One fixed positive Fourier density gives two real multisets with identical power traces at every order and different numbers of simple points. Their entrywise overlaps recover the missing count. Independent copies amplify the difference to simple-point fractions ranging from 1/5 to 2/5 while preserving all normalized power traces.
These are original finite constructions, with ordinary proofs and the scoped Lean checks stated below. No novelty claim or improved asymptotic statement about zeta zeros is made.
1. What the spectrum already determines
Let p be an even, nonnegative, compactly supported probability density, positive almost everywhere on some open interval. Set
K(z) = integral p(u) exp(-2 pi i u z) du.
Take a finite conjugation-invariant complex multiset, counting multiplicities. Let R be its number of distinct real locations, k its number of distinct nonreal conjugate pairs, N its total multiplicity, and s its number of simple real locations. Let A be the real signed operator of FINITE-THEOREM.md, Section 10. Write nu_+ and nu_- for its positive and negative eigenvalue counts, including spectral multiplicities. Then
nu_+ = R+k, nu_- = k, rank A = R+2k. (1)
To prove this, the functions sqrt(p(u)) exp(-2 pi i u z) for distinct z are linearly independent over the complex numbers. A linear relation would give an exponential polynomial vanishing almost everywhere on the interval where p is positive. Continuity makes it vanish throughout that interval. Its derivatives at an interior point form an invertible Vandermonde system, so all coefficients vanish.
Passing to the real feature vectors for real points and the real and imaginary feature vectors for conjugate pairs preserves independence. In this basis the coefficient form defining A is diagonal: each real location contributes one positive entry, and each nonreal pair contributes one positive and one negative entry. All entries are nonzero because multiplicities are positive. Sylvester's law of inertia proves (1).
Thus the full spectrum already determines the number of distinct locations and whether any are nonreal. Multiplicity capacity then gives
N >= s + 2(R-s) + 2k = 2nu_+ - s,
and consequently
s >= max(0, 2nu_+ - N). (2)
This is sharp using only N and the inertia. If R>0 and R+2k <= N <= 2R+2k, make exactly N-(R+2k) real locations double, keep the others simple, and give every nonreal pair multiplicity one. If N >= 2R+2k, make all real locations multiple and assign any surplus to one of them. If R=0, the feasible total N is even and s=0. Arbitrary distinct locations realize these inertia counts by the independence argument above. This sharpness statement does not assert that every full spectrum permits every such allocation.
In particular, a positive occurrence Gram matrix of rank N already forces all N points to be simple and real. The earlier four-point full-rank example therefore demonstrated loss of a statistic, but did not demonstrate loss of the simple-point count. The construction below does.
2. A fixed density and two different counts
On [-1/2,1/2] take
p(u) = 1 + cos(2 pi u) + (1/4) cos(4 pi u),
and set p to zero outside. With x=cos(2 pi u),
p(u) = 1/4 + (1/2)(1+x)^2 >= 1/4.
It is even and has integral one. Integer-frequency orthogonality gives
K(0)=1, K(1)=K(-1)=1/2, K(2)=K(-2)=1/8, K(n)=0 for every integer |n|>2. (3)
Use the two multisets
Z_A = (0,0,0,3,6), multiplicities (3,1,1), s_A=2, Z_B = (0,0,1,1,4), multiplicities (2,2,1), s_B=1.
Both have N=5 and three distinct real locations. If C is the kernel Gram matrix on distinct locations and D is the diagonal matrix of their multiplicities, the nonzero spectrum of the occurrence Gram matrix is the spectrum of H=sqrt(D) C sqrt(D). In these two cases,
[3 0 0] [2 1 0] H_A = [0 1 0], H_B = [1 2 0]. [0 0 1] [0 0 1]
Both have eigenvalues 3,1,1. Each five by five occurrence Gram matrix also has two zero eigenvalues. Therefore the complete raw cycle moments agree:
M_j(Z_A) = M_j(Z_B) = 3^j+2 for every integer j>=1. (4)
Their different simple counts prove that the full sequence of raw moments does not determine s, even with one specified Fourier kernel. Equation (2) gives s>=1 for this spectrum, and Z_B attains it.
3. The overlaps distinguish the actual count
For the occurrence Gram matrix G define
S = sum_i (sum_j G_ij^2)^2, Q = sum_(i,j) G_ij^4,
and let D_4 be the ordered four-cycle sum over pairwise distinct occurrence indices. Distinct indices may have the same location. Exact values are
| Multiset | N | s | M_2 | M_3 | M_4 | S | Q | D_4 |
|---|---|---|---|---|---|---|---|---|
| Z_A | 5 | 2 | 11 | 29 | 83 | 29 | 11 | 0 |
| Z_B | 5 | 1 | 11 | 29 | 83 | 26 | 19/2 | 9/2 |
For Z_A, the Gram matrix is a three by three all-ones block and two isolated ones. No nonzero cycle can use four distinct indices. For Z_B, the first four occurrences form two duplicate pairs with cross entries 1/2, and the fifth is isolated. Eight four-cycles alternate between the pairs, each contributing 1/16. The other sixteen contribute 1/4 each. This gives D_4=9/2.
There is also a universal count classification within this exact spectral class, not just a comparison of the two examples. By (1), a configuration with nonzero spectrum 3,1,1 has three distinct real locations and N=5. Its only multiplicity patterns are (3,1,1) and (2,2,1).
For distinct real x,y, positivity of p on an interval implies
|K(x-y)| < 1. (5)
Equality in the triangle inequality for its Fourier integral would require the phase to be constant almost everywhere on that interval, impossible for x-y != 0. Thus, in any real configuration,
M_2-Q = sum_(locations a!=b) m_a m_b K(x_a-x_b)^2 [1-K(x_a-x_b)^2] >= 0,
with equality exactly when all distinct-location features are orthogonal. Then their weighted Gram eigenvalues are precisely their multiplicities. In particular, the shared spectrum with Q=M_2=11 forces s=2.
The other case is exact as well. Since H has spectrum 3,1,1, H=I+2vv^T for a real unit vector v. The diagonal is the multiplicity vector. For multiplicities (2,2,1), this forces v_i^2=(1/2,1/2,0), so the only nonzero cross-location kernel value has absolute value 1/2. Consequently Q=19/2 and s=1. Throughout this entire spectral class,
s = (2Q-16)/3 = (S-23)/3. (6)
4. The count difference survives arbitrary scale
Fix a positive integer h. Translate h independent blocks by 10j, j=0,...,h-1, and choose t of them to be type A and the other h-t to be type B. Distinct blocks have integer separation at least four, so (3) makes every cross-block kernel entry zero. Their occurrence Gram matrices form an exact direct sum. Hence
N = 5h, M_j = h(3^j+2) for every j>=1, s = h+t, Q = (19h+3t)/2, S = 26h+3t, D_4 = (9/2)(h-t). (7)
All raw moments remain fixed as t varies. The simple-point fraction ranges from 1/5 to 2/5, so this loss of counting information does not disappear when the multisets grow.
The all-A member has Q=M_2 and exactly 2h simple points. In fact, any configuration with its nonzero spectrum 3 repeated h times and 1 repeated 2h times, together with Q=M_2, has those multiplicities and hence s=2h. However, a bound using only that full spectrum cannot exceed h, because the all-B member has exactly the same spectrum and s=h. The overlaps can therefore improve a finite count beyond the best bound from the spectrum alone.
5. S recovers the count throughout the amplified spectral class
The exact formula involving S is stronger than a property of the chosen block family. Suppose the nonzero spectrum consists of h copies of 3 and 2h copies of 1. Inertia again forces all points to be real. The weighted location Gram matrix H is positive definite and satisfies
H^2 = 4H-3I.
Its diagonal entries are the integer multiplicities m_a. The spectral bounds force m_a to belong to {1,2,3}. An occurrence at location a has squared-row sum
sum_j G_ij^2 = (H^2)_aa/m_a = 4-3/m_a.
Writing R=3h and T_1=sum_a 1/m_a gives
S = sum_a m_a(4-3/m_a)^2 = 16N-24R+9T_1.
For any multiset of multiplicities from {1,2,3}, direct separation of the three possible values gives
s = 3T_1 - (5R-N)/2.
Combining these identities proves, throughout this whole spectral class,
s = S/3 - 29N/6 + 11R/2 = (S-23h)/3. (8)
The broader overlap bound is developed separately in OVERLAP-BOUND.md. Equation (8) is an exact theorem for the stated spectral class; it is not a claim that S determines s for arbitrary spectra.
6. Scope and reproduction
The density has endpoint jumps. Its rescaling q(u)=L p(Lu) preserves every Gram entry when every point is multiplied by L, and makes frequency support as narrow as desired. This finite rescaling does not identify any multiset with actual zeta zeros or supply a smooth height-cutoff transfer theorem.
counting_overlap.py uses exact Fraction arithmetic. It checks the expanded five by five matrices, the matrix recurrence G^3=4G^2-3G that proves (4) at all orders, direct cycle enumeration, the overlap values, and separated amplifications. Separate 60-digit Fourier quadrature checks all integer differences from zero through 26. The script saves its results beside the source. Run it from the repository root:
.venv/bin/python hunts/cycle_moments/counting_overlap.py
CountingOverlap.lean (CountingOverlap.lean) checks the two occurrence matrices against the same integer kernel, computes their simple counts and overlaps, and proves equality of power traces for every natural exponent. OverlapBand.lean (OverlapBand.lean) checks (8) from the explicit matrix polynomial relation and diagonal multiplicity hypotheses, in every finite dimension. The Fourier realization, inertia argument, and passage from eigenvalues to those hypotheses are ordinary proofs. Exact source hashes and AXLE receipts are in FORMAL-CHECKS.json (FORMAL-CHECKS.json).