--- title: "The algebraicity threshold for orthogonal tensor multiplicities" date: 2026-09-08 version: "0.1.0-candidate" doi: 10.5281/zenodo.22660867 pdf: https://github.com/ipitchford/orthogonal-tensor-algebraicity/releases/download/v0.1.0-candidate/orthogonal-tensor-algebraicity-0.1.0-candidate.pdf repository: https://github.com/ipitchford/orthogonal-tensor-algebraicity archive: https://zenodo.org/records/22660867 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # The algebraicity threshold for orthogonal tensor multiplicities ## Summary How complicated is the series that counts a chosen symmetry type inside repeated tensor products? This candidate gives a single dividing line for complex orthogonal groups: every fixed nonzero genuine representation has an algebraic counting series in dimensions one, two and three, and none does in higher dimensions. All these series nevertheless satisfy linear differential equations with polynomial coefficients. The consequence is stronger than an invariant-only result: changing the chosen genuine representation cannot restore algebraicity above dimension three. This is an unrefereed written classification candidate, assembled from explicitly attributed classical inputs. Historical priority is not established. ## Summary for specialists Let $V=\mathbb C^m$ be the defining representation and let $W\ne0$ be a fixed finite-dimensional algebraic representation of $O(m,\mathbb C)$, with $m\ge1$. Put $$P_W(t)=\sum_{N\ge0}b_W(N)t^N,$$ $$b_W(N)=\dim\operatorname{Hom}_{O(m)}(W,V^{\otimes N}).$$ The candidate proves that $P_W$ is D-finite for every $m$, and algebraic over $\mathbb C(t)$ if and only if $m\le3$. The quantifier includes reducible genuine representations, but excludes virtual subtraction and a representation varying with $N$. For the symmetric determinantal quotient, put $$S=\operatorname{Sym}(\operatorname{Sym}^2\mathbb C^\infty),$$ $$B_m=S/I_{m+1}.$$ The invariant specialization gives algebraicity of its **symmetric-sequence ordinary Hilbert series** exactly for $m\le3$ (including the trivial $m=0$ case). This is not a claim about the usual finite-variable commutative Hilbert series. ## Technical account The low-dimensional side gives explicit formulas for all types of $O(1)$, $O(2)$ and $O(3)$. For $O(3)$, the half-line tensor-product recurrence must retain the exceptional boundary at spin zero; a parity projection then distinguishes the two extensions from $SO(3)$. For the obstruction, character integration over the full compact orthogonal group isolates the two maximal trace points, $I$ and $-I$. If the central parity of $W$ is $\varepsilon$, the compatible coefficients satisfy $$b_W(2n+\varepsilon)\sim K_W m^{2n}n^{-\alpha},$$ $$K_W=\dim(W)m^\varepsilon c_m,$$ $$\alpha=\frac{m(m-1)}4,$$ where $$c_m=2^{1-m}(m/2)^\alpha\pi^{-m/2}\prod_{j=1}^m\Gamma(j/2).$$ The scalar constant is already supplied by Regev's work. The manuscript also derives the normalization directly and explains why the two maximal points add rather than cancel on the compatible parity. Integer exponents force logarithmic singularities after differentiation. Half-integer exponents force a leading amplitude incompatible with algebraicity above dimension three. Mixed central parity is handled by extraction. D-finiteness follows through finitely generated covariants, bounded row support and the Sam–Snowden theorem. The genuine-representation qualification is substantive. In dimension four the trivial and determinant multiplicities are $(C_n^2+C_n)/2$ and $(C_n^2-C_n)/2$. Their virtual difference is $C_n$, so subtraction cancels the high-rank obstruction. This illustration is not advertised as a new Catalan identity. ## Evidence, assurance and limitations The package contains a thirteen-page integrated manuscript, precise source comparison, response matrix, preserved substantive reviews and five confirmation reports. Internal editorial confirmation accepted the scoped candidate with attribution and priority qualifications; it does not constitute independent journal peer review or an externally certified star rating. Producer replay compares 463 tableau shapes, checks 841 weight and 841 spin multiplicities, verifies parity and dimension identities, and rejects a deliberately corrupted hook divisor under normal and optimized Python. Fresh extraction and Linux CI validate the shipped finite diagnostics. They do not prove the universal theorem. No formal verification or independent reimplementation is claimed. The theorem does not classify arbitrary modules over twisted commutative algebras, nonreduced thickenings, growing representation types or all virtual characters. D-finiteness alone supplies neither a useful differential-equation order nor an efficient coefficient algorithm. ## Relationship to earlier work Regev (1981), Theorem 3.3 and Formula 4.5.1, already gives the scalar asymptotic including its constant. Almkvist, Dicks and Formanek (1985), Theorem 2.1, gives the relevant tensor-algebra character integral; their Example 5.10 gives the $SO(3)$ invariant radical whose even part is the orthogonal invariant series. The package records the published erratum. General fixed-type tensor asymptotics also have classical antecedents, including the Biane result discussed by Tate and Zelditch. The presented contribution is the unified all-fixed-genuine-type orthogonal classification, with explicit low-dimensional formulas, parity treatment and the no-restoration consequence. It is not a claim to have originated the scalar asymptotic, the rank-three radical, character integration or the classical algebraicity obstruction. The supplied Regev and Almkvist–Dicks–Formanek papers were directly inspected. Cohen–Regev (1988) was unavailable and is explicitly marked uninspected; no claim about its detailed contents is made. The bounded comparison does not establish historical priority. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Representation theorists | Know when every fixed orthogonal type admits an algebraic ordinary series. | Fixed genuine type and characteristic zero are essential. | | Invariant theorists | Transfer the invariant classification to symmetric determinantal series. | Distinguish symmetric-sequence series from finite-variable Hilbert series. | | Symbolic-computation researchers | Choose D-finite rather than algebraic ansatzes above dimension three. | No effective differential-order or runtime guarantee is supplied. | ## Why the problem matters Algebraic series form a much narrower class than D-finite series. A sharp classification explains why a successful radical formula in low dimension cannot simply be extended by choosing a different genuine symmetry type in higher dimension. It also identifies cancellation, rather than an improved invariant formula, as the issue when virtual representations are allowed. These are mathematical consequences, not measured practical or productivity benefits. ## How to inspect or reproduce the recorded checks Download the versioned ZIP, compare its SHA-256 with `SHA256SUMS`, and extract it into a fresh directory. With Python 3.10 or later, run `python3 replay.py` and `python3 -O replay.py`. Both must report `PASS`; the receipt identifies the exact checks and their finite boundary. No third-party Python library is required. The manifest is checked before the diagnostics. ## The most valuable next projects 1. Obtain an unaffiliated proof audit concentrating on compact-group normalization, the singularity obstruction and finite generation of fixed covariants. 2. Determine explicit annihilating differential equations and useful order bounds for selected higher-dimensional types. 3. Classify virtual character combinations whose singularities cancel; the genuine theorem does not settle this problem. ## Who might contribute Useful scrutiny would combine orthogonal representation theory, analytic combinatorics and twisted commutative algebra. Separate historical work could assess priority beyond the directly inspected antecedents. No endorsement by the cited authors is implied. ## What is in the evidence package The immutable candidate release includes the PDF and Markdown manuscript, Python verifiers and semantic controls, complete manifest, replay receipt, claim index, source audit, review history, editorial decision, citation metadata and component licences. Original prose is CC0; original code is MIT. Supplied full third-party papers are not redistributed. Historical paths in preserved reports are provenance, not dependencies needed to run the package. ## Open directions for follow-up research - Determine explicit differential equations and useful order bounds for further fixed representation types. - Classify which virtual character combinations cancel high-rank singularities. - Obtain unaffiliated proof scrutiny and a broader historical-priority assessment. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:symmetric-determinantal-hilbert-threshold - Attempt and metric receipts: ep-attempt:symmetric-determinantal-hilbert-threshold-remaining-publication: published / positive; scope publication-only; target Complete accurately attributed unrefereed candidate publication through immutable archives and final canonical readback.; active forecast 100 minutes (60-180); Fermi components Package gates and confirmation: 1 x 20/30/50 minutes low/central/high (Integrated manuscript and diagnostics already available.); Public archives and media: 1 x 20/35/65 minutes low/central/high (Existing authenticated APIs and house scripts.); Composite CI, deploy and readback: 1 x 20/35/65 minutes low/central/high (Two controlled site publication cycles.); positive-signal/closure probabilities 0.9/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 29/unknown/unknown/0/0/8; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -71 minutes; ratio 0.29; inside interval false; positive-signal/target-closure Brier scores 0.01/0.0225; missing telemetry activeHumanMinutes: Human effort not instrumented.; computeMinutes: No complete substantive compute meter; ordinary publishing builds are not research compute.; deduplicatedModelTokens: Root-only unique response records available in local receipt, but complete reviewer-tree usage cannot be established; no sum of cumulative snapshots.; uncachedInputTokens: Root-only partial counter available; reviewer-tree total unavailable.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Opening 1 retained; terminal 6. (reason: Opening one-coordinator snapshot preserved; terminal count includes five differentiated confirmation reviewers.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Terminal approximately eight minutes of operational rework; no mathematical claim correction. (reason: Opening zero snapshot preserved; later schema, formula-layout and shallow-clone repair subtotal recorded separately from scientific work.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, communication - Decision object: reusable-method — Uniform orthogonal algebraicity classification and parity-resolved singularity obstruction. Scope: Fixed nonzero genuine finite-dimensional algebraic O(m,C) representations; characteristic zero. - Reusable methods: Structural compression (structural-compression); Exact regime stitching (regime-stitching); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Character integrals identify the exact tensor multiplicities; parity-separated coefficient asymptotics obstruct algebraicity. Finite diagnostics inspect formulas only. Remaining risks: Written universal proof is not formally verified.; Historical priority remains unestablished.; Virtual subtraction falls outside the genuine-representation quantifier.. - Human judgement gates: Assess universal proof and normalization of compact-group integrals.; Check prior-work attribution and contribution scope.; Distinguish finite replay from all-case proof.; Preserve creator, rights and assurance boundaries. - Next assurance action: Separate unaffiliated proof scrutiny, replay and priority assessment. - Claim ceiling: Unrefereed written classification candidate for fixed nonzero genuine algebraic O(m,C) representations. No arbitrary tca-module classification, virtual-character extension, historical priority, external specialist validation or formal verification is claimed. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable orthogonal algebraicity classification candidate in Producer-coordinated mathematical publication; design none; comparator No matched comparator.; estimand No acceleration or impact effect estimated.; no real-world effect evidence asserted ## Verification status Unrefereed written classification candidate for fixed nonzero genuine algebraic O(m,C) representations. No arbitrary tca-module classification, virtual-character extension, historical priority, external specialist validation or formal verification is claimed. ## References 1. Regev (1981), Asymptotic values for degrees associated with strips of Young diagrams: Theorem 3.3 and Formula 4.5.1 already supply the scalar asymptotic. 2. Almkvist, Dicks and Formanek (1985), Hilbert series of fixed free algebras and noncommutative classical invariant theory: Theorem 2.1 and Example 5.10. 3. Rains (1998), Increasing subsequences and the classical groups, Section 3. 4. Sam and Snowden (2018), Hilbert series for twisted commutative algebras, Theorem 6.1 and Example 7.11. 5. Tate and Zelditch (2004), Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers, Theorem 8. 6. Bostan, Salvy and Singer (2025), On deciding transcendence of power series, discussion of classical asymptotic obstructions.