Einstein Audit for Faster‑Than‑Light Hidden Sectors (MIN): Causality Indices, Delta N_eff Guardrail, PTA Coherence & Ringdown GR Tests

Short description. The Einstein Audit for Faster‑Than‑Light (FTL) Hidden Sectors is an analysis‑only toolkit that turns classic FTL thought experiments into a concrete constraint space tied to real observables. We define population‑level causality indices from the tachyon antitelephone geometry, compress the FTL sector into a universal small parameter \epsilon_{\text{FTL}}^2 = \langle(W-1)^2\rangle, and combine this with a ΔN_eff guardrail auditor, a PTA Probability of Consistency Index (PCI), and a pooled ringdown deviation parameter ε to form an “Einstein distance’’ D_E. The condition D_E \le 1 becomes a no‑free‑FTL‑lunch inequality that bounds the RMS FTL excess, paradox wedge occupancy, and any FTL channel capacity allowed by current or future cosmology and gravitational‑wave data. The pack ships templates, minimal Python runners, and a LaTeX mini‑note so others can reproduce or stress‑test the audit. 2.3 Long description Einstein Audit for Faster‑Than‑Light Hidden Sectors (MIN) Status: testable framework / analysis‑only mini‑kit. This deposit does not claim real faster‑than‑light travel or violations of General Relativity (GR). Instead, it provides a reproducible “Einstein audit’’ that tells you how much room is left for any FTL‑like hidden sector, given cosmology and gravitational‑wave constraints. Core idea Tachyon antitelephone thought experiments show how superluminal signaling can generate closed causal loops. Here we turn that kinematic geometry into population‑level causality indices and connect them directly to ΔN_eff, pulsar timing arrays (PTAs), and black‑hole ringdowns. Causality layer. We define a paradox strength P(\beta,W) for each pair of relative speed \beta=v/c and FTL factor W=w/c, and identify a paradox wedge in the (\beta,W) plane where closed loops are possible. From a distribution \rho(\beta,W) over the hidden‑sector kinematics we build: Q[\rho]: wedge occupancy (probability that a random configuration lies in the paradox wedge), J[\rho]: mean‑square paradox severity, C[\rho]=\sqrt{J/Q}: conditional RMS paradox strength. In the near‑light regime W=1+\delta with \delta\ll 1, all of these collapse to a single universal parameter \epsilon_{\text{FTL}}^2=\langle\delta^2\rangle: Q\simeq\frac{1}{2}\epsilon_{\text{FTL}}^2, J\simeq\frac{1}{6}\epsilon_{\text{FTL}}^2, and C\to 1/\sqrt{3}. ΔN_eff guardrail. The hidden sector’s stochastic GW background is modeled as a smooth broken power law (SBPL) \Omega_{\rm gw}(f). A runnable ΔN_eff auditor (template JSON + Python snippet) maps the SBPL parameters into an extra‑radiation contribution \Delta N_{\rm eff} = \frac{8}{7}\!\left(\frac{11}{4}\right)^{4/3}\! \int \Omega_{\rm gw}(f)\,d\ln f and reports Ω_gw at 3 mHz and 25 Hz. We define an energy fraction \varepsilon_{\rm energy} = \Delta N_{\rm eff}/\Delta N_{\rm eff}^{\max} with a default guardrail \Delta N_{\rm eff}^{\max}=0.3. PTA and ringdown gates. Cross‑PTA agreement enters through a Probability of Consistency Index (PCI) computed from NG15/EPTA/PPTA “knee’’ posterior draws under a common prior (one‑column CSV templates included). Ringdown deviations from GR are summarized by a pooled parameter \varepsilon\pm\sigma_\varepsilon derived from per‑event QNM deviation posteriors (template CSV included). Einstein distance and no‑free‑FTL inequality. The pack combines these ingredients into a three‑component Einstein constraint vector \vec X = \Big(S/S_0,\ 1-\mathrm{PCI},\ \varepsilon/\sigma_\varepsilon\Big), where S=\varepsilon_{\rm energy}J is the causality–cosmology action and S_0 is a reference scale (default S_0=1/6). The Einstein distance is D_E = \|\vec X\|. The condition D_E\le 1 defines a unit “Einstein ball’’ of models that are jointly compatible with the ΔN_eff guardrail, PTA coherence, and GR‑consistent ringdowns. In the small‑excess limit this yields a no‑free‑FTL‑lunch inequality for the universal parameter, \epsilon_{\text{FTL}}^2 \le \frac{\sqrt{1-(1-\mathrm{PCI})^2-(\varepsilon/\sigma_\varepsilon)^2}}{\varepsilon_{\rm energy}}, which directly bounds the RMS FTL excess, wedge occupancy Q\simeq\frac{1}{2}\epsilon_{\text{FTL}}^2, and paradox severity J\simeq\frac{1}{6}\epsilon_{\text{FTL}}^2. Information‑capacity bound. A compact Monte Carlo pack (capacity CSV + notes) provides a toy FTL channel whose capacity obeys C(\varepsilon)\sim \varepsilon\log(1/\varepsilon) as the wedge occupancy \varepsilon\to0. Given the Einstein‑limited occupancy Q_{\max}, the toolkit returns an “Einstein capacity bound’’ C_{\rm FTL}^{\max} on the per‑use information capacity of any FTL‑like hidden sector that remains inside the Einstein ball. What this is and is not This is an analysis‑only, pre‑theory framework. It does not claim evidence for FTL signals or violations of General Relativity. Instead, it offers a concrete way to ask: if some exotic hidden sector allowed FTL‑like signaling, how large could its RMS excess \epsilon_{\text{FTL}} and channel capacity be without conflicting with ΔN_eff, PTA, and ringdown constraints? As real PTA posteriors, ΔN_eff limits, and ringdown tests improve, the Einstein ball shrinks and the allowed region in (\varepsilon_{\rm energy}, \epsilon_{\text{FTL}}) space becomes correspondingly smaller. Files in this MIN pack This MIN version is intentionally minimal and analysis‑only—no real PTA or ringdown data are included. It ships: \texttt{FTL_Chronon_Toy_Framework.tex}: LaTeX mini‑note with the full formalism and inequalities. \texttt{DeltaNeff_Audit_TEMPLATE_MIN_…json} + \texttt{DeltaNeff_Audit_RUN_MIN_…py}: SBPL → ΔN_eff guardrail auditor and multiband Ω_gw checkpoints. \texttt{NG15/EPTA/PPTA_knee_COMMON_PRIOR_TEMPLATE_MIN_…csv} + \texttt{pci_from_knee_samples_MIN_…py}: PTA PCI template and demo PCI runner. \texttt{ringdown_with_errors_TEMPLATE_MIN_…csv}: ringdown ε pooling template. \texttt{FTL_Thought_Experiment_Pack/}: core antitelephone formulas, wedge threshold CSVs, and worldline diagrams. \texttt{capacity_bound_vs_epsilon.csv} + \texttt{MONTE_CARLO_NOTES.txt}: information‑capacity bound curve and derivation notes. \texttt{README_Last_Mile_MIN_…md}: step‑by‑step instructions for running the audit and interpreting the Einstein distance. Researchers can drop in real PTA knee posterior CSVs and ringdown ε estimates, run the provided scripts, and obtain concrete bounds on \epsilon_{\text{FTL}}, Q_{\max}, J_{\max}, and C_{\rm FTL}^{\max} under their chosen ΔN_eff guardrail and Einstein‑distance threshold.