TB: Signal Detection under Background Uncertainty

• Write data density as f = η·f_s + (1−η)·f_b with signal shape f_s known, background f_b unknown. Expand f_b against any postulated background g in a basis containing the normalized score S† ∝ f_s/g − 1. Only the coefficient δ = ∫S† dF_b (the compensator) affects inference on η; all orthogonal error is irrelevant.
• With a background-only control sample, δ is a plain sample mean → asymptotically normal test with control-sample uncertainty propagated, and results nearly invariant to the choice of g (even a uniform stand-in for a steeply falling power law loses no power).
• Without a control sample, δ is unidentifiable; a dominating "diffused bump" injected into g guarantees δ ≤ 0 and thus valid-but-conservative inference, tuned by sensitivity analysis.
• LRT-based safeguard/spurious-signal methods break the χ²₀₁ boundary asymptotics: backgrounds that look visually excellent can still give stochastically inflated test statistics (δ > 0) and unavoidable false discoveries.
• Because the true background is unknown, toy-MC calibration is impossible in principle — valid asymptotics are the only game in town.

• Bump hunting with unknown signal location is handled by a supremum over a Gaussian process, calibrated via the Euler-characteristic heuristic — the statistician's formalization of the look-elsewhere effect.
• The framework directly critiques the data-driven background-estimation literature (density ratios, optimal transport — including Manole, Bryant, Alison, Kuusela & Wasserman 2024 on HH→4b) for not propagating background-estimate uncertainty into the test; the compensator supplies exactly that propagation.

Worked out while refereeing the binned-data companion (STAPRO-D-26-00599). Full note: ~/ClaudeBrain/projects/compensator-vs-hemisphere-mixing.md.

• The mapping. The CMS ZZ/ZH→4b background systematic (arXiv:2403.20241, §7) is structurally a compensator machine: postulated background g ↔ the JCM×FvT nominal prediction; background-only sample from F_b ↔ the 15 hemisphere-mixed synthetic datasets (signal-free, and — crucially — available IN the signal region and at high classifier score, where real 4b data is too thin); the compensator δ ↔ the residual of the mixed models vs. the prediction.
• Signal-adapted vs. signal-agnostic basis. The paper's first basis axis IS the signal shape S†, so it isolates the one scalar (δ) that can counterfeit signal. ZZ/ZH instead expands the residual in a signal-AGNOSTIC Fourier/Legendre basis and carries all coefficients as constrained nuisances.
• Measure-and-subtract vs. bound-and-null-test (the key difference). The compensator MEASURES δ on the background-only sample and SUBTRACTS its central value from the point estimate. ZZ/ZH does NOT form the S† projection: its variance step is an ensemble spread in a generic basis, its extrapolation step a generic-basis mean discrepancy, and its spurious-signal step a PASS/FAIL F-test on the signal direction. A detected signal-direction bias would be treated as an uncertainty to WIDEN, never a bias to REMOVE. That inflate-vs-subtract choice is the compensator's actual methodological contribution.
• The profile likelihood already projects out f_b^⊥ — but only in variance. In a Combine template fit (μ + background nuisances), the Fisher coupling between μ and nuisance a is ⟨s, d_a⟩ = the overlap of the nuisance direction with the signal template; a background nuisance orthogonal to the signal shape moves neither μ̂ nor σ_μ. So the fit is provably blind to f_b^⊥ — the same geometry as η depending on f_b only through its S† projection. BUT this controls σ_μ (variance), whereas the compensator δ is a MEAN. Un-parameterized signal-direction mismodeling flows into μ̂ as a bias with no warning inflation. Profiling neutralizes δ only if the nuisance basis spans S†, its assigned magnitude is large enough, and — the piece only external data supplies — its central value is informed by a control-sample measurement of δ. The compensator fills exactly those last holes.