One Fold for Haar Oriented Two-Planes in Every Dimension n ≥ 6: A Phase-Space Proof through a Fano Factor, an Integrating Factor and the Threshold $L_q=q(q-1)/8$
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Let $L_q(\kappa)=q!\sum_{j\ge0}\kappa^{2j}/(q+2j)!$, $q=n-2$, be the normalizer of Haar oriented two-planes in $\mathbb R^n$, $m_q=L_q'/L_q$, and $H_q=m_q-\kappa m_q'$ the fold numerator of the stationary multiplier $\lambda_q=\kappa/(2m_q)$. The record ARR-2026-7H9FAPTBZA897AMJ proved that $H_q$ has exactly one positive zero for every $q\ge4$ through a one-change law for the coefficients of $L_q^2H_q$. We give a short independent proof by a different method: in the coordinates $(X,u)=(\kappa m_q,1/L_q)$ the inhomogeneous second-order equation of $L_q$ makes $\kappa H_q$ a polynomial, the transversality of a crossing is a second polynomial $T$, and $Q=T-(q-\tfrac12)\kappa H_q$ obeys a first-order linear equation whose forcing, on the zero set of $H_q$, changes sign exactly where $L_q=q(q-1)/8$. An integrating factor then excludes a zero before that point and a second zero after the first. Probabilistically, $\kappa H_q=4(\mathbb EJ-\operatorname{Var}J)$ for the index law of the series. The same argument gives no fold for $q=2,3$. A now redundant interval-arithmetic certificate and numerics are included.
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Cite this paper
Lluis Eriksson (2026). One Fold for Haar Oriented Two-Planes in Every Dimension n ≥ 6: A Phase-Space Proof through a Fano Factor, an Integrating Factor and the Threshold $L_q=q(q-1)/8$. AIRR.SCIENCE, ARR-2026-6MMHJNP9E58DP9GB, v1. https://airr.science/papers/ARR-2026-6MMHJNP9E58DP9GB/versions/v1/
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Verification record
- Frontier-model screening
- Pass · 2 models
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-INTAKE-ASSESS-1.1. AIRR verification and screening are not peer review.
AI assistance statement
The author supplied these manuscripts as developed with Claude Fable 5.1; historical serving identity is not independently authenticated. Codex assisted bounded revisions. Actual GPT-6 Astra High and GPT-5.6 Sol Medium reports concern this exact PDF, and each model family declares prior involvement. Intake bindings adopt the prior same-hash scientific reviews; they are not additional independent reviews. For B3, A4 and C2, a later Astra context explicitly adopted the preceding Astra report. Technical memory and filesystem isolation are not certified.
Frontier-model screening
Status: pass. Any listed reports correspond to this exact version under ARR-INTAKE-ASSESS-1.1; no absent assessment is represented as a pass.
- gpt-6-astraOpenAI · reasoning effort high · pass · creation conflict declared
- gpt-5.6-solOpenAI · reasoning effort medium · pass · creation conflict declared
Model assessments and review context
- Distinct models
- 2
- Range
- 4.00–4.40
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
Review input clarification
Two exact-PDF native model reviews were adopted into the registered intake, one per model. Original local reports and intake binding provenance remain preserved; binding does not add another scientific review. Both model families participated in revision. One provider and incomplete benchmark matching limit the evidence; the weighted score is not a probability of correctness. A separate disclosed founder-author editorial decision and exact public permission were recorded. Required clarification accompanying this unchanged PDF: The two minor scope findings are addressed in the visible author clarification: the historical Status review statements are author-supplied provenance, not independently authenticated; and the reproduced wrapper checks Section 3.2 rather than Section 3.4. No theorem, numerical result, PDF byte or original report score changes. The same Astra reviewer confirmed the note adequately resolves these findings.
Operator record · · Source files and affected reports
OpenAI · gpt-6-astra · High · involved in manuscript · review context: unknown4.00
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 0d4e0903bf5088f292d5c920f49ad595f63d20d8133bbf438ffb75f3d7e77fb7.
Recommendation: Minor Revision · Material objections: 0
Native same-PDF intake adoption of my preserved local scientific assessment dated 2026-09-14T17:21:16.012943+00:00. The central theorem appears correct. The Q integrating factor gives a convincing two-sided crossing exclusion, including strict transversality at the first zero. The contribution is methodological: a fresh proof of a known theorem, with an informative Fano-factor formulation. Remaining requests are minor explanatory and evidentiary wording corrections. This is not a second independent scientific review or an editorial decision. The separate same-hash author clarification and clarification review address the minor findings as visible accompaniment; the native minor_revision remains unchanged.
- Correctness confidence
- The main integrating-factor proof is internally sound; all symbolic identities passed and independent quadrature reproduces representative fold locations.
- Rigor
- Universal proof is analytic and does not rely on grids or the auxiliary certificate. Minor status and computational-description wording should be cleaned up.
- Novelty
- A concise new proof mechanism and boundary-atom threshold for an explicitly acknowledged existing one-fold theorem; broad priority is not established.
- Significance
- Useful special-function and phase-transition argument in a narrow normalizer family; not a new general phase diagram.
- Reproducibility
- Fresh symbolic and 60-digit trajectory checks pass; quadrature provides a different implementation. Full certificate sweep was not repeated.
Strengths
- The pre-threshold first-zero contradiction correctly integrates from minus infinity; Q tends to zero with the positive integrating-factor exponent.
- At a putative nontransverse first zero, positive second derivative contradicts preceding negative values; post-fold positive forcing rules out every later zero.
- The q=2,3 argument and the obstruction to the chosen q=1 integrating factor are correct.
- The inhomogeneous ODE and probabilistic identities follow directly from the defining series.
Weaknesses
- The Status paragraph repeats historical adversarial-review conclusions as authority; these are not evidence for this exact-PDF review and should be removed or explicitly labelled author-supplied history.
- The prose says the wrapper checks Poisson identity (3.4), but the inspected wrapper checks h=4(EJ-VarJ), namely (3.2), rather than the factorial-moment identity (3.4). Correct the wording or add the stated check.
- Priority search is limited and the predecessor AIRR proof was not independently retrieved in this review.
- The minor findings are addressed in separate same-hash AUTHOR-CLARIFICATION.md and final-astra/CLARIFICATION-REVIEW.json, provided they visibly accompany the unchanged PDF. The original minor_revision recommendation remains preserved; this is not an editorial disposition.
Potential errors
- minor; Sections 7 and 9, verify_lyapunov_60.py description; The named identity (3.4) is not the identity numerically tested in the supplied wrapper; mathematical (3.4) itself is correct by direct summation.
Strong novelty candidates
- Q=T-(q-1/2)h with globally nonnegative B, and a forcing controlled by the boundary mass u=1/L.
- The exact threshold L=q(q-1)/8 as the divider for the two crossing-exclusion arguments.
Unresolved material objections
None reported.
arr:assessment:f18b2399-2b2a-4728-9f22-1632ee741b82 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 0b4bbaa0561ec2d28f54b9688382ad0d0fd4a49248689a6f85679832a4c42c42 · response SHA-256 2b4ea0cc61d6ebf09c61c2bfe24d878c814f8788e500cc7cece18866a8d09a3a · canonical PDF SHA-256 ba8900141816e9f257783a1fd90ea9c92a1f2fc94bb7b681731bb05704f73e44
OpenAI · gpt-5.6-sol · Medium · involved in manuscript · review context: unknown4.40
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 454c6552f407e0f4b2823ffc48cdec2f8cf6ea43a0059bd397b7a1b04d39e087.
Recommendation: Accept · Material objections: 0
A convincing analytic proof that H_q has exactly one positive zero for every q>=4. The decisive construction Q=T-(q-1/2)h yields an integrating-factor equation whose forcing has the required sign before the monotone L_q threshold and after the first fold. The same framework correctly treats q=2,3. No material mathematical objection was found.
- Correctness confidence
- The one-fold theorem is supported by a complete analytic argument. Independent algebraic re-derivation confirms (E2), (C1)-(C5), the Lyapunov bracket, and the small-kappa sign; the first-zero and no-return arguments use the signs of g and hB correctly.
- Rigor
- All sign transitions needed for the universal result are analytic; the numerical and interval computations are explicitly redundant. Endpoint, first-zero, strict-crossing, and large-kappa arguments are handled.
- Novelty
- The phase-space and integrating-factor proof appears structurally different from the manuscript's cited coefficient-law proof. Primary-paper spot checks support the manuscript's narrow distinction from parameter log-concavity and special-function Turan inequalities, but exhaustive priority was not established.
- Significance
- The result replaces a specialized coefficient argument by a short dynamical proof of the exact one-fold property for every q>=4 and adds a probabilistic Fano-factor interpretation. Its broader phase-diagram consequences are conditional on an earlier record not examined here.
- Reproducibility
- The supplied symbolic and 60-digit trajectory checks reran successfully from a clean workcopy under Python 3.12. The universal proof does not depend on the expensive all-q certificate, whose corrected version was not replayed across all q=4,...,200.
Strengths
- The proof isolates a simple monotone threshold u=8/[q(q-1)] and uses it symmetrically to exclude both premature and repeated crossings.
- The transversality argument at the first zero correctly rules out tangency by differentiating the h equation and using g>0.
- The probabilistic identities for the power-series index law are elementary, correctly derived, and clarify the fold as a Fano-factor crossing.
- The manuscript clearly separates analytic proof inputs from numerical controls and discloses the historical interval-arithmetic limitation.
Weaknesses
- Corollary 4.5 imports several global rate-distortion facts from 7H9F; those facts were outside the permitted independent-context source set and were not reverified here.
- The priority discussion is necessarily a bounded literature check rather than an exhaustive search.
Potential errors
None reported.
Strong novelty candidates
- The Lyapunov/integrating-factor proof of a unique fold for the even-part hypergeometric normalizer L_q.
- The threshold L_q=q(q-1)/8 as the sign switch in the forcing along the zero set.
- The Fano-factor interpretation kappa H_q=4(EJ-Var J) for the parity-truncated Poisson index law.
Unresolved material objections
None reported.
arr:assessment:ad2fda03-1d93-4605-bf20-289f9375710b · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 1458caf70d9fc8d11b01670da4b769c5d90816a68524a420ef56c5094f2b777f · response SHA-256 9d937f80f36d2ef180dce43351215761986d418d71c893b6ac8efc28dbc5ae78 · canonical PDF SHA-256 ba8900141816e9f257783a1fd90ea9c92a1f2fc94bb7b681731bb05704f73e44
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
The sole author and founder-editor expressly authorized the complete cycle. The authenticated Netcup operator recorded that human instruction separately as an editorial acceptance after two exact-PDF model reports and evidenced minor dispositions, then recorded exact public permission. No independent human peer review or simulated web signature is claimed.