Most mathematical papers announce what they prove. The Maypole–Lattice Framework paper — A Conditional Approach to the Riemann Hypothesis via Simultaneous Diophantine Approximation of Prime Logarithms (Zenodo DOI: 10.5281/zenodo.21280549) — does something rarer and arguably more valuable: it announces, with equal precision, fifteen things that cannot be proved by existing methods, and explains exactly why each one fails.

This is not a paper that retreats to vagueness. Each barrier is named, located, and dissected. The obstruction is identified not as a gap in cleverness but as a structural feature of the mathematical landscape. That distinction matters.

What the Framework Is Actually Doing

At its core, the paper introduces the Maypole constant C(K) — a measure of how well the logarithms of the first K primes can be simultaneously approximated by rationals. Because log p₁, log p₂, … are rationally independent (a consequence of the fundamental theorem of arithmetic), the vector of prime logarithms is never exactly rational. The Maypole constant asks how badly approximable that vector is, with a precise exponent.

Why does this matter? Because moments of Dirichlet polynomials built from primes — objects that appear throughout analytic number theory — behave very differently depending on whether the prime log vector is well-approximated by rationals or not. When it is poorly approximated (large C(K)), diagonal terms dominate the moment computation and life is tractable. When approximation is good, off-diagonal interference terms pile up and the computation becomes intractable.

The paper's central result (Theorem 7.1) identifies the exact threshold: diagonal dominance holds if and only if the Baker constant B(K) satisfies B(K) = o(K log K). This is a sharp, unconditional equivalence — not an implication, not a sufficient condition. An if and only if.

The problem: current technology gives B(K) = exp(Θ(K log K)). The threshold is polynomial growth. Every known bound is super-exponential. The gap is not a matter of working harder with existing tools.

Three Things the Paper Does Prove

Before reaching the barriers, the framework establishes three unconditional results worth pausing on.

Equidistribution floor (Theorem 3.2). Using Vinogradov–Korobov bounds on prime exponential sums, the paper shows that only finitely many primes are needed to defeat any fixed finite set of rational denominators. The number of primes required grows as exp(C(log Q)^(3/4)) — subexponential in Q. This resolves an open growth condition and clarifies exactly where the conditional chain begins.

The sharp threshold itself (Theorem 7.1). Already described above. The point worth emphasizing is that this is unconditional — it doesn't assume the Riemann Hypothesis or Conjecture D. It simply characterizes when diagonal dominance holds, in terms of the Baker constant.

Single-zero invisibility (Proposition 8.1). This one is philosophically striking. If you take any single zero of the Riemann zeta function and move it off the critical line, every positive-order moment of the zeta function on the critical line changes by a factor of 1 + o(1). In other words, moments cannot see individual zeros. Any strategy that tries to prove RH by controlling moments must somehow extract information that moments, by design, wash out. The difficulty doesn't disappear — it relocates into the conversion step.

The Fifteen Barriers: A Taxonomy of Failure

The paper's most distinctive contribution is its systematic classification of fifteen approaches to the remaining open problems — and its proof that each one terminates at a specific, identifiable obstruction.

A few examples illustrate the depth of the analysis:

Baker–Matveev linear forms represent the state of the art in transcendence theory. Applied here, they yield B(K) bounds that are exponential in K — intrinsically, because the method relies on a Siegel auxiliary polynomial whose degree grows with K. This is not a gap in the Baker–Matveev approach; it is the approach faithfully reporting the difficulty of the problem.

Kleinbock–Margulis extremality establishes that almost every smooth curve on a manifold is extremal — meaning its points are not well-approximable by rationals in a quantitative sense. This is a beautiful theorem. It is also an almost-everywhere result. Whether the prime log vector specifically belongs to the measure-zero exceptional set is precisely what cannot be determined constructively from the theorem.

Harper–Soundararajan multiplicative chaos gives upper bounds on moments of the zeta function in the critical strip — powerful, state-of-the-art bounds. But the Maypole framework needs lower bounds on Diophantine linear forms, which is a different (and harder) question. The tools point in the right direction but answer a different question.

Evertse S-unit counting bounds the number of integers where a linear form is nearly zero — at most 3·7^(K+1) of them. But the paper needs to bound the depth of the smallest value, not count how many times near-zero values occur. The distinction between cusp count and cusp depth is exactly the gap the method cannot bridge.

What makes this classification genuinely useful — rather than merely discouraging — is that it is complete and principled. The fifteen barriers cover the space of available techniques. Knowing which approaches terminate, and where, tells future researchers exactly what kind of new idea would be required. This is obstruction theory doing its proper job.

The Conditional Connection to RH

The paper includes a conditional path toward the Riemann Hypothesis, constructed via the Gonek–Hughes–Keating factorization: ζ(s) = P_K(s) · Z_K(s) · (1 + ε_K(s)), where P_K captures the contribution of the first K primes and Z_K captures the zeros.

At leading asymptotic order, the head and tail factors are independent — meaning their contributions to moments multiply cleanly. The numerically observed anti-correlation between head and tail resides in subleading terms, which the Maypole lattice does not yet control.

Conditionally on Conjecture D (that the smallest denominator q(K) defeating the K-prime Maypole condition satisfies q(K) ≤ K^A for some fixed A), the Maypole constant C(K) approaches its maximal value of 1/2, and the full conditional chain reduces the Riemann Hypothesis to a single open problem: prove the subleading head–tail anti-correlation is characterized over the integers.

Computational evidence over K ≤ 300 primes and denominators up to 2×10⁷ is consistent with Conjecture D at exponent A = 5, with C(K) exceeding 0.42 across all tested values.

Why This Approach to Mathematics Matters

The Maypole–Lattice Framework exemplifies a style of research that is undervalued in popular accounts of mathematics: the systematic study of why hard problems are hard. Proving that a problem is difficult, at a precise and technical level, is a genuine contribution. It constrains the search space for future proofs, prevents rediscovery of dead ends, and occasionally reveals that an apparent approach to problem A would actually require solving problem B — which we already know is harder.

The three open problems the paper leaves on the table (the subleading anti-correlation at K=1, Conjecture D itself, and the limiting value of C(K)) are not loose ends. They are precisely located targets, with the territory around them mapped.

That is what rigorous obstruction theory looks like in practice.

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The Maypole–Lattice Framework paper is available at Zenodo: 10.5281/zenodo.21280549. The Signal Carries Everything series, which applies information-theoretic thinking to related questions about formalism limits in mathematics and physics, is documented separately in the series overview.