Negative on/off interaction: no recovery from the paper's inputs

Disposition

No positive portion of the ordered-gap floor follows from the paper's existing unconditional zero-side inputs. The obstruction is quantitative, not merely a missing sign argument.

The sharp universal coefficient multiplying extra on-line cross mass is zero. The interaction 2 tr(PQ) itself has no finite lower bound from rank, trace of P, and positive index of Q: replacing Q by -tP sends it to negative infinity while keeping n_+(Q)=0.

This is an information statement. It does not assert that a zeta-zero block realizes the families below. It says that the data passed to the paper's rank-trace step do not exclude them, so those data alone cannot yield the required strengthening.

Pinned inputs

• Local repository state at the start of this audit: 21ad0d49720a288b1d428d46664d623a0f6c4282.
• Paper PDF SHA-256: 6792988e6cd0e17690621ce898abd5d534f98407741bc7cb14bbe7d07c77d72f.
• Lean companion: anthropics/zeta-23-lean, commit 3635e74826a4c1fcece7d1cd2b6fa75e43a00510, tag v1.0.
• Upstream Mathlib commit: 51e6992efd06126df61a496bebf8f49482a4e129.

The decisive interfaces are Zeta23/ZeroSide.lean:470-555 and Zeta23/LinAlg/RankTrace.lean:157-195 at the pinned commit. They supply:

P is positive semidefinite
rank(P) <= r
tr(P) <= s
Q is Hermitian
n_+(Q) <= b
A = P + Q.

No hypothesis there couples an off-line hyperbolic block to the ordered on-line configuration.

Sharp inequality available at this interface

For every c > 0, the pinned rank-trace lemma gives

||A||_F^2
  >= c tr(P) - c^2 r/4 + 2c tr(Q) - c^2 b
   = 2c tr(A) - c tr(P) - c^2(r/4+b).

When r+4b > 0, optimizing this one-parameter family gives

||A||_F^2 >= ((2 tr(A)-tr(P))_+)^2 / (r+4b).

At the paper's c=2 this is

||A||_F^2 >= 4tr(A) - 2tr(P) - r - 4b.

Using only tr(P) <= s and r <= s reduces it to the paper's census form

||A||_F^2 >= 4tr(A) - 3s - 4b.

None of these expressions contains the positive on-line cross quantity

R(P) = tr(P^2) - sum_j ||u_j||^4
     = sum_(i != j) |<u_i,u_j>|^2.

The omission cannot be repaired with a positive universal coefficient.

Exact scalar family

For an integer m >= 1, put

n = 2m^2 + 2.

Take n simple on-line labels, each with scalar evaluation vector 1, and one off-line conjugate pair with evaluation vectors im and -im. Then

P = n
Q = (im)^2 + (-im)^2 = -2m^2 = -(n-2)
A = P+Q = 2
R(P) = n(n-1)
2tr(PQ) = -2n(n-2).

All entries are Gaussian integers. The paper's c=2 census bound has slack

||A||_F^2 - (4tr(A)-3n-4) = 3n.

Consequently any proposed universal strengthening

||A||_F^2 >= 4tr(A)-3n-4 + theta R(P)

must satisfy

theta <= 3/(n-1) = 3/(2m^2+1).

The right side tends to zero. Therefore every fixed theta > 0 fails for an explicit integer m. Even if the actual rank rank(P)=1 and actual positive index n_+(Q)=0 are inserted, the c=2 slack is only 2n-3, and the same zero-coefficient conclusion follows.

Matching the paper's prime-side moments

The scalar family has deliberately extreme moments. A direct sum removes that possible objection.

Let M = kn. Add:

• M orthogonal positive off-line blocks, each with eigenvalue 2+1/k;
• L orthogonal unit on-line blocks.

Each positive block is an exact off-line hyperbolic block, represented for example by the conjugate pair of real algebraic vectors sqrt((2+1/k)/2). The resulting quantities are

N = n + L + 2(1+kn)
tr(A) = 2 + kn(2+1/k) + L = N
||A||_F^2 = 4 + kn(2+1/k)^2 + L
paper-bound slack = 3n + n/k
R(P) = n(n-1).

For any fixed target C in (1,2), choose L as the nearest nonnegative integer to

(||A_0||_F^2 - C N_0)/(C-1).

Then ||A||_F^2/N approaches C. Taking k large drives the paper-bound slack per zero to zero. After that, taking m large makes R(P)/N exceed any prescribed positive floor.

The exact checker uses the paper's printed second-moment target C = 1327499296/10^9, with m=10, k=100000. Nearest-integer dilution gives

N                         = 123360900
L                         = 82960696
||A||_F^2/N               = 8188075400101/6168045000000
distance from C           = 122617/154201125000000
paper-bound slack/N       = 49591/10095000000
R(P)/N                    = 6767/20560150
twice the old LP floor    = 14371/500000000.

Numerically, these last three densities are

slack/N                   = 0.000004912431897...
R(P)/N                    = 0.000329131839991...
twice the old LP floor    = 0.000028742.

Thus the paper's trace identity, its printed Frobenius ratio, and more than the entire proposed additive gap floor can coexist with failure of the strengthened rank-trace inequality. The actual-rank version has still less slack, 2n+n/k-3 before normalization.

Missing invariant

The missing datum is a signed on/off incidence law. It must retain the joint distribution of:

• an ordered on-line zero or gap word;
• an off-line conjugate pair;
• the pair's horizontal displacement from the line;
• its ordinate relative to the on-line word;
• the signed kernel value 2 Re(B(x,z)^2);
• reuse of the same off-line pair across overlapping on-line words.

Separate on-line gap statistics and aggregate off-line counts cannot recover this information. Trace and Frobenius moments see only totals after positive and negative blocks have already cancelled.

Next configuration hierarchy

No decimal search should run before level 1 below has an unconditional zeta constraint that excludes the obstruction family.

The first useful theorem schema is not another unmarked gap floor. It is a conditional signed-incidence statement of the form

joint on/off incidence constraints
  => 2tr(PQ) + theta R_selected >= -E N

with exact theta > 0 and E < theta Delta. The zeta application may begin only when every incidence constraint on the left is already available unconditionally from the pinned paper or a separately established input.

Reproduction

From the repository root:

.venv/bin/python hunts/frontier_math/interaction_obstruction.py
.venv/bin/python -m pytest -q -o addopts='' \
  hunts/frontier_math/test_interaction_obstruction.py

Both commands use integer or rational arithmetic. No optimizer, sampled grid, or floating feasibility tolerance enters the obstruction.