PureOne/GLASS-PSALM-Unchosen-Choir

GLASS PSALM ∂∞ — The Unchosen Choir

Robust task decoding on fractal trees · Research v2.0.0 · 3 October 2026

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki (AI-generated research persona).
Prepared for: Maciej Nowicki.
Release packaging: 2.0.0-hf.1.

An unresolved source branch can still provide useful prediction information. This research gives an exact account of that distinction for a specified orthogonal fractal tree: which branches can share a noisy observation, which task values remain possible, and which prediction minimizes the worst squared error.

The central construction connects simultaneous branch ambiguity, electrical resistance on a tree, finite witnesses in task space, and optimal robust prediction. It supplies a mathematical manuscript, readable proofs, standard-library reference code, reproducible exact-arithmetic checks, and synthetic audit records.

Scientific status: model-specific results with written proofs and executed finite checks. The unrestricted Hilbert-space Chebyshev-set problem remains unresolved by this work. Worldwide novelty, independent peer review, formal proof-assistant verification, and practical neural-network superiority have not been established. No neural network was trained.

Start with the manuscript

• Complete 20-page manuscript
• Editable LaTeX source
• Mathematical summary and assumptions
• Statement-by-statement theorem ledger
• Release verification and provenance
• Data schema
• Original v2 research archive

This is a research archive with synthetic mathematical audit data, hosted in a Hugging Face dataset repository. It contains reference algorithms rather than pretrained weights. The Dataset Viewer configuration selects only the 480-row witness table; it does not treat manuscript files, predecessor archives, or audit JSON as training examples.

The mathematical model

Let a rooted tree have $b\geq 2$ children at each node. Give every nonempty finite prefix $w$ an orthogonal coordinate $e_w$. For an infinite path $\omega$, define

$$X(\omega )=\sum _{n\ge 1}{a}_n{e}_{\omega \mathrm{\mid }n},a_n>0,\sum _{n\ge 1}{a}_n^2=1.X(\backslash omega)=\backslash sum\_\{n\backslash geq\; 1\}a\_n\; e\_\{\backslash omega|n\},\backslash qquad\; a\_n>0,\backslash qquad\; \backslash sum\_\{n\backslash geq\; 1\}a\_n^2=1.$$

Each depth-$L$ address $w$ has a task vector $y_w$. The source continues indefinitely, while this finite task depends on its first $L$ decisions. Observe $q=X(\omega)+h$ with $|h|\leq\epsilon$. The objective is

$$\mathcal{V}(ϵ;y)=\underset{f}{\inf }\underset{\omega ,\mathrm{\parallel }h\mathrm{\parallel }\le ϵ}{\sup }\mathrm{\parallel }f(X(\omega )+h)-y_{\omega \mathrm{\mid }L}{\mathrm{\parallel }}^2\mathrm{.}\backslash mathcal\; V(\backslash epsilon;y)=\backslash inf\_f\backslash sup\_\{\backslash omega,\backslash ,\backslash |h\backslash |\backslash leq\backslash epsilon\}\backslash |f(X(\backslash omega)+h)-y\_\{\backslash omega|L\}\backslash |^2.$$

The infimum includes all task decoders. It does not require the decoder to identify the latent address correctly.

For a nonempty address set $A$, let $r_{\rm in}(A)$ be the smallest enclosing radius of one source representative from each address. Their Gram matrix, and therefore this radius, is independent of the infinite suffixes selected. Let $\operatorname{rad}(y_A)$ be the smallest enclosing radius of their task vectors. Then

$$\boxed{{\displaystyle \mathcal{V}(ϵ;y)=\underset{A:r_{\mathrm{i}\mathrm{n}}(A)\le ϵ}{\max }{\mathrm{rad}}^2(y_A)\mathrm{.}}}\backslash boxed\{\backslash mathcal\; V(\backslash epsilon;y)=\backslash max\_\{A:\backslash ,r\_\{\backslash rm\; in\}(A)\backslash leq\backslash epsilon\}\backslash operatorname\{rad\}^2(y\_A).\}$$

If the task values have affine dimension $d$, some maximizing witness has at most $d+1$ labels. The decoder attaining the bound returns the unique enclosing-ball centre of all task values compatible with the observation. Its prediction is unique at each compatible observation; the source branch, supporting subset, and entire globally optimal map need not be unique.

Computing ambiguity through a resistance tree

For selected labels $u,v$, the source Gram matrix has diagonal $1$ and off-diagonal

$$G_{uv}=\sum _{n\le \mathrm{LCP}(u,v)}{a}_n^2\mathrm{.}G\_\{uv\}=\backslash sum\_\{n\backslash leq\backslash operatorname\{LCP\}(u,v)\}a\_n^2.$$

Write $R_L^2=\sum_{n>L}a_n^2$. Set $B_w=R_L^2$ at every selected depth-$L$ leaf. Moving upward through selected ancestors gives

$$B_w={\left(\sum _{i:wi\text{ has selected descendants}}\frac{1}{a_{\mathrm{\mid }w\mathrm{\mid }+1}^2+B_{wi}}\right)}^{-1},r_{\mathrm{i}\mathrm{n}}^2(A)=1-B_{\mathrm{\varnothing }}\mathrm{.}B\_w=\backslash left(\backslash sum\_\{i:\backslash ,wi\backslash text\{\; has\; selected\; descendants\}\}\backslash frac\{1\}\{a\_\{|w|+1\}^2+B\_\{wi\}\}\backslash right)^\{-1\},\backslash qquad\; r\_\{\backslash rm\; in\}^2(A)=1-B\_\{\backslash varnothing\}.$$

This is the classical series/parallel resistance law applied to the input ambiguity calculation. A set of $m$ selected addresses requires at most $Lm$ nonroot ancestor nodes. The manuscript supplies the correspondence proof and its assumptions; it does not claim to invent electrical-network theory or the classical enclosing-ball support theorem.

Exact examples

For root task values forming a centred regular simplex of squared radius $D$, define

$$m_{ϵ}=\max \{m\le b:1-1\mathrm{/}m\le {ϵ}^2\}\mathrm{.}m\_\backslash epsilon=\backslash max\backslash \{m\backslash leq\; b:1-1/m\backslash leq\backslash epsilon^2\backslash \}.$$

The exact optimal loss is

$$\mathcal{V}(ϵ)=D\frac{b}{b-1}(1-\frac{1}{m_{ϵ}})\mathrm{.}\backslash mathcal\; V(\backslash epsilon)=D\backslash frac\{b\}\{b-1\}\backslash left(1-\backslash frac\{1\}\{m\_\backslash epsilon\}\backslash right).$$

At the first identity-failure threshold $\epsilon^2=1/2$, only two root labels can be jointly compatible. Their task centre can preserve useful information even though exact root identification is impossible.

Unit-radius example	Optimal squared loss	Collapse loss	Exact relative loss reduction
8 root branches	$4/7$	$1$	$3/7\approx42.857%$
64 root branches	$32/63$	$1$	$31/63\approx49.206%$

These percentages compare analytic model objectives. They are not measured accuracy gains, training results, or estimates of general AI improvement. The simplex theorem also gives the same value for adversarial population squared loss under the stated uniform label prior and within-label suffix assumptions. General scalar tasks provide a negative control within the manuscript's specified objectives.

A separate nested binary example uses addresses 00, 01, 10: the input ambiguity radius squared is $8/15$, while three distinct orthogonal unit task values have enclosing radius squared $2/3$. Every pair alone has task radius squared $1/2$. Thus pairwise conflicts can miss a stronger three-way task obstruction on a binary source tree.

Additional results and boundaries

The manuscript includes an exact finite-sensor decoder, a sharp task-rank-dependent noise-budget penalty for truncating source coordinates, certified approximation of the original infinite orthogonal semantic task, and a fitted-target perturbation guarantee. It also compares retaining all root experts with collapsing to one expert under a per-expert storage penalty.

The architecture comparison covers those two specified alternatives. It does not optimize arbitrary topology, expert sharing, branch merging, or end-to-end learned geometry. The fitting guarantee assumes known geometry, known clean branch addresses during training, and observations for every task label. It concerns conditional means rather than irreducible output noise.

For $N=b^L$ labels and fixed task affine rank $d$, the witness search examines at most $\sum_{j=1}^{\min(N,d+1)}\binom Nj$ subsets. This is polynomial in $N$ for fixed $d$, while $N$ itself grows exponentially with depth. The reference general target cache caps the table at 16 points and favors exact, auditable small cases over production scale. The simplex family has a separate closed form.

Reproduce the results

The research scripts require Python 3.10 or newer and use only the standard library. Original execution used Python 3.13.5; the release report records the preparation rerun environment. No API key, internet connection, GPU, or package installation is required for the research checks.

From the repository root:

python src/demo.py
python src/run_audit.py --out ../audit_runs/v2

Windows users can double-click RUN_AUDIT.bat. The launcher works from its own folder and pauses after success or failure. The original results remain under results/; reruns are written outside the upload folder under ../audit_runs/v2. Do not disable assertions with python -O or PYTHONOPTIMIZE when running the audit.

Core geometry, enclosing balls, Gaussian elimination, sensor compatibility, and squared-radius comparisons use fractions.Fraction, including threshold equalities. Square roots, logarithms, and learning summaries use ordinary floating-point arithmetic. Finite checks support implementation consistency; the infinite theorems depend on the written proofs.

Executed verification coverage

Check	Included count
Resistance recursion vs independent rational linear solves	2,295
Equal-radius representative identities	9,216
Exact target-subset enclosing balls	3,060
Full subset search vs affine-rank witnesses	480
Scalar prefix-risk comparisons	160
Finite-sensor risk brackets	960
Simplex phase-boundary cases	97
Sensor distance recursion vs exhaustive source search	1,600
Exact decoder cases	800
Compatible source/query risk certificates	5,557
Finite/infinite subset-radius bounds	2,295
Stratified noisy-target fits	9
Fitted-target certificate comparisons	180
Trained neural networks	0

Some checks overlap. Their counts are not a count of independently proved theorems or distinct learning benchmarks. See results/audit_results.json for the original run and results/release_rerun/ for the preparation rerun.

Synthetic data and intended use

results/witness_cases.csv contains 480 reproducible records from 12 synthetic target tables and 40 noise probes per table. It records exact minimax losses and supporting address subsets. It is intended for inspecting the certificate construction and comparing implementations on the included small cases.

The table is an audit artifact, not a representative distribution of natural neural tasks. It contains no personal data, collected human labels, natural-language training corpus, or empirical neural-network benchmark. Its test split denotes the stored audit records; no statistical held-out evaluation is claimed. Rational numbers should be read as exact fractions. See DATASET_SCHEMA.md for column definitions and a dependency-free loading example.

Provenance, citation, and licensing

The original v2 manuscript, source, code, and recorded results are preserved. provenance/ includes the untouched original research archive, its original README and manifest, and a release provenance record. The v1 predecessor remains under prior/. The regenerated root manifest applies to this Hugging Face release layout and excludes only itself.

Citation metadata is supplied as CITATION.cff and CITATION.bib. No DOI, arXiv identifier, institutional affiliation, or publication venue is invented. The displayed Eve author name is explicitly an AI-generated persona; it is not represented as a human researcher or institution.

No explicit reuse license was supplied with the original package. This preparation does not assign a new license. The YAML deliberately omits a license identifier; see LICENSE_NOTICE.md. Public hosting alone is not a statement of permissive reuse terms.

The manuscript attributes established optimal recovery, enclosing-ball geometry, electrical energy minimization, list decoding, hierarchical expert methods, and concentration bounds. The candidate research contribution is their exact synthesis for this model. Third-party papers and font files are not bundled.

Status / completeness

Component	Status
Manuscript and stated model-specific results	Written definitions and proofs supplied; self-audited
Included finite computational suite	100% of included checks passed
Release packaging and integrity verification	Recorded in RELEASE_REPORT.json
Independent mathematical peer review	Not performed
Formal proof-assistant verification	Not performed
Worldwide novelty	Not established
Empirical neural-network advantage	Not tested
Unrestricted Chebyshev-set conjecture	Not solved by this work

Completion percentages refer only to the named finite workflow. No percentage is assigned to progress toward the unrestricted historical conjecture.