A sharp deletion bound and exponential reliability separation for homometric integer arrays
Let S ⊂ ℤ be finite, with distinct positions and unit sensor weights. For d > 0, let wS(d) count pairs {x, x+d} in S. Two equal-size sets are homometric when these counts agree for every positive d.
Fix a nonempty set L of positive lags represented in both arrays. Define ρL(S) as the smallest number of deleted sensors that eliminates at least one lag in L. The guaranteed deletion tolerance is ρL−1.
A = {0, 1, 5, 7, 8, 10, 12}
B = {0, 1, 2, 5, 7, 9, 12} L = {1, 2, 3, 4, 5}
For d = 1,…,12, both positive-difference histograms are
Nevertheless, their per-lag minimum deletion sizes for L are respectively (2,2,2,2,2) and (1,2,2,1,2). Thus ρL(A) = 2 and ρL(B) = 1. Reflection of A is not B, so this is not a trivial translated or reflected pair.
In A, deleting one sensor cannot hit both distance-one pairs. In B, deleting position 1 hits both. At lag 4, A has (1,5),(8,12); B has (1,5),(5,9). Other protected lags require two deletions in either array.
Lemma. For a fixed positive lag d, form the graph on S with edges {x,x+d}. If its maximal d-step runs have vertex counts ℓ1,…,ℓr, then
Proof. Every component is a finite path. Erasing the lag means deleting a vertex cover. A path with ℓ vertices contains ⌊ℓ/2⌋ disjoint edges, forcing that many deletions. Alternating vertices attain this number. Finally, losing any required lag is minimization over its individual deletion problem. ∎
This uses classical path matching and vertex-cover facts. Neither those facts nor sensor-failure analysis is claimed as new.
Theorem 1. If S,T are homometric finite integer arrays and L is a nonempty common represented positive-lag set, then
Proof. Write wd for their common lag multiplicity and m = mind∈Lwd. A vertex covers at most two lag-d edges, whereas choosing one endpoint of every edge always covers them. Consequently, for either array,
The ratio bound follows. ∎ The proof only needs matching multiplicities on L; full homometry is a stronger condition satisfied by our examples.
Sharp family. For every integer q ≥ 1, put
Their generating polynomials each acquire the same factor ∑jz18j, so full homometry is preserved. The nearest cross-block separation is 6 > max L. Lag graphs for L therefore consist of disjoint copies of the base graphs. Hence
Each array has 7q sensors and aperture 18q−6. The tolerance counts are 2q−1 and q−1; the factor-two claim is about minimum destructive cuts, not their tolerance ratio.
Let PS(ω)=∑x∈Seiωx. Expanding |PS(ω)|² expresses it entirely in the cardinality and difference histogram, so it agrees for each homometric pair at every real ω.
If sensors independently survive with common probability s=1−p, then
Diagonal terms use EIx=s; off-diagonal terms use EIxIy=s². Thus mean damaged spectra also agree. This does not assert equality of damaged-spectrum distributions, phase, or higher moments.
Let RS(q,p) be the probability of losing at least one protected lag in the q-block family. Fix p ∈ (0,1). A two-vertex edge has no surviving pair with probability u=2p−p². A three-vertex path has no surviving edge with probability v=p+p²−p³: either its middle vertex fails, or its middle survives and both endpoints fail.
Set a=u² and b=uv. Disjoint path components give these one-block lag-absence probabilities:
| Array / lag | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| A | a | b | a | a | b |
| B | v | b | a | v | b |
Since u−v=p(1−p)² and v−a=p(1−p)³, we have 0 < b < a < v < 1.
Proof. A lag disappears globally exactly when it disappears in all q blocks. Block independence raises its base absence probability to the qth power. A single lag event and the union bound give
Divide the inequalities and take qth roots. ∎ No independence between different lag events is used. Both absolute risks tend to zero. The unbounded ratio is not a claim of growing absolute failure probability.
| Sensors | RA at p=0.01 | RB at p=0.01 | Ratio |
|---|---|---|---|
| 7 | 1.17322939801 × 10−3 | 2.038322840599 × 10−2 | 17.37 |
| 14 | 5.09964704186222 × 10−7 | 2.04116734490999 × 10−4 | 400.26 |
| 21 | 1.98362629055186 × 10−10 | 2.06005027978686 × 10−6 | 10,385.27 |
Displayed risks are probabilities, not percentages; rounded from exact rational arithmetic. At 21 sensors, these are roughly one in 5.04 billion and one in 485,425 under this structural model.
For nonempty T ⊆ L, let hS(T,p) be the probability all lags in T are absent in one block. Inclusion-exclusion gives
All 31 terms come from the 128 base deletion states. In general this is not RS(1,p)q.
Homometry and nonuniqueness of autocorrelation are classical [4]. Sparse-array essentialness, fragility and destructive failure families were developed by Liu and Vaidyanathan [1,2]. Shared-sensor dependencies are explicitly discussed in recent array-failure work [3]. Those concepts are not contributions of this note.
Candidate scope. The result assembled here is the explicit full-homometry construction with a sharp cut ratio and a fixed-p exponential risk ratio. The cut bound itself is an immediate degree-two graph argument; the probability strengthening is elementary block amplification. The bounded review found no explicit match in the inspected sources, but identified a strong objection: this may be an example-level synthesis of established homometry and failure-family theory. Several foundational full texts and forward citations remain unchecked. The fixed-p amplification was derived separately and is not novelty-cleared. Specialist assessment is required before a novelty claim.
The accompanying dependency-free JavaScript programs run under modern Node.js:
node verify-homometric.js node verify-risk-amplification.js
The first checks all 128 deletion masks of each base array, complete difference histograms, cuts and ten expanded families. The second uses exact BigInt arithmetic and compares inclusion-exclusion against a separate 32-state intersection convolution. It also checks all 16,384 deletion masks of each 14-sensor array at p=0.01, 0.5 and 0.9. Finite computations test examples; the proofs establish the infinite statements.
Prepared through AI-assisted exploration initiated by Spooky (@5p00kyy). An AI assistant running in OpenClaw generated the construction, arguments, code and exposition; separate AI sessions challenged the mathematics and searched prior work. Retained reports distinguish these checks from human peer review. No proof-assistant formalization, human specialist endorsement or journal submission is claimed. This draft accompanies the public Equal Spectra explainer. No researcher contact was made for this release.