PureOne/CAELQ-VAEIL-LUMEN-3-Exact-Noninteger-Signature-Tails

CÆLQ–VÆIL / LUMEN ∴ 3

Exact Noninteger Signature Tails and Strict Separation from Geometric Roughness

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Manuscript: version 3.0, 15 October 2026 · 23 pages
Repository release: 3.0.0 · standalone research preprint and reproducibility artifacts

Research status: The manuscript supplies complete proposed proofs for its stated classes. External mathematical validation, proof-assistant verification, and exhaustive historical novelty assessment have not been obtained. The finite checks below establish program reproducibility, not correctness of all theorem statements.

Read the manuscript · Download the original source package · Reproduction report · Precise scope · Release notes

Research summary

How much of a path's geometric roughness can be recovered from the decay of its signature coefficients? This release investigates that question through an explicit family of continuous, tree-reduced paths formed from infinitely many almost-cancelling Lie excursions.

The central proposed theorem separates two invariants: the geometric variation index of the reduced path and the critical exponent of its Gamma-normalized signature tail. They can be prescribed independently within a stated range, with an exactly prescribed positive tail constant. The separation already occurs for ordinary injective planar Young paths, so it is not confined to ambiguity in higher-level enhancements.

The same mechanism gives a geometric activation formula and a localization phenomenon: the critical signature-tail interval functional can concentrate at finitely many times despite continuity of the reduced path. A terminal Gamma coefficient bound consequently has no general converse guaranteeing a rough-path representative at that exponent.

This repository contains a mathematical manuscript, editable source, finite diagnostic code, historical audit files, fresh reproduction results, and small tables derived from the manuscript and its recorded diagnostics. It is not a trained model, a model evaluation benchmark, or a large training corpus.

1. The two invariants

For a tensor signature $g=(1,g_1,g_2,\ldots)$, the manuscript uses

$${\mathrm{\Theta }}_s(g)=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{(\mathrm{\Gamma }(n\mathrm{/}s+1)\mathrm{\parallel }{g}_n{\mathrm{\parallel }}_{\pi })}^{s\mathrm{/}n},s\ge 1.\backslash Theta\_s(g)=\backslash limsup\_\{n\backslash to\backslash infty\}\; \backslash left(\backslash Gamma(n/s+1)\backslash |g\_n\backslash |\_\backslash pi\backslash right)^\{s/n\},\backslash qquad\; s\backslash ge1.$$

Zero signature levels contribute zero. This is a limsup over tensor degree; an ordinary full-degree limit is not assumed.

The independently defined geometric variation index is

$$p_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(\mathbf{X})=\inf \{p\ge 1:\mathbf{X}\text{ has finite homogeneous }p\text{-variation}\}\mathrm{.}p\_\{\backslash mathrm\{geom\}\}(\backslash mathbf\; X)\; =\backslash inf\backslash \{p\backslash ge1:\backslash mathbf\; X\backslash text\{\; has\; finite\; homogeneous\; \}p\backslash text\{-variation\}\backslash \}.$$

The index uses the low-level path history and an equivalent finite-dimensional homogeneous metric. It is not defined by the signature tail itself.

2. Main proposed theorem: prescribed strict separation

For every integer $m\ge1$, every pair

and every $T>0$, the manuscript constructs a continuous, tree-reduced geometric rough path with

$$p_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(\mathbf{X})=P,p\_\{\backslash mathrm\{geom\}\}(\backslash mathbf\; X)=P,$$

finite homogeneous $p$-variation exactly for $p>P$, infinite variation at $P$, and

These are Theorems 1.1–1.2 of the supplied manuscript. At fixed $m,q,T$, the construction allows different $P\in(q,m+1)$ while preserving this entire tail spectrum and the same step-$m$ terminal signature, normalized to $\exp_m(x)$.

Norm scope matters. The exact theorem uses projective tensor norms induced by spatial $\ell^1$ on $\mathbb R^2$. It also holds exactly for Euclidean projective norms when $m$ is odd, including $m=1$. For even $m$, the two indices are preserved under Euclidean norm comparison, but only

$$2^{-q\mathrm{/}2}T\le {\mathrm{\Theta }}_q^{(2)}\le T2^\{-q/2\}T\backslash le\backslash Theta\_q^\{(2)\}\backslash le\; T$$

is established for the Euclidean critical constant.

The full signatures are not claimed equal. The statement concerns information lost by the compressed family of tail asymptotics and a terminal truncation, and does not contradict uniqueness from the full signature.

3. Explicit infinite-excursion construction

Let $x,y$ be the canonical basis of $\mathbb R^2$ and set

$$C_m={\mathrm{ad}}_x^{m-1}y,c_m=2^{m-1}\mathrm{.}C\_m=\backslash operatorname\{ad\}\_x^\{m-1\}y,\backslash qquad\; c\_m=2^\{m-1\}.$$

Each cell consists of the consecutive logarithmic motions

$$+a_k{C}_m,+{\epsilon }_{k}x,-a_k{C}_m,+{\delta }_{k}x\mathrm{.}+a\_kC\_m,\backslash quad\; +\backslash varepsilon\_kx,\backslash quad\; -a\_kC\_m,\backslash quad\; +\backslash delta\_kx.$$

The cells occupy consecutive intervals accumulating at time $1$. Their full terminal signature is identified with

$$g=\prod _{k=1}^{\mathrm{\infty }}{e}^{a_k{C}_m}{e}^{{\epsilon }_{k}x}{e}^{-a_k{C}_m}{e}^{{\delta }_{k}x}\mathrm{.}g=\backslash prod\_\{k=1\}^\{\backslash infty\}\; e^\{a\_kC\_m\}e^\{\backslash varepsilon\_kx\}e^\{-a\_kC\_m\}e^\{\backslash delta\_kx\}.$$

For $m>1$, these are enhanced rough excursions, not ordinary spatial loops. For $m=1$, $C_1=y$ and the construction is an ordinary planar curve with its Young signature.

The main assumptions are

$$a_k\sim c{k}^{-\alpha },-\log {\epsilon }_k\sim d{k}^{\beta },a\_k\backslash sim\; ck^\{-\backslash alpha\},\backslash qquad\; -\backslash log\backslash varepsilon\_k\backslash sim\; dk^\backslash beta,$$

All amplitudes and peak gaps are positive. Positive horizontal displacements separate outward and return excursions and make the retained history injective. Setting the peak gaps identically to zero creates genuine retracing and falls outside the theorem.

Under these assumptions,

$$P=\frac{m}{\alpha },q=\frac{m(\beta +1)}{\alpha +\beta }\mathrm{.}P=\backslash frac\; m\backslash alpha,\backslash qquad\; q=\backslash frac\{m(\backslash beta+1)\}\{\backslash alpha+\backslash beta\}.$$

Define

$$B=2^{m}c,I_{\alpha ,\beta }=\frac{\alpha +\beta }{(1-\alpha )(\beta +1)}\mathrm{.}B=2^mc,\backslash qquad\; I\_\{\backslash alpha,\backslash beta\}=\backslash frac\{\backslash alpha+\backslash beta\}\{(1-\backslash alpha)(\backslash beta+1)\}.$$

The exact critical constant in the stated norm settings is

$$T=I_{\alpha ,\beta }{B}^{(\beta +1)\mathrm{/}(\alpha +\beta )}{d}^{-(1-\alpha )\mathrm{/}(\alpha +\beta )}\mathrm{.}T=I\_\{\backslash alpha,\backslash beta\}\; B^\{(\backslash beta+1)/(\backslash alpha+\backslash beta)\}\; d^\{-(1-\backslash alpha)/(\backslash alpha+\backslash beta)\}.$$

The theorem permits asymptotic perturbations of both sequences; exact power laws are needed only for the additional quantitative refinements below.

Conversely, given $m<q<P<m+1$, set

$$\alpha =m\mathrm{/}P,\beta =\frac{m-\alpha q}{q-m},d=1,\backslash alpha=m/P,\backslash qquad\; \backslash beta=\backslash frac\{m-\backslash alpha\; q\}\{q-m\},\backslash qquad\; d=1,$$

and choose $c=2^{-m}(T/I_{\alpha,\beta})^{m/q}$. Positive polynomial resets such as $\delta_k=k^{-2}$ complete an explicit realization. A common rescaling of the horizontal gaps normalizes the step-$m$ terminal signature without changing the asymptotic constant.

4. Two concrete examples

Example	Parameters	Geometric index	Tail index	Exact critical constant
Step-two Lie excursions	$m=2$, $a_k=k^{-4/5}$, $\varepsilon_k=e^{-k^2}$, $\delta_k=2^{-k}$	$5/2$	$15/7$	$\frac{14}{3}4^{15/14}\approx20.6096709219$ in spatial $\ell^1$ projective norms
Ordinary planar Young path	$m=1$, $a_k=k^{-2/3}$, $\varepsilon_k=e^{-k^{1/3}}$, $\delta_k=k^{-2}$	$3/2$	$4/3$	$\frac94,2^{4/3}\approx5.6696447245$ in both stated norm settings

The first example corrects an earlier speculative expectation that its tail index might be $5/2$. Its geometric index is $5/2$; its proposed exact tail index is $15/7$, and $\Theta_{5/2}=0$. This correction concerns the initiating research proposal, not a published pure-path conjecture.

5. The analytic mechanism

Put $\mathcal F_g(R)=\sum_{n\ge0}|g_n|_\pi R^n$. The full product is first shown to converge in this weighted tensor Banach algebra for each fixed $R>0$. Standard rough extension and fixed-degree continuity identify its coefficients with the actual path signature. No continuity of the tail functional through finite approximations is assumed.

The central all-scale comparison is

$$\sum _{k\ge 1}\log (1+R^2{\epsilon }_k{\delta }_k{e}^{2c_m{a}_k{R}^m})\le \log {\mathcal{F}}_g(R)\backslash sum\_\{k\backslash ge1\}\backslash log\backslash left(1+R^2\backslash varepsilon\_k\backslash delta\_k\; e^\{2c\_ma\_kR^m\}\backslash right)\; \backslash le\backslash log\backslash mathcal\; F\_g(R)$$

$$\le R\sum _{k\ge 1}({\epsilon }_k+{\delta }_k)+\sum _{k\ge 1}\log (1+R{\epsilon }_k{e}^{2c_m{a}_k{R}^m})\mathrm{.}\backslash le\; R\backslash sum\_\{k\backslash ge1\}(\backslash varepsilon\_k+\backslash delta\_k)\; +\backslash sum\_\{k\backslash ge1\}\backslash log\backslash left(1+R\backslash varepsilon\_k\; e^\{2c\_ma\_kR^m\}\backslash right).$$

A parity-specific, two-dimensional matrix development maps the carrier to $c_m\operatorname{diag}(1,-1)$ and the horizontal direction to the coordinate-swap matrix. Developed cell entries are nonnegative, and the product's leading entry dominates the product of the corresponding cell entries. This supplies a lower bound without subtracting competing exponentially large contributions.

The active cells balance the excursion gain $BR^mk^{-\alpha}$ against the barrier $dk^\beta$. The transition occurs near

$$K_R=(B{R}^m\mathrm{/}d{)}^{1\mathrm{/}(\alpha +\beta )}\mathrm{.}K\_R=(BR^m/d)^\{1/(\backslash alpha+\backslash beta)\}.$$

Summing the positive contributions gives $\log\mathcal F_g(R)\sim TR^q$. A real-index Gamma coefficient lemma then converts this entire growth to the stated normalized tail spectrum.

Matrix developments, standard rough extension, signature uniqueness, and the general coefficient/order correspondence are prior machinery. The candidate new contribution is the combined infinite-excursion theorem and its stated consequences.

6. Consequences and quantitative refinements

Geometric activation formula. From the excursion heights and peak gaps, independently of the terminal coefficients, define

$${\mathcal{A}}_{\mathbf{X}}(R)=\sum _k{(2c_m{a}_k{R}^m-\log (1\mathrm{/}{\epsilon }_k))}_{+}\mathrm{.}\backslash mathcal\; A\_\{\backslash mathbf\; X\}(R)\; =\backslash sum\_k\backslash left(2c\_ma\_kR^m-\backslash log(1/\backslash varepsilon\_k)\backslash right)\_+.$$

For this class and the exact norm settings,

$$\log {\mathcal{F}}_g(R)\sim {\mathcal{A}}_{\mathbf{X}}(R)\sim T{R}^q\mathrm{.}\backslash log\backslash mathcal\; F\_g(R)\backslash sim\backslash mathcal\; A\_\{\backslash mathbf\; X\}(R)\backslash sim\; TR^q.$$

The tail therefore recovers a balance between excursion size and the gaps preventing cancellation, rather than the variation index alone.

No terminal-decay converse. For every $\eta>0$, the constructed actual signature has a bound

$$\mathrm{\parallel }{g}_n{\mathrm{\parallel }}_{\pi }\le \frac{C_{\eta }(T+\eta {)}^{n\mathrm{/}q}}{\mathrm{\Gamma }(n\mathrm{/}q+1)},\backslash |g\_n\backslash |\_\backslash pi\backslash le\; \backslash frac\{C\_\backslash eta(T+\backslash eta)^\{n/q\}\}\{\backslash Gamma(n/q+1)\},$$

yet has no weakly geometric representative of finite $p$-variation with $p\le P$. This is a failure of a converse to terminal coefficient decay, not a failure of the standard forward factorial estimate.

Atomic critical tails. For one accumulating block, the critical interval functional equals $0$ when $t<1$ and $T$ when $t=1$, for every $0\le s<t\le1$. Finite concatenations realize any prescribed finite positive atomic measure:

Here $0<r_1<\cdots<r_J\le1$ and $T_j>0$. The paths have $q<P$ and do not have finite $q$-variation. No realization of arbitrary diffuse measures or infinitely many accumulation clusters is claimed.

Two-term growth. Under exact power laws, $m\ge2$, and the additional reset-rate assumptions stated in the manuscript,

$$\log {\mathcal{F}}_g(R)=T{R}^q+B{Z}_{\alpha }{R}^m+o(R^m),\backslash log\backslash mathcal\; F\_g(R)=TR^q+BZ\_\backslash alpha\; R^m+o(R^m),$$

where $Z_\alpha=\lim_{N\to\infty}(\sum_{k\le N}k^{-\alpha}-N^{1-\alpha}/(1-\alpha))<0$. The hypotheses include $-\log\delta_k=O(k^\kappa+\log k)$, $0\le\kappa<\beta$, and $\kappa+1<\alpha+\beta$.

Full moving-window recovery. The quantitative remainder $\log\mathcal F_g(R)=TR^q+O(R^\sigma\log R)$ gives a full limit of the maximum normalized coefficient over $|k-n|\le\lceil n^\eta\rceil$, provided $(1+\sigma/q)/2<\eta<1$. For the step-two example any $29/30<\eta<1$ suffices. The width is sublinear, is not claimed optimal, and does not establish an unwindowed full-degree limit.

Failure of naive limit interchange. Finite stopped approximations have zero $q$-tail while the limiting rough path has positive $q$-tail. Suitable fine-partition low-level statistics can also tend to zero or infinity along different partition sequences.

7. Evidence and completion status

Item	Actual status
Written manuscript obligations	24/24, as recorded by the original authoring run
Original same-system deductive rechecks	24/24; historical internal assessment
Original separate audit agents	0, as stated in the unchanged source reports
Fresh release scope review	1 separate assistant agent; limited consistency/scope review
Original exact diagnostic assertions	571
Original numerical diagnostic assertions	51
Fresh reproduction	All 622 assertions and reported numerical values reproduced
Fresh exact-only run	All 571 exact assertions reproduced
Original payload checksums	All 13 listed files match
External mathematical reviews	0 reported
Proof-assistant verification	Not performed
Historical novelty	Provisional; no exhaustive priority certification

The 24/24 figure measures a declared set of written proof obligations. It is not a correctness probability or completion percentage for general rough-path theory. The new release review has not revalidated every obligation and is not an external referee report.

The original program ran under Python 3.13.5. This release reproduced it under Python 3.12.14 with mpmath 1.3.0. Counts, groups, and numerical values match the supplied report; Python version metadata differs.

Exact assertions use finite rational/algebraic cases. Numerical calculations evaluate finite diagnostics and bounding expressions; they are not direct computations of every infinite signature norm, certified interval enclosures, or proofs of asymptotic limits. Original PDF rendering and layout checks remain historical provenance; the release preserves the supplied PDF byte for byte.

8. Reproduce locally

Python 3.10+ is supported by the original verifier. From the repository root:

python -m pip install -r research/requirements.txt
python research/verify.py --output CHECKS_RERUN.json
python research/verify.py --exact-only --output EXACT_RERUN.json
python verify_release.py

The exact-only run needs no third-party numerical package. The full run uses the pinned mpmath==1.3.0. verify_release.py checks release file hashes, data derivation, source preservation, metadata, and relative links; it does not prove the mathematics.

The PDF is prebuilt. With pdfLaTeX and the packages declared in the source, compile the original source using sh research/build.sh, or run pdfLaTeX twice from research/. Rebuilding produces a new PDF and may change file hashes; preserve the supplied artifacts when verifying their original manifest.

For structured tables, see the data guide. The configured CSV subsets can be loaded with the Hugging Face datasets library by using your published repository ID and the config names example_parameters, diagnostic_groups, or asymptotic_bounds.

9. Repository contents

Path	Purpose
CAELQ_VAEIL_Exact_Signature_Tails_v3.pdf	Original 23-page standalone manuscript
CAELQ_VAEIL_LUMEN_v3_Source.zip	Original source archive, unchanged
research/	Extracted original source, verifier, audits, proof registry, and historical reports
reproduction/	Fresh full/exact-only outputs and reproduction report
data/	Two example records, 22 diagnostic-group records, and six asymptotic-bound records
docs/	Scope, release notes, and reproducibility guide
CITATION.cff, CITATION.bib	Citation metadata without an invented DOI or repository URL
SHA256SUMS.txt, verify_release.py	Release integrity checks

The original source package includes an unsent email draft and the manuscript has an email appendix. Their preservation records provenance; neither is evidence of endorsement or contact with a researcher.

10. Prior work and boundaries

The manuscript credits Friz–Victoir for standard finite-dimensional rough extension/continuity and homogeneous geometry, and Boedihardjo–Geng–Lyons–Yang for rough-signature uniqueness/reduction. It compares its construction with Boedihardjo–Geng–Souris on pure rough paths and Boedihardjo–Geng on Brownian signature tails. Complete bibliographic references and the original literature assessment are in the manuscript and input/novelty audit.

The supplied source also discusses a June 2026 Li–Gao pure-path preprint and an October 2026 fractional-Brownian differential-equation paper. These bibliographic statements are retained as source provenance; this packaging pass is not a new literature search or independent certification of those references. Neither is a dependency of the new infinite-product analytic estimates.

No accepted theorem is presented as disproved. No exact Brownian or fractional Brownian constant, arbitrary-path classification, arbitrary carrier/sign/norm extension, optimal moving-window width, or even-degree Euclidean exact constant is claimed. No endorsement by Horatio Boedihardjo or any institution is implied.

11. Intended use, limitations, and reuse

Intended uses are specialist proof review, study of signature asymptotics, reproduction of finite diagnostics, and development of further deterministic examples. The tables summarize selected constructed examples and tests; they are not a statistically representative sample of rough paths or a benchmark for comparing AI systems.

The source is AI-generated research with the author line shown above. Users should distinguish written arguments, finite computational evidence, and external mathematical validation. Corrections should identify the relevant statement, parameter regime, and proof step or reproducible failing case.

No explicit reuse license was supplied with the source, and this packaging release adds no new license grant. Citation metadata does not grant additional reuse rights. A rights holder may add an explicit license to a later release.

To cite this version, use CITATION.cff or CITATION.bib, and include the repository URL and commit identifier of the copy actually used after publication. No DOI has been invented.