Why this file exists. The goal is a novel critical line / deep structure in the primes. Hard lesson from the June-2026 sessions: the genuinely deep open problems here resist empirical attack by their nature (the parity barrier; criticality). That is the qualifier for a worthy target, not a reason to drop it. You don't brute-force these — you take one and hunt for a synthesis that throws a computable shadow of it, then read the shadow. This file gathers the frontier conjectures, their honest status, the barrier that makes each hard, and every angle we have to get a clue.
- A straight line is cheap (poles, constants, telescoping sums are all linear). Nothing counts until a matched twin fails to reproduce it (Cramér / shuffle).
- Raw pointwise prime transforms are barren: PNT-linear or Weyl-flat.
- The primes' additive structure is fully consistent with Hardy–Littlewood at computable scale — the unexplained residual is not there.
- The open frontier not closed by H–L: multiplicative randomness (Chowla / Sarnak), and family phenomena (murmurations) which need labeled data.
- You cannot name a novel finding in advance; maximize (model-incomplete) × (under-computed) × (clean null), and look without a prior.
- Tooling:
scripts/explore.mjs,scripts/hunt.mjs(gauntlet + self-improving generator),logs/bias.md, the 5-bar test, the Cramér twin.
All nontrivial zeros of ζ have real part ½. Status: open. Elementary
equivalents (no ζ needed): Mertens M(x)=O(x^{1/2+ε}); ψ(x)−x=O(√x log²x);
Robin's inequality σ(n)<e^γ n log log n; Lagarias. Criticality: de
Bruijn–Newman Λ=0 ⟺ RH, and Λ≥0 is proven (Rodgers–Tao 2018) — the primes
sit with zero slack at the edge of a phase transition. Barrier: the zeros /
the explicit formula; criticality means no margin.
Our no-ζ findings: cancellation exponent θ=½ recurs (Mertens, Chebyshev);
a different critical exponent θ≈¼ lives in the squarefree/divisor residual
(Dirichlet-divisor family); cos-sum-over-primes shows an α-resonance transition
(known Vinogradov); the "primon gas" critical temperature is just ζ's pole in
disguise; Weil/Hasse gives a hard spectral critical envelope |a_p|≤2√p.
For distinct shifts h₁,…,h_k: (1/x)Σ_{n≤x} λ(n+h₁)···λ(n+h_k) → 0, i.e. every
sign pattern in {±1}^k has natural density exactly 1/2^k. (λ(n)=(−1)^Ω(n).)
Status (checked, 2026-06):
- k=1 ⟺ PNT (proven).
- natural-density equidistribution for k≥2: OPEN (this is the heart).
- 2-point log-averaged Chowla: proven (Tao 2016); odd-order log-averaged: proven (Tao–Teräväinen 2017); even-order ≥4 log-averaged: OPEN.
- Occurrence of all patterns: length ≤3 positive density (Matomäki–Radziwiłł–Tao 2015), length 4 positive density + conjectured log-density (Tao–Teräväinen 2017), length 2 equidistribution unconditional (2022). [So "k=4 occurrence" is CLOSED; exact natural-density equidistribution is the open part.] Barrier: the parity problem — sieve methods provably cannot determine the parity of Ω(n). Structure here is invisible to the field's main tool. Synthesis / clue angles → see the section below.
Use Möbius, not Liouville. Over F_q[t], Σ_{deg=n} μ_q = 0 for n≥2 (mean-zero),
whereas λ_q carries an exact ±q^{n/2} degree-parity oscillation from
ζ_q(u²)/ζ_q(u). So λ needs square-divisor bookkeeping; μ is the clean object.
Recover λ via λ(n)=Σ_{d²∣n} μ(n/d²).
Target — uniform power-saving binary Chowla (correct quantifiers). Do NOT write
≪_h and "uniform in h" together; they pull opposite ways. Correct form:
∀η>0 ∃δ(η)>0: sup_{1≤|h|≤N^{1−η}} |Σ_{n≤N} μ(n)μ(n+h)| ≪_η N^{1−δ(η)}.
"Nailing δ" is ill-posed (any δ ⇒ all smaller); the testable question is whether the
true scale is N^{1/2+o(1)}.
The engine — the unshifted Type-II form is FALSE. Not
|Σ_{m∼M}Σ_{n∼N} α_m β_n μ(mn)| ≪ X^{1−δ} for arbitrary |α|,|β|≤1:
take α_m=μ(m), β_n=μ(n); since μ(mn)=μ(m)μ(n)1_{(m,n)=1} the sum counts coprime
squarefree pairs ≍ X (for λ every term is +1). Same counterexample over F_q[t],
so this is not what Sawin–Shusterman prove. The parity-sensitive engine needs the
shift inside the form (a shifted bilinear / cut-norm):
sup_{|α|,|β|≤1} |Σ_{m∼M}Σ_{n∼N} α_m β_n μ(mn+h)| ≪_η X^{1−δ}, MN≍X, M,N≥X^η.
Caveat: not literally equivalent to binary Chowla — also needs Type-I estimates,
decompositions, shift-uniformity, local-factor control.
Binary Chowla ≠ twin primes. The prime parity-breaker is a shifted-prime Möbius
estimate Σ_{n≤x} Λ(n) μ(n+h) = o(x) plus Elliott–Halberstam (Murty–Vatwani).
What F_q[t] actually gives (corrected anchors).
- Carmon–Rudnick 2014: large-q, fixed-n, normalized correlation
O(q^{−1})(Pellet + Weil RH for curves). Not the fixed-q,n→∞regime that mirrorsN→∞. - Sawin–Shusterman 2022: power-saving μ-cancellation on polynomial sequences, quadratic Bateman–Horn, near-level-1 distribution; mechanism = μ mimics a quadratic character on special subspaces (sheaf/trace-function geometry) — not the arbitrary-coefficient Type-II form.
- Keating–Rudnick reformulation:
Σ_{deg f=n} μ(f)μ(f+1) = |Y_n(F_q)| − q^n(point count on a double cover) — the geometric source of cancellation.
Route A (ergodic) — correct strength. Frantzikinakis–Host: log-Furstenberg systems
of μ have no irrational spectrum, ergodic components = Bernoulli × ∞-step nil; ergodicity
of the Liouville system ⟹ log-Chowla. This yields o(N) (qualitative); a power saving
N^{1−δ} is a separate quantitative mixing input not delivered by structure alone.
Computational program (corrected objects). (i) shifted bilinear cut norms
sup_{|α|,|β|≤1}|Σ α_m β_n μ(mn+h)| — measure the scale; (ii) block entropy / spectral
measure of the Liouville Furstenberg model (Route A shadow); (iii) exact F_q[t]
analogues (μ_q correlations; the Y_n point counts). Goal is not to "prove δ" but to
test N^{1/2+o(1)} and expose the secondary/local structure any proof must reproduce.
Σ_{n≤N} μ(n) f(n) = o(N) for every f from a zero-entropy dynamical system.
Status: many cases proven; general open. Chowla ⟹ Sarnak. Clue: the
λ/μ subshift — its complexity and entropy (see angle 1).
First H–L: prime constellations counted by the singular series (unproven). LO–S (2016): consecutive primes' residues mod q are biased — explained by the H–L secondary term; extended to sums of two squares (2021). A Liouville analog appears under-explored but is likely null (λ is conjecturally genuinely random, with no residue-class constraint to create the bias). Barrier: H–L itself unproven; the 2nd H–L conjecture is incompatible with the 1st (Hensley–Richards) and believed false.
Spacings of ζ-zeros match GUE random matrices, with arithmetic lower-order corrections (Bogomolny–Keating). The multiplicative dual is Chowla. Uses zeros.
Maier proved primes in windows (log x)^λ violate the Cramér model. M–S
conjecture a precise short-interval variance with log corrections + Gaussian
fluctuations. Computable; the model is provably broken — the right model is the
open part.
He–Lee–Oliver–Pozdnyakov (2022–23), Zubrilina (density). An oscillating
correlation between Frobenius traces a_p and rank, averaged over families of
elliptic curves ordered by conductor; invisible in any single object and to a
linearity detector. Key lesson: it needs labeled-family data (rank,
conductor) — information not present in a bare prime sequence. To hunt this class,
the instrument must ingest (object, invariant) pairs (LMFDB export, or
function-field families where labels are computable from point counts).
The principle: λ is conjecturally random; pair "is λ random?" with a field that has a computable test for randomness, and read the deviation.
| # | Field fused | Computable shadow | The clue |
|---|---|---|---|
| 1 | symbolic dynamics (Sarnak) | block-complexity p(k), block-entropy of λ | deficit below 2^k = hidden structure |
| 2 | additive combinatorics | Gowers U², U³ norms of λ on [N] | the phase λ leans toward |
| 3 | machine learning | predict λ(n) from residues/digits/prior λ; holdout acc vs 50% | any persistent edge + feature importance |
| 4 | information theory | LZ / entropy-rate of the λ bitstream | a compressibility gap |
| 5 | function fields (PROVEN) | exact λ-correlations over F_q[t] | the mechanism/rate of real cancellation |
| 6 | spectral (Σλ/n^s=ζ(2s)/ζ(s)) | empirical power spectrum of {λ(n)} | any peak/atom = the zeros' shadow |
| 7 | M–R short intervals | H-window averages of λ | the exceptional (non-cancelling) set |
| 8 | Elliott / cross-corr | λ vs other multiplicative fns / characters | surprising non-vanishing = dependency |
First moves: angle 3 (ML predictability) + angle 2 (Gowers U²/U³), anchored by angle 5 (function-field ground truth). Angles 1 and 6 are the deepest bridges if the empirical ones light up.
- ~1,900 no-ζ specs hunted; 0 survived all 5 bars → the primes' additive residual is empty (consistent with H–L). Logged in KNOWLEDGE.md as CLOSED-NO-SURVIVOR.
- Hardened
hunt.mjs: order-sensitive chips (dyexp/diff) need a Cramér twin, not the (rigged) shuffle null;updaterewards bar-5 survival, not promotion. - Murmuration reproduction from scratch failed for the right reason: signal lives in the rank-parity split; bare family average ≈ twin (needs labeled data).