CNT Risk Corridor Gauge Theory: Field–Connection Forecasting with Renormalized Couplings (Proof Battery v1)

This Proof Battery evaluates a CNT-style risk corridor gauge theory in which hazards are forecast as calibrated probability corridors rather than precise event times. We model a base risk field pΔ(t)=P(hazard within next Δ)p_\Delta(t)=P(\text{hazard within next }\Delta)pΔ(t)=P(hazard within next Δ) and transport it to multi-horizon corridor probabilities via a learned gauge connection pwin(t;H)≈1−c(H) S(t;H)κ(H)p_{\text{win}}(t;H)\approx 1-c(H)\,\mathcal{S}(t;H)^{\kappa(H)}pwin(t;H)≈1−c(H)S(t;H)κ(H), where S\mathcal{S}S is a survival term and (κ,c)(\kappa,c)(κ,c) are horizon-dependent connection parameters. We validate the framework across two space-weather observables—GOES X-ray flux (5-minute cadence) and GFZ Kp/ap (3-hour cadence)—using calibrated probabilistic scoring (Brier Skill Score vs climatology), ablations, out-of-era tests (Kp), and circular-shift surrogate nulls that preserve autocorrelation while destroying alignment. Results show statistically significant corridor predictability in both domains, scale-dependent “running coupling” behavior under renormalization (Δ changes), and strong multi-hour X-ray corridor transport when using Δ=15 minutes and path survival transport, with empirical surrogate p-values as low as ~1/5001 at 2–3 hour horizons. Methods (tight, publishable) Data Kp/ap: GFZ “Kp_ap_since_1932.txt”, subset to 2010–2025 (3-hour cadence), hazard thresholds defined by training-set quantiles (e.g., q=0.93q=0.93q=0.93, Kp≈3.667Kp\approx 3.667Kp≈3.667). X-ray: 5-minute resampled GOES-like X-ray flux series (log-flux used), with hazards defined either by an existing hazard_flag label or by training-set quantiles on log-flux for universality scans (e.g., q=0.95q=0.95q=0.95). Features (signal-first) Rolling and exponential-memory features built causally (trailing windows only), including rolling mean/std, slopes, EWMA deviations, and first differences.For Kp, ap features are included (rolling stats + differences).(Where hazard-memory features were used in earlier experiments, ablation analyses show signal-derived drift features dominate hazard nowcasting in Kp.) Two gauges (two target definitions) Exact / timing gauge: yexact(t;H)=hazard(t+H)y_{\text{exact}}(t;H)=\text{hazard}(t+H)yexact(t;H)=hazard(t+H) Corridor / window gauge: ywin(t;H)=1[∃ hazard in (t,t+H]]y_{\text{win}}(t;H)=\mathbb{1}[\exists\,\text{hazard in }(t,t+H]]ywin(t;H)=1[∃hazard in (t,t+H]] Field + connection Train a calibrated base field pΔ(t)=P(ywin(t;Δ)=1)p_\Delta(t)=P(y_{\text{win}}(t;\Delta)=1)pΔ(t)=P(ywin(t;Δ)=1) at a natural base scale Δ. Transport to horizon H using a survival term S(t;H)\mathcal{S}(t;H)S(t;H) and learned (κ(H),c(H))(\kappa(H),c(H))(κ(H),c(H)): pwin(t;H)≈1−c(H) S(t;H)κ(H).p_{\text{win}}(t;H)\approx 1-c(H)\,\mathcal{S}(t;H)^{\kappa(H)}.pwin(t;H)≈1−c(H)S(t;H)κ(H). Kp transport: block-style survival is adequate (slow dynamics). X-ray transport: path survival is required: S(t;H)=∏j=0m−1(1−pΔ(t+jΔ)).\mathcal{S}(t;H)=\prod_{j=0}^{m-1}\big(1-p_\Delta(t+j\Delta)\big).S(t;H)=j=0∏m−1(1−pΔ(t+jΔ)). Calibration Base models are calibrated using Platt scaling (logit calibration) on a held-out calibration tail of the training set. Evaluation Brier Skill Score (BSS) vs a climatology baseline (constant train corridor rate for each horizon). Out-of-era split (Kp): train through 2020-01-01, test afterward (and additional earlier era splits in the battery). Circular-shift surrogate tests: hold predictions fixed, circularly shift labels to form an autocorrelation-preserving null; report empirical p-values with +1 correction. Renormalization / coupling flow Repeat the pipeline across base scales Δ (Kp: 3h/6h/12h; X-ray: 5m/15m/30m) and measure: coupling capacity κ∞(Δ)\kappa_\infty(\Delta)κ∞(Δ) (running coupling), saturation time TcT_cTc (e.g., κ reaching 63% of its rise), shape milestones s50,s80s_{50}, s_{80}s50,s80 in scaled time s=H/Tcs=H/T_cs=H/Tc. Figure captions (for the plots in your ZIP) kp_bss_vs_horizon_hazard.pngBrier Skill Score (BSS) of Kp hazard corridor forecasts vs horizon (hours). Demonstrates long-horizon corridor persistence and the slow decay/half-life behavior in the Kp domain. kp_bss_vs_horizon_onset_clean.pngBSS of Kp clean-onset corridor forecasts vs horizon (hours). Highlights the distinction between hazard corridors and onset corridors, with onset predictability weaker but measurable. xray_micro_bss_vs_horizon.pngX-ray hazard corridor BSS vs micro horizons (minutes to hours). Shows fast decay at short horizons (spark class) and the corridor/regime behavior under window labeling. kp_vs_xray_hazard_overlay.pngCross-domain overlay of hazard corridor BSS vs horizon-hours for Kp (slow “tide” class) and X-ray (fast “spark” class). Illustrates stark separation of echo half-lives and motivates gauge/renormalization framing.