Three consecutive ones for odd-order multiplicative functions

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proof_sketch: An end-to-end sketch for the odd-k triple question: realization and sieve arguments, the unresolved aggregate cofactor estimate, scoped tests of statistical and finite-sums mechanisms, and a conditional spectral route for every odd order using an explicit quotient by the bounded-distance core.

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This lead's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI.

Target

The target is the second question of Problem 436: whether $\Lambda(k,3)$ is finite for every odd $k$. The lead works with the equivalent statement T(k): every completely multiplicative function from the positive integers into the $k$th roots of unity takes the value $1$ at three consecutive integers. The problem's growth question is outside this lead. The problem page records the status search and the accepted partial results: the pair theorem of Hildebrand and the value $\Lambda(3,3)=23532$.

Standing

The canonical proof sketch is author-recorded and unreviewed. No claim card has been made and the full target is unresolved. Its supported parts are:

• $\Lambda(k,3)=\sup_fN(f)$ for odd $k$, where $N(f)$ is the first start of three consecutive ones. The argument proves independence of the prescribed prime values, applies Chebotarev, and uses compactness to pass from existence for each function to a bound for each fixed $k$.
• Every $\mu_3$-valued completely multiplicative function has $N(f)\le23532$, from the accepted cubic result and the realization bridge.
• If $f$ weakly pretends to a Dirichlet character, its triples of ones have positive upper natural and logarithmic densities. Section 4 derives the needed multiform probability approximation from the fundamental sieve lemma, with its summed errors. Bounded pretentious distance gives positive lower densities.
• Product-based logarithmic correlation cancellation reduces the other cases to monochromatic triples, with a relative condition for composite $k$. The cited correlation theorem is used at the statement standing of its quotation in the held source; its primary proof was not read.
• The three-point logarithmically averaged Elliott conjecture would imply the full assertion for every odd $k$. The exact aggregate estimate needed by that conditional argument is stated in Section 10.
• The spectral hypothesis (QSP) in Section 12 would also imply the full assertion for every odd $k$. It concerns an explicit quotient of $f$ by a pointwise root of its bounded-distance core. This quotient need not be multiplicative, but it is invariant under multiplication by $k$th powers. If one of its logarithmic Furstenberg systems has no nonzero rational eigenvalue, polynomial recurrence transfers to consecutive triples and composes with the core's positive lower density on the same scales. The spectral assertion is unproved. A simpler multiplicative factorization covers squarefree orders conditionally on (SP); its required Dirichlet root can fail, as an order-nine example shows.

The remaining gap is positive mass for the relative diagonal of three consecutive values. Pairwise uniformity does not fill it. A digit-twisted stationary process with odd $k\ge7$ satisfies all the specific rotation, pair, dilation and correlation-isotypy identities tested in the sketch but has no monochromatic consecutive triple. This limits those identities, and is not a counterexample multiplicative function. Its colour process has the rational eigenvalue $1/3$. Section 12 shows that a process with a Haar position and the conditional self-similarity identities can avoid monochromatic consecutive triples only if its colour process has a nonzero rational eigenvalue. For $k=5$, the sketch proves that an independent linear-phase construction with arc colours cannot have this behavior if it obeys the conditional identities at $2$ and $5$; more general processes remain outside that argument.

The cofactor route remains open (sketch Section 11). Its exact weighted aggregate has a total-variation error summed over all characters, and its common-scale cancellation would compose with the positive bounded-distance core on the same selected population. Independence of the cofactor term from the entire small-prime vector is a stronger sufficient assertion, not the required estimate. The digit-twisted process defeats deductions from the label law and listed stationary identities alone; its formal cofactors are not values of one multiplicative function.

Other proposed mechanisms have narrower obstructions. Every polynomial family of consecutive values given by products of affine forms, in any number of parameters, depends on one affine form and so is a local pattern; the digit-twisted process gives probability zero, for odd $k\ge7$, to all families with two factors per value, and for odd $k\ge9$ to all with three. Hildebrand's special-set method for pairs cannot yield triples from first moments, since one colour class of an even-order function avoids every scaled consecutive triple in any finite set. The finite-sums strengthening of the pair theorem also leaves a selection problem: explicitly constructed completely multiplicative functions, both at bounded distance from $1$ and in Case C, admit a homogeneous finite-sums sequence for which every normalized predecessor has any prescribed nontrivial odd-order value (Proposition F). These functions have a triple of ones elsewhere; the examples refute the proposed universal reflection constraint, not an argument using a global absence of triples.

Next investigation

The quantitative analytic test is (AC) of Section 10, or the selected population version (CF) of Section 11. A proof must control the actual joint cofactor expectations in their weighted sum, with the summed error and one common sequence of scales. No assertion of independence from every label is needed. A one-sided estimate beating the negative of the positive core would already suffice for existence; its threshold is stated in Section 11.

The qualitative alternative is (QSP) of Section 12, for the actual quotient $u_f$ defined there. A proof of its spectral assertion would cover every odd order without assuming a Dirichlet root of the core. The quotient has uniform one-point marginals on residue classes, but this does not imply the process independence used in transferring recurrence and combining it with the core. The spectral input remains unproved. With trivial bounded-distance subgroup, $u_f=f$ and the hypothesis specializes to (SP); a weaker condition at one recurrence step can then suffice. The related spectral question for the Liouville function remains open in the cited source.

The function equal to a fixed primitive $k$th root at every prime remains a necessary test of any proposed full route: it requires three consecutive integers whose numbers of prime factors, counted with multiplicity, are all divisible by $k$, for every odd $k$. The finite-sums alternative would need a method for selecting a favorable sequence, since arbitrary homogeneous refinements can have nontrivial predecessor colours. Local Fourier uniformity or additional polynomial identities are possible inputs, but no necessity or sufficient arithmetic consequence of those proposed additions is proved here. Further model tests alone would not settle the target.