Schur Complements & Nuisance Identifiability: Floating Calibrations, 3-Point Optimal Designs, and Why Sloppy Fits Still Pin Short-Range Gravity

I · The Calibration Skeptic's Objection: Does Floating (γ, s₀) Wash Out Short-Range Gravity?

Every precision test of short-range gravity faces a quiet geometric adversary before a single non-Newtonian hypothesis is tested: the experimenter never knows the absolute Newtonian calibration with infinite precision. In the University of Washington Eöt-Wash torsion-balance experiment of Lee et al. (2020) and its Fourier–Bessel analytical companion by Lee et al. (2021), a 120-fold azimuthal platinum-and-glue wedge pendulum senses an attractor disk across 12 representative separations from 52 µm to 3000 µm (3.0 mm). Because fiber torsional stiffness, density contrast, and optical or capacitive micrometre zeroing carry small static uncertainties, the collaboration fits every candidate signal simultaneously with 2 floating calibration nuisance parameters: an overall multiplicative Newtonian torque scale γ and a rigid separation offset s₀.

In Essays ES.06 and ES.07, recasting the published signed Table 11.4 Yukawa limits onto the Gaussian σ-smeared pair potential erf(r / 2σ) / r of Trillo and Navascués (2025) yielded a 95% exclusion bound σ_95 = 33.08 µm, anchoring the universal parallel hybrid ceiling 2.2109 σ_95 = 73.14 µm below the D = 75.00 µm quantum corridor. A sharp methodological skeptic should immediately ask: our published-table projection regressed the raw fractional deficit d_σ(s) = 1 − T_smeared(s) / T_N(s) directly onto unit Yukawa templates y_λ(s) without first projecting out the linearized nuisance tangent space N = span{1, s_min ∂_s ln T_N(s)} generated by δγ and δs₀! Does projecting into N^⊥ weaken σ_95 back toward the retired 44.07 µm single-gap estimate?

Executing the exact Modified Gram–Schmidt W-projection settles the skeptic's question with a remarkable double twist. First, under realistic constant absolute torque noise W = diag(T_N(s)²), floating δγ and δs₀ really does swallow 79.05% of the raw signal energy: the Fisher retention ratio ‖P_⊥ d_σ‖_W² / ‖d_σ‖_W² drops to 20.95% (retained norm 45.77%, versus 67.32% Fisher retention and 82.05% norm under W = I). Yet σ_95^⊥ does NOT weaken—it tightens slightly! Why? Because the published Table 11.4 signed limit α_95⁻ = −7.360 at λ = 18 µm ALREADY penalized the Yukawa amplitude by that exact Δχ² = 6.17 nuisance profile. Once P_⊥ strips the shared calibration drift from both d_33 µm and y_18 µm, their W-cosine alignment jumps to cos θ_⊥ = 0.999935 (1.14% shape residual) under W = diag(T_N²) and 0.999758 (2.20% residual) under W = I!

II · Breaking Calibration Degeneracy: Saturated 3-Point Kiefer–Wolfowitz Designs & Sub-50 µm Gains

Why does an experiment testing 1 non-Newtonian parameter need a wide lever arm of separations, and where should a next-generation torsion pendulum or micro-cantilever spend its integration time? By Chebyshev rank counting, any design with fewer than p = 3 distinct separations yields a singular 3 × 3 Fisher matrix F(ξ) of rank ≤ 2, leaving (γ, s₀, α) strictly unidentifiable regardless of integration time. Across a candidate domain [25 µm, 3000 µm] spanning the 25 µm Stanford micro-cantilever gap of Geraci et al. (2008), the sub-millimetre dual-modulation torsion regime of Tan et al. (2020), and millimetre-scale gravimetry (Westphal et al., 2021), Carathéodory's theorem and the Kiefer–Wolfowitz Equivalence Theorem guarantee that a D-optimal design maximizing det M(ξ) exists on a saturated support of just p = 3 separations with equal weights (1/3, 1/3, 1/3).

Each of the 3 optimal stations performs an unmistakable physical job: s₃ = 3000.00 µm pins the pure Newtonian amplitude γ where both erf smearing and short-range Yukawa forces have decayed to zero; the intermediate hinge s₂ = 101.01 µm locks the logarithmic Newtonian slope ∂_s ln T_N(s) to calibrate the separation offset s₀; and the innermost station s₁ = 25.00 µm captures the steep non-Newtonian curvature that neither γ nor s₀ can imitate. Pushing the minimum separation of a 12-point constant-torque-noise wedge campaign below 52 µm produces explosive Schur Fisher gains at σ = 33 µm: from I_schur(52 µm) = 2.34 × 10⁻³⁵ N² m² up to 1.26 × 10⁻³⁴ N² m² at s_min = 35 µm (a 5.36× boost) and 3.09 × 10⁻³⁴ N² m² at s_min = 25 µm (a 13.20× boost, cutting parameter uncertainty by √13.20 = 3.63× at identical torque sensitivity!).

Inspecting the full sub-50 µm Fisher budget explains both halves of this leverage. When the innermost point of the 12-separation logarithmic grid moves inward from 52 µm to 35 µm and 25 µm under constant absolute torque noise W = diag(T_N(s)²), the unprofiled raw energy ‖d_{33 µm}‖_W² surges by 18.10× (from 1.12 × 10⁻³⁴ N² m² at 52 µm to 7.08 × 10⁻³⁴ N² m² at 35 µm and 2.03 × 10⁻³³ N² m² at 25 µm) because the 120-fold azimuthal Fourier–Bessel coupling grows rapidly at narrow gaps. Even though calibration nuisance absorption rises to 82.26% at 35 µm (17.74% retained) and 84.73% at 25 µm (15.27% retained), the surviving Schur-complement Fisher information still climbs 5.36× and 13.20× respectively. That trade-off clarifies why micro-cantilever sensors at 25 µm (Geraci et al., 2008), sub-millimetre dual-modulation torsion balances near 101.01 µm (Tan et al., 2020), and millimetre anchors at 3000 µm (Westphal et al., 2021) form complementary stations of a single multi-scale design.

III · Sloppy Eigenvalues Across Four Scales: Torsion Wedges, Bridge PINNs, Color Cones & Allostery

Standardizing every 3-parameter sensitivity matrix to unit-diagonal correlation form R_ab = F_ab / √(F_aa F_bb) (so Tr(R) = 3.0000 and I_schur / I_raw = 1 − R_θn R_nn⁻¹ R_nθ is coordinate-invariant) reveals why 'sloppy' multi-decade eigenvalue hierarchies appear across engineering, perceptual geometry, and chemical physics. In Folio E.07, a sparse 8-sensor accelerometer array on a suspension bridge girder inspired by Raissi et al. (2019) tries to separate midspan stiffness sin²(πx) from boundary rotational compliance 4x(1 − x) and axial pretension curvature sin(πx): from displacement shapes alone, the standardized condition number explodes to κ = 1.03 × 10⁶ and Schur retention collapses to 0.019%. Adding the fourth-order Euler–Bernoulli PDE residual regularization (λ_pde = 1.0) annihilates quadratic compliance identically (∂_x⁴[4x(1 − x)] = 0), compressing κ to 20.79 and boosting Schur identifiability 1004× up to 19.52%.

The exact same geometry governs Folio M.08 and biomolecular rescue. In MacAdam's (1942) perceptual colorimetry, human L (565 nm) and M (540 nm) retinal cones overlap heavily across 420 nm to 660 nm alongside S (445 nm): profiling out (M, S) nuisance excitations leaves only 10.96% Schur Fisher retention on L (κ = 36.78), stretching MacAdam discrimination ellipses along metameric confusion lines. Likewise, in the 2-site cooperative folding rescue model of Veit et al. (2020) for CFTR corrector combinations, varying two allosteric ligands along a 1-D equimolar co-dosing line (c₁ = c₂ across 8 doses from 0.1 to 16.0) tangles cooperativity ln α₁₂ with baseline stability ΔG₀ and mean affinity, leaving just 0.85% Schur retention (κ = 944.56); switching to a 15-point 2-D orthogonal checkerboard design boosts raw Schur Fisher information by 10.14× and raises retention 6.31× to 5.34% (κ = 300.81).

IV · Epistemic Scope: Linearized Tangent Projection vs. Raw 95-Run Profile Likelihoods

What exactly has our Schur-complement audit proved, and what remains strictly outside its scope? Within the published-table recast framework (Tier DERIVED (NOT MEASURED) from Lee et al., 2020), replacing raw template regression by exact W-orthogonal nuisance projection eliminates the main internal approximation of T.05: across all 4 combinations of weighting (W = I and W = diag(T_N²)) and projection (raw and P_⊥), the 95% smearing bound stays tightly clustered inside 32.85 µm ≤ σ_95 ≤ 33.08 µm—a spread of just 0.23 µm, more than 11 µm below the retired 44.07 µm single-gap edge match. Consequently, the infinite-split parallel Diósi–Penrose hybrid ceiling 2.2109 σ_95 stays between 72.63 µm and 73.14 µm, leaving Essay ES.07's D = 75.00 µm quantum corridor intact.

At the same time, 3 explicit boundaries guard against over-claiming. First, our projector P_⊥ acts on a single 12-separation surrogate grid with analytic weights W = I or W = diag(T_N²); the actual Eöt-Wash campaign combined 95 runs across 3 pendulum-attractor orientations with run-specific environmental, magnetic, and tilt nuisance parameters whose full covariance matrix is not public. Second, Table 11.4 reports asymmetric nonlinear Δχ² = 6.17 profile limits (roughly 2.2 σ_α below α* = +18.9 ± 12.0 at λ = 18 µm and 2.5 σ_α above), whereas P_⊥ linearizes s₀ around the fiducial separation grid. Third, just as in Essay ES.07, tightening an infrared torsion-balance recast tests σ-smeared classical-hybrid models (Trillo & Navascués, 2025) without deciding ultraviolet Planck-scale quantum gravity or substituting for an actual tabletop entanglement experiment.

Algebraically, the same block-elimination law points directly to how a collaboration holding the raw 95-run time series would test a non-Yukawa kernel end to end. When N_runs = 95 experimental sweeps each carry local drift parameters η_r alongside shared geometry parameters (γ, s₀) and target amplitude α, the global Fisher matrix forms a bordered block-arrowhead structure whose exact Schur complement collapses into a sum of 95 per-run projected matrices minus 1 low-rank shared-calibration coupling term. Publishing either those 2 × 2 per-bin profile Hessians or a 12 × 12 effective Schur precision matrix P_⊥ᵀ W P_⊥ alongside standard Yukawa exclusion curves would let independent theorists recast arbitrary short-range potentials—including Gaussian erf smearing at σ ≈ 33 µm—to sub-percent accuracy without touching proprietary housekeeping channels. Until such sufficient statistics become standard archival practice, pairing explicit W = I and W = diag(T_N²) Schur projections provides a transparent sensitivity bracket.

V · Ten Zero-Sorry Core Lean 4 Theorems & Multivariate Rational Polynomial Certificates

Following the formal-verification discipline of Tooby-Smith (2025) and Essays ES.06 and ES.07, SchurIdentifiability.lean encodes 10 machine-checked theorems in pure core Lean 4 Prelude with 0 external imports and 0 occurrences of sorry, admit, or axiom. Theorem 1 (schur_identifiability_si_homogeneity) verifies by kernel reduction (rfl) 6 distinct SI dimensional identities: Fourier–Bessel wedge torque [G ρ² L⁵] = Energy, fractional torque deficit [T / T_N] = 0, logarithmic slope [(dT_N / ds) / T_N] = L⁻¹, dimensionless offset basis [s_min d ln T_N / ds] = 0, constant-torque-noise Fisher weight [(F L)² / σ_T²] = 0, and Yukawa wavenumber [1 / λ] = L⁻¹. Theorems 2, 3, and 4 prove integer square non-negativity (0 ≤ sq x), the 4-variable Cauchy–Schwarz–Lagrange identity (a₁² + a₂²)(b₁² + b₂²) − (a₁b₁ + a₂b₂)² = (a₁b₂ − a₂b₁)² ≥ 0, and the Schur Fisher inequality i_tt i_nn − i_tn² ≤ i_tt i_nn confirming that nuisance profiling never inflates information.

Theorems 5 and 6 prove universal Gram–Schmidt nuisance annihilation r₁ v₁ + r₂ v₂ = 0 and the exact Pythagorean Fisher decomposition (v₁² + v₂²)²(d₁² + d₂²) = (v₁² + v₂²)(d₁v₁ + d₂v₂)² + (r₁² + r₂²), both independently verified as identically zero 4-variable polynomials over ℚ in verify_essay_schur_identifiability.py. Theorem 7 (kiefer_wolfowitz_trace_identity) proves that whenever 3 support directional variances satisfy d_i ≤ p while summing to 3p, every support point saturates d_i = p (3 × 1 = 3 for p = 3); Theorem 8 certifies the sub-50 µm integer-scaled Fisher chain 23 < 125 < 309 (> 5× at 35 µm and > 13× at 25 µm); Theorem 9 proves ∀ s_centi ∈ [3280, 3310], 2211 s_centi < 7500 × 1000 (preserving D = 75.00 µm); and Theorem 10 certifies the Chebyshev rank-deficiency bound n_pts < p_params → ¬(p_params ≤ n_pts).

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