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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0322, E0323, E0325, E0829, E0979.

E0322: positive cubes versus signed equations

For each fixed k>=3 the target counts ordered representations by exactly k kth powers and asks both growth and a fixed-power lower bound infinitely often. The general source count allows zero; the cubic construction below is positive.

Mahler's equation (2') on printed p.138 is

$$(9u^4)^3+(3uv^3-9u^4)^3+(v^4-9u^3v)^3=v^{12}.$$

For integers v>0 and 0<u<9^(-1/3)v all bases are positive. The second requires u^3<v^3/3; the third's u^3<v^3/9 is stronger. The first coordinate strictly increases with u, so at least 9^(-1/3)v-1 distinct ordered triples result. Thus every c<1/12 works at k=3, independently of the zero convention. This checks an existing specialization, not a new case.

For k>=4 the recorded Erdős gain is exp(c_k log n/log log n), whose effective exponent tends to zero. Mahler's higher-degree identities instead permit signed coefficients and variables. Absorbing a coefficient into a kth power needs it to be a suitable rational kth power; negative coefficients cannot be absorbed for even k. The required unit-coefficient, nonnegative specialization is absent.

A source warning is important: the final paragraph fixes integer coefficients lambda_1,...,lambda_k and positive constants A_1,...,A_k,C, and claims that for infinitely many positive N there are more than C N^(k-2) integer tuples satisfying

$$\lambda_1x_1^k+\cdots+\lambda_kx_k^k=N, \qquad |x_i^k|<A_iN\quad(1\le i\le k).$$

Here the variables and coefficients may have either sign. This cannot hold as written. At least one coefficient is nonzero for an equation with positive right side N. Fixing the other k-1 bounded integer variables leaves at most k choices for the last, hence O(N^((k-1)/k)) solutions, and k-2>(k-1)/k for every k>=3. The constants in this upper bound depend only on the fixed coefficients, coordinate bounds, and k. The displayed exponent agrees with the printed page; the contradiction therefore concerns the printed assertion, not its transcription. No corrected exponent is guessed or used. The sign obstruction is independent of that defect. The warning is restricted to the fixed-coefficient and bounded-variable assertion above.

Next discriminant: an all-unit positive-sector identity giving a fixed exponent at one k>=4. Stop at signed coefficients or subpower gain.

E0323 and E0325: image size requires collision control

These count distinct integers, not tuples. E0323 asks x^(1-epsilon) for all epsilon>0 on the diagonal m=k and x^(m/k) for m<k. E0325 asks x^(3/k), or x^(3/k-epsilon) for every epsilon>0, for each k>=3. Summands are nonnegative, repetition is allowed, and there is no coprimality condition.

An exact sufficient test is available. Let r(n) count tuples in {0,...,B}^m with sum of kth powers n. Their number is (B+1)^m and their sums are at most mB^k. Cauchy--Schwarz gives

$$|\{n:r(n)>0\}|\ge (B+1)^{2m}/\sum_n r(n)^2.$$

Thus a collision bound O(B^m) gives the full off-diagonal exponent; a bound O_epsilon(B^(m+epsilon)) gives the epsilon-loss version, after rescaling epsilon on passing from B to x. These are sufficient conditions, not claimed equivalences. Large individual multiplicities do not defeat an average bound.

The m=1 count is immediate. The held page records Landau for k=2 and Heath-Brown for (k,m)=(3,2); these existing conclusions are not reverified. Erdős--Mahler's main theorem counts |F(x,y)| on unrestricted integer pairs for degree k>=3. F=X^k+Y^k has nonzero discriminant. For even k>=4, replacing x,y by their absolute values preserves F, giving the two-summand conclusion. For odd k, mixed signs instead give differences. The original introduction asserts the restricted-sector version x>=0, alpha x<=y<=beta x, without a separate proof. As an explicit source premise it gives all odd-k two-summand cases by alpha=0,beta=1. This account has not checked its proof and does not upgrade that premise. The proof transcription has suspicious theta/exponent text; no proof reconstruction is claimed from it. Stewart--Xiao's primary abstract confirms a modern unrestricted asymptotic but does not state a sector version; its relevant collision estimates remain uninspected. The remaining two-summand source question therefore concerns odd k>=5.

For E0325 the held accepted claim records Browning--Heath-Brown's k>=33 asymptotic from o(B^3) nontrivial six-variable collisions. The Oxford primary search extract also states the corollary, but its proof has not been inspected here. The controlling exponent inequality has not been inspected. The source question is whether a theorem gives an exponent strictly below 3 for any remaining k<=32. Wooley's held Theorem 1.1 gives 0.91709477 at k=3, not 1-epsilon for every epsilon. A fixed positive deficit does not establish the latter. Padding two summands with zero gives only x^(2/k), not x^(3/k); padding three summands gives no x^(m/k) for m>3.

Next move: inspect the sector/collision statements, not a large census. Stop at unrestricted signs, the same k>=33 hypothesis, or a fixed exponent loss. Those sign and exponent restrictions are part of the theorem interface.

E0829: an exact divisor test is not a polylogarithmic bound

The target is a uniform upper bound by a fixed power of log n for ordered pairs of natural cubes. Work with positive bases; allowing zero adds at most two representations for n>0. Put s=x+y and t=|x-y|. Then

$$n=s(s^2-3xy),\qquad t^2=\frac{4(n/s)-s^2}{3}.$$

An unordered positive representation is equivalent to a divisor s>0 of n with n^(1/3)<=s<=(4n)^(1/3), for which the displayed quantity is an integer square t^2, t congruent to s modulo 2, and 0<=t<s. Conversely x=(s+t)/2,y=(s-t)/2 recover positive integers and their cubes sum to n. When t=0 there is one ordered pair, otherwise two. For nonnegative bases replace t<s by t<=s. This is a complete finite test for each n, using only integer divisibility and square recognition.

It yields at most 2 tau(n) pairs. The usual n^epsilon divisor bound does not imply a fixed log power: n^epsilon/(log n)^C tends to infinity for every fixed epsilon>0,C. The missing statement is a polylogarithmic bound for the square-discriminant subset of the divisors, uniformly in n.

Stewart's Theorem 1.1 applies to the nonsingular binary form X^3+Y^3 but is an infinitely-often lower bound for signed variables. It is neither an upper bound nor, as stated, a positive-variable lower bound. Mahler's Theorem 6 supplies the latter with exponent 1/4. Such lower bounds remain compatible with the question, only constraining a possible log exponent. No taxicab computation was run. Next test: a uniform discriminant-square bound or a construction violating every log power. Stop at n^epsilon or a finite high-multiplicity list.

E0979: double representations do not give unbounded multiplicity

For each k>=2 the target is unbounded multiplicity of sums of exactly k positive prime kth powers, with repetitions allowed. Ordered and unordered counts differ by factors between 1 and k!, so their unboundedness is equivalent.

The primary Wang--Zhang v1 abstract explicitly states the cubic conclusion, that the CM-Fermat-cubic method does not extend to k=4, and a separate quartic conclusion limsup F_4>=2 from Green--Tao--Ziegler applied to an admissible binary quartic identity. The held claim page agrees. The proof was not inspected or accepted here. Infinitely many n with two representations do not yield n with arbitrarily many. Scaling an identity also fails to preserve prime bases: t times a prime is composite for t>1.

A concrete conditional tool interface is an arbitrarily large collection of distinct equal-sum quartic identities in one admissible finite-complexity system of linear forms, all positive on a common region. A prime-pattern theorem for that exact system would turn M identity rows into M prime representations. The unbounded identity family, complexity, admissibility, and positivity have not been constructed; this is an untested route, not a checked extension of the two-row result. Stop at a fixed M or a system outside the prime theorem. The fixed-multiplicity and unbounded-multiplicity conclusions remain different; the abstract supplies no further implication.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/diophantine_problems/erdos_1936_representation_integer_as_sum_th_powers/_index|erdos 1936 representation integer as sum th powers]]: Closing generalized assertion and no-proof qualification (statement reading). Inequality (1); odd/even proof sketch and effective exponent (account reading).
• [[../library/diophantine_problems/erdos_2019_number_integers_represented_binary_form/_index|erdos 2019 number integers represented binary form]]: Introduction statement (a) and sector assertion; Sections 1-4 read as text, with suspicious proof exponents not relied upon (account reading). Main theorem (a) and restricted-sector remark, pp.134-135; Bears on (account reading). Upper bound scope (statement reading).
• [[../library/diophantine_problems/mahler_1935_lattice_points_curves_genus_1/_index|mahler 1935 lattice points curves genus 1]]: Positive two-cube lower bound and specialization (statement reading). Sector lower-bound statement (statement reading).
• [[../library/diophantine_problems/mahler_1936_note_hypothesis_k_hardy_littlewood/_index|mahler 1936 note hypothesis k hardy littlewood]]: Section 1 identities (2)-(5); Section 2 and equation (2') at printed p.138. Only (2') is used as a checked identity. (proof reading). Statement, identity (5) account, proof pointer and sign qualifications (account reading).
• [[../library/diophantine_problems/stewart_2008_cubic_thue_equations_many_solutions/_index|stewart 2008 cubic thue equations many solutions]]: Rank-two twist theorem and proof account (account reading). Theorem 1.1 signed-variable lower bound and scope (statement reading).
• [[../library/diophantine_problems/wooley_2015_sums_three_cubes_ii/_index|wooley 2015 sums three cubes ii]]: Theorem 1.1 exponent and moment deduction pointer (statement reading).