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teal-sea / zeta-labstate of record · compiled 02 Nov 2026 · revision b09fa4dd · source

Library · hunts/mobius_orbits/RESULTS.md

Multi-prime Mobius orbits: exact cancellation, still an open RH target

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Status: ordinary argument with independent model challenge completed. See the review and its limitations. This is not human peer review or a kernel-checked statement. The finite tests check the identities, not an unbounded estimate. There is no RH proof or disproof here, and no claim of novelty. Base: 8941a80. The earlier p=2 note is not edited or regraded.

1. Target and attempted construction

The parent target, from the factorization note, is

D_N = S_smooth(N) + Sigma1(N) + Sigma2(N), |D_N| = O_epsilon(N^(1/2+epsilon)) for every epsilon > 0.

Its relation to RH depends on that note's full reduction. Nothing here re-establishes the reduction or supplies the joint bound. In particular, controlling Sigma2 alone does not control cancellation with the other terms.

For integer N >= 4, set H=floor(N/2), U=floor(sqrt(H)), and V_b=floor(H/b). Write

T_b = sum_{U<a<=V_b} mu(a) [1-floor(N/(ab))], Sigma2 = sum_{b=2..U} log(b) T_b.

This is the integer-kernel branch inherited from the parent note. Empty sums are zero. The extension attempted here keeps the sign relations for several primes simultaneously instead of one p=2 pair. It is compared to a separate exact level-set expansion, not assumed better because it is newer.

2. An exact orbit identity with all boundaries

Let P>1 be squarefree. Every term not killed by a squared prime dividing P has a unique representation a=dm with d|P and gcd(m,P)=1. Then mu(a)=mu(d)mu(m), including zero when m is not squarefree. Thus, with

J_m = {d|P : U<dm<=V_b}, K_m = sum_{d in J_m} mu(d) [1-floor(N/(dmb))],

one has exactly

T_b = sum_{m<=V_b, gcd(m,P)=1} mu(m) K_m. (1)

No cutoff term has been discarded. The sums are finite. Terms divisible by a squared prime from P contribute zero to the original sum and are absent on the right for that reason only.

An orbit is complete precisely when U<m and Pm<=V_b. For those m, all d|P are active, and elementary inclusion-exclusion gives

K_m = -Phi_P(floor(N/(mb))), (2) Phi_P(x) = #{j in Z : 1<=j<=x, gcd(j,P)=1}.

Indeed sum_{d|P}mu(d)=0, and sum_{d|P}mu(d)floor(x/d)=Phi_P(x). This is an ordinary finite combinatorial proof, not an inference from numerical agreement.

The complete-orbit kernel has a fixed negative sign. The surviving oscillation is mu(m), not cancellation inside Phi_P. Setting m=1 in a complete orbit is impossible because U>=1; the formula itself still holds as an inclusion-exclusion identity for any positive x.

3. A specified majorant and its fixed-P asymptotic

Define A_P(N), explicitly discarding |mu(m)|<=1, by

sum_{b=2..U} log b [ sum_{U<m<=V_b/P, gcd(m,P)=1} Phi_P(floor(N/(mb)))

Then |Sigma2|<=A_P(N). Complete orbits use (2); incomplete ones use |1-floor(N/(ab))|<=N/(ab), valid on this range since ab<=H<=N/2.

Candidate ordinary asymptotic, with P fixed: if c_P=phi(P)/P, then

A_P(N) = (c_P^2/48) N(log N)^3 + O_P(N(log N)^2). (4)

This is not asserted uniformly for P growing with N.

Proof details to be challenged:

  1. Periodicity gives Phi_P(x)=c_P x+O_P(1). Partial summation of the periodic coprimality indicator gives, uniformly for B>=U>=1, sum_{U<m<=B,gcd(m,P)=1}1/m = c_P log(B/U)+O_P(1/U).
  2. An incomplete orbit satisfies m<=U or Pm>V_b. Every active a=dm then lies in (U,PU] or (V_b/P,V_b], intersected with (U,V_b]. The representation is unique, so its total majorant is bounded by the sum of N/(ab) over these two intervals. Their harmonic mass is O_P(1). Consequently all incomplete orbits together cost O_P(N sum_{b<=U}(log b)/b)=O_P(N(log N)^2).
  3. The exact geometric cutoff for a potentially nonempty complete interval is B=floor(H/[P(U+1)]). Coprimality may still leave no m. Set L=H/(PU); then B=L+O_P(1) for fixed P. For the complete part, the upper m cutoff is floor(H/(Pb)). Its harmonic evaluation gives the main term

c_P^2 N sum_{2<=b<=L} (log b)/b log(L/b).

Extending the exact integer endpoint to L adds only the endpoint errors covered by the same estimate. Floor and harmonic errors cost O_P(N(log N)^2); the O_P(1) error per coprime-count term also sums to O_P(N sum_{b<=U}(log b)/b).

  1. Integral comparison yields sum_{2<=b<=L}(log b)/b log(L/b)=(log L)^3/6+O((log L)^2). For fixed P, log L=(log N)/2-log(P sqrt(2))+O(N^(-1/2)). This yields (4). Finitely many small N are absorbed in the fixed-P remainder constant; none is omitted from the exact identities.

At P=2, the proposed coefficient is 1/192, matching the previously recorded leading coefficient. P=6 gives 1/432, P=30 gives 1/675, and P=210 gives 4/3675. The script computes these fractions rather than estimating them from a short scan.

Thus this SPECIFIED fixed-P absolute-value estimate changes the leading constant, not the exponent or logarithmic power. It supplies no RH-strength bound. It does not obstruct a signed estimate, a growing-P construction with uniformly controlled boundaries, or a different argument entirely.

4. An independent exact floor-level representation

Let M(t)=sum_{a<=t}mu(a) be the Mertens function and Q_b=floor(N/[b(U+1)]). Since

floor(N/(ab))-1 = sum_{k=2..floor(N/(ab))}1,

interchanging finite sums yields

T_b = -sum_{k=2..Q_b} [M(floor(N/(bk)))-M(U)]. (5)

For k>=2 the upper a cutoff is at most floor(N/(2b))=V_b, so there is no missing V endpoint. For k<=Q_b it is at least U+1; if Q_b<2 both sides are empty.

A useful uniform transfer, with its input visible: suppose

|M(t)| <= t E_U for every integer t>=U.

Then, for each nonempty b-row,

|T_b| <= (2N/b) E_U sum_{k=2..Q_b}1/k. (6)

Both Mertens arguments are at least U, and U<=floor(N/(bk))<=N/(bk), which proves (6) without assumptions on where floor jumps occur. For U>=2, summing gives the coarse bound

|Sigma2| <= 2N E_U (log U)(1+log U) log N. (7)

For U=1 the sum is empty. The finite test uses the exact rational envelope max_{U<=t<=floor(H/2)} |M(t)|/t, which covers all arguments in (5), not an infinite-range estimate. It checks (6) row by row with Fractions.

Inserting an independently supplied classical envelope E_U=C exp(-c sqrt(log U)) into (7) gives a subexponential saving over N log^3 N, with constants retained. This corollary is explicitly conditional on that input as stated here; the underlying analytic-number-theory bound is not re-proved in this hunt. The exact identity (5) and transfer (6) address the endpoints and uniformity that were left unpriced in the earlier factorization note. They do not give the N^(1/2+epsilon) joint estimate.

5. What arithmetic has and has not been removed

For fixed P and Re(s)>1, absolute convergence of the Euler product gives

sum_{gcd(n,P)=1} mu(n)n^(-s) = (1/zeta(s)) product_{p|P}(1-p^(-s))^(-1). (8)

The finite product is analytic and nonzero on Re(s)>0, because |p^(-s)|<1 there. Its meromorphic continuation cannot cancel a pole of 1/zeta at an off-critical zero. This is a conditional statement about such a zero, not a claim that one exists. Removing finitely many Euler factors alone has not removed the remaining zero-exclusion problem.

Equation (8) is not a theorem that every multi-prime approach fails. An estimate exploiting the remaining mu(m) signs could still supply the missing information; (3) discards precisely those signs.

6. Evidence and controls

probe.py independently computes a Mobius sieve and a trial-division Mobius function, compares (1) and (5) to the direct integer sum in every b-row, checks the rational bound (6), and compares the inclusion-exclusion count to direct gcd enumeration. pilot.json and results.json retain the finite rows. The full run covered 1196 (N,P) cases and 8412 b-rows: N=4..300, plus 1000 and 10000, at P=2,6,30,210. Every direct/orbit/level comparison had exact zero defect. The Mobius sieve matched independent trial division at all 5000 inputs and coprime counts matched direct gcd counts in 4004 checks. At P=210, N=10000 there are no complete orbits: all of Sigma2 is boundary. A favorable local density must not erase it. Log-weighted totals are ordinary floating-point diagnostics, not sign enclosures or proofs of (4).

Dropping incomplete orbits and reversing the complete-orbit sign are intentional lesions. Both must have an exact nonzero witness or the script fails. No floating-point precision setting is needed for these integer and rational checks; the float totals are outside their truth claims.

Davenport-Heilbronn is not run as a rival here: no structural implication from shared analytic axioms to RH is claimed. The construction explicitly uses the genuine Mobius function and does not establish its required uniform cancellation. Nor does a surrogate passing inclusion-exclusion refute the elementary identity. No zeta zero census is being inferred.

7. Disposition

RH remains unresolved in this attempt. The exact construction survives the finite checks only at their stated scope. The handwritten asymptotic is subject to the separate review record; it is not a formal proof. Nothing is published, and no prior manuscript or accepted result is changed.

The doors

Completed on landing, 2026-10-21. The three bullets above are the hunt's text of 2026-09-26. They name no active constraint and predate the energy continuation and its pause, so the parts below complete the section from the hunt's own record. No claim, run or grade changes.