Priority and significance assessment

Henry Zweiman. September 16, 2026. P47 revision 1.13, developed as Q55.

Contribution and internal decision

The manuscript proves the complete Hilbert-space case of Nazarov-Shcheglova Conjecture 4.14, for every derivative order n>=2: the sharp derivative-L1 norm is twice the sharp point norm, with precisely the same midpoint-symmetric extremizers. It also proves the constant identity and coincidence of maximizing sets for the relaxed measure endpoint p=1. Revision 1.1 extends this to an order-dependent open interval around p=2; Revision 1.2 settles endpoint symmetry, the sharp constants and all knots in order four. Revision 1.3 completes the measure endpoint in every order. The full all-exponent conjecture remains unresolved.

Provisionally admit this as one substantial research preprint under the revised program standard. The main advance is an all-order rearrangement principle for a clamped Green quadratic form after the zero-mean constraint is imposed. Its Jacobi spectrum becomes a positive mixture of spherical Poisson interactions. This supplies a global optimizer and equality classification, rather than a finite-order calculation or local optimality test. The measure argument gives a complementary mechanism through moment circuits and unimodal splines. These related results form one paper.

The explicit open conjecture, uniformity in derivative order, and reusable load-rearrangement principle support significance. Possible extensions to other constrained Green operators are research directions, not established applications or citation forecasts. The p=2 result is still one exponent of a larger conjecture. An editor may judge the breadth insufficient for Advances in Mathematics or Transactions of the AMS. Admission is an internal, revisable judgment, not journal acceptance or independent certification.

Revision 1.13: all-order endpoint regularity

Theorem 19 proves |u^(n)|=1 almost everywhere for every normalized uniform-endpoint variation maximizer, in every derivative order n>=2. Quotient duality and a higher-order conserved quantity give endpoint residual magnitude nV(n,infinity). A one-signed-derivative zero lemma and explicit treatment of residual zero intervals prove |c|<1. Repeated level-set derivative arguments then rule out positive-measure zero sets of both the residual and u'. The highest derivative is constant near each endpoint.

An accumulation-point argument then proves finite switching: vanishing derivatives in the conserved quantity force the other function to be nonzero nearby, making the first a nonzero polynomial there. A spline zero count proves all critical points and residual roots simple, with exactly n+r-1 residual roots for r critical points. Compactness at p=infinity, combined with the published continuation above one and compact-exponent theorem, gives an implicit bound on r uniform over all 1<p<=infinity for each n>=3. This does not resolve the remaining higher-order variation conjecture. Two exact local order-four stationary solutions, at q=1 and q=2, show that the additional mixed derivative terms prevent deriving the order-three pointwise residual bound from the local conserved quantity alone. They are not clamped maximizers and do not disprove the conjecture or a prospective global bound.

The final primary survey was refreshed at Conjecture 4.14, Remark 4.15 and the endpoint convention. Bounded searches did not establish exhaustive priority for the structural result. The proof was internally audited, with exact algebra used only to verify the explicit local examples. No independent expert certification or additional paper is claimed.

Revision 1.12: the complete order-three case at every exponent

Theorem 18 classifies every variation extremizer at p=infinity in derivative order three. Its third derivative is, up to a scalar and null sets, the four-piece control +1,-1,+1,-1 with knots a,1/2,1-a, where a=(1-1/sqrt(2))/2. The sharp variation constant is (2-sqrt(2))/48 and the height constant is half that value. The height constant and its midpoint result are credited prior work; the new conclusion is the complete variation equality classification.

The proof applies to an arbitrary endpoint maximizer. L1 quotient duality, a strict bound on the mean parameter and an absolutely continuous conserved quantity prove finite simple critical points and the sign control without assuming them. The positive action is extended to q=1 using a freshly calculated depth derivative and separate boundary-face estimates. It excludes every nodal count at least four. Reflection and exact moment equations handle the remaining three-interval profile and show its value is strictly smaller than the explicit one-peak competitor. The two-interval control is uniquely determined by its moments.

Together with the complete measure endpoint in Theorem 9 and the finite-exponent result in Theorem 17, this closes the order-three case of Nazarov-Shcheglova Conjecture 4.14 for every exponent. Ordinary p=1 nonattainment retains the measure interpretation stated in the survey and manuscript. The all-order conjecture remains open for the remaining cases in n>=4. These revisions form one paper.

The current final survey was refreshed at Conjecture 4.14 and Theorems 4.5 and 4.12. No new external theorem beyond standard quotient duality and the preceding manuscript results is needed. Exact computations are supplemental checks of displayed algebra, not a substitute for the hand proof. Internal review does not certify independent verification, exhaustive priority or extraordinary significance. The broader goal remains active.

Revision 1.11: the full finite-exponent variation theorem in order three

Theorem 17 proves V(3,p)=2C(3,p) for every finite p>1 and identifies all nonzero variation extremizers with the unique midpoint-symmetric height extremizer up to multiplication by a scalar. This closes the n=3 finite-exponent case of Nazarov-Shcheglova Conjecture 4.14. Its corollary proves the constant identity at p=infinity by passage to the limit. The complete variation equality classification at that endpoint is not asserted.

The universal proof uses a conserved residual quantity and the quadratic structure particular to order three. Every possible global maximizer is reconstructed from nodal vertex depths and a mean parameter. A positive action is strictly concave in the depths and strictly convex after optimizing in the mean parameter. Appending two repeated depths increases the action; the optimized comparison then constructs an actual admissible function with two fewer intervals and a strictly better quotient. The earlier two-switch theorem excludes the remaining three-interval case. No finite branch enumeration or symbolic check substitutes for this proof.

The current final survey still states the all-order conjecture and records the prior n=2 case in Remark 4.15. This makes the n=3 extension a substantive special-case resolution, while the full n>=4 problem remains open. The new theorem also makes the order-three length restriction in Theorem 16 unnecessary for classifying global finite-exponent maximizers; it is retained as an earlier independent necessary-condition argument. The lower-count competitors and the zero-mean condition are both proved, rather than assumed from a formal shooting model.

The scope audit compared the current survey formulation and retained the earlier mismatch with nonlinear eigenvalue results that lack the additional mean constraint or the limiting L1 forcing. No new external theorem is imported. Bounded literature discovery does not establish absolute priority; internal mathematical review is not independent expert certification. The broader goal remains active pending further conjecture-level work and audit. This remains one paper.

Revision 1.10: an explicit nodal-length restriction in order three

Theorem 16 proves that a normalized variation maximizer in W0^(3,p), for every finite p>1, has at most two nodal intervals of u' with energy per unit length below (3p-1)/(p^2-1). For 1<p<3 these intervals occupy more than p(3-p)/(3p-1) of the domain. Thus the total length outside two intervals is explicitly O(p-1) as p decreases to one.

The proof is a direct finite-dimensional variation: stretch each nodal interval and rescale its amplitude to preserve u'' at the joins, impose total length and zero signed area, and use the constrained Hessian. The positive rank-one term and the second derivative of the signed-area constraint are retained. A dimension argument permits at most two positive diagonal coefficients. No new external theorem is imported beyond the manuscript's finite-switch theorem and elementary Sobolev gluing and constrained calculus.

This quantitatively restricts possible remaining maximizers in order three. It does not exclude extra short intervals, bound their number, establish a new full exponent interval, or solve the full variation conjecture. In the exponent intervals already covered by the manuscript the complete single-peak theorem is stronger. No claim of optimality of the new length bound is made. Bounded literature discovery does not certify priority; internal proof review is not independent verification. The broader conjecture goal remains active and this remains one paper.

Revision 1.9: the arbitrary derivative-weight extension is false

Theorem15 gives an explicit positive bounded piecewise-constant derivative weight in order3 for which V(a)^2>967/10^6 while4C(a)^2<900/10^6. A rational Bernstein certificate covers the entire evaluation interval; it is an exact finite polynomial proof, not an extrapolation from sampled points. Two separately implemented calculations verify the printed coefficient bounds and the exact two-switch energy. The gap persists under smoothing, so positive smooth derivative weights also give counterexamples.

This closes the unrestricted weighted Hilbert shortcut proposed in the revision1.8 research notes. The Holder freezing transfer for1<p<2 remains correct, but any successful application must use the special self-consistent relation between the original maximizer and the weight. The example has not been shown to satisfy that relation and does not disprove the original Nazarov-Shcheglova conjecture. It also does not alter the constant-weight two-switch exclusion theorem.

Hindov and Lokharu, arXiv:2605.25637v1, Theorem1, study a weight on the function's L1 norm with an unweighted derivative energy. The present variation problem also has an extra mean constraint after passing to u'. A formulation-level primary-source comparison therefore does not provide the missing weighted shortcut. Bounded searches do not establish exhaustive priority. The broad conjecture goal remains active; this remains one paper.

Revision 1.8: affine motion and symmetric three-switch exclusion

Theorem 14 excludes every midpoint-symmetric global variation maximizer with exactly three interior critical points, for n>=3 and every finite exponent 1<p<infinity. It does not prove that arbitrary variation maximizers have midpoint symmetry. It also gives a necessary condition for any remaining three-switch maximizer: the unique minimum of the auxiliary response to dilation about the middle switch must lie strictly outside the two outer switches.

The proof derives the response to a general affine velocity v(t)=A+Bt. Homogeneity of the truncated-power kernel cancels the parameter and spatial derivatives into a polynomial. A normalization term with coefficient B((n-1)q+1) cancels the remaining signed residual power. This gives an exact weighted-polynomial differential equation, endpoint values, an orthogonality identity and the Hessian action. For dilation about the middle of three switches, the weighted Rolle count makes the auxiliary response negative and strictly decreasing then increasing. If its minimum lies between the outer switches, their two Hessian contributions are positive and the fixed middle switch contributes zero. Midpoint symmetry forces precisely this location.

The argument is a hand proof and uses only the manuscript's regularity and differentiability results. A separate exact order-three Hilbert calculation isolates the symmetric stationary root and certifies positive translation and centered-dilation second variations by rational interval arithmetic. This is a diagnostic of one stationary family and is not the proof of the universal theorem.

A primary-source comparison with Elias and Pinkus, Proceedings of the Royal Society of Edinburgh A 132 (2002), 1333-1359, DOI 10.1017/S0308210500002134, inspected the initial formulation and main Theorem 1.3 in the author's hosted final-form PDF. That result requires strictly positive signed-power exponents and matched homogeneous scaling, with two-point boundary conditions. Our sign-forced equation with an additional mean constraint does not meet those hypotheses; its nonlinear eigenvalue ordering cannot be imported directly. The author's list and publisher metadata agree on 2002, despite the publisher's later 2007 online date. No theorem from this source is a dependency of Theorem 14, and the full proof and appendix were not audited here.

The research notes also contain a conditional transfer of 1<p<2 maximizers to a weighted Hilbert problem and a bounded twelve-weight diagnostic. The required weighted single-peak theorem remains unproved, and no such inequality is asserted in the manuscript. Bounded searches and originating-assistant review do not establish independent verification or exhaustive priority. Nonsymmetric three-switch maximizers, all counts at least four and the p=infinity variation endpoint remain open. This is one paper; the broad conjecture-resolution goal remains active.

Revision 1.7: two-switch exclusion at every finite exponent

Theorem 13 excludes every global variation maximizer with exactly two interior critical points, for every n>=3 and every 1<p<infinity. No symmetry or restriction on the locations of the two switches is assumed. Together with Theorem 12, this leaves only the known single-peak class or configurations with at least three switches. This answers the two-switch part of the manuscript's remaining stability question, but does not resolve the full all-exponent variation conjecture or its p=infinity endpoint.

The proof translates both switches simultaneously. The load derivative cancels its spatial derivative, leaving a polynomial after subtraction of the best approximating polynomial. The corresponding auxiliary potential has highest derivative equal to a positive weight times that polynomial. Its endpoint signs and a weighted Rolle count force strict unimodality. Polynomial orthogonality and the equal endpoint residual magnitudes force the auxiliary values at the two switches to agree. The switches consequently straddle its unique peak, which gives a strictly positive translation second variation and contradicts maximality.

All steps are hand arguments. The earlier order-three Hilbert diagnostic is consistent with this universal two-switch theorem, but is not used as a proof. The weight may be singular when q<2: the proof uses L1 differentiation, absolutely continuous primitives and integrals between their zeros, never differentiation of the weight. A repeated interior zero is counted only using the continuous first derivative; no unavailable higher derivative at a residual root is assumed.

The full primary survey's Conjecture 4.14 and Remark 4.15 were refreshed. Bounded targeted searches for variation maximizers, critical-point counts, translation instability and higher-order Sobolev extremals did not identify a matching two-switch exclusion in the inspected results. Searches were noisy, and this is not exhaustive priority certification. The new proof has no external dependency beyond the manuscript's established results. It has received originating-assistant review only.

An exact auxiliary four-node example in the research notes shows why strict unimodality and an alternating value sum alone cannot extend this sign argument to all higher switch counts. It is not a counterexample to the conjecture. An additional affine-dilation cancellation is recorded for the next investigation. No distinct paper is added and the broader goal remains active.

Revision 1.6: global regularity and a justified stability matrix

Theorem 12 proves, for every n>=3 and 1<p<infinity, that every normalized global variation maximizer has finitely many nondegenerate interior critical points and nonzero highest derivatives at both endpoints with explicit normalization. If r is its critical-point count, the dual residual has exactly n+r-1 simple interior zeros. The count is uniformly bounded for p in each compact subinterval of (1,infinity), with n fixed; the bound is not explicit.

This removes the unproved finite-switch and root-regularity hypotheses from the proposed stability route. Section 11 differentiates the best-polynomial normal equations and obtains the exact switching Hessian and a necessary matrix inequality for every maximizer. It does not prove that every r>=2 configuration violates that inequality. The all-exponent variation conjecture remains open, including the separate p=infinity endpoint issue. No new distinct paper is added, and this reduction does not fulfill the broader conjecture-resolution goal.

The proof is self-contained after the manuscript's duality and compactness results. Endpoint integrations by parts use an absolutely continuous residual power and do not differentiate the sign function. A two-sided perturbation supported where the highest derivative is bounded away from zero excludes a degenerate critical point for every p>1. Repeated Rolle counting then gives residual simplicity, which justifies differentiation even when q<2. Compactness through the highest derivative gives the uniform count bound.

A research-only exact order-three Hilbert diagnostic independently checks every Hessian entry by symbolic differentiation. The symmetric two-switch stationary configuration is a saddle: its translation eigenvalue is strictly positive. This verifies one family's signs and normalizations and does not establish a universal exclusion. A numerical root filter was replaced by the exact stationary quartic and its exact root count; no floating-point root claim is used in the hand proof.

Bounded searches for Sobolev variation extremals, simple zeros and perfect-spline stability found no matching primary regularity or universal switch-exclusion theorem in inspected results. The abstract of Nazarov's arXiv:1308.2259 concerns constant extremizers in other embedding inequalities and supplies no dependency here; its full body was not reviewed in this audit. No exhaustive priority certification or independent expert review is claimed.

Revision 1.5: variation continuation immediately above one

Theorem 11 proves the full variation identity and unique midpoint-symmetric extremizer classification for 1<p<1+eta_n, with eta_n>0 depending on the derivative order. This extends the completed measure endpoint into the ordinary Sobolev range and complements the earlier neighborhood of two. Neither radius is estimated, and overlap is not established; the all-p conjecture remains open.

The proof starts with arbitrary normalized variation maximizers as p tends to one. Sharp-constant convergence follows from the measure bound and a fixed smooth recovery function. The unique endpoint equality class identifies every measure limit. Equality of the limiting total masses also gives weak-star convergence of the absolute measures. The loads determined by derivative signs converge to the midpoint sign load, and varying-exponent best approximation converges to the unique uniform residual. Its nonzero endpoint values fix the highest-derivative signs in endpoint neighborhoods despite concentration.

For n>=4, continuous twice-differentiated kernels give uniform curvature convergence at the nondegenerate B-spline peak. For n=3, the symmetric four-knot measure has a knot-free central interval; convergence of absolute measures gives uniform convergence of the cumulative signed measures there. These two arguments supply the curvature control that C1 convergence alone would not provide. A contradiction-sequence argument then forces every nearby variation maximizer to be single-peaked. Theorem 10 completes symmetry and all equality cases.

The primary survey's final Conjecture 4.14 and Remark 4.15 were refreshed. The B-spline peak nondegeneracy used in the proof was rechecked in Foucart's author text at Proposition 6, retaining the existing author-version versus final-publisher-body qualification. The remaining inputs are proved in this manuscript or supplied directly by elementary compactness and duality arguments. Bounded successor searches were noisy and found no matching primary continuation theorem in the inspected results. This does not certify priority. No numerical experiment is used as evidence for the new universal statement, and no independent expert review has occurred.

This is a further interval of the named conjecture, not a resolution at every exponent. The goal remains active. The next global issue is to exclude a different multiple-peak variation maximizer at an interior boundary of the set of valid exponents; compactness of the known branch is insufficient. No distinct paper is added.

Revision 1.4: finite-exponent height symmetry

Theorem 10 proves strict midpoint maximization of the evaluation norm for every n>=2 and 1<p<infinity, with unique height extremizers up to scale, symmetry, positivity and a nondegenerate single peak. Together with the measure endpoint and the credited p=infinity theorem, this settles the k=0, target-L-infinity slice of Nazarov--Shcheglova Conjecture 4.3. It does not settle all k and target exponents in that conjecture, or the all-p variation identity of Conjecture 4.14.

The proof first establishes exactly n simple zeros of the best L^q residual using its moment equations and a one-knot zero count. This gives a positive norming potential with one nondegenerate peak and differentiability for every q>1, including the integrable singular weight when q<2. For W=dot u+u'-ell u, the nth derivative is a positive weight times a one-knot spline. Its endpoint conditions and a repeated Rolle count force W' to be negative at the peak. Hence the peak moves with velocity less than one. Reflection and the envelope identity give the global midpoint comparison. This proof does not depend on the B-spline peak monotonicity theorem used for the measure endpoint.

The final 2026 survey was read at Conjecture 4.3, Theorems 4.5 and 4.12, Remark 4.6, and Conjecture 4.14. The order-two and order-three results are credited. The primary Watanabe et al. 2009 publisher PDF, pp1-3, was inspected: Theorem 1.1 and Lemma 1.3 cover orders 1-3; Remark 1.4 explicitly identifies the failure of their symmetrization proof in orders at least four. The new argument crosses that order restriction for midpoint height symmetry without claiming closed-form constants for all orders. The credited best-approximation formula was rechecked in Garmanova--Sheipak 2024, Theorem 2 and Corollary 1. The separately archived even-order Garmanova--Sheipak paper concerns Hilbert input exponent, as its abstract confirms.

Twenty-four floating-point quadrature cases in orders 3,4,6, at evaluation 1/4 and 1/2, with dual exponents 3/2,2,3,5 checked moment equations, peak velocity, and two independent endpoint identities. Root duplication at the kernel knot and singular-weight roundoff were corrected in the diagnostic; a small numerical weight cutoff remains explicitly disclosed. These are consistency checks without certified error bounds and are not premises of the theorem.

This is an internally audited preprint. Bounded primary-source comparisons and noisy successor searches do not establish exhaustive priority. No independent expert verification or extraordinary-significance certification has occurred. The broader research goal remains active, and the revision adds no distinct paper.

Revision 1.3: all-order endpoint symmetry

Section 8 settles the endpoint symmetry question left open after revision 1.2 in every derivative order. The point-evaluation profile is strictly increasing to the midpoint and strictly decreasing thereafter. Its normalized maximizing measure is unique up to sign, with a positive symmetric B-spline potential. Together with Theorem 4, this gives the full constant identity, coincidence and symmetry assertions at p=1 and rules out ordinary Sobolev attainment. General closed-form knot coordinates and constants are not supplied; the fourth-order formula remains the explicit example.

The main new step is a contact-motion inequality. A zero count makes all contacts regular. Differentiating the uniform-approximation equations produces a one-knot spline with prescribed zeros and endpoint signs; a multiplicity and parity argument gives t_i'(x)<1. The peak of the optimizing B-spline then moves strictly more slowly than evaluation, proving the midpoint comparison. No finite-order extrapolation, numerical global certificate, conjectured centroid bound or generic knot-reflection inequality is used.

Foucart's 2005 B-spline peak monotonicity is a credited dependency. Its primary author preprint, dated February 24, 2005, was accessed and the relevant statements and proof in Sections 2-3 were read, specifically Lemma 3, Formulae 5 and Proposition 6. The cited theorem concerns interior knots. The manuscript explicitly proves the needed endpoint extension and accounts for differentiation of the support-length factor. Crossref and the author's publication list confirm Journal of Approximation Theory 135(1), 1-21, DOI 10.1016/j.jat.2005.03.001. The 24-page author preprint was read for the dependency; the 21-page final publisher body was not obtained (403 / access errors). No final-body equivalence audit is claimed.

The preceding source search also found de Boor's 2003 course notes, printed page 74, containing the centered cubic Chebyshev-spline example. The reference is now added explicitly: the symmetric dual cubic itself is prior work. The fourth-order contribution claimed here is unrestricted sharpness and equality classification. The new all-order argument has different scope from that example and from the published B-spline interlacing theorem alone.

Bounded searches combined the source authors, endpoint Sobolev symmetry, truncated-power approximation, contact motion and B-spline interlacing. No matching all-order endpoint solution was identified in the primary statements inspected. The searches were noisy and do not establish exhaustive priority. The originating assistant audited the full written argument, including the zero-at-knot exception and endpoint monotonicity extension. Eight high-precision contact-system checks support algebraic consistency only. Independent expert review has not occurred.

This is a substantially broader resolution of a manuscript open question than the previous single-order extension. It completes one endpoint of the named conjecture, not the full all-p statement. Exceptional significance and journal suitability remain judgments requiring external assessment; the active goal is not marked complete merely on this internally audited endpoint result. The revision remains part of P47 and adds no distinct paper.

Revision 1.2: the fourth-order measure endpoint

Section 7 proves C(4,1)=1/1296 and V(4,1)=1/648. The unique normalized extremizing measure, up to sign, has weights (1/9,-1/4,5/18,-1/4,1/9) at (0,1/6,1/2,5/6,1). The ordinary clamped W(4,1) constants agree, but are unattained. This answers the order-four endpoint symmetry question left open in the earlier manuscript. It is an additional case of the larger conjecture, not an all-order endpoint solution or a distinct paper.

The proof classifies every global maximizing extreme measure by uniform best-approximation duality. Alternating contact at five knots, stationarity of the potential and an explicit elimination reduce all possibilities to the symmetric knot set. A short polynomial sign estimate excludes the asymmetric alternatives. Krein-Milman then classifies every maximizing measure, not just the extreme ones. The derivation and sign estimate are in the manuscript; the research workspace's symbolic check merely corroborates the displayed algebra. A stronger reflection inequality proposed during exploration is still unproved and is not a proof input.

The final Nazarov-Shcheglova publisher statement was refreshed on September 16. Theorem 4.5 already includes the lower-order point cases, and Remark 4.15 credits the full order-two variation result. The final Garmanova-Sheipak highest-derivative result has different scope. Targeted searches for the fourth-order L1 sharp constant, the value 1296 and the survey authors found no matching primary result in the inspected material. These searches were noisy and bounded; no exhaustive citation-chain or priority certification is claimed. The proof uses no new unread source theorem.

The originating assistant checked the global-maximizer reduction, possible peak at the middle knot, all divisions, exclusion of the other algebraic root, constants and extension from extreme points. This is internal review. The significance of one further endpoint order does not by itself establish that the renewed exceptional-conjecture goal has been met. That goal remains active.

Revision 1.1: local continuation in the exponent

For every fixed n>=3, Section 6 proves the full constant identity and midpoint-symmetric extremizer classification throughout an open interval about p=2. This is a partial extension of the existing conjecture, not its full all-p resolution. The proof starts from the published Hilbert theorem and supplies its own duality, highest-derivative compactness, explicit Hilbert profile, endpoint sign preservation, polynomial sublevel estimates and finite-dimensional implicit-function argument. It derives continuity of the point-profile second derivative for exponents on both sides of two without assuming simple zeros of a best-approximation residual.

The exact final Conjecture 4.14 and Remark 4.15 were reread in current publisher HTML, alongside arXiv v3 Conjecture 2 and Remark 10. The order n=2 was already known for every exponent and is not a new claim. The polynomial-approximation methodology and Hilbert diagonal remain credited prior inputs. The primary MathNet abstract for Kazimirov--Sheipak 2024 was also checked: it concerns uniform-norm point evaluation, not the variation identity near p=2. Its full text was not accessed and it is not a proof dependency.

Focused searches covered the named conjecture and authors, higher-order total variation, perfect splines, polynomial approximation and continuation near p=2. No matching result was identified in the inspected primary statements; this bounded search is not exhaustive priority certification. The originating assistant found no remaining gap in the new written proof. Independent expert review has not occurred.

A global extension cannot be inferred from the local theorem. A competing multiple-peak maximizer may become globally optimal at another exponent; compactness of the known branch alone does not exclude this. Section 6 records the precise continuation criterion and this remaining obstacle. The interval is order dependent and not numerically specified. The active exceptional-conjecture goal remains unfulfilled by this partial result and is not marked complete.

Primary open statement and prior work

Nazarov-Shcheglova, final survey, Conjecture 4.14, was published online March 19, 2026. The final statement agrees with Conjecture 2 in arXiv:2101.10752v3, still the latest listed version on September 15, 2026. Final Remark 4.15 already settles n=2 for every p. No novelty is claimed for that order alone. The new Hilbert assertion is uniform in all orders n>=3 as well.

The sharp point-evaluation diagonal and its central maximum are credited to Kalyabin 2010, Theorem 2. All six publisher pages and the affine normalization were checked. Garmanova-Sheipak's 2024 final paper supplies prior polynomial-approximation methodology. Its Theorem 3 concerns the highest derivative at p=1 into Linfinity, a narrower statement than the displayed early-preprint formulation. The 2021 final paper already contains the corresponding pointwise maxima in their June 2026 arXiv upload. Upload date is not treated as a new result date.

Hindov-Nitzan-Olsen-Rydhe prove the sharp L1 embedding of the function itself. Both their nine-page author preprint and final publisher PDF were read. This result does not impose the derivative's zero-mean condition used here. Their interpolation construction is prior work.

Dependencies and current discovery

Burchard-Hajaiej, Theorem 2 explicitly covers spheres and gives strict equality conditions for the positive strictly decreasing kernels used here. The final revised author text, including the equality proof, was checked. Publisher metadata agrees with JFA 233 (2006), 561-582; a publisher-body comparison was not available. The withdrawn O'Neill preprint arXiv:1810.06813v2 is excluded as a dependency.

The normalized Poisson formula is from Axler-Bourdon-Ramey, second edition, author PDF revision July 17, 2020. The endpoint spline normalization is checked against de Boor's Genocchi-Hermite formula (52). Classical Brunn-Minkowski is used through Gardner-Hug-Weil's introductory formula (3) at p=1. These classical facts are credited and not claimed as new.

Eleven arXiv API feeds were screened, with totals and truncation flags retained in the research records. The final targeted 2025-September 2026 query returned all 22 hits. Searches also combined the source authors, Sobolev constants, total variation, Jacobi, zero mean and spherical rearrangement. Primary versions of close matches were inspected. No matching resolution was identified; this does not certify absolute priority or exhaustive coverage. Semantic Scholar's citation API returned 429.

Scope and review limits

All functions and measures are real. No all-p variation theorem or closed-form knot coordinates in every order is claimed. Height symmetry is now proved for all finite p. Measure-endpoint symmetry and ordinary nonattainment are proved in every order. The originating assistant audited its own proof; no independent human referee or mathematical expert has reviewed this manuscript. Exact rational checks support normalization only. Source and artifact checks are separate from mathematical validity.

Revision 1.14: endpoint Hessian and algebraicity

Theorem 20 closes the explicitly listed endpoint-Hessian question. Its root weights and polynomial projection are derived directly, including the case where a residual zero equals a load switch. The exact order-three two-switch profile confirms the factors and both Hessian eigenvalue signs; that calculation is corroboration, not a universal exclusion. The rank bound supplies a negative direction, which does not contradict maximality.

Theorem 21 applies the finite-switch theorem to a rational semialgebraic description of the total variation of a fixed-count control spline. Classical real-closed-field quantifier elimination is credited to Basu, Theorem 2.1. The sharp endpoint constant is algebraic in every order and admits an algebraic maximizing spline. No general closure of semialgebraic functions under integration is asserted; the proof explicitly lists the finitely many polynomial critical points.

These are structural results within P47. They do not settle the remaining n>=4 variation conjecture, compute all sharp constants, prove all maximizing knots algebraic, or give an effective stopping count. The source refresh read the relevant Basu statements and the final survey conjecture; bounded searches are not exhaustive priority certification. Internal review is not independent expert verification.

Revision 1.15: endpoint switch exclusions

Theorem 22 closes the endpoint two-critical-point and midpoint-symmetric three-critical-point exclusion questions, and proves the full necessary response-location restriction for other three-critical-point candidates. The new argument approximates the atomic linear response by positive-density boundary moment problems and recovers strict second variation using separated zeros. A flat-interval knot count and an order-four boundary sign count handle the degeneracies that prevented direct transfer of the finite-exponent proof.

The clamped separated-zero count is classical, credited to Caminati's Lemma 2.4 and Proposition 2.5, and reproved in the manuscript. The affine response, limiting argument and application to these extremizers are distinguished from that dependency. Only the relevant source sections were audited; bounded searches and the unchanged final survey conjecture do not establish absolute priority.

The result strengthens the same paper and completes these exclusions for every exponent above one. Nonsymmetric three-critical-point maximizers and configurations with at least four critical points remain possible obstructions in higher orders. No full all-order conjecture solution, independent expert verification or new paper count is claimed.