Remaining questions and retained approaches

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The unresolved targets

The supplied cubic theorem, from Beyer de Ryke's unreviewed manuscript, claims positive lower density of the complement for $r=3$; it is a pending claim on its claim page. The quartic argument proposes the analogous $r=4$ result, pending whole-conclusion review of the prose. No argument here establishes infinitude for $r\ge5$, or decides density zero versus positive upper density of any $S_r$, $r\ge3$.

Unordered count congruences

The last new attempt was divisibility of representation counts, not another local-solubility test. A proved criterion would give a positive-density complement if, on a progression of relative mean $\mu$, the unordered count were divisible by an integer $m>\mu$ outside a density-zero exception. No such congruence was established.

The exact Newton and prime-adic layer identities retain the five-distinct-part count. The mixed layers have order-$X$ mass; at the prime five, the all-prime-free five-part family and one specific four-free-plus-one-divisible family even have equal leading means. They cannot be discarded. Repeated prime-free kernels and nontrivial target-preserving changes supported on finitely many fixed primes affect only $o(X)$ source tuples, so those carry or valuation actions supply no generic extra orbit.

One bounded computation through $100000$, reproducible by the preserved script, found witnesses against every residue class for six specified pairs of target and count moduli. Its conclusion is finite: it rules out the corresponding congruences holding throughout a residue class, not eventual congruences or untested moduli.

Renewed attempts at bounded typical multiplicity for $x^3+2y^3+4z^3$, and at descent on $(r+1)p^r$ or $(r+1)q^{rk}$, also supplied no target result. In particular, the ray arguments did not force a common root divisor. No new unconditional critical estimate was obtained from the source checks. The tentative finite-curve direction was not completed or adopted as a theorem.

The strongest target conclusions are unchanged: the complements have positive lower density for $r=3,4$; complement infinitude for $r\ge5$ and every density question remain unresolved here. An integer-fiber estimate, a deletion comparison, or a new cross-kernel count symmetry would be additional arithmetic input. These are provisional possibilities, not demonstrated routes. The proof is incomplete.

Composition, shifted fibers, and cube differences

Neither of the two algebraic statements below resolves a target. Both are proved and author-recorded.

The rational-composition theorem excludes every total-degree-two polynomial identity $F(C(\mathbf x,\mathbf y))=F(\mathbf x)F(\mathbf y)$ for a positive diagonal ternary cubic. It handles the possible pure quadratic terms by the absence of a nonconstant rational map from $\mathbb P^2$ to a smooth plane cubic, followed by two bidegree comparisons. Consequently every fixed rational composition has only $O(N^{17/3})$ positive integral input pairs in balanced boxes. This does not control arbitrary product representations or maps varying with the scale.

The joint-shift calculation treats $F_j(\mathbf x_j)=n+h_j$ for any number of distinct shifts. Its natural complete intersection is geometrically integral and normal, with dualizing sheaf $\mathcal O(-r-1)$, independent of the number of shifts. Thus increasing that number does not create general type. No point-counting contradiction to cofiniteness follows; the affine critical scale remains $B^r$. The finite-core consequence of cofiniteness requires a fixed tuple only along an unbounded sequence of scales, not necessarily at every scale.

For actual cubes, difference inversion has vanishing average logarithmic multiplicity, by the existing binary-cubic collision theorem. Nevertheless $\omega(|w^3-z^3|)/\log\log N\to2$ in probability. Every positive-mass pair restriction therefore retains an unbounded logarithmic divisor majorant. This identifies a loss in that attempted entropy argument, not a bound for the weighted incidence among three independent differences.

No bounded-fiber theorem for a fixed critical diagonal form, and no arithmetic void comparison for powerful sums, was obtained. Density remains undecided here for every $r\ge3$; complement infinitude for $r\ge5$ remains unproved.

Three auxiliary results with explicit limitations

None of the three auxiliary results below resolves a target. Each is retained with its limitations explicit.

The metric countermodel is one increasing integer sequence $f(n)=n^r+O((\log n)^{1+o(1)})$. It agrees eventually with actual powers modulo every fixed modulus and obeys the same uniform power-difference bound, yet every positive rational coefficient tuple has a density-zero value set, while almost every real rounded tuple has positive upper density. This disproves extraction from those metric and congruence inputs alone. It does not concern actual powers or preserve every growing-modulus estimate. No tight arithmetic coefficient average or finite-dilation decorrelation estimate for actual powers was obtained.

For an odd prime $p$, the prime-degree geometric form $\sum_{i=0}^{p-1}p^i x_i^p$ is exactly uniform modulo every $p^m$. Its first digit box is bijective modulo $p^p$, and a box of side $B=p^k$ is uniform modulo $pB$. Neither statement reaches the modulus $B^p$ needed to control integer fibers. The valuation $v_p((u+p^jv)^p-u^p)=j+1+v_p(v)$ for unit $u$ explains why characteristic-$p$ digit interleaving cannot simply be iterated over the integers. No carry correction or positive-density induction was proved; numerical box counts are not retained as certificates.

The affine-fiber Brauer calculation proves $\operatorname{Br}(U)=\operatorname{Br}(\mathbb Q)$ for $U:\sum_{i=1}^r d_i x_i^r=n$, with $r\ge5$ and all $d_i,n$ nonzero. Thus ordinary Brauer--Manin on the full affine fixed-core fibers provides no additional obstruction to already existing local points. This is not an integral local-global theorem. Punctured opens and height restrictions are outside the calculation; the attempted punctures supplied no uniform estimate over the other root variables.

An elliptic/Selmer transfer was also unsuccessful. For $x^3+y^3=m=n-az^3>0$, the exact substitution

$$u=\frac{12m}{x+y},\qquad v=\frac{36m(x-y)}{x+y}$$

gives $v^2=u^3-432m^2$. Its index is $m$, which changes with $z$, not the fixed full target $n$. No bounded-multiplicity map indexed only by $n$, nor a height-sensitive average over these linked indices, was established. Likewise, renewed quadratic-fiber and exact-covering attempts did not improve the critical integer count.

The best target conclusions are unchanged: complement infinitude is established here for $r=3,4$ only, and no density question is settled. Possible future work must add exact-polynomial, finite-height input to one of the counting or void mechanisms; that is a provisional direction, not a demonstrated route. The proof is incomplete.

Birational maps, dilation, and large primes

No target case was resolved. Three additional arguments have been checked and retained.

For $r\ge4$, the birational-similitude theorem classifies every birational rational transformation satisfying $F\circ T=mF$: it is monomial. The proof applies de Fernex's checked superrigidity theorem to the generic degree-$r$ level in $\mathbb P^r$, then uses its anticanonical embedding and a $t$-adic valuation argument. This excludes composition laws birational in one input, not arbitrary rational correspondences. The same note proves that ordinary products of proper powerful subsums with at most $r$ expanded summands have count $O_r(X^{1/2}(\log X)^{k-1})$.

The global dilation countermodel has precisely one missing target, a nonnegative primitive decomposition, monotonicity under every powerful multiplier, and any prescribed mean $\lambda\ge1$. Thus even the exact mean and these global dilation identities do not imply the missing Palm comparison. The model is not an additive representation function.

The large-prime secant lemma excludes a target prime $p>2HK$ from secant content, without assuming that direction coordinates are units. A prime-target example rules out automatic height-decreasing secant descent. For the special cubic, the direction factor $J(\mathbf k)=\gcd(F(\mathbf k),|k_1k_2k_3|)$ has bounded logarithmic mean under uniform primitive-direction sampling. Actual collisions weight those directions by the unknown quadratic-fiber counts; replacing that weighting by uniform sampling is the unresolved step.

Two further alternatives also fall short. Coprime binary traces give simultaneous-square conditions in a quotient interval of length $B$, with expected count $B^{-1}$ for each of $B^4$ parameter choices. The available pointwise divisor bound is only $B^\epsilon$, not the required averaged saving. An entropy-inverse argument from binary energy cannot extract positive input mass from a slowly divergent ternary entropy deficit: the asymmetric energy remains far below its maximal scale. Neither is a multiplicity theorem.

Bounded numerical tests at the natural small-prime moduli for $r=6,8,10$ gave very large representation means, not deficits. They are not adopted as certificates or as all-modulus results. There is still no density resolution and no infinitude proof for $r\ge5$.

Exact scope of the fixed-field route

The attempted fixed-field route turned out to refine the previous Eisenstein parametrization, not to evade its obstruction. The expanded proof classifies all integral multipliers of a primitive Eisenstein parameter, gives the exact height scale $B^4/g$, and proves that a content-tail bound $B^6/D$ with logarithmic losses in $B^3/D$ would suffice for critical cubic energy. The local permutation calculation and the K3 common-zero geometry are also checked. The endpoint $D\asymp B^3$ already contains the original primitive collision problem; no needed estimate there was established.

Integer translation averaging does converge to ambient Haar measure and gives the genuine bound $A_{N,H}\ll N^3+N^6H^{-2+\varepsilon}$ for $H\ge N$. It works in the previously controlled short-box range, not at the bounded aspect ratio required for density extraction.

The covering attempt proves local anti-concentration of proper powerful subsums. Cassels bases satisfy even stronger estimates and are nevertheless additive bases. This rules out a universal argument based only on those estimates, bounded-error counting, and a finite mean. It does not rule out using the exact root tracks or multiplicative closure.

The split-prime normal-order theorem shows that typical inputs to $x^3+2y^3+4z^3$ have $(1/6+o(1))\log\log N$ distinct, exactly-once prime factors splitting completely in the splitting field of $T^3-2$. Restricting this number to be bounded therefore loses almost all input mass. No assertion that the local classes yield small positive integral partners was established.

The all-three-traces construction gives arbitrarily many interior representations of cube targets with all three families of pairwise trace ratios of full height. It rules out unrestricted deterministic trace pigeonholing. No primitivity is claimed, and the constructed family has zero input mass, so a statistical estimate remains possible.

The renewed coefficient-separation and prime-denominator attempts also did not resolve a target. A fixed large coefficient only reduces the number of possible competing roots to a constant multiple of $N$, not to a bounded number. On a missing prime ray, the case of $r$ summands all prime to the denominator still requires the uncontrolled index-one hypersurface count and a size-sensitive core tail.

Possible further work on the content-tail endpoint, integral heights in local descent classes, or exact arithmetic covering is provisional. None is an established route to completion. The density cases and the infinitude cases $r\ge5$ remain unresolved here.

Trace slices and bounded core statistics

Three additional lemmas are proved and author-recorded, without resolving a target case.

The binary-trace slice bound shows that, for $x^3+y^3+cz^3$, each prescribed reduced ratio $(u+v)/(x+y)$ permits only $O_{c,\varepsilon}(N^{2+\varepsilon})$ nontrivial collisions, uniformly in the ratio. Consequently all ratios of height at most $N^{1/2-\eta}$ can be removed at a cost of $o(N^3)$ input mass, also in the logarithmic moment. In particular the common-trace stratum has a power saving. The unrestricted trace-ratio tail is not controlled.

The canonical entropy identity shows that mixing the infinitely many canonical forms contributes only $O_r(1)$ conditional entropy under uniform representation sampling. Thus competition between cores is tight relative to the selected form's self-multiplicity. Integer-coefficient averaging has not supplied an estimate for that self-multiplicity; useful coefficient ranges growing with the scale have vanishing canonical input mass.

The Palm truncation inequalities show that the bounded comparison statistic changes by at most a constant times the first-moment tail. This would transfer a finite-family arithmetic comparison to all powerful coefficients. No such comparison was proved. Individual representation events at different target values are negatively correlated, while the positive association on prime exponents along a missing ray may put all its void mass at the original seed. Neither supplies the required new arithmetic input.

The attempted local quadratic and varying-trace constructions also did not close the density problem. Exact quadratic cancellation is available on a cubic trace slice, but interactions between different slices remain uncontrolled. For degree at least four, positive fourth directional derivatives prevent an exact affine quadratic reduction on a nonconstant positive patch. These are limitations of the tested constructions, not impossibility theorems for a positive-density solution.

The open density cases and the infinitude cases $r\ge5$ remain unresolved here.

A different proposed mechanism: deleting a sampled representation

The Palm/deletion strategy changes the question from collision moments to a bounded void-probability statistic. Compare a uniform target's fiber with the fiber of a uniformly sampled representation after that representation is deleted. A proved one-point comparison criterion would force a positive density of missing targets at any finite mean, including means above the quintic first-moment floor. Compound-Poisson clustering is compatible with it.

The arithmetic comparison is unproved; no deletion coupling has been constructed. The complementary lower comparison is equivalent to the positive-density problem, not an independent theorem. An explicit covering by convex power tracks shows why spacing and critical cardinality alone cannot establish the proposed comparison. The unresolved targets above are unchanged.

Checked additions from the secant and grouping approaches

The local secant-content formula identifies the good-prime valuation of $\gcd(F(\mathbf k),3\sum a_i x_i k_i^2)$ as $\min(v_p(n),2\min_{i<j}v_p(x_i k_j-x_j k_i))$, under primitivity of both endpoints. Two exact squarefree-output examples show that this content can be nontrivial and that the third secant point can even be integral. Output squarefreeness alone does not remove the uncontrolled secants; their alignment indices still need an averaged estimate.

The powerful-subsum theorem gives, uniformly for $|h|\le X$, an upper bound $O_{r,k,\varepsilon}(X^{k/r-\delta_r+\varepsilon})$ for ordered $k$-tuples of $r$-powerful integers with total at most $X$ whose total plus $h$ is $r$-powerful, where $\delta_r=(r-1)/(r^2(2r+1))$. Its proof uses the checked coefficient-uniform plane-curve bound, not the primitive ternary theorem. A local sieve also shows that grouping given summands into fewer blocks of any fixed powerful exponent at least two retains only $o(X)$ of the original $r$-tuple input mass. Neither result counts arbitrary targets, and the quantitative bound is still much larger than the population of near-powerful targets.

The polynomial missing-ray lemma shows that a linear-denominator polynomial identity on $t m^r$, with $t\notin S_r$, has a fixed nontrivial denominator content. Finitely many fixed univariate polynomial identities cover only $O(M^{1/2})$ prime denominators up to $M$. Representations outside such fixed families remain uncontrolled.

These additions do not change the unresolved targets above. In particular, no unconditional positive-density theorem or higher-exponent gap theorem has been established.

Comparison with the independent record

The independent input has been compared with the topical proofs. Its canonical-core equivalence, energy criterion, cubic quadratic-fiber reformulation, Hessian argument and norm obstruction agree with existing work; the existing canonical tail bound is stronger, attaining the exponent $1/[r(r+1)]$ without the independent argument's loss. The current higher-exponent progression theorems, especially the quintic certificate, also go beyond its large-good-prime limitation.

Checked additions are the homogeneity counterexample and polynomial-summand restriction, the prime-adic injectivity obstruction, and the sharper nonzero-target local bound. The quartic comparison verifies the paired-family obstruction and its global density bound, and explains how the current proof bypasses it. One restricted optimization failure is certified exactly. The independent record's statement that quartic infinitude was not proved is superseded by the current proof; none of its checked obstructions invalidates that proof. The unresolved targets above are unchanged.

Comparison with the second independent record

The later independent input does prove the same quartic complement theorem already established here. Its random omission completion has been checked and retained as an alternative to the finite bounded-budget construction. New useful inputs are the two-sided quadratic prime-harmonic bound and an explicit radical-size Gaussian modulus. The local trace, principal-character elimination, and lattice-density passage agree with the existing proof.

Its finite-mask warning applies to the earlier equal-budget construction, not the successful weighted finite distribution. The quartic comparison records that distinction and checks its mixed-principal, valuation-class, and simultaneous-splitting examples. Its fixed-form equivalence is already known here and its coefficient tail is weaker; its additional growing unrestricted tuple constant is correct but yields no distinct-value bound.

The record's named supplementary files were not supplied in this research folder. The checks use the arguments in the record and the relevant full source passages, not its reports of internal review. Its numerical Euler products remain noncertified diagnostics. Its cubic and higher-exponent conclusions are compatible with the more detailed established collision, profinite, and first-moment notes; none supplies the missing integer-fiber estimate or an additional target case. The unresolved targets are unchanged.

Comparison with the third independent record

The new independent input includes its auxiliary working proofs, unlike the second record. Its conservative statement that the assembled quartic argument had not been finally certified is historical, not a disagreement with the established theorem. The analytic assembly is checked and retained as the universal-selector proof.

One infinite negatively biased set of primes controls every positive squarefree determinant and every cutoff interval simultaneously. The maximal Hoeffding estimate and its countable union bound are valid; the two-sided quadratic estimate bounds the infinite character mixture. Removing all coefficient primes, including those absent from the squarefree determinant, costs only an exponential in the radical's number of prime factors. A growing deterministic support cutoff then gives a single summable majorant for both regular and exceptional tuples. The resulting moduli and residues can be compatible and nested, but the conclusion is still only the existing quartic complement theorem.

The local check yielded an additional refinement. Prescriptions are needed only at primes occurring to odd valuation in some coefficient, even for the mixed principal-character cases. The weighted exceptional tail with such a prime exceeding $Y$ is $O(Y^{-3/2})$. The finite-family note proves this for the whole exceptional union; the independent record gave it only for square products. That note also records the harmless $81$ versus $81e$ logarithmic-exponent discrepancy and the precise failure of iterating an unconditioned negative one-prime mean.

The Euler note retains the valid interpolation counting principal constituents. Its regular case is already established, with a slightly sharper defining ideal than the independent $(32R)$. The quadratic harmonic note includes the independent elementary Landau nonvanishing proof, so it needs no fixed-character prime-distribution input. The supplementary lower-Hecke-$L$ route is unnecessary: neither the baseline nor either random completion requires lower bounds for Gaussian Hecke $L$-functions.

The finite-core equivalence, tuple-count formula, cubic energy criterion and Perazzo mismatch agree with existing, more detailed arguments. The truncated numerical Euler products are not adopted as certificates. The independent higher-exponent limitation is weaker than the established uniform progression floor and quintic obstruction. No density question or infinitude case $r\ge5$ is resolved.

Positive density and energy

For fixed positive coefficients $\mathbf a=(a_1,\ldots,a_r)$, let

$$V_{\mathbf a}(N)=\left\{\sum_i a_i x_i^r:1\le x_i\le N\right\}, \quad E_{\mathbf a}(N)=\#\left\{\mathbf x,\mathbf y\in[1,N]^r: \sum_i a_i x_i^r=\sum_i a_i y_i^r\right\}.$$

Cauchy–Schwarz gives $|V_{\mathbf a}(N)|\ge N^{2r}/E_{\mathbf a}(N)$. Thus a uniform estimate $E_{\mathbf a}(N)\ll N^r$ for even one fixed positive coefficient tuple would give positive lower density of its value set and, after a fixed dilation, of $S_r$. A bound along an unbounded sequence would already give positive upper density.

The finite-core equivalence shows that letting coefficients grow with $N$ does not bypass the fixed-form issue: outside any sufficiently large fixed canonical core, the total candidate tuples are at most an arbitrarily small constant times the target range.

Wang's Theorem 1.3 gives the needed bound for three cubes only under Conjectures 1.2, 1.4, and 1.5. The theorem statement and hypotheses have been checked in the full supplied paper. This is a conditional negative answer to density zero, not an unconditional resolution. The signed almost-all results elsewhere in that paper do not supply positive summands.

These notes isolate a weaker target: bounded multiplicity on a positive fraction of input tuples is equivalent to positive upper density, and bounded average logarithmic multiplicity is sufficient. The complete statements and proofs also show convergence of the good-prime energy product for every $r\ge3$. For cubes, the associated real quadric-fiber volumes sum to the expected $O(N^3)$, but a pointwise lattice bound at that volume scale is false because of permutation families. No averaged bound removing this obstruction was proved.

The subsequent real-coefficient theorem achieves the log-free near-collision bound after averaging over a real coefficient box. It proves positive upper density of rounded values for almost every real vector. All rational vectors may lie in its exceptional set, so this is not a counterexample to the density-zero question. Atomic coefficient measures cannot satisfy the proof's general inverse-width slab bound, and any successful countable weighted energy estimate would already imply the requisite estimate for each supported form.

The special cubic $x^3+2y^3+4z^3$ eliminates rational zeros at infinity and coordinate-permutation identities, but does not eliminate split positive fibers: an explicit two-parameter identity demonstrates this. Its ruling directions through a fixed nonzero-coordinate diagonal point lie on a genus-two curve. Finiteness at each fixed point is established; a uniform estimate over varying points is not.

Fixed linear changes of variables do not remove the coefficients

Suppose $r\ge3$, $F_{\mathbf a}=\sum_i a_i x_i^r$, $F_{\mathbf b}=\sum_i b_i x_i^r$, and

$$F_{\mathbf b}(T\mathbf x)=\lambda F_{\mathbf a}(\mathbf x),\qquad T\in\operatorname{GL}_r(\mathbb Q),\quad\lambda\ne0.$$

Then $T$ is monomial. Indeed, the Hessian determinant of a nondegenerate diagonal form is a nonzero constant times $(x_1\cdots x_r)^{r-2}$. The Hessian transformation rule forces the product of the row linear forms of $T$ to be a constant times $x_1\cdots x_r$. Unique factorization makes each row a scalar coordinate. Consequently an invertible rational linear transfer to equal coefficients requires all the $a_i$ to lie in one rational $r$-th-power class. This is a proved limitation of that specific transfer mechanism, not a theorem excluding all positive-density constructions.

Higher exponents need more than the present first-moment method

For $r\ge5$, the uniform lower-bound theorem gives $c_r>0$ such that every fixed progression has normalized ordered representation mean at least $c_r$. The good-prime error is summable even at zero targets and at arbitrary prime powers. This rules out arbitrarily small first moments, the precise conclusion used in the cubic and quartic arguments.

For $r=5$ the quintic note proves more: every unordered progression first moment exceeds $385732347/156250000>2.468$, strengthening the earlier $429/400$ bound. The finite coefficient family, exact small-prime convolutions, rational product comparisons, and analytic tail estimates are preserved with the proof. Consequently neither complete ordered nor complete unordered representation counting can directly produce a deficit below one on a fixed progression. This does not exclude a method that controls or removes repeated representations. No comparable below-one obstruction is asserted here for every $r\ge6$.

No monotonicity of the required sumsets transfers the quartic complement theorem to $r\ge5$: although $r$-powerful integers are 4-powerful, the allowed number of summands increases with $r$.

There is a stronger obstruction to transferring the quartic proof by identities. The quadratic-compression proposition excludes any rational-function representation of a positive diagonal degree-$r$ form in $r\ge5$ variables by a fixed positive quadratic form in at most four variables, even with arbitrary denominators. For $r\equiv2\pmod4$, it excludes every smaller quadratic dimension. This is a limitation of the compression mechanism, not an infinitude theorem.

The quintic first-moment obstruction is not confined to fixed moduli. The uniform lattice inequalities show it persists for $M=o(X^{1/5})$, and on intervals $(A,B]$ of length $H\gg M B^{4/5}$ with a sufficiently large fixed constant. For $H/(MB^{4/5})\to\infty$, the normalized unordered mean has liminf at least $429/400$. Distinct values might still leave gaps in these intervals.

A qualitative simultaneous-approximation proposal also fails: for every finite collection of fifth-power cores, the root fractional parts $(n/d)^{1/5}$ are equidistributed along a suitably chosen represented ray $n=(b^5+4)m^5$. Thus every nonempty open pattern of finitely many axis intercepts occurs among represented targets. This does not exclude shrinking-target conditions or arguments using the full lattice shell.

A scaling obstruction cannot exclude an entire positive rational homothetic ray either. Every $q\in\mathbb Q_{>0}$ is $d x^r$ with $d$ a canonical core and $x\in\mathbb Q_{>0}$: reduce every prime valuation modulo $r$ using the canonical exponents. Clearing denominators turns any fixed positive rational decomposition into a powerful-number decomposition after a common $r$-th-power dilation. In particular, every positive integer $n$ has the $r$-powerful multiple $n^{r+1}$. This rules out nonrepresentability over positive rationals as that particular obstruction; it does not rule out missing integers along a ray.

Bounded tests retained only as rejected ideas

An initial quartic optimization restricted to character profiles determined by coefficients modulo $16$ did not produce a deficit. With canonical Euler factors truncated at primes $10^6$, the ordered-to-unordered normalization gave a minimum near $87.14$ for the class $8\pmod{16}$, and near $59.84$ after also imposing $v_3(n)=1$. These are exploratory approximations, not certified constants; the restricted profile family was abandoned. The successful proof uses a different distribution and does not depend on these computations.

The tempting sequence $8\prod_{p\le Y,\ p\equiv3\pmod4}p$ is not generally excluded. For example,

$$8\cdot3\cdot7\cdot11\cdot19 =35112=625+1296+16384+16807,$$

and every summand is 4-powerful. This exact certificate defeats that proposed direct construction.

Suggested next work remains provisional: a special fixed diagonal form with controlled collision count, or a genuinely nonlocal argument for gaps when $r\ge5$. Neither has yet been supplied.

Exact status of the controlled cubic ranges

The controlled cubic ranges are bounded at $O(N^3)$: coordinate-zero differences, differences of size at most $N^{2/3-\eta}$, and primitive difference directions of size at most $N^{1/2-\eta}$. The last two bounds have power savings away from coordinate-zero differences. The proof uses a coefficient-uniform elementary ternary quadratic bound. For $x^3+2y^3+4z^3$, what remains in this energy approach consists entirely of nonsingular fibers with large primitive direction. The range $\|\mathbf k\|\asymp N$, $t=1$ is still uncontrolled. Neither the inspected singular-cubic counting theorem nor an unjustified average over binary ideal classes removes this gap.

The linear identity rank theorem gives a precise limitation of redundancy constructions. A positive nontrivial identity with $r$ degree-$r$ terms on each side has parameter rank at most $r-1$. If a nonproportional relation survives, the rank is at most $\lfloor r/2\rfloor+1$. Thus a fixed finite collection removes only $o(X)$ representation pairs. Even when identity heights grow, linear mass requires at least order $X^{1/r}$ families; for genuinely nonproportional quintic families the necessary threshold is $X^{2/5}$. No collection achieving the required mass was found.

Scaling along a ray: an exact identity, not a descent

For fixed $F_{\mathbf d}=\sum_{i=1}^r d_i x_i^r$ and a positive integer $t$, let

$$A_{\mathbf d,t}(h)=\#\{\mathbf v\in\mathbb N^r: F_{\mathbf d}(\mathbf v)=th^r,\ \gcd(h,v_1,\ldots,v_r)=1\}.$$

If $R_{\mathbf d}(n)$ counts positive integral representations by this form, then exactly

$$R_{\mathbf d}(tm^r)=\sum_{h\mid m}A_{\mathbf d,t}(h). \tag{1}$$

Given a representation, take $g=\gcd(m,x_1,\ldots,x_r)$, $h=m/g$, and $\mathbf v=\mathbf x/g$. Conversely scale a primitive tuple by $m/h$. These maps are inverse.

If $t\notin S_r$, then $A_{\mathbf d,t}(1)=0$ for every canonical core tuple. At a prime $p$, (1) therefore leaves precisely the primitive denominator-$p$ points on

$$\sum_i d_i v_i^r=t p^r.$$

It does not force a representation to descend to $t$. For exactly $r$ summands this is a smooth degree-$r$ hypersurface in $\mathbb P^r$, of index one; no estimate sparse enough on prime denominators was established. There are two additional gaps in a proposed prime-ray argument: the global core tail $O(XT_r(Y))$ cannot control a population of only order $X^{1/r}/\log X$, and fewer-than-$r$ summands cannot be discarded on that ray merely because they have global density zero. The analogous denominator identity holds for those shorter tuples too, but no sufficient count was proved.

Any next step is provisional. The critical large-direction cubic average and nonparametric control of distinct higher-degree values remain possible targets; none of the foregoing results settles them.

Divisibility-minimal multipliers on a missing ray

The following elementary structure statement is proved. It does not produce an unrepresented target.

Fix $t\notin S_r$ and put

$$M_t=\{m\ge1:tm^r\in S_r\},\qquad B_t=\{b\in M_t:\text{no proper divisor of }b\text{ belongs to }M_t\}.$$

The set $M_t$ is upward-closed under divisibility: multiply every summand in a representation by $q^r$ to pass from $m$ to $mq$. Also $1\notin M_t$.

Lemma. If $M_t$ contains every sufficiently large prime, then $B_t$ consists of those sufficiently large primes and finitely many additional elements. In particular this holds if $M_t$ is cofinite.

Proof. Every prime in $M_t$ is a minimal element, since its only proper divisor is $1$. Choose $P$ such that every prime greater than $P$ belongs to $M_t$. A composite minimal element cannot have a prime divisor greater than $P$, since that prime would be a proper divisor lying in $M_t$. The composite minimal elements are therefore supported on finitely many primes. Their exponent vectors form an antichain under coordinatewise comparison, so there are finitely many by Dickson's lemma. One elementary proof of the needed fact is that every infinite sequence in $\mathbb N^k$ has an infinite coordinatewise nondecreasing subsequence: successively extract such a subsequence in each coordinate, using either an infinitely repeated value or an increasing subsequence. Two distinct elements of an antichain cannot belong to such a sequence. $\square$

Thus either infinitely many prime nonmembers of $M_t$ or infinitely many composite members of $B_t$ would prove infinitely many missing targets on the ray. Neither has been established. A bound $\sum_{b\in B_t}b^{-1}<\infty$ would suffice, because it precludes every sufficiently large prime being a generator, but no such bound follows from the available canonical-core estimates.

Nor would proving density one for $M_t$ settle its cofiniteness. This is an actual limitation of the divisibility-ideal argument: choose an infinite set $\mathcal Q$ of primes with convergent reciprocal sum, and let $M$ be the set of integers divisible by some prime outside $\mathcal Q$. Then $M$ is upward-closed and misses every prime in $\mathcal Q$. Nevertheless it has density one. For any finite set $E$ of primes outside $\mathcal Q$, the upper density of its complement is at most $\prod_{p\in E}(1-1/p)$ by CRT; these products tend to zero because the reciprocal sum over primes outside $\mathcal Q$ diverges. This example concerns abstract divisibility ideals, not a construction of the actual set $M_t$.

Cube-free estimates and near-neighbor implications

The cube-free estimates retain positive input mass but do not bound distinct-value multiplicity. Two additional observations are checked in the energy note: unrestricted multiplicities for $x^3+2y^3+4z^3$ satisfy $r_F(2n)=r_F(n)$ exactly, and a vanishing proportion of inputs with a neighbor on the same level surface at distance at most $\varepsilon N$ would imply positive lower density. The latter is a conditional implication; the needed near-neighbor estimate is unproved. The existing short-difference estimates do not reach the required fixed-relative-distance scale.

Direct completion attempt: still unresolved

A further attempt to close the two main gaps did not produce a solution. For cubic near neighbors, even primitive differences of size $\varepsilon N$ leave the uncontrolled critical quadratic-fiber average; choosing only a nearest neighbor supplies no proved arithmetic descent. The exact carry formulation likewise leaves the bounded-multiplicity question intact. Maynard's 2026 survey, whose introduction was checked, also identifies typical multiplicity as the unresolved issue for three positive cubes and states Wang's positive-density result only conditionally (Section 1). This does not establish that every possible special-coefficient approach must fail.

For the higher-exponent prime-ray route, one additional reduction is exact. Let $r<t<2^r$ and let $p>t$ be prime. If $tp^r$ has a representation by at most $r$ positive $r$-powerful summands, every summand divisible by $p$ is exactly $p^r$. Indeed its $p$-adic valuation is exactly $r$, since $p^{r+1}>tp^r$, and its quotient by $p^r$ is $r$-powerful and less than $2^r$, hence is $1$. Remove the $k$ such terms. The residual relation is

$$a_1+\cdots+a_s=(t-k)p^r,\qquad 2\le s\le r-k,\quad p\nmid a_i,\quad\gcd(a_1,\ldots,a_s)=1.$$

There is at least one residual term because $t>r$, and there cannot be just one because it would be divisible by $p$. Their gcd is $r$-powerful, coprime to $p$, and divides $t-k<2^r$, so it is $1$.

This includes the elementary missing seed $t=r+1$, but the dominant case $k=0,s=r$ remains. Its fixed-core projective equation $\sum_i d_i v_i^r=tp^r$ has degree $r$ in $\mathbb P^r$, hence index one, not general type. No estimate at the necessary $P/\log P$ scale for prime denominators, and no suitable coefficient-tail estimate on that ray, was obtained. These observations are not an infinitude or density theorem.

Nor does the size restriction exclude every prime $p>r+1$. Already

$$6\cdot7^5=2^5+2^{12}+3^8+2^7 3^5+3^{10}=100842$$

is a representation by five $5$-powerful integers prime to $7$. This exact identity refutes that stronger proposed prime-ray exclusion; it says nothing about infinitely many other prime denominators.

Open questions from the moment, carry, and descent approaches

No unresolved part of the target problem is settled by the following checked results and failures:

• Primitive cubic fibers: removing common dilations costs $O(X)$ in the logarithmic moment; a uniform bound for primitive logarithmic multiplicity is equivalent to one for all representations. Low-height projected collision planes contribute $o(N^3)$ to that moment. Every curve over $\mathbb Q$ on $x^3+2y^3+4z^3=nw^3$ has degree divisible by three, but varying higher-degree curves and primitive large fibers remain uncontrolled.
• Dyadic carries: exact local uniformity and an exact lift recurrence hold, but a proposed uniform decay of primitive high-conductor carry characters is false. The explicit obstruction is the persistent rational resonance $1/9$. A spectral argument retaining these resonances remains an idea, not an established estimate.
• The Perazzo transfer: adding the required cubic term changes a quadratic-character Fourier transform into the explicit Salié-sum expression (14). The needed cancellation in that sum was not proved; the singular-cubic theorem cannot simply be substituted.
• Prime-ray descent: for every fixed core and all sufficiently large good primes there are nonsingular zero residues with no proper zero subsum, and these lift and combine by CRT. Thus denominator factorization alone cannot force a shorter relation. The Archimedean size restriction still has not been exploited sufficiently.
• Adaptive coefficient averaging: a bounded logarithmic moment plus positive mass on a fixed finite arithmetic coefficient set would extract a successful fixed form. Neither the required arithmetic average nor its discrete tightness was established.

Possible next steps are therefore provisional: control primitive large-fiber input mass, obtain a resonance-aware carry estimate, or find an arithmetic gap argument using size as well as congruences. The present notes contain no theorem furnishing any of those missing estimates.

Two cubic estimates, no resolution

The coordinate-zero refinement now gives $O(N^{1+\sqrt3}(1+\log N)^4)=o(N^3)$ off-diagonal coordinate-sharing collisions for $x^3+2y^3+4z^3$. Its proof uses the checked uniform diagonal surface estimate of Salberger. Deleting all participating inputs changes the logarithmic moment by only $o(N^3)$. More generally, coordinate-sharing near neighbors contribute $O(\varepsilon N^3)+o(N^3)$ for every positive ternary diagonal cubic, including the weighted-swap cases. Thus the near-neighbor criterion now leaves only all-nonzero coordinate differences.

The major-arc estimate proves a sharp $Q^{-2/3}$ tail for rational resonance weights and an asymptotic $N^3\mathfrak J\mathfrak S+O(N^{11/4})$ on major arcs of denominator at most $N^{3/8}$. Their contribution to the input mass over fibers of size greater than $K$ is $O(N^3/\sqrt K)$. The complementary Fourier correlation is not bounded; this is not a full high-fiber tail estimate.

The higher-exponent search found no main-term duplication or size-sensitive gap argument. In particular, no infinitude case $r\ge5$ and no density question has been resolved here.

Large-prime input trial

The last distinct approach restricted each cubic root to $\mathcal A_\theta(N)=\{x\in(\delta N,N]:P^+(x)>N^\theta\}$, for fixed $0<\delta<1$ and $1/2<\theta<1$. The complete argument proves

$$|\mathcal A_\theta(N)| =\bigl((1-\delta)\log(1/\theta)+o(1)\bigr)N.$$

This is a positive-mass subset of inputs to a fixed integral form, not an extraction of rational coefficients from a real metric theorem. Writing $x=mp$ gives bilinear cubic sums, but neither their critical sixth moment nor a sufficient logarithmic moment was bounded.

The negative conclusions are specific. A large prime dividing an input root need not divide a competing root; all-unit nonsingular local solutions remain available. Cauchy in the cofactor variable followed by termwise absolute values leaves a factor equal to its range in the pointwise bound. Restricting cofactors to a subpower range loses positive input density. These statements defeat that divisibility argument and that Cauchy estimate, not every joint bilinear argument.

Two checks address proposed rough-target filters. For every prescribed roughness threshold and number of target prime factors, the note constructs represented targets with all root primes in one fixed finite set. Separately, for fixed $r\ge5$, $k\ge1$, and $0<\eta<1/2$, a checked application of the lower linear sieve gives

$$\#\left\{\mathbf z\in(Z,2Z]^r: P^-\!\left(\sum_i z_i^{rk}\right)\ge Z^\eta\right\} \gg_{r,k,\eta} Z^r/\log Z.$$

Their canonical roots $z_i^k$ are comparable and have no prime factor above $2Z$. This tests a stated range of power-scale roughness as well. It counts input tuples, not distinct outputs; for $k>1$ the input family is sparse at the root scale. Neither construction excludes a statistical roughness argument.

An Eisenstein height-counting attempt

A separate attempt used a rational parametrization rather than local Fourier estimates. Here is the exact arithmetic issue it left open. Put $K=\mathbb Q(\sqrt{-3})$, $\delta=\sqrt{-3}$, and, for fixed positive integers $a_1,a_2,a_3$, define

$$A(\mathbf w)=\sum_{i=1}^3 a_iw_iN_{K/\mathbb Q}(w_i).$$

For $z_i=(x_i-y_i)+\delta(x_i+y_i)$, $\operatorname{Re}A(\mathbf z)=4\sum_i a_i(x_i^3-y_i^3)$. Conversely, when $A(\mathbf w)\ne0$, the vector $\mathbf z=\delta\overline{A(\mathbf w)}\mathbf w$ has $A(\mathbf z)=3\delta N(A(\mathbf w))^2$, so $\operatorname{Re}A(\mathbf z)=0$. Scaling $\mathbf w$ by $\alpha\in K^\times$ scales $\mathbf z$ by the positive rational number $N(\alpha)^2$.

Take $\mathbf w\in\mathcal O_K^3$ primitive as an ideal vector, let $H=\max_i|w_i|$, and let $c$ be the integer content of $A(\mathbf w)$ in an integral basis of $\mathcal O_K$. Since this ring is a PID, the common ideal divisor of the coordinates of $\mathbf z$ is exactly $(\delta\overline A)$. Multiplication by $\delta$ has determinant three on the integral basis, so the integer content of $\delta\overline A$ is between $c$ and $3c$. The rational inverse linear map from $\mathbf z$ to $(\mathbf x,\mathbf y)$ has fixed denominators. Consequently the resulting primitive rational point has height

$$B\asymp H|A(\mathbf w)|/c.$$

Positivity removes the small-$A$ complication: it forces each $w_i$ to lie within angle $\pi/6$ of $A(\mathbf w)$. Hence $|A(\mathbf w)|\ge\cos(\pi/6)\sum_i a_i|w_i|^3\asymp H^3$, and $B\asymp H^4/c$ on the relevant cone.

A bound of the proposed shape $O(H^6/G)$ for primitive parameters with $c\ge G$ would address the collision count. It was not proved in the necessary endpoint range $G\asymp H^3$. There, $q=A/c\in\mathcal O_K$ ranges over a fixed finite set; fixing $q$ imposes

$$\operatorname{Im}(\overline q\,A(\mathbf w))=0.$$

The additional sign is $\operatorname{Re}(\overline q A)>0$. The fixed rational rotation $\mathbf w\mapsto\delta\overline q\mathbf w$ turns this back into the critical positive cubic collision equation. Small-modulus divisibility counts do not bound this large-content range. Thus the parametrization and its checked height formula do not supply the missing energy or multiplicity estimate.

The Cartesian-extraction theorem and the entropic sieve

The Cartesian-extraction theorem applies to every fixed degree-$r$ diagonal form in $r\ge3$ variables. An output-defined restriction keeps $1-o(1)$ of all input tuples, but every Cartesian rectangle inside it has volume at most $N^r\exp(-c_r\log N/\log\log N)$. The cubic specialization has vanishing relative entropy from the product law while removing an exponentially large product-sieve expectation. This checks a concrete attempt to use the new weighted entropic sieve of Croot, Mao, Pohoata, Sheffer, and Yip: its independent-coordinate and pointwise bounded-fiber hypotheses cannot simply be replaced by high input mass or small relative entropy. The theorem does not prove that the actual bounded-fiber input set is small, and it gives no density conclusion.

The subsequent joint congruence calculation proves a second-moment bound uniform in every modulus, bounded profinite entropy, and quantitative convergence of the local densities for $x^3+2y^3+4z^3$. It also gives exact descent by 27 and the complete $3$-adic distribution. These do not control integer-box fibers: the required comparison between complete residue boxes and roots of size $N$ at output scale $N^3$ remains unproved. No additional target case was settled here.

The requested proof is incomplete. The established infinitude cases remain $r=3,4$; the density question for every $r\ge3$ and infinitude for $r\ge5$ are unresolved in this work. The new collision, resonance, and sieve arguments have been documented with their hypotheses and scope, and none is asserted to supply the missing global estimate.

Possible further work is provisional: all-nonzero near-neighbor existence, minor-arc correlation with heavy fibers, a joint estimate for the bilinear sums before the lossy Cauchy step, or a different size-sensitive higher-exponent argument. No such argument is claimed here.

Further checks of the structural and Fourier routes

No target case was resolved here. The following additional statements are proved and have been checked.

The extended quintic certificate raises the uniform unordered progression mean above $2.468$ by allowing canonical five-parts in one coefficient. A checked third-moment gap inequality is useful only below mean two under Poisson moments; above that threshold a positive-supported law matches all three moments. This does not estimate the actual collision moments or prove gaps.

The congruence-preserving sampling theorem constructs synthetic integer multiplicities with the actual cubic marginals modulo $N^2$ and every fixed modulus, the same joint profinite limit, and all the established pointwise and integrated major-arc estimates. Their support is only $O(N^2\log N)$ although their mass is $N^3$, and their logarithmic multiplicity is at least $\log N-\log\log N-O(1)$. The missing minor-arc correlation can carry essentially the entire mass. These are not polynomial representation functions: their Fourier transform need not factor into three cubic sums. The theorem pinpoints the need for additional factorization or integer-fiber information, rather than settling the density question.

The nearest-neighbor Pell family gives reciprocal nearest positive cubic neighbors whose third secant point has height more than six times the endpoint height asymptotically. It excludes a universal secant descent but not an averaged argument. Also, the constant-field norm obstruction rules out an identity expressing the diagonal form as a norm from a fixed nontrivial number field, even by rational functions. Reduced norms and varying fields are outside that statement.

A cofinite abstract model with the available dilation properties

The proposed dilation-only proof of higher-exponent gaps has a precise obstruction. Fix an $r$-th-power-free integer $t>1$, and define

$$R(n)=\sum_{d^r\mid n}\mathbf1_{n/d^r\ne t} =\tau_r(n)-\mathbf1_{\{n=tm^r\}},\qquad \tau_r(n)=\prod_p\left(\left\lfloor v_p(n)/r\right\rfloor+1\right).$$

This abstract nonnegative count vanishes only at $t$. Indeed, $\tau_r(tm^r)=\tau(m)$, and hence

$$R(tm^r)=\tau(m)-1=\sum_{h\mid m}A(h), \qquad A(1)=0,\quad A(h)=1\ (h>1). \tag{2}$$

Its mean has the strong estimate

$$\sum_{n\le X}R(n) =\sum_{d\le X^{1/r}}\lfloor X/d^r\rfloor -\lfloor(X/t)^{1/r}\rfloor =\zeta(r)X+O_r(X^{1/r}). \tag{3}$$

Here the tail $X\sum_{d>X^{1/r}}d^{-r}$ and the floor errors are both $O_r(X^{1/r})$. For every nontrivial $r$-powerful $k$, at least one prime exponent increases by $r$ or more, so $\tau_r(nk)\ge\tau_r(n)+1$. This offsets any increase of the subtracted indicator, proving $R(nk)\ge R(n)$.

Thus an initial missing seed, bounded mean, closure under every powerful multiplier, and the exact primitive-denominator convolution are compatible with a cofinite support. In this model the minimal represented multipliers on the missing ray are precisely all primes. It is not the actual representation function of powerful sums, and asserts nothing about that function's support. A higher-exponent proof needs arithmetic information not contained in these structural properties.

A controlled secant stratum, no target resolution

The secant-slice estimate is uniform and has no logarithmic or epsilon loss. For a primitive direction $k$ of height $K$, a fixed step $t$, and one prescribed value of $B/C$, there are $O(1+N/K)$ collision endpoints. The proof intersects an affine plane of lattice covolume comparable to $K^2$ with a nonsingular conic and counts parallel lattice lines.

Its bounded-denominator consequence bounds short collisions by $O(D^2N^2H/\eta)$ when the reduced denominator of $B/(tC)$ is at most $D$ and $|F(k)|\ge\eta K^3$. This yields bounded energy and a vanishing near-neighbor existence limit on each fixed $D,\eta$ stratum. There is no estimate for the complementary input mass as $D\to\infty$ and $\eta\downarrow0$. Thus it does not close the positive-density criterion.

The binary-norm calculation checks a separate failed simplification. For $Q=x^2-xy+y^2$, the number of split prime factors is asymptotic to $\log\log N$ in probability on a root box, forcing unbounded logarithmic norm multiplicity on every positive-mass input restriction. This excludes discarding the binary trace condition and then selecting a bounded norm-divisor subfamily; it does not estimate the actual cubic fibers.

What cofiniteness would force on a missing ray

For this statement use all canonical cores and every number of summands $1\le j\le r$, not just a fixed form. Fix $t\notin S_r$, and let $R_t(m)$ be the number of unordered powerful-sum representations of $tm^r$. In each summand use its unique canonical expression $d_i x_i^r$. Let $A_t(h)$ count such representations of $th^r$ satisfying $\gcd(h,x_1,\ldots,x_j)=1$. Every count is finite. Extracting $g=\gcd(m,x_1,\ldots,x_j)$ gives the exact identity

$$R_t(m)=\sum_{h\mid m}A_t(h),\qquad A_t(1)=0. \tag{4}$$

Canonical uniqueness makes this operation well-defined also for unordered representations.

If $S_r$ were cofinite, then $A_t(p)=R_t(p)\ge1$ for every sufficiently large prime $p$. Hence, for some fixed $P_0$,

$$R_t(m)\ge\omega_{>P_0}(m),\qquad \sum_{m\le M}R_t(m)\ge M\log\log M-O_t(M). \tag{5}$$

The mean bound follows by summing $\lfloor M/p\rfloor$ over $P_0<p\le M$ and using the prime harmonic sum. Also $\omega_{>P_0}(m)$ tends to infinity in density: for any fixed finite prime set its divisibility indicators become independent in the limit by CRT, with variance at most their mean; the means over increasing finite prime sets diverge, so Chebyshev applies. Thus cofiniteness would force divergent typical multiplicity on this ray.

This gives no contradiction to the available estimates. In particular, if one hypothetically had $\sum_{h\le H}A_t(h)\sim cH$ with $c>0$, then partial summation in (4) would give

$$\sum_{m\le M}R_t(m) =M\sum_{h\le M}\frac{A_t(h)}h +O\left(\sum_{h\le M}A_t(h)\right) =cM\log M+o(M\log M), \tag{6}$$

which is compatible with (5). No asymptotic for these actual primitive counts is asserted. An $o(M\log\log M)$ upper bound would disprove cofiniteness, but it is not proved and is not the consequence of a linear primitive-point asymptotic. No additional infinitude case was obtained from this route.

The unresolved targets remain unchanged: no density conclusion for any $r\ge3$, and no infinitude conclusion for $r\ge5$, was established here. Further use of the secant slices or of the exact trace--norm condition is provisional; neither supplies the missing uniform tail estimate.

Direct completion attempt: the small-prime ray fails

The missing seed $t=r+1$ does not generate missing targets by repeatedly scaling by a small prime. In fact, for every prime $q\le r+1$,

$$(r+1)q^{rk}\in S_r\qquad(k\ge1).$$

Write $r+1=\sum_j c_jq^j$ in base $q$. Since $r+1\ge q$, its digit sum is strictly less than $r+1$, hence at most $r$. Thus

$$(r+1)q^r=\sum_j c_jq^{r+j}$$

is a sum of at most $r$ positive $r$-powerful integers. Multiplication by $q^{r(k-1)}$ proves the assertion. This excludes that specific fixed-ray construction, not missing targets on other rays.

For larger denominator primes, the unresolved issue is still the exact bounded-root equation, not complete congruence counts. The attempted positive-density construction likewise still requires the cross-coordinate cubic correlation or a sufficient weaker multiplicity estimate. Neither was established. No new target case was resolved, and there is no complete proof.

Checked norm-ratio, transformation, and analytic statements

These results add no target resolution. For $F=x^3+y^3+cz^3$, the binary norm-ratio estimate bounds nontrivial collisions of reduced norm-ratio height at most $D$ by $O_{c,\varepsilon}(DN^{2+\varepsilon})$. Thus heights at most $N^{1-\eta}$ can be removed with negligible input mass and logarithmic moment. There is no tail bound up to the possible height $N^2$. The exact entropy identity separates the already controlled binary multiplicity from the uncontrolled quantity $H(Z\mid X^3+Y^3+cZ^3)$ for independent uniform inputs $X,Y,Z\in\{1,\ldots,N\}$. Neither that conditional entropy nor the corresponding shifted four-cube correlation is estimated here.

The balanced-cone observation locates nearly vanishing directions near coordinate-swap directions, but the available thin-sector count is too large at fixed relative distance. It does not extend the exact coordinate-sharing estimate. Also, choosing different positive cubic coefficients cannot give the generic collision quadric a rational section: specialization to positive differences would make it positive definite. Passing to $\mathbb Q(\sqrt{-3})$ produces a point but returns to the existing integral-height obstruction.

Two rational-transformation results are also proved. A fixed rational self-map preserving a positive diagonal form acts on only $o(N^r)$ positive integral inputs unless it is a monomial symmetry. The nonpolynomial case has the explicit bound $O_{T,\varepsilon}(N^{r-1+\varepsilon})$. Separately, rational linear similitude multiplier classes form a finite group of order dividing $r$. Their integer representatives have summable reciprocals, and such multiplicative saturation preserves density zero. These results exclude fixed-map amplification and a nonsummable linear dilation family, not families of transformations of growing complexity.

The proposed extensions of the quintic first-moment certificate to additional primitive or subsum restrictions were not completed and are not adopted as results. The established quintic floor and its stated scope remain unchanged.

The following checked statements also settle no target case. The composition-law argument excludes rational composition laws linear in one input, not arbitrary nonlinear constructions. The fifth-moment inequality would force gaps under five Poisson factorial-moment asymptotics of mean at most $3.1$; neither the required upper bound on the actual progression mean nor those asymptotics is proved. The special target-class estimate removes the targets $2^e m^3$, $e=0,1,2$, for the special cubic at negligible input and logarithmic-moment cost. No uniform counting consequence for the remaining surfaces, including from exploratory Picard-rank considerations, is adopted here.

The new analytic attempts also leave precise gaps. The square-class reformulation has elementary mass and moment bounds, but its grouped coefficient at square class $3$ is exactly the unknown majorant $T_N$. Ignoring this coefficient in a quadratic large-sieve application is invalid; an explicit abstract weight example shows that the two proposed moment bounds alone would not suffice. The truncated restriction theorem controls only high Fourier amplitudes, and its combination with the available lower moments retains a power of $N$. The packing calculation likewise controls only fibers of polynomially growing size, not fixed large size. None of these calculations excludes a sharper estimate using the additional arithmetic structure.

For the higher-exponent rays, the exact identity $6\cdot7^5=2^5+2^{12}+3^8+2^7 3^5+3^{10}$ also excludes a proposed uniform obstruction for all primes larger than the seed. Attempts with moduli exceeding the root scale did not establish either a below-one full unordered representation mean or a tail estimate uniform on the selected progression. No global basis obstruction or large/composite-exponent positive-density construction was proved.

Further work, if undertaken, would need a new estimate for primitive integer fibers or their unrestricted ratio tails, or a genuinely size-sensitive higher-exponent argument with the coefficient tail included. These are provisional ideas, not claims that the present reductions can be completed. The density question for every $r\ge3$ and infinitude for $r\ge5$ remain unresolved here. The proof is incomplete.

New multiplicative and analytic trials: checked obstructions

The matrix-composition attempt embeds $a x_1^3+b x_2^3+a^2b x_3^3$ as the determinant of an explicit three-dimensional matrix family. This includes the special cubic. It avoids the fixed number-field norm obstruction, but ordinary products can be conjugated back only on two proportional-coordinate loci. Their product values up to $T$ number $O(T^{2/3}(\log(2T))^2)$, even with arbitrary input-dependent conjugating matrices.

A modified product does remove both trace obstructions. Its determinant is an explicit sextic $H$, not the product of the original determinants. Even scalars chosen separately at every positive integer input cannot repair this on a substantial product set: coefficient-uniform Bombieri–Pila bounds give $O_\varepsilon(T^{17/18+\varepsilon})$ admissible input pairs with product at most $T$. The proof includes all unbalanced root heights. The matrix identities were checked by an exact symbolic certificate. These statements exclude this composition mechanism, not arbitrary nonlinear constructions or positive density of the original value set.

The generating-function calculation tests cofiniteness through positivity after subtracting an integer-tail pole. All fixed finite tests at rational or irrational limiting angles, including $O(t)$ angular offsets, reduce precisely to the progression first moments. The quintic floor makes their limiting matrices strictly positive, so even lower-order terms cannot contradict cofiniteness in such a fixed test. Increasing-complexity tests remain outside the result.

The local-repair lemma is another checked restriction: for fixed nonzero $h$, only $o(X)$ targets $n\le X$ admit representations of both $n$ and $n+h$ whose matched physical roots are within $B^{1/(r-1)-\eta}$, where $B=X^{1/r}$. Its proof sums a fixed-core binary norm-divisor bound and removes the core tail at the target-existence level. Cofiniteness does not force such small jumps, and the elementary cap-packing scales do not meet the counting estimate.

No additional target case was resolved. Density for every $r\ge3$ and complement infinitude for $r\ge5$ remain open here. A different composition operation, growing-complexity analytic test, or arithmetic root-coherence argument would require new input; these are provisional possibilities, not reductions claimed to solve the problem.