Internal proof review

Henry Zweiman. September 16, 2026.

This is the originating assistant's audit of its own argument, not independent peer review. No subagents or outside mathematicians were used in this stage.

Spaces and normalization. For n>=2, differentiation is an energy isometry from the clamped order-n space onto the zero-mean subspace of the clamped order-(n-1) space. The final endpoint condition on u is exactly the integral constraint on u'. Real scalars are specified.
Jacobi identity. In the 2m integrations by parts, the derivative orders in each boundary product sum to 2m-1, so at least one is below m and vanishes. The leading coefficient gives (k+2m)!/k!, with the sign canceled by the leading coefficient of x^m(1-x)^m. Energy orthogonality and polynomial density give a complete basis, not just a formal expansion.
Constraint and sphere dimension. The mean representer is w/(2m)!. Deleting its mode yields Q_m with coefficients M/lambda_k. On S^(2m+2), the affine coordinate density is proportional to x^m(1-x)^m. The normalized zonal harmonic basis matches the Jacobi basis without an additional multiplicity factor.
Poisson mixture. The beta integral equals k!/(k+2m)!. Tonelli applies to nonnegative spectral terms. Subtracting the constant harmonic is essential and is preserved at fixed beta-weighted measure. No interchange of a conditionally convergent Green kernel series is needed.
Rearrangement and equality. Burchard-Hajaiej Theorem 2 applies to two identical indicators, strictly supermodular product integrand, and positive bounded strictly distance-decreasing Poisson kernels. Integrated equality forces equality at some radius. Zonal invariance and the nonzero cap centroid fix the cap axis up to reversal. Null sets for beta measure and Lebesgue measure coincide in the open interval.
Duality and optimizer. Layer cake represents any load in [-1,1] as an average of signs; convexity justifies restricting the supremum. Constants are annihilated, giving the factor four. An endpoint interval represents point evaluation after integration. The known point diagonal has its unique maximum at 1/2. For an actual extremizer, the sign of u' has both signs on positive-measure sets. The equality chain and Riesz uniqueness then give the full classification.
Moment geometry. Norm below one excludes extremality by scaling; zero is excluded separately. n+2 positive-measure pieces give a nontrivial perturbation satisfying n moments and one variation constraint. Vandermonde excludes <=n nodes. Only necessity of the n+1-node description is used. Divided-difference weights and the n! factor agree with the simplex density normalization.
Endpoint compactness. The potential map is weak-star to uniform continuous for n>=2. Variation is a supremum of continuous linear functionals, since the integrated bounded load kernel H_g is continuous, including n=2. The closed convex variation sublevel set contains the extreme points, so Krein-Milman proves the bound. Height compactness supplies attainment; upper semicontinuity of variation is not assumed.
Endpoint coincidence. Extreme points of the exposed maximizing face are extreme in the original moment ball. Scaling a shorter support to the full interval increases height by L^(1-n) at fixed derivative-measure norm. All resulting full-support splines share orientation by their initial derivative signs. Root concavity gives strictly signed derivatives outside the maximum interval; disjoint maximum intervals contradict the common norming load. Pairwise intersecting closed intervals have a common point, and weak-star continuity of evaluation extends this point to the face.
Recovery. Mollification of the zero extension and compression with amplitude (1+2 epsilon)^(1-n) preserve the bound on the nth derivative's L1 norm. Smooth endpoint jets vanish, while the height and variation converge. This proves equality of ordinary and relaxed constants, without asserting ordinary attainment.

No mathematical gap was identified in this audit. The 42 exact rational normalization cases are corroborating evidence, not a substitute for any all-order argument. At the original revision, broader exponents and endpoint symmetry remained open; subsequent advances are audited below. Artifact checks are recorded separately in artifact-check.json.

Revision 1.1: local continuation audit

Moment constraints: n-fold integration identifies the clamped derivative space with the annihilator of polynomials of degree <=n-1. Best Lq approximation supplies the exact dual functional and Holder equality supplies the highest derivative of every variation maximizer.
Existence: compactness through order n-1 preserves variation, and weak Lp compactness supplies the admissible highest derivative. Positive optimal value forces norm one.
Strong compactness: the dual loads and minimizing polynomials are uniformly Lipschitz for p in a compact subset of (1,infinity). The optimal values stay positive. Signed powers therefore make the highest derivatives uniformly equicontinuous. This assertion concerns maximizers, not arbitrary Sobolev sequences.
Hilbert profile: the integral in (6.7) is a clamped piecewise polynomial with the correct single delta load. Its derivative is strictly signed on either side of the midpoint, its peak has negative second derivative, and its endpoint highest derivatives do not vanish. These are derived explicitly rather than inferred from weak unimodality.
Endpoint persistence: n-fold integration of the sign of the highest derivative controls u' near the clamped endpoints. Cn convergence controls the strict signs elsewhere and the simple midpoint critical point. This proves that every nearby-exponent variation maximizer has exactly one peak.
Equality: a single peak yields variation twice height; the universal reverse inequality then yields the sharp constant identity and coincidence of both maximizing sets.
Profile regularity: the residual has nonzero polynomial pieces near the base point. Lagrange interpolation proves a uniform sublevel estimate, which makes negative powers uniformly integrable when q is near two. This justifies differentiating the normal equations and applying the implicit function theorem. Multiple residual zeros are allowed.
Symmetry: the Hilbert diagonal has a unique, nondegenerate midpoint maximum. Uniform convergence excludes distant maxima and continuous negative second derivative excludes competing nearby maxima. Reflection gives the unique maximum at the midpoint; strict convexity gives the unique normalized extremizer and its symmetry.
Scope: no uniform-in-order interval, explicit radius, full all-p result, or measure-endpoint symmetry is claimed. Local continuation is not promoted to a global theorem. No numerical proof substitute or independent review is asserted.

Revision 1.2: fourth-order endpoint audit

Global existence and reduction. The measure ball is weak-star compact; its potential map is uniformly continuous. A maximizing evaluation exposes a compact face with an extreme point. The earlier moment argument forces exactly five nodes. Stretching a shorter support strictly increases height, so the outer nodes are 0 and 1.
Orientation and peak location. The divided-difference weights give a positive cubic spline. Its derivatives at the first and last inner knots have strict opposite signs. Reflection permits a<x<=b<c, including a peak exactly at b.
Dual contact. Hahn-Banach and Riesz identify the evaluation norm with distance to the cubic polynomials. A best polynomial exists. Equality against every nonzero signed atomic weight forces the alternating values. The residual is C2 even at x, so derivative contact remains valid when x=b.
Right cubic. Its derivative has roots b and c and positive leading factor. Equal values at b and 1 force b=1-3h/2 and c=1-h/2. Neither a cubic of lower degree nor a vanishing factor can satisfy the alternating values.
Left contact and peak stationarity. Both equations in (7.8) follow by direct integration. Dividing by positive factors yields 3/2<v<=2. Independently U'(x)=0 gives (7.10); R(0)=R(1) gives (7.11). Only the two leftmost nodes contribute to U'(x), also when x=b.
Elimination. Equations (7.12)-(7.14) preserve solutions because a,h,v and v-1 are positive. The L,M,N subtraction proves (7.15) without a black-box resultant. Ordinary expansion verifies its factorization and (7.17) excludes every zero of Pi in the allowed interval. At v=2 the second root h=2/3 violates a>0.
Sharpness. The proof identifies all global maximizing extreme measures, whose existence was established first. Thus the computed height is the unrestricted optimum. It is not inferred from local stationarity or optimization inside symmetric knots.
All equality cases. Every extreme point of an exposed height-maximizing face is one of the two signed measures. The face's positive exposed value permits only one sign; Krein-Milman makes it a singleton. The earlier coincidence theorem transfers the classification to variation.
Ordinary endpoint. The unique relaxed derivative measure is atomic. It cannot be the absolutely continuous fourth derivative of a nonzero ordinary Sobolev extremizer. Mollification and rescaling still recover both sharp constants.
Limits. The all-p conjecture, higher-order endpoint symmetry and the stronger knot-reflection inequality remain open. The exact symbolic checks are algebra diagnostics only. No independent referee review or exhaustive priority audit has occurred.

Revision 1.3: all-order contact-motion audit

Zero count. The Descartes argument treats missing Taylor coefficients: paired sign variations across a gap of h degrees contribute at most h. The two independent top coefficients add at most two. If all shared coefficients vanish at the knot, there is only one distinct zero; this case is explicitly excluded where multiple contacts are required.
Contacts and dual measure. Moment duality forces n+1 alternating contacts. A constant residual piece would permit only two alternating values. The derivative zero count gives exactly n-1 interior contacts, nonzero endpoint derivatives, and simple interior critical points. Equality forces every norming measure onto this contact set; Vandermonde rank gives uniqueness for the whole face.
Low-order and knot exceptions. For n=3, a critical contact at the kernel knot would make a derivative piece identically zero. For n>=4, multiple critical contact at that knot exceeds the multiplicity bound. Thus the second derivative at each contact is defined and nonzero.
Parameter regularity. The contact-value block in polynomial coefficients and amplitude is invertible by alternation. The derivative-contact block is diagonal with nonzero entries. The system is C1 at kernel/contact collisions when n>=4. Compactness and uniqueness identify the implicit branch with actual best approximation and prevent contact coalescence.
Auxiliary spline. In V=dot R+R'-ell R, the differentiated kernel cancels because its two partial derivatives sum to zero. V is a one-simple-knot spline, not a double-knot spline. Its endpoint signs are inherited from R', and both pieces are nonzero.
Parity. V has n-1 prescribed distinct zeros and a multiplicity budget of n. The endpoint signs prohibit either one extra simple zero or one doubled contact; restoring parity exceeds the budget. The all-common-coefficients-zero exception has only one zero and is impossible. Hence V' and R'' have the same alternating signs at contacts, proving 1-t_i'>0. No lower bound on t_i' is assumed.
Credited peak theorem. Foucart Proposition 6 gives strict interior-knot monotonicity and uses nondegenerate B-spline peaks. Lemma 3 and Formulae 5 were checked in the author text. The final publisher body was unavailable and is not represented as read.
Endpoint extension. Repeating an endpoint keeps the support length fixed in the insertion identity. Its first-order differential equation and the opposite endpoint boundary value give the integral formula. Bounding the original spline by its peak makes the repeated-knot value strictly smaller there. The second-derivative identity then has the required negative sign. Differentiating the original support-length normalization contributes a multiple of B' at the peak, which is zero.
Mode velocity. Positivity for all knot derivatives and translation equivariance make their sum one. Fixed endpoint velocities and t_i'<1 give s'(x)<1, even if some interior contact velocity were negative. Reflection fixes s(1/2)=1/2. This proves the peak/evaluation ordering without the centroid conjecture.
Envelope and equality. Testing neighboring unique optimizers justifies A_n'=U_mu_x'(x). The spline has a unique peak with strict derivative signs, so the profile is strictly monotone on each side of the midpoint. The unique midpoint norming measure gives all global height equality cases; the earlier coincidence theorem transfers them to variation. Atomicity excludes ordinary attainment while recovery preserves constants.
Evidence and scope. Eight contact systems in orders 3,4,6,8 at x=1/4 and 1/2 were solved to 65-digit working precision, including differentiated contact equations. These finite checks are not a proof of the all-order theorem. No full all-p solution, unproved centroid/reflection inequality, independent review or absolute priority certification is claimed.

Revision 1.4: finite-exponent proof audit

Notation and scope. q is the Holder dual of input p. The result addresses k=0 and target L-infinity in Conjecture 4.3. It does not identify target q with this dual exponent, assert the full symmetry conjecture, or prove the remaining variation identity.
Residual zeros. The n moment conditions force n sign changes. A zero piece would leave a fixed-sign monomial on the other side. The one-knot multiplicity bound then gives exactly n simple interior roots and nonzero endpoint values; the two-monomial exception cannot supply n>=3 roots.
Norming potential. The unnormalized highest derivative is |R|^(q-2)R. Its moments impose all clamped conditions, and evaluation equals integral |R|^q>0. Rolle counting with endpoint multiplicities excludes an interior zero of u, two critical points, or a degenerate critical point. The norming potential is positive and its unique peak has negative second derivative.
All-q regularity. Transverse residual roots persist under coefficient and evaluation perturbations, including root/kernel collision since n>=3. The singular weight has locally uniform integral O(h^(q-1)) on residual sublevels for 1<q<2. This justifies normal-equation differentiation and a positive-definite finite Gram matrix. The coefficient map is C1; f is C1 into L1 and spatially absolutely continuous.
Auxiliary function. u is not normalized; ell=dot u(s)/u(s) fixes W(s)=0 without differentiating ell in t. The identity partial_a K+partial_t K=0 makes dot R+R' polynomial. Thus D^n W=|R|^(q-2)[(q-1)H-ell R] and the bracket has one simple knot. W belongs to W^(n,1), hence C^(n-1), enough to use W'' for n>=3.
Endpoint data. Differentiating the clamped conditions gives W^(j)=0 at both endpoints through j=n-2, and W^(n-1)=f there. Positivity of u fixes f(0)>0 and (-1)^n f(1)>0, so W is positive near zero and negative near one.
Weighted Rolle count. Failure of W'(s)<0 supplies three interior zeros counted with multiplicity: an upward crossing plus neighboring roots, a tangency plus another root, or a triple root. Along with endpoint multiplicities n-1 this gives 2n+1 zeros. After n-1 ordinary Rolle steps, at least n+2 distinct zeros of W^(n-1) remain. Each intervening zero integral forces a separate sign change of T, exceeding its n-zero bound. No classical nth derivative at the singular-weight points is required.
Zero-piece exception. If T vanishes on one side, the other is a monomial, and W^(n-1) is nonzero constant on one side and strictly monotone on the other. It has at most one zero. If T is identically zero, it is constant nonzero. Both contradict the required n+2 zeros.
Peak comparison. W'(s)=u''(s)(1-s')<0 gives s'<1. Reflection fixes s(1/2)=1/2. The envelope identity and strict peak signs give strict profile monotonicity and all height equality cases.
Remaining variation gap. A variation residual comes from an unknown multi-switch load and need not have one knot. The argument does not classify arbitrary variation maximizers. The full all-p variation conjecture remains open.
Diagnostics. All 24 corrected quadrature cases passed the specified tolerances. These are floating-point evidence only. The numerical weight is floored at 1e-14 near roots for the singular Jacobian; roots within 1e-8 at the shared polynomial interface are merged. Initial failed runs are documented, not represented as theorem failures or successful certificates.

Originating-assistant review only; not independent mathematical certification.

Revision 1.5: endpoint-continuation proof audit

Arbitrary sequences. The proof chooses arbitrary normalized variation maximizers at exponents tending to one. It does not follow only the known midpoint branch. Existence and the dual formula are already established in Section 6.
Sharp-constant limit. The interval has length one, so the L1 derivative norm is at most its Lp norm and V(n,p)<=V(n,1). For the reverse limit inferior, one fixed smooth recovery function is tested before the exponent limit, then its recovery error is sent to zero. No uniform recovery rate is assumed.
Measure limit and normalization. Weak-star compactness preserves all moment constraints. Joint continuity of the potential and first-derivative kernels gives C1 convergence for n>=3. The variation limit therefore maximizes the endpoint problem, whose complete equality class is given by Theorems 4 and 9. Lower semicontinuity forces the total masses to tend to one.
Absolute measures. Any subsequential limit of |mu_j| dominates |mu_n| by continuous-test-function variation duality. Both masses equal one, so the positive measures agree. This is weak-star convergence of absolute measures, not total-variation-norm convergence to an atomic measure.
Dual load limit. The limiting derivative has strict signs except at endpoints and midpoint. C1 convergence implies almost-everywhere sign convergence, then L1 convergence by boundedness. The load formula gives uniform convergence to 2K_mid minus a polynomial.
Varying exponent. Best-polynomial coefficients are bounded using their L1 norms, without a norm-equivalence constant depending on q. For each fixed finite r, pass to the limit in Lr<=Lq; only then let r tend to infinity. Uniform best-approximation uniqueness identifies every limit. No interchange of an unbounded exponent with an uncontrolled residual is used.
Endpoint signs. The limiting uniform residual has nonzero values at both endpoints with the correct signs. Uniform convergence fixes signs on whole neighborhoods. The dual equation transfers them to the highest derivatives; repeated integration gives the first-derivative signs. Boundedness of the highest derivatives is neither assumed nor needed.
Central curvature for n>=4. The twice-differentiated kernel is continuous exactly in the stated order range. Uniform convergence and the credited nondegenerate B-spline peak give a fixed interval of negative curvature.
Order three. Reflection symmetry and four distinct knots give a central gap. The limiting second derivative is a negative constant there. Convergence of absolute measures removes mass from a closed subinterval in that gap, and convergence of the signed cumulative measure at its left end yields the uniform bound in (10.7). Relative boundary {c} avoids the endpoint atom at zero; continuous approximation is justified using the absolute-measure limit.
Single peak and equality. Endpoint signs, intervening compact signs and negative central curvature cover the interval. Every sufficiently late member of any sequence is single-peaked. The contradiction-sequence argument gives one radius valid for all maximizers. Variation equals twice height and Theorem 10 supplies the exact midpoint equality class.
Scope. Neither eta_n nor overlap with the neighborhood of two is estimated. The full all-p variation theorem remains open. No new numerical diagnostic is used to support the universal compactness result. This is originating-assistant review only, not independent certification.

Revision 1.6: switching regularity and Hessian audit

Starting regularity. The dual representation gives R in W^(n-1,infinity), continuous f=u^(n), and u in C^n. No finite-switch hypothesis is used at this point. The sign of u' is chosen arbitrarily on its zero set.
Endpoint integration. The chain rule for |u'| gives the two identities in (11.3), including on its zero set. All boundary terms in n-2 integrations by parts involve clamped derivatives below order n. The signs yield integral R'f=0 and integral tR'f=(n-1)V. The absolutely continuous function |R|^q then gives (11.5) and the endpoint normalization (11.1). Neither g nor f is differentiated.
Admissible perturbation. Nonzero endpoint f supplies intervals on both sides of any interior point where |f| is bounded below. If evaluation against the truncated kernel vanished on the kernel of the moment map, finite-dimensional factorization would represent it by one polynomial on both intervals. Its vanishing on the right and nonvanishing on the left contradict that representation. Integrating the resulting smooth moment-annihilating perturbation gives a clamped h with h'(a) nonzero.
Degenerate critical point. The norm of f+epsilonh^(n) is C2 near zero for every p>1 because the perturbation is supported where |f| is positive. Averaging the inequalities for both signs of epsilon cancels the linear term. If u'(a)=u''(a)=0, the exact absolute-value identity and u'(t)=o(|t-a|) give a gain cM*epsilon^2 for arbitrary fixed M and sufficiently small epsilon. This contradicts one fixed quadratic upper bound. The order of quantifiers is essential.
Finiteness. Nondegenerate interior critical points are isolated. The clamped endpoint Taylor expansions and nonzero endpoint highest derivatives exclude accumulation at either endpoint. Any remaining accumulation would give a degenerate interior critical point. Thus the count is finite and at least one.
Residual zero count. Moment orthogonality gives at least n sign changes. Repeated mean values force both signs in the highest residual derivative, so its affine offset c lies strictly between -1 and 1. Every residual piece then has full degree n-1. Repeated Rolle arguments from u' and from R squeeze the residual count to n+r-1. Any extra double zero would add one to the upper count and is impossible. A zero at a switching knot is included because R is C1 even in order three.
Uniform bound. An unbounded sequence of critical-point counts on a compact exponent interval has a C^n-convergent subsequence of normalized maximizers by Section 6.5. Nonzero endpoint derivatives and the limiting maximizer's finitely many nondegenerate critical points force the count eventually to be identical to that of the limit. This proves a bound without estimating it.
Normal-equation derivative. Simple residual roots make |R|^(q-2) locally integrable for every q>1 and give uniform transverse-root sublevel estimates in a neighborhood. The polynomial Gram matrix is positive definite. Implicit differentiation therefore gives the weighted projection formula; no smoothness across a multiple root is assumed.
Hessian normalization. The moving load is 2sum sigma_iK_tau_i modulo the polynomial space. Thus the Gram term has coefficient 4q(q-1), and the diagonal curvature term has coefficient 2q*sigma_i. The unnormalized potential U=I^n(|R|^(q-2)R) is used consistently. For n=3, the second kernel derivative is an indicator; dominated integration introduces no boundary term.
Maximality and scope. Nearby ordered switches still define sign loads of magnitude one, whose dual norm is at most the sharp variation norm. Hence the Hessian must be negative semidefinite. The matrix inequality is necessary, not sufficient, and no general exclusion of r>=2 is proved. The exact two-switch calculation verifies the formula and a saddle in one Hilbert family only. This is originating-assistant review, not independent certification.

Revision 1.7: translation exclusion audit

Scope and normalization. Theorem 12 applies to arbitrary global maximizers. Scaling to norm one and choosing the sign do not change critical-point counts. For two switches the derivative signs are +,-,+. The n+1 simple residual roots and equal endpoint magnitudes give F(1)=(-1)^(n+1)F(0), with F(0)>0.
Admissible motion. Small simultaneous translations keep both switches ordered and strictly interior. The associated loads still have magnitude one. No boundary-condition change to the original variational space is made.
Polynomial cancellation. In the representative H_s=2K_(a+s)-2K_(b+s), partial_s H+partial_t H=0. Thus T=dot R+R'=-dot P-P' has degree at most n-1. The omitted actual-load term is a polynomial and does not affect the best-approximation residual.
Singular weight. Theorem 12 gives simple roots; Section 11 justifies L1 differentiation of |R|^(q-2)R under switch motion. Spatial differentiation has the same integrable weight. Hence W=dot U+U' lies in W^(n,1) and satisfies its differential identity almost everywhere for all q>1. No derivative of |R|^(q-2) is taken.
Boundary values. All U_s satisfy every clamped condition by moment orthogonality. L1 differentiation preserves these moments and gives the clamped conditions for dot U. Adding U' leaves only W^(n-1)(0)=F(0), W^(n-1)(1)=F(1) nonzero. The signs of W' near the two endpoints are respectively positive and negative.
Nonzero polynomial. If T vanished identically, W would be a polynomial of degree <=n-1 with 2n-2 endpoint zeros counted with multiplicity. Since n>=3 this forces W=0, contradicting its nonzero endpoint derivative. The root count therefore applies to a nonzero polynomial.
Weighted Rolle count. W' has endpoint multiplicity n-2 at each end. N selected interior zeros, with any zero where W'' vanishes counted twice, lead after n-2 Rolle steps to at least n-2+N distinct zeros of W^(n-1). Between consecutive zeros its integral derivative has zero integral; positivity of the weight forces a root of T. This gives N<=2. Selected zeros suffice, so an infinite zero set is also excluded.
Strict unimodality. Opposite endpoint signs require a sign-changing zero. A second sign-changing zero would require a third to recover the endpoint signs; a tangency plus the crossing has multiplicity at least three. Both contradict the bound. Thus there is exactly one zero and strict derivative signs away from it. This conclusion does not require the crossing itself to be nondegenerate.
Boundary integration. The only surviving term in n-1 integrations by parts is [RF]_0^1. It vanishes by the established endpoint magnitude identity. The polynomial moment integral is zero, and the constant in the highest residual derivative multiplies W(1)-W(0)=0. The result is exactly W(a)=W(b), with no missing endpoint term or factor two.
Strict Hessian sign. Equal values at distinct points of a strictly unimodal function place those points strictly on opposite sides of its peak. Differentiating the envelope formula gives J''=2q(W'(a)-W'(b))>0. This contradicts a local maximum of the dual norm's qth power. No assumption of midpoint symmetry is used.
Limits of the result. Three or more switches produce an alternating sum instead of equality at two points. The research-only four-node example refutes the latter inference from the auxiliary conditions alone. The full conjecture and p=infinity endpoint remain unresolved. Internal review and artifact checks are not independent expert certification.
Typesetting correction. Equation (10.6) had the literal letters quad after R_(1/2), owing to a missing backslash. The spacing command is now restored. This affects notation rendering, not the limiting statement or proof.

Revision 1.8: affine response and three-switch audit

Affine kernel identity. For fixed v(t)=A+Bt, v(tau)-v(t)=B(tau-t). Combining the parameter and spatial derivatives of K_tau gives mB*K_tau. Thus T_v=dot R+vR'-mBR is a polynomial of degree at most m, including the omitted polynomial part of the load.
Normalization coefficient. Differentiating vU' n times gives vF'+nB F. With d=mq+1, n-d=-m(q-1). Therefore W_v=dot U+vU'-BdU has highest derivative (q-1)|R|^(q-2)T_v, with no remaining F term. This coefficient is essential.
Regularity. Simple roots from Theorem 12 justify L1 parameter and spatial derivatives of the signed residual power for every q>1. W_v is W^(n,1), so the weighted Rolle argument and integrations by parts are valid without differentiating a potentially singular weight.
Endpoint response. Every U_s and dot U is clamped. Derivatives of W_v through n-2 vanish at both endpoints. Its derivative n-1 is vF, since all remaining Leibniz terms contain a clamped derivative below order n. The endpoint identity gives exactly the boundary term B|R(0)|^q.
Hessian action. The envelope gradient is 2qsigma_iU'(tau_i). Differentiation along the straight switch path gives the stated Hessian action. W_v' has the additional term B(1-d)U', which vanishes at each switch. Since the path is affine in the parameter, no acceleration term occurs.
Three-switch parity. After sign normalization, the switching signs are +,-,+. Exactly n+2 simple residual roots give F(1)=(-1)^n F(0). For v=t-b, the auxiliary derivative is negative near zero and positive near one. This parity differs from the two-switch case and was checked separately.
Unique minimum. The endpoint derivatives are nonzero because 0<b<1. The polynomial T_v cannot vanish identically. Applying the Section12 weighted Rolle count gives at most two interior zeros counted with its stated convention. Opposite endpoint signs exclude a second zero or tangency. Hence the auxiliary response has one strict minimum and is negative in the interior.
Closed-interval criterion. If the minimum z lies in (a,c), both outer Hessian summands are positive. If z=a or z=c, only one summand vanishes and the other stays positive. Thus a maximizing configuration requires z<a or z>c, excluding the closed interval, not only its interior.
Symmetry of the whole motion. Centered dilation preserves the symmetric switch set. The moved sign load is antisymmetric; reflection modulo the polynomial space and uniqueness of the best Lq approximation give residual parity (-1)^n. Clamping uniqueness then gives symmetry of every U_s and therefore of dot U and W_v. This is a consequence of the assumed symmetry of u, not an assumption about all maximizers.
Positive centered dilation. Symmetry puts the unique minimum at 1/2. The middle switch has zero velocity, and both outer switches have sigma=+1. The second variation is 4q(1/2-a)W_v'(1-a)>0, contradicting maximality.
Diagnostic and literature limits. The rational Hilbert calculation checks one stationary three-switch family only. Elias-Pinkus Theorem1.3 does not cover the sign-forced mean-constrained equation; no eigenvalue-ordering shortcut is used. The weighted Hilbert route remains conditional. The full conjecture, nonsymmetric three-switch case and higher counts remain unresolved. This is internal review, not independent expert certification.

Revision 1.9: weighted obstruction audit

Admissible weight. The reciprocal weight is explicitly positive and bounded: 1/100<=rho<=1500000+1/100. Consequently a=1/rho gives a norm equivalent to the ordinary third-derivative Hilbert norm. Rectangle endpoints affect no integrals.
Dual normalization. Point evaluation uses (x-t)_+^2/2. The profile D uses twice that kernel, so max D=4C(a)^2. The two-switch sign load +,-,+ has primitive h_b-h_c modulo a quadratic, with no missing factor of two.
Endpoint conditions. Projection in L2(rho) makes rhoR annihilate all quadratics. Integrating rhoR three times with zero initial data therefore supplies all three terminal conditions.
Actual lower witness. This constructed function has energy J and load pairing J; Holder duality then gives V(a)^2>=J. The proof does not assume that the chosen load equals the sign of the constructed derivative.
Uniform upper bound. The moment matrix is strictly positive definite because rho>0 on an interval and monomials are independent. On every constant-weight interval, the moment vector has degrees at most 3,4,5 and the raw squared-kernel integral has degree at most5. Thus D has degree at most10.
Exact certificate. The table is a rational lower bound on all degree10 Bernstein coefficients of 9/10000-D, not sample values of D. Fourteen closed intervals cover [0,1], including every weight breakpoint. Nonnegative basis functions sum to one, so positivity of these coefficients proves a uniform strict bound.
Alternate check. A second implementation adds each rectangle contribution to the background independently and evaluates exact integrals at eleven rational points. The already-established degree bound makes interpolation an exact identity. It verifies the degree10 coefficients against every printed table bound and computes the mixed-kernel lower energy by a different integral expansion. Both implementations use SymPy rational matrix inversion; this is a distinct computational check, not independent expert review.
Strict comparison. The exact fraction for J is greater than967/10^6, while max D<900/10^6. Both comparisons use exact rational arithmetic. The floating-point search is a discovery diagnostic only; the proof does not rely on it.
Smoothing. Nonnegative convolution preserves the positive lower and upper bounds and gives L1 convergence of rho. Uniform boundedness of kernels makes moments and D converge uniformly; the inverse moment matrices converge since the limiting matrix is positive definite. The strict gaps persist. Compactness ensures a weighted maximizer exists, and a single-peaked function cannot reach a norm strictly above twice the height constant.
Freezing scope. The Holder transfer for1<p<2 remains valid. Its arbitrary-weight single-peak shortcut is false, but the special self-consistent weights from original maximizers have not been excluded. No counterexample to the original constant-weight conjecture is claimed.
Primary-source scope. Hindov-Lokharu2026 Theorem1 has a weight on the L1 function norm and an unweighted derivative energy. Replacing u by u' introduces an additional mean constraint absent from that result. Its formulation does not cover the assertion disproved here. The arXiv HTML formulation and main theorem were inspected; no PDF visual or full proof certification is claimed.

This is an internally audited explicit counterexample proved by exact finite polynomial arithmetic. The original all-exponent variation conjecture, nonsymmetric three-switch exclusion and higher-switch exclusion remain open. No independent expert review or exhaustive priority certification is claimed.

Revision 1.10: nodal-length concentration

Finite decomposition. Theorem 12 supplies finitely many simple interior zeros of u'. Together with the two outer endpoints they partition the whole domain into intervals of constant nonzero sign. Both signs occur because the integral of u' is zero.
Positive local energies. Vanishing energy on one interval would make u' affine there. Its two zero endpoint values would force it to vanish identically, a contradiction.
Admissible changes. On an interval stretched by s_j, multiplying u' by s_j preserves its first derivative at the joins. The value is zero on both sides of each join. Piecewise W2p gluing therefore gives a global W2p first derivative. Integrating it gives all three zero initial and terminal traces precisely when the signed-area constraint holds.
Exact functionals. Area scales by s_j squared; the p-energy of the second derivative of u' scales by s_j to the power 1-p. The quotient is smooth as a function of finitely many positive scale factors. Differentiability of the Sobolev-valued deformation is not used.
Regular constraints. The length gradient is positive and the signed-area gradient has both signs; they are independent. The implicit function theorem gives a smooth local constraint manifold with the stated two linear tangent conditions, including the zero-dimensional case of two intervals.
First variation. Summing the multiplier equations gives nu=(2+1/q)V. Its tangent contraction gives A dot x=-V(E dot x)/(2q). No additional stationarity or symmetry hypothesis is assumed.
Second variation. Differentiating both the numerator and denominator gives a cross term. Combining it with the positive square term yields V(p-1)(E dot x)^2/q^2. This positive term must not be dropped or assigned the opposite sign. The signed-area constraint contributes its nonzero second derivative. The length constraint has zero second derivative.
Dimension argument. If three diagonal entries were strictly positive, a vector supported on those three entries could satisfy the two homogeneous tangent conditions. The diagonal quadratic form would be positive and the remaining square term nonnegative, contradicting local maximality. Equality in the threshold is deliberately not excluded.
Length estimate. For 1<p<3 the sum of the diagonal entries is positive, so at least one is positive. There are at most two. The remaining intervals each have energy at least d/(p+1) times length, while their total energy is strictly less than one. Subtracting their length bound from one gives the asserted strict bound.
Scope. The energy-density restriction holds for every finite p>1 in order three. The nontrivial length conclusion holds only for p<3. It does not bound the count of short additional intervals, prove the full variation conjecture, or extend the gluing argument to higher orders. The numerical stationary-branch checks and exact scalar algebra check are supplemental diagnostics, not substitutes for the hand proof or independent review.

The proof is internally audited. No independent expert review or exhaustive priority certification is claimed. This is an additional necessary condition in one existing paper; the main conjecture remains open.

Revision 1.11: the complete finite-exponent result in order three

Stationary normalization. The unnormalized potential satisfies v''=phi_q(R), R''=sign(v)-c, v=v'=0 at the endpoints, and integral v=0. The residual is C1 and quadratic between the finitely many simple switches supplied by Theorem 12. The parameter satisfies -1<c<1. All later scalings are stated for this unnormalized potential.
Conserved quantity. R'v'-R''v-|R|^q/q is constant on each nodal interval and continuous at switches because v vanishes there. Its endpoint value is -A^q/q. Integrating gives A^q times total length=(2q+1) times residual energy. The identity holds before imposing the mean constraint.
Interior strictness. At an interior residual extremum away from a switch, R''v>0 forces |R|<A. A hypothetical equality at a switch has R'=0 and opposite nonzero curvatures; the residual then passes monotonically through that value, contradicting its global bound. Hence every interior value is strictly below the endpoint magnitude.
Vertex parametrization. The Hamiltonian fixes the sign of R' at every switch. The outer signs follow from the clamped traces. Each nodal interval therefore has one quadratic vertex, and integration of v*v'' proves its residual sign is opposite to v. Its depth lies strictly between zero and A.
Reconstruction. Matching residual values and slopes gives the exact half-length formulas. The scalar nodal equation gives both zero values of v. Strict monotonicity of Psi gives the correct sign throughout each interval. The same conserved quantity matches v' at all joins; the outer derivatives vanish. The remaining signed-area equation is precisely the third terminal trace. Consequently the smaller-count competitors are actual admissible functions, not abstract lengths and energies.
Positive action. Each edge action is maximized at its matched residual value. The derivative in the join variable has a strictly decreasing bracket and a positive factor. All edge and outer contributions are positive. The maximizing join is a maximum, which is essential to the subsequent convexity argument.
Depth derivatives and boundary faces. The gradient is q times the nodal equations. The boundary contribution at an internal join is f(z)/s, with the two changing half-lengths counted together. The outer boundary contribution vanishes. At fixed half-length, the depth derivative of Psi is strictly negative. The Hessian is symmetric, negative and strictly diagonally dominant. Its integrals remain finite when 1<q<2 because residual roots are simple. Uniform inward gradient signs exclude every face of the closed depth cube.
Unique depth solution. Strict concavity gives one interior maximizer and no other critical depth vector. A separate maximum-difference argument proves uniqueness using monotonicity alone, avoiding differentiation through a residual root.
Mean parameter. For fixed depths and a fixed joining value, the positive integrals are multiplied by (1-c*sigma)^(-1/2), which is convex. Maximizing first over the join and then over the depths preserves convexity. Positive outer terms give strictness. Fixed positive-depth tests give divergence at both ends of the parameter interval, including odd counts whose negative intervals are internal.
Envelope and zero mean. The optimized depth Hessian is invertible, giving differentiable dependence locally. Join and depth derivative terms vanish at their critical points. The remaining derivative of the optimized action is q times the signed area. Its unique interior minimum therefore supplies the required zero-mean profile for every N>=2.
Count comparison. Appending two copies of the last depth preserves the old outer terms and all old edge terms while adding two positive edges. This proves a strict comparison at every fixed mean parameter. Evaluating at the larger-count minimizing parameter and then minimizing the smaller-count action preserves the strict direction. The smaller-count profile has a strictly larger variation quotient after unit-length normalization. This is a comparison of reconstructed functions, not a literal cut-and-paste of the original maximizer.
Full finite-exponent conclusion. Every global maximizer is regular by Theorem 12. Counts N>=4 are excluded by the action comparison; N=3 is excluded by Theorem 13; N=1 contradicts zero mean. Thus N=2, and Theorem 10 supplies the complete midpoint height/variation equality classification. The argument covers every q>1, equivalently every finite p>1, in derivative order three.
Uniform endpoint. Height constants converge as p increases to infinity by weak Ls compactness for each fixed finite s and uniform convergence through the second derivative. A limit with every finite Ls norm at most one has L-infinity norm at most one. Passing the proved inequality to that limit gives the endpoint constant identity. This does not classify every uniform-endpoint equality case.
Evidence scope. The finite branch, off-stationary derivative and scalar algebra checks are supplemental diagnostics. None is used to prove the universal comparison. The final survey's Conjecture 4.14 and Remark 4.15 were refreshed; the survey records the prior n=2 case. Bounded searches and formulation comparisons do not certify exhaustive priority. This is internal review, not independent expert verification.

The proof audit supports Theorem 17 and its uniform-norm constant corollary. The finite-exponent conjecture in derivative order three is resolved within this manuscript; the all-order conjecture and the uniform-endpoint variation equality classification remain open. These additions form one revised paper.

Revision 1.12: complete uniform-endpoint equality in order three

Arbitrary maximizer. The proof starts with any normalized endpoint maximizer. Quotient-space duality in L1 modulo quadratics gives a best residual and equality in its pairing with the actual third derivative. The control equals sign(R) only where R is nonzero at this stage. No bang-bang, finite-switch, extreme-point or convergence assumption is made.
Strict mean parameter. For c>=1 the residual is concave. If positive somewhere, its control pattern and the two outer clamped traces force the first derivative to be nonpositive throughout, contradicting zero mean. If it is nonpositive and has a zero interval, the residual equation forces c=1 and a positive first derivative there. Continuity against the explicit negative outer quadratic pieces forces that zero interval to fill the domain, contradicting nonzero residual norm. Negation gives c>-1. Zero intervals are addressed explicitly instead of discarded as a null set.
Conserved quantity without sign differentiation. The expression R'v'-|v|+cv-|R| is absolutely continuous. Its derivative vanishes where R is nonzero by dual equality, and almost everywhere on the zero set because R'=0 there. Integration gives endpoint magnitude A=3J>0. No derivative of sign(R) or sign(v) is taken.
Regularity before parametrization. The conserved quantity bounds |R| by A. A degenerate interior zero of v would have residual magnitude A; the locally constant control would then make v a signed quadratic and force |R| above A. Thus every interior zero is simple. Nonzero endpoint residuals fix the signs near the boundary, so compactness excludes infinitely many zeros. Strict interior residual magnitude follows from the opposite curvatures at a simple zero. These steps justify every later finite-interval argument at p=infinity.
Bang-bang control as a conclusion. Each nodal interval now has a nonzero constant curvature, one vertex of opposite residual sign, and finitely many simple residual roots. Only at this point is the entire third derivative known to be sign(R) almost everywhere. The first derivative is C1,1; C2 regularity is not asserted at control switches.
The q=1 derivative is recalculated. The exact Psi formula contains a positive-part squared term. Its depth derivative includes the additional contribution from the moving residual zero. Naively setting q=1 in the finite-exponent formula would lose this term. The explicit endpoint formula is continuously differentiable and strictly decreasing in depth.
Boundary faces. The zero-depth positivity proof is separately supplied in the two ranges t<sqrt(2h/a) and t>=sqrt(2h/a). This replaces the finite-q bound, whose small-depth term would not tend to zero at q=1. The upper-depth face uses strict decrease and a positive lower bound on the half-lengths.
Action and reconstruction. Depth concavity, the unique interior depth solution, optimized strict convexity and coercivity in c all remain valid for the nonnegative integrand 1-|r|. The derivative is the signed area. The reconstructed control has magnitude one almost everywhere, preserves all endpoint traces, and gives an actual admissible competitor. The unit-length quotient is 1/(3 Lambda squared).
Higher counts. The two-copy insertion comparison is unchanged and strictly excludes N>=4. This is a universal action comparison, not a numerical enumeration.
Three intervals. Reflection preserves the +,-,+ phase and c. Depth uniqueness therefore makes the residual and control symmetric. Four simple residual roots give two unknown roots on the first half. The zeroth and centered second moments determine them exactly. The explicit first critical point lies in the stated control interval, and antisymmetry of u makes its variation four times its first height. Rational inequalities show this value is strictly smaller than the explicit two-interval competitor. No extension of Theorem 13 to p=infinity is assumed.
Two intervals. Reflection sends c to -c, so optimized strict convexity gives c=0. Equal vertex depths make the residual antisymmetric. Three residual roots and the moment equations determine the four-piece control uniquely. Its midpoint critical point, positivity on the first half, height and total variation are calculated explicitly. Multiplication by a scalar gives every nonzero extremizer.
Coincident equality sets. Every height maximizer has variation at least twice its height and at most the sharp variation constant, so it belongs to the classified variation set. This proves full coincidence, not just the existence of one symmetric maximizer. Together with Theorems 9 and 17, the order-three case is complete at every exponent with the measure interpretation at p=1.
Source and evidence scope. The height constant and midpoint height results are credited to the final survey. The new step is classification of all variation maximizers at the uniform endpoint. The exact algebra program checks the two explicit controls' moments, critical points and values; it does not prove regularity or universal exclusion. Bounded primary-source discovery does not certify exhaustive priority. This is internal proof review, not independent expert verification.

Revision 1.13: all-order endpoint regularity

Scope and normalization. Every maximizer is covered; no extreme-point selection is made. Compactness gives attainment, and positivity plus homogeneity fixes highest-derivative norm one.
Duality. The dual of L1 modulo degree-m polynomials is their L-infinity annihilator. Its load norm is bounded above by the variation constant and below by the given maximizing pairing. This proves equality for any best polynomial and controls f only off the residual zero set initially.
Regularity available before proof. R and v are W^(m,infinity); their derivatives through m-1 are absolutely continuous. This suffices for products, chain rules, traces and integration by parts in the conserved expression. The m=1 empty sum is handled by direct differentiation.
Signs in the telescoping expression. The signs are (-1)^(m-1-i). Differentiation leaves (g-c)v'+R'f. Subtracting |v|-cv+|R| gives zero, including on residual zeros because R'=0 almost everywhere there. No derivative of sign is assumed.
Integrated normalization. Each of m-1 summands integrates to -J by i integrations by parts; all boundary traces vanish. The two remaining integrals are -J each. Thus the conserved level is -(m+1)J=-nJ, with endpoint magnitudes nJ.
Zero lemma. The divided difference uses m+1 distinct zeros. The Hermite-Genocchi simplex measure has a density positive on the entire open convex hull of distinct knots. A one-signed mth derivative with zero integral against that density is zero there almost everywhere; the resulting degree-(m-1) polynomial has too many zeros. It is not sufficient merely to assert a pointwise version of Rolle's theorem for a weak derivative.
No-zero-interval case. If c>=1, the zero lemma gives at most m zeros. A polynomial with roots at sign-changing residual zeros has degree at most m and has positive pairing with f. Non-sign-changing roots need not be inserted. This contradicts the clamped moment conditions.
Zero-interval case. A maximal zero interval has zero derivative traces through m-1. Signed Taylor integral formulas give fixed strict signs outside; equality outside would extend the zero interval. On the interval c=1 and v>0 almost everywhere. The two negative outer monomials and continuity force the interval to fill [0,1], contradicting positive residual norm. Negation handles c<=-1.
Positive-measure zero sets. Repeated absolutely continuous level-set derivative statements apply outside a finite union of null sets. On a positive-measure residual zero set, its mth derivative equation and |c|<1 force c=0 and v=0. The conserved expression would then be zero almost everywhere there, contradicting -nJ. Saturation subsequently makes a positive-measure zero set of v impossible.
Endpoint monomials. Continuity and nonzero endpoint residual values first give fixed signs near both ends. Saturation then gives constant controls and the explicitly signed monomials; endpoint values of an arbitrary L-infinity representative are not asserted.
Local obstruction. Both explicit order-four examples solve their local stationary equations with positive v and R>A and a negative conserved level. Their identities and interval sign bounds were checked exactly. Neither satisfies clamped global conditions; neither is a counterexample to the conjecture or to a prospective global bound. This limits the method, not the conjecture.
Finiteness requires a separate argument. At an accumulation of zeros of either function, its derivatives through order m-1 vanish by repeated Rolle arguments. The conserved quantity makes the other function nonzero there. The first function then agrees with a degree-m polynomial near that point, contradicting accumulation. Endpoint accumulation is excluded by the fixed endpoint controls. This applies to arbitrary maximizers, including possible initial infinite switching.
Status. The proof is a structural advance within P47. No independent expert verification, exhaustive priority certification, new paper count, or complete all-order conjecture solution is asserted. The source survey still states Conjecture 4.14 and credits n=2; bounded searches do not establish absence of a prior structural result.
Multiplicity count. A zero with vanishing derivatives through m-1 forces the other function to be nonzero by the conserved quantity, so the first is a degree-m polynomial locally and the multiplicity is exactly m. Other multiplicities are classical and at most m-1. The spline zero bound follows after m-1 Rolle steps from a continuous piecewise affine derivative with nonzero slope on every piece. Counting the two endpoint zeros of v with multiplicity m gives m+M<=s<=Z<=m+r0<=m+M. Equality forces every zero of both functions to be simple and sign-changing, with exactly n+r-1 residual zeros.
Uniform bound near infinity. Any sequence of normalized maximizing functions with p tending to infinity, or already at infinity, has a subsequence whose highest derivative has every finite Ls norm at most one in the limit. Its limiting highest derivative is L-infinity bounded by one. Uniform convergence of u' proves convergence of the sharp constants and endpoint maximality. The limit is covered by the new theorem.
Preservation of the count. The endpoint limit has finitely many simple critical points. Convergence in C1 of the first derivatives controls the interior critical points, while uniform convergence of the best residuals fixes the derivative sign on common endpoint neighborhoods. Finite-exponent controls follow (6.6), endpoint controls follow (18.3). Best polynomials stay bounded in coefficients, and their Lq comparison passes to L1 as q tends to one. Counts are eventually equal to the limiting count. This rules out unbounded counts near infinity.
Entire exponent range. The already published one-peak interval above p=1 and compact-exponent regularity theorem, together with the new argument near infinity, give a finite count bound for each fixed n>=3 over all 1<p<=infinity. The bound remains implicit and does not prove the count is one in higher orders.

Revision 1.14: internal audit of endpoint Hessian and algebraicity

Root weight. The sign jump is 2sign(R'), while the zero velocity is -h/R'. Moving integration endpoints therefore give the positive weight 2/|R'|. No ordinary sign derivative is used.
Enough nodes. Theorem 19 gives s=n+r-1>=n simple residual zeros. Polynomial data of degree at most n-1 have a positive definite atomic Gram matrix.
Root-switch coincidence. For n>=3 the load kernel is jointly C1 in its parameter and integration variable. The simple-root IFT applies even at a load switch. This is why the Hessian theorem is stated for n>=3.
Persistence of all roots. Small neighborhoods have derivatives bounded away from zero; the complementary compact set has residual magnitude bounded away from zero. The local branch has exactly the same roots and nonzero endpoint values.
Best-polynomial branch. The derivative of the signed-moment equations in polynomial coefficients is minus the Gram matrix. IFT supplies a C1 branch, and zero signed moments give global L1 optimality by convexity.
Uniqueness. Equal best residuals must have the same sign almost everywhere. Both vanish at every simple zero of the first. Their polynomial difference has at least n distinct zeros and degree at most n-1, so it is zero.
Envelope and C2 regularity. Differentiated normal equations give R_i=2sigma_i B_i. Differentiating U'(tau_i) gives the atomic Gram term and moving-evaluation term. For n=3 an indicator kernel is integrated by dominated convergence. Finite-root formulas make the gradient C1.
Factors and normalization. At p=infinity, U=I^n(sign R)=u for the normalized maximizer. The Hessian is 4G-2diag(D); negative semidefiniteness is 2G<=diag(D), not G<=diag(D).
Rank and scope. The node-space polynomial complement has dimension s-n=r-1. A nonzero vector in the Gram kernel gives a negative Hessian direction, which is compatible with a maximum. No universal multi-peak exclusion follows from this fact.
Exact diagnostic. The already classified order-three two-switch profile was checked using radicals. Direct weighted polynomial projection agrees with the independent barycentric representation. The translation eigenvalue is positive and the opposite eigenvalue negative. This checks one configuration only.
Clamped parameterization. The alternating control moment equations are rational polynomial equations. Their vanishing is equivalent to every right clamped trace; the left traces vanish by integration. Each piece has highest derivative +1 or -1.
Semialgebraic graph. Each derivative piece has degree n-1 and nonzero leading coefficient. There are finitely many possible lists of interior critical points; a quantified formula requires these to be all the roots. Variation is then a finite sum of absolute polynomial differences. No false general claim about integration of semialgebraic functions is used.
Attainment at a fixed count. Theorem 19 supplies one finite control count containing a global maximizer. Every admissible configuration at that count has variation no larger. Hence the attained maximum at that count equals the global sharp constant, even though the strict knot domain is not compact.
Algebraic maximum. A finite endpoint of a nonempty semialgebraic subset of the line defined over Q is algebraic; otherwise all defining nonzero polynomial signs persist on a neighborhood. This applies to the attained maximum.
Algebraic optimizer. Impose the algebraic sharp value as a coefficient. Quantifier elimination transfers the existence statement from R to the real closed field of real algebraic numbers and supplies algebraic knots. It does not force every real optimizer to have algebraic coordinates.
External source. Basu's author survey explicitly includes real algebraic numbers as a real closed field and states quantifier elimination in Theorem 2.1. Only these relevant passages were read, not the entire survey. The source and download hash are archived locally, without redistribution in the paper package.
Open scope. There is no effective stopping count, explicit degree bound, or new resolution of the general n>=4 conjecture. The original assistant performed this review; neither the alternate algebra calculation nor artifact checks constitute independent mathematical certification.

Revision 1.15: internal endpoint-exclusion audit

Arbitrary maximizing input. Theorems 19 and 20 supply saturation, finite simple residual roots, all critical points simple, and a C1 best-polynomial branch. No initially finite-switch ansatz is imposed.
Affine cancellation. The kernel identity gives T=dotR+hRprime-(n-1)BR in P_(n-1). At q=1 the potential response is W=dotU+h uprime-nB u. The nBf derivative term cancels exactly.
Atomic derivatives. Df=mu Rprime and dotf=mu dotR are distributional identities from the finite sign jumps and moving roots. R mu=0 gives D^n W=mu T. They are not ordinary pointwise sign derivatives.
Trace regularity. The response is C^(n-2), piecewise polynomial of degree at most n-1. Its derivatives through n-2 vanish at both endpoints; its next traces are h(0)f(0) and h(1)f(1). These are one-sided values on fixed-control neighborhoods.
Moment uniqueness. Testing D^n W against P_(n-1) leaves just the two highest endpoint traces. The positive definite atomic polynomial Gram matrix makes the response polynomial unique.
Boundary pairing. Integration of R against the derivative measure is zero because R vanishes at each atom. The first integration by parts has the BV highest response derivative; subsequent steps are absolutely continuous. The only surviving boundary term is [Rhf]. Thus 2sum sigma_i W(tau_i)=-nVB.
Hessian normalization. At a critical point Wprime=dotUprime+h udoubleprime. The response formula is h-vector-transpose Jsecond h-vector=2sum sigma_i h(tau_i)Wprime(tau_i), with no missing power or factor.
Positive approximation. Mollified atoms plus a vanishing positive background converge weakly to mu. Their finite Gram matrices and inverse matrices converge, so the polynomial coefficients converge. Boundary moments remain exact. Uniform convergence of integrated continuous kernels gives convergence through derivative n-2.
Unimodality uses only the linear problem. The positive-weight Rolle argument depends on the polynomial degree and clamped response traces. It does not require an actual finite-exponent continuation of the endpoint stationary profile. The two nonzero endpoint signs select a unique peak or minimum for each smooth approximation.
Weak limits do not imply strict slopes. The limit is only weakly unimodal. Flat intervals and isolated zero slopes are handled separately; strictness is not asserted from convergence alone.
Separated-zero count. The clamped spline count is credited to Caminati's strong Rolle lemma and Proposition 2.5 and reproved by induction. Choosing extrema with nonzero values ensures derivative-zero separation, including from the endpoint zeros. Zero intervals and lower-degree pieces are allowed.
Six derivative zeros. Four nonzero extrema between the endpoints and three interior zeros, plus any two tangency points, give six separated zeros of the derivative. Constancy between any consecutive chosen derivative zeros would contradict an intervening original zero or nonzero extremal value.
Flat response count. Vanishing of Wprime on a knot-free open interval gives a nonzero signed measure, including the nonzero endpoint coefficient, annihilating P_(n-2). Vandermonde interpolation forces at least n-1 residual knots on each side. Vanishing atomic coefficients cannot weaken the requirement.
Two critical points. For n>=4, n+1 knots cannot support a flat interval. Equal response values place the two points on opposite sides of the strict peak. If both slopes vanished, Wdoubleprime would have six separated interior zeros, exceeding its bound four. The n=3 case is already classified by Theorem 18.
Full three-point condition. The dilation response is weakly decreasing then increasing. For n>=5 the knot count excludes every flat interval. In n=4 the two boundary moment nullspaces give atomic signs +,-,+ in each of two disjoint root triples, requiring four roots of a polynomial of degree at most three. Thus the response is strictly decreasing to a unique minimum and strictly increasing afterward, without symmetry assumptions.
Null Hessian direction and strict location. If that minimum belongs to [a,c], the dilation quadratic form is nonnegative. Negative semidefiniteness at a maximizer makes it zero and therefore makes the Hessian times the direction zero. The component response identity forces zero slopes at all three critical points, including when the minimum is a boundary point of [a,c]. At least two slopes are tangencies of Wprime; four intervening nonzero extrema plus these tangencies give six separated zeros of Wdoubleprime, exceeding the bound five. Under midpoint symmetry the unique minimum is at the midpoint, giving the symmetric exclusion.
Scope and evidence. Both exclusions are hand proofs; no numerical search or finite enumeration is used as proof. The theorem establishes the nonsymmetric three-point endpoint response-location restriction but does not exclude every such candidate or solve the full n>=4 conjecture. The source review covers the needed Caminati definitions and proofs, not its later applications. The originating assistant performed this review; independent expert verification and exhaustive priority certification are not claimed.