Exact distinct-cycle corrections and information lost by power traces

The repeated-index corrections below are exact. Applying separate bounds to their terms can lose information, and power traces alone do not determine every corrected cycle statistic. A finite example proves that distinction. Neither the identities nor the example establish an improved zeta proportion. No novelty claim is made here.

1. Definitions, including multiplicities

Let K be a complex symmetric N by N matrix with K_ii=1. Set

M_j = tr(K^j), S = sum_i (sum_j K_ij^2)^2, Q = sum_(i,j) K_ij^4,

and let D_j be the sum of products

K_(i_1,i_2) K_(i_2,i_3) ... K_(i_j,i_1)

over ordered tuples of pairwise distinct indices. Products and powers are ordinary complex products, not absolute squares. The identities do not assume that individual terms are nonnegative.

Indices label occurrences. Distinct indices can have the same underlying zero location or feature vector. Thus this convention retains multiplicity and matches the distinct-index convention of the correlation theorem; replacing it by distinct locations would require falling-factorial multiplicity weights.

2. The third and fourth corrections

The exact formulas are

D_3 = M_3 - 3 M_2 + 2N, (1)

D_4 = M_4 - 4 M_3 - 2S + 10 M_2 + Q - 6N. (2)

For a partition F of the cycle positions, identify the indices within each block, leaving indices in different blocks free. Its inclusion-exclusion coefficient is

mu(F) = product_(B in F) (-1)^(|B|-1) (|B|-1)!.

For three positions, the three pair identifications give M_2 and the single triple identification gives N. This proves (1). For four positions, the full partition calculation is:

For example, identifying the first and third positions gives sum_(i,j,k) K_ij^2 K_ik^2=S. Identifying both opposite pairs gives sum_(i,j) K_ij^4=Q. These weighted overlaps differ from ordinary power traces. Replacing every identification of the same partition size by one power trace would therefore give an incorrect simplification.

There is a useful further identity. Define

V_3 = sum_(i,j,k pairwise distinct) K_ij^2 K_ik^2.

Separating j=i, k=i and j=k in S gives

S = 2M_2 + Q - 2N + V_3.

Consequently the quartic correction in FINITE-THEOREM.md is

Delta_4 = 2N - 5M_2 + 4M_3 - M_4 = -D_4 - 2V_3 + M_2 - Q. (3)

Together with Delta_3=D_3, equation (3) expresses that score in terms of exact distinct-cycle corrections. No inequality has yet been applied to any term.

3. Same full spectrum, different corrected fourth cycles

Take the following four unit vectors in R^2:

v_1=(1,0), v_2=(0,1), v_3=(c,s), v_4=(-s,c), c^2+s^2=1.

They form the union of two orthonormal bases. Let V be the two by four matrix with these columns, and K=V^T V their Gram matrix. Since VV^T=2I,

K^2 = V^T(VV^T)V = 2K.

Induction gives K^j=2^(j-1)K for j>=1, and therefore

M_j = 2^(j+1), j>=1. (4)

Equivalently the eigenvalues are 2,2,0,0. Every power trace is independent of the rotation. Also (K^2)_ii=2, so S=16. In contrast,

Q = 4 + 4(c^4+s^4) = 8 - 8c^2s^2.

Substituting N=4, M_2=8, M_3=16, M_4=32 and S=16 into (2) gives

D_4 = -8c^2s^2. (5)

There is also a direct check: a nonzero distinct four-cycle must alternate between the two bases. There are eight such ordered cycles, each with product -c^2s^2. Every other ordered four-cycle contains a zero inner product.

Two exact instances are:

Thus even the complete sequence of power traces does not determine D_4. The entrywise overlap Q distinguishes these frames. This demonstrates an actual information loss when a finite configuration is summarized by its spectrum. It does not identify an error in exact inclusion-exclusion. The example is a finite unit-vector family, not an asserted realization by the zeros of zeta or by one specified Fourier kernel.

SAME-KERNEL-EXAMPLE.md strengthens this construction: after adding a small identity component, both Gram matrices are realized by distinct real point sets for one explicit positive Fourier density.

4. What the corrections do to Fourier support

For K(z)=hat(p)(z), let p be a nonnegative compactly supported frequency profile. The unrestricted four-cycle kernel is

f_4(z_1,z_2,z_3,z_4) = K(z_1-z_2)K(z_2-z_3)K(z_3-z_4)K(z_4-z_1).

Its Fourier transform lies on sum xi_j=0, with density

Phi_4(xi) = integral p(t+xi_1)p(t+xi_1+xi_2) p(t+xi_1+xi_2+xi_3)p(t)dt. (6)

Suppose p>0 on (-lambda,lambda) and lambda>1/4. Choose 1/2<d<2lambda. Then

Phi_4(d,-d,d,-d) = integral p(t+d)^2 p(t)^2 dt > 0, sum_j |xi_j| = 4d > 2. (7)

This explicitly locates nonzero Fourier density outside the region furnished by the restricted-support correlation theorem. The same conclusion holds for the symmetrized four-cycle kernel: symmetrization adds nonnegative densities.

For nonzero score parameter t, the coefficient of M_4 in the quartic score is -t^2/D_t. The other power-trace terms have degree at most three in the counting measure. To see why those terms cannot algebraically cancel (7), replace the measure by sum_i w_i delta_(z_i) and extract the coefficient of w_1w_2w_3w_4. Only M_4 contributes. The lower-index corrections contribute nothing to that coefficient. Integer multiplicities suffice: an identity for every choice of nonnegative integer w_i is a polynomial identity.

Therefore exact reorganization by (1)-(3) does not itself remove the beyond-support four-variable term. It can move the unknown contributions among different statistics: Q uses the kernel K(z)^4, while S uses K(z_1-z_2)^2 K(z_1-z_3)^2. At a near-full-width p, these kernels also reach beyond the permitted Fourier region. They cannot simply be assigned the already-known pair or triangle constants.

This is an algebraic support statement about the displayed reorganization. It does not exclude additional arithmetic cancellation in expectations for actual zeta zeros, or a different counting inequality that retains entrywise information.

5. Relation to the source theorem and checks

Rudnick and Sarnak, Section 4 (https://www.math.tau.ac.il/~rudnick/papers/nlevelDuke.pdf) uses the partition-lattice coefficient in equation (4.4), unrestricted partition sums in (4.6), and exact inversion in (4.9). Their equation (4.14) pushes frequencies to block sums v_B=sum_(i in B)xi_i. The triangle inequality then preserves an already-valid bound sum|xi_i|<2. It does not assert that a larger original support shrinks into that region. Their Theorem 4.1 states the determinant identity on the same restricted region. The final passage also explains how the smooth version follows from their unconditional Theorem 3.1.

distinct_cycles.py checks direct enumeration against both independent partition inversion and the reduced formulas (1)-(2), using exact fractions. Its controls include a constant kernel, a nonconstant Gram matrix with a repeated vector, and both frames in Section 3. It checks K^2=2K, which pins all the traces in (4), not just a finite list of moments. The saved results are in distinct_cycle_results.json.

DistinctCycles.lean proves (1) over any commutative ring, retaining the all-equal add-back. IsospectralCycles.lean proves the complete finite unit-Gram example in Section 3, including every power trace and direct distinct-fourth enumeration. Both passed Lean 4.33.0 through AXLE with standard axioms only. The fixed-Fourier-kernel extension and support argument remain ordinary mathematical proofs.

The mathematical distinction is now explicit: exact inversion retains the overlap terms; a spectral summary can discard them. Whether retaining those terms improves a zeta bound is a separate question left open here.