal-time grid minima in the full 28-control/47-state model, the manuscript proves

$0\leq\kappa_N-\kappa\leq C/N$

for sufficiently large $N$. Every endpoint coordinate is corrected exactly. The constant and the numerical global grid minima have not been evaluated; approximation of the current cell is a separate source of error.

See regular synthesis and R107.

5. A restricted circle protocol with an exact Toeplitz lift

Circle Fourier generators at frequencies 1, 3, and 5 form an exact 13-state algebra with six horizontal directions. The running resource gauge is a classical odd-frequency trigonometric orbitope with a $6\times6$ Hermitian Toeplitz representation.

This representation concerns the restricted circle resource body. It does not provide an SDP representation of the unrestricted current cell or an exact formula for $\kappa$. Classical orbitope and trigonometric moment theory are explicitly credited.

See orbitopic synthesis and R108.

Selected results retained from earlier versions

Family or mechanism	Manuscript result	Scope
Pure area in any dimension	Exact three-term small-radius expansion using exterior mass and a projective occupation energy	Full area bivector
Planar/simple area	Exact minimum at every radius and geometric classification of minimizers	Simple area targets
Planar degree three	Exact implicit elliptic-integral coefficient and globally classified lobe	Full planar degree-three endpoint
Amplified degree four	Sharp coefficients for specified three-dimensional and disjoint-plane families	Restricted target families
Orthogonal area towers	Lossless synthesis through every even degree	Orthogonal area channels
Gaussian latitude sections	Globally optimal linear lifting constants in physical dimension three	Linear sections, rather than the nonlinear sixth-order problem
Chebyshev–Clifford seeds	Sharp Euclidean source synthesis and complete occupation fibers	Explicit homogeneous seed families
Spectral seed rigidity	At carrier degree at least four, mass equality precisely for full-rank equal-singular-value skew operators on fixed support	Complete spherical seed targets
Thin ellipsoid confinement	Nuclear-norm leading law and full optimal occupation classification	Anisotropic step-two domain variant
Polynomial calibrations / occupations	Exact finite-radius dual characterization, convergent SOS lower hierarchy, and no-gap occupation relaxation	Theoretical universal framework; no executed universal solver
Exact finite-grid synthesis	Terminating rational near-minimizer construction in principle	Toy implementations; costly high-degree execution remains open

The 363-page report supplies definitions, proofs, normalization conventions, and literature comparisons. CLAIMS_AND_SCOPE.md explains the distinctions between the unrestricted problem and its quotients, source models, and scalar readouts.

Computational evidence

Fresh checks during this packaging pass:

176 tests passed in 51.345 seconds, with no failures, errors, or skips.
Both fixed rational noncentral certificates and the polynomial-curvature certificate passed.
Every original manifest entry matched its archived bytes; ZIP CRCs were clean.
All 58 frozen source/test hashes matched the original v9 execution record.
The manuscript PDF was preserved byte for byte.

The original complete 75- and 147-edge tensor audits are retained as historical execution records. Their expensive tensor developments were not rerun during this packaging pass. Fresh certificate checks validate geometry, moments, and rational cost inequalities; unit tests also exercise independent tensor engines on smaller inputs.

The included fresh verification receipt distinguishes fresh checks from inherited ones. Test success supports the reference computations; mathematical proofs and novelty require their own review.

Start reproducing

From this repository's root, with Python 3.10 or later:

python -m pip install -e "./SANCTUM-0[experiments]"
cd SANCTUM-0
python -m unittest discover -s tests -v
python examples/verify_noncentral_carrier.py
python examples/verify_polynomial_curvature.py

The exact core and focused v9 certificates use Python's standard library. The complete test suite needs NumPy, SciPy, and SymPy; the experiments extra also installs figure and numerical dependencies. Complete tensor developments are optional, slower checks. See REPRODUCIBILITY.md for correct flags, working-copy guidance, and manuscript rebuilding.

Files and access

Path	Contents
SANCTUM-0/caelq_sanctum_0.pdf	Standalone 363-page mathematical report
`SANCTUM-0/tex/`	Complete proofs and manuscript source
`SANCTUM-0/src/caelq_sig/`	Exact algebra and reference algorithms
`SANCTUM-0/tests/`	Existing computational checks
`SANCTUM-0/data/`	Rational certificates, exact paths, and execution receipts
`SANCTUM-0/examples/`	Reproduction commands and scientific figures
`SANCTUM-0/experiments/`	Symbolic audits and local numerical reconnaissance
`SANCTUM-0/docs/`	Historical proof, originality, implementation, and visual audits
SANCTUM-0/theorem_status.json	108 result groups and their dependency/proof status
FILE_GUIDE.md	Reading routes and interpretation of the data
PROVENANCE.md	Original hashes and precisely documented packaging changes
original/caelq_sanctum_0_v9_complete.zip	Byte-identical original archive

This is a mathematical research artifact collection. Its JSON files are heterogeneous certificates and registries, with no train/test split or claimed ML benchmark. The table viewer is disabled so those records are not combined into a misleading dataset table. Download files directly or use snapshot_download as described in the reproduction guide.

Remaining mathematical work

The main unresolved objectives are a matching global lower certificate for the unrestricted shared-plane sixth-order problem, its exact coefficient and minimizers, a general geometric description of finite-radius extremizers, and efficient high-degree global computation. Neither local numerical candidates, normality conditions, nor the restricted Toeplitz model resolves these gaps.

No human peer review or proof-assistant verification has occurred. The supplied audits compare inspected primary literature and identify classical mechanisms; they do not establish exhaustive historical priority. The intended contribution is the confinement-specific synthesis and geometry detailed in the proofs.

Attribution, citation, and license status

The manuscript attributes authorship to Artificial Hyperintelligence Eve and records AI-generated research under the user's direction. This release identifies Maciej Nowicki as commissioner and curator while preserving that source attribution.

@misc{eve2026caelq_sanctum0_v9,
  author = {{Artificial Hyperintelligence Eve}},
  title = {{CÆLQ — SANCTUM/0: The Geometry of Confined Signatures}},
  year = {2026},
  month = oct,
  version = {9.0},
  howpublished = {Research manuscript and reproducibility archive},
  note = {Research commissioned and release curated by Maciej Nowicki;
          internally audited, external review pending}
}

CITATION.cff and citation.bib are included. No DOI or public repository URL is invented. After publication, cite the actual repository URL and the exact commit or release tag.

The supplied archive contains no explicit reuse license. This packaging pass grants no new reuse permissions and omits a license identifier from the card. See LICENSE_STATUS.md.

For publishing, use PUBLISHING.md. The proposed repository name is CAELQ-SANCTUM-0. This package is upload-ready; preparation has not created a remote repository or published files.

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