Beyond Smooth Baselines: A Falsifiable Search for Universal Patterns

Beyond Smooth Baselines: A Falsifiable Search for Universal Patterns Matthew J. Hall∗1 1Independent Researcher, Wilmington, DE, USA September 17, 2025 Abstract Background: After standard baselines are removed, faint patterns sometimes linger across very different settings — from quantum interference and precision QED shifts (Casimir, Lamb) to cosmological spectra such as the CMB and the stochastic gravitational-wave background (SGWB). These recurring hints raise a simple question: are they traces of real physical structure, or just statistical noise? Methods: To address this, we outline a model-agnostic framework for searching such patterns. Candidate signals are tested with Lomb–Scargle periodograms and Whittle likelihood methods, combined with explicit trials correction and model-selection criteria. The design emphasizes reproducibility, portability across datasets, and pre-registration to make outcomes unambiguous. Results: We sketch a compact table-top experiment using time-domain double-slit interferometry as a direct quantum probe, and include a toy re-analysis on mock data. The toy exercise shows that the method returns clean nulls when no structure is present, yet reliably recovers weak injected patterns. Existing public datasets — Casimir, Lamb, CMB, and SGWB — offer immediate opportunities to apply the approach more broadly. Conclusion: The framework is built to be falsifiable. Consistent nulls would reinforce community baselines, while the recovery of common parameters across independent domains would call for deeper investigation of scale-invariant phenomena. Either way, the outcome moves us toward clearer limits on what current models can (and cannot) explain. 1 Introduction Subtle structures that remain after precision measurements are compared against well-established baselines have often provided the first hints of overlooked physics. From laboratory interference experiments to cosmological surveys, these “leftover” patterns are now being examined more carefully as possible indicators of hidden structure. Classic electron and photon double-slit experiments demonstrated interference as a direct manifestation of quantum coherence [1–3], while attosecond “time-slit” experiments showed that coherence can also be controlled in the temporal domain [4,5]. These systems set clean baselines: any unexplained modulation is immediately testable against well-understood predictions. Vacuum fluctuation observables such as the Casimir effect and Lamb shift extend this logic to precision QED. Both have been measured with extraordinary accuracy, from Casimir’s original proposal [6] to modern torsion-balance and microresonator studies [7], and from the first Lamb shift observations in hydrogen [8,9] to advanced QED calculations [10]. Here too, the baselines are so well characterized that even faint departures can be quantitatively isolated. ∗ORCID: 0009-0001-7066-2558 1 On the largest scales, the cosmic microwave background (CMB) has been mapped to subpercent precision [11–13], while pulsar timing arrays and ground-based interferometers probe the stochastic gravitational-wave background (SGWB) across complementary frequency bands [14,15]. Next-generation observatories such as LISA and DECIGO [16,17] promise to push these baselines even further. Collectively, these datasets represent the most stringent community-wide tests of scale-invariant structure. Searching for faint, repeating patterns in such data requires robust statistical methods. Lomb– Scargle periodograms [18,19], Whittle likelihood estimators [20], and information criteria such as AIC and BIC [21,22] provide well-tested tools for identifying weak signals in noisy contexts. At the same time, safeguards such as trials correction and pre-registration of analysis steps are essential to avoid false positives [23–25]. In this work, we present a model-agnostic framework to test whether shallow log-periodic patterns appear consistently across independent domains. The guiding hypothesis is that any genuine structure would emerge as repeating features in ln Xwith a common spacing parameter β, regardless of whether Xdenotes delay time, plate separation, frequency, or wavenumber. By bringing together laboratory probes (time-domain double-slit), precision QED measurements (Casimir and Lamb), and cosmological datasets (CMB and SGWB), the framework defines a deliberately falsifiable test: a consistent βacross systems would motivate new theoretical inquiry, while robust nulls would sharpen current baselines. 2 Scope, Positioning, and Falsifiability We emphasize from the outset that the present framework is not a proposal for new particles or fields, but a phenomenological residual template to be tested against accepted baselines. Specifically, we consider shallow oscillations in ln Xwith spacing β, R(X)∼cosβln(X/X0), as a generic functional form that can be applied across domains where smooth theoretical expectations are well established: quantum interference, vacuum fluctuation observables, and cosmological backgrounds. The positioning of this study is methodological. We consolidate statistical tools already standard in astronomy and physics—Lomb–Scargle periodograms for uneven sampling [18,19], Whittle likelihoods for stationary time series [20], and information criteria such as AIC and BIC for model selection [21,22]. Our contribution is to pre-register the use of these methods to search for a common βacross otherwise unrelated physical contexts. Falsifiability is built into the framework. A successful outcome requires a consistent, globally significant βacross independent domains, robust under data splits and null tests. Failure occurs under any of three conditions: (i) no peaks remain after correcting for trials factors and look-elsewhere penalties, (ii) inconsistent values of βemerge between domains beyond quoted uncertainties, or (iii) the signal is unstable under jackknife tests or disappears under randomized controls. We emphasize the need for pre-registration of search ranges, thresholds, and significance criteria to avoid hindsight bias, in line with best practices for modern data analysis [25,26]. 2.1 Quantum Interference and Time-Domain Double-Slit Interference phenomena remain one of the most direct illustrations of quantum coherence. Electron double-slit experiments, beginning with Tonomura’s classic visualization of single-electron build-up [1], have established that matter waves interfere with themselves even when particles traverse the apparatus one at a time. Analogous results with photons and neutrons reinforce the universality of the principle [2,3]. 2 Recent advances have extended the concept from spatial to temporal domains. In attosecond physics, “time-domain double-slit” experiments generate two well-separated ionization bursts by controlling the electric field of a driving laser [4,5]. The resulting photoelectron spectra display interference fringes whose spacing encodes the temporal separation of the slits. Such experiments have opened a new window for probing coherence in the femtosecond-to-attosecond regime. The theoretical modeling of visibility in both spatial and temporal slits typically invokes coherent superposition of two amplitudes with relative phase determined by path (or delay) difference [27]. Decoherence sources—finite coherence length, environmental noise, and spectral bandwidth—enter as envelope functions that reduce fringe contrast without altering the fundamental sinusoidal form. These models provide a reliable baseline against which any additional structure in residuals must be assessed. 2.2 Vacuum Fluctuations: Casimir and Lamb Quantum electrodynamics predicts measurable consequences of vacuum fluctuations, two of the best known being the Casimir effect and the Lamb shift. The Casimir force between parallel conducting plates was first derived as a consequence of altered zero-point modes in confined geometries, yielding an attractive force inversely proportional to the fourth power of separation L[6]. Modern precision experiments, including torsion-pendulum and microresonator setups, have verified this scaling to sub-percent accuracy while also quantifying corrections from finite conductivity, temperature, and surface roughness [7,28]. The Lamb shift, originally observed as a small splitting between the 2S1/2and 2P1/2levels of hydrogen, provided one of the earliest confirmations of vacuum fluctuations [8,9]. It is now calculable in QED with extraordinary precision, incorporating radiative, recoil, and vacuum polarization effects [10]. These two cases provide well-established baselines where residuals beyond the known theoretical corrections can be clearly defined and quantitatively tested. 2.3 CMB and Stochastic Gravitational-Wave Background On cosmological scales, vacuum fluctuations during inflation are believed to seed the temperature anisotropies observed in the cosmic microwave background (CMB). The Planck 2018 analysis remains the reference baseline, offering sub-percent level characterization of the CMB angular power spectrum [11]. Ground-based instruments such as ACT and SPT have extended these measurements to smaller angular scales, providing independent cross-checks and high-ℓ residual analyses [12,13]. In the gravitational-wave sector, pulsar timing array (PTA) collaborations have recently reported common-spectrum signals consistent with a stochastic gravitational-wave background (SGWB) [14]. At higher frequencies, LIGO, Virgo, and KAGRA have placed stringent isotropic SGWB upper limits across the 20–100 Hz band [15]. Future observatories such as LISA and DECIGO aim to bridge the frequency gap, probing mHz to deci-Hz windows with orders-ofmagnitude greater sensitivity [16,17]. Together, these datasets form the current community baselines for cosmological and gravitational-wave fluctuations. 2.4 Statistical Tooling Used by the Community The search for weak, structured signals in noisy datasets relies on well-established statistical methods. Lomb–Scargle periodograms provide a standard approach for unevenly sampled data, widely used in astronomy and physics [18,19]. For stationary time series, the Whittle likelihood furnishes an efficient approximate likelihood in the frequency domain [20]. Model selection and evaluation typically employ information criteria such as the Akaike Information Criterion (AIC) [21] and the Bayesian Information Criterion (BIC) [22]. In all such searches, care must be taken to correct for the “look-elsewhere” effect and to ap3 ply trials factors, ensuring that nominal significances reflect the true probability of spurious detections. These practices constitute the methodological baseline against which any claims of periodic residuals must be judged. 3 Methods The proposed framework is structured to allow reproducible searches for shallow log-periodic residuals across laboratory and cosmological datasets. We describe below the experimental protocols, re-analysis procedures, and statistical tools that form the methodological basis of this work. 3.1 Laboratory Protocols Two complementary table-top implementations provide direct experimental access to residuals in the time domain. The first is a photon-based time-domain double-slit, where two electrooptic or acousto-optic modulators define transmission windows separated by a variable delay ∆t. Interference visibility V(∆t) is measured in the far field using SPAD or sCMOS detectors, and residuals are computed relative to a calibrated baseline V0. The second is a distancedependent double-slit geometry, in which slit separation is varied in controlled steps, enabling tests for logarithmic periodicity in spatial interference. Trade-offs between photon and electron implementations are summarized in Table 1, and a schematic of the time-domain apparatus is shown in Fig. 3. 3.2 Re-Analysis of Existing Data The same residual template can be applied directly to established high-precision datasets: (i) Casimir force measurements as a function of plate separation L, (ii) Lamb-shift compilations across principal quantum numbers, (iii) CMB power spectrum residuals as a function of multipole ℓor wavenumber k, and (iv) SGWB posteriors as a function of frequency f. In each case, the accepted smooth baseline B(X) (QED theory, ΛCDM cosmology, or interferometer noise model) is used to normalize the observable Y(X), and residuals are defined as R(X)=Y(X)/B(X)−1. Negative results in any domain are reported as upper limits on the amplitude a1(β). 3.3 Statistical Framework Residuals are reparameterized onto a logarithmic grid x= ln(X/X0), and spectral searches are performed for oscillatory modulations of the form cos(βx +ϕ). For uneven sampling, Lomb– Scargle periodograms [18,19] are employed; for stationary series, Whittle likelihood methods [20] are used. Information criteria (AIC, BIC) guide model selection [21,22], and global p-values are computed via simulations to control the look-elsewhere effect [23]. Bootstrap and jackknife resampling provide empirical covariance estimates, while injection–recovery tests quantify sensitivity to shallow modulations. The overall analysis pipeline is summarized in Fig. 4. 3.4 Pre-Registration and Controls To avoid hindsight bias, analysis ranges, binning choices, and significance thresholds are specified in advance. Laboratory protocols include single-slit controls, gate-order reversal, and bandwidth variation to isolate systematics. For archival datasets, null tests are performed by phase-scrambling or segmenting data, ensuring that any significant βdetection is robust to surrogate analyses. A simulated recovery of an injected modulation is shown in Fig. 5. 4 Residual Template and Joint-Consistency Principle For each dataset, we consider an observed spectrum or measurement Y(X) and an accepted smooth baseline B(X) derived from community-standard theory or calibration. The normalized 4 residual is defined as R(X)≡Y(X) B(X)−1, x ≡ln(X/X0),(1) where X0is an arbitrary reference scale. This reparameterization maps multiplicative scalings in Xto additive translations in x, facilitating the search for periodic modulations in log-space. The residuals are modeled as R(x) = M X m=1 amcosm βx +ϕm+ϵ(x),(2) where amand ϕmare the amplitude and phase of the mth harmonic, and ϵ(x) denotes noise contributions modeled by an empirically calibrated covariance C[29]. This form corresponds to shallow log-periodic modulations superimposed on otherwise smooth spectra. A central prediction of this framework is the existence of a common spacing parameter βacross independent domains, such as quantum interference, vacuum fluctuation observables, and cosmological backgrounds. While amplitudes and phases may vary due to instrument response and physical context, the recovered value of βshould be statistically consistent if the underlying phenomenon is real. Inconsistency of βacross domains, or its disappearance under null and jackknife tests, constitutes a decisive falsification. This approach echoes log-periodic residual searches already used in critical phenomena and earthquake precursors [30], as well as harmonic analysis in cosmology where oscillatory features are sought in the primordial power spectrum [31]. By formalizing a cross-domain jointconsistency requirement, we provide a falsifiable criterion that separates universal structure from domain-specific systematics. 5 Pre-Registered Predictions (P1–P4) To ensure clarity and falsifiability, we state explicit pre-registered predictions (P1–P4). Each prediction specifies the observable, the expected form of residual structure, and the corresponding null condition. •P1 (Interference): In a time-domain double-slit experiment, the normalized fringe visibility V(∆t), after dividing by a smooth envelope function, is analyzed as a function of logarithmic delay ln ∆t. If the residual template is correct, shallow periodic modulations with spacing βshould be recovered. The experimental baseline is that visibility decays monotonically with increasing ∆tdue to finite coherence length [4,5]. Any log-periodic structure must therefore appear above this well-established trend. •P2 (Vacuum): After subtraction of standard QED predictions for the Casimir force and Lamb shift [7,10], residuals plotted against ln L(plate separation) or ln n(principal quantum number in hydrogenic states) should exhibit the same βas in P1. If no consistent βis present, the phenomenology is disfavored. •P3 (CMB ×SGWB): In cosmology, the residual power spectrum after subtraction of best-fit ΛCDM baselines [11] can be examined for oscillatory features in ln k. Likewise, stochastic gravitational-wave background analyses yield posteriors for strain power as a function of ln f[14,15]. A decisive prediction is that both domains recover consistent β values within joint uncertainties. Disagreement constitutes a falsification. •P4 (Nulls and Controls): Any genuine signal must survive robustness tests. Phasescrambled or surrogate datasets should eliminate the periodicity, confirming it is not an artifact of spectral leakage or binning. Conversely, jackknife resampling and split-sample 5 analyses should preserve the recovered βif it reflects underlying physics. Failure under these controls invalidates the claim. Together, these pre-registered predictions ensure that the framework can be decisively tested. Positive detection requires cross-domain consistency, while negative outcomes are equally informative as they rule out hidden log-periodic structure at the sensitivity achieved. 6 Experimental Protocol: Time-Domain Double-Slit (Table-Top) 6.1 Apparatus Two complementary implementations are considered, both based on standard interferometry components. In the first, a narrowband photon (or electron) source is directed through two fast temporal gates (electro-optic or acousto-optic modulators for photons; pulsed gating for electrons). These define two temporal apertures separated by a controllable delay ∆t. The spatial path is held fixed, and far-field interference is recorded by a single-photon avalanche diode (SPAD) array or equivalent detector. In the second approach, the apparatus is a conventional spatial double-slit interferometer with a coherent photon source (e.g. attenuated laser or single-photon emitter). The novelty lies in mounting the detection plane on a calibrated translation rail, allowing the slit–screen distance L to be varied systematically from the near-field to the far-field regime. This reframes the screen as a tunable probe of temporal evolution, converting successive positions into effective “frames” of photon dynamics [32]. 6.2 Measurement Procedure 1. Temporal gating variant: Calibrate a reference visibility V0at ∆t= ∆t0. Sweep ∆t on a logarithmic grid (e.g. multiplicative steps ×1.1) spanning several decades. Compute residual visibility R(∆t) = V(∆t)/V0−1, set x= ln(∆t/∆t0), and perform Lomb–Scargle and Whittle-likelihood scans in βwith pre-registered trials control. 2. Distance-dependent variant: Record interference fringes at multiple, evenly spaced detector positions L. Extract fringe visibility V(L) as a function of Land normalize to a short-distance reference. Classical optics predicts monotonic decay of V(L) with L, while the structured-time hypothesis predicts non-monotonic breathing and revivals with period ∆LT[32]. 6.3 Controls and Systematics Robust null tests are critical. For the temporal-gating method: operate with only one gate open (single-slit control), reverse gate order, vary source bandwidth, split the dataset by detector segment or acquisition time, and inject synthetic signals to confirm recovery. For the distance-dependent method: replicate with multiple photon wavelengths, vary slit separation, and perform runs under different environmental conditions (air vs vacuum). Standard optics predicts that wavelength or medium only affects fringe spacing, not visibility revivals. Observation of distance-periodic breathing independent of wavelength would be distinctive. Together, these protocols provide two independent probes—temporal gating and distancedependent detection—that share the same falsifiable criterion: monotonic decay supports conventional optics, while reproducible log-periodic revivals support structured temporal dynamics. 7 Re-Analysis Playbook for Existing Data We outline a reproducible workflow for applying the residual template to four well-studied domains: (i) Casimir force vs. plate separation L, (ii) Lamb-shift compilations vs. principal 6 quantum number, (iii) CMB power-spectrum residuals vs. ln k, and (iv) SGWB posteriors vs. ln f. The goal is to enable transparent re-analyses that either detect a common spacing βor set upper limits on the leading amplitude a1(β). Negative (null) results are equally valuable and should be reported as constraints. 7.1 Common Workflow 1. Data ingestion and baseline: Load measurements Y(X) and adopt the community baseline B(X) (QED/Casimir theory; Planck/ACT/SPT best-fit ΛCDM; PTA/LIGO SGWB baselines).[7,10–15,28] 2. Residual construction: Form normalized residuals R(X)=Y(X)/B(X)−1; define x= ln(X/X0). 3. Spectral search: Compute Lomb–Scargle periodograms for uneven xgrids and/or fit the cosine template with Whittle likelihood.[18–20] 4. Trials control: Pre-register the βscan range and harmonic content m≤M. Convert local to global significance via Monte Carlo or Gross–Vitells approximations for lookelsewhere effects.[21–23] 5. Joint consistency: If one domain yields a candidate ˆ β, test consistency across other domains by a joint likelihood or meta-analysis on βwith nuisance amplitudes/phases. 6. Robustness: Apply jackknife/split tests (time/segment/instrument), null surrogates (phase-scramble, bootstrap), and injection–recovery to validate sensitivity. 7. Reporting: Provide either (a) a globally significant common βwith uncertainties and goodness-of-fit, or (b) upper limits on a1(β) across the scan, including data and code for reproducibility. 7.2 (i) Casimir Force vs. L Dataset: Precision plate–plate force measurements with quantified corrections (finite conductivity, temperature, roughness).[7,28] Baseline: Theoretical B(L) from QED with accepted corrections. Analysis: Build R(L) and evaluate Rvs. x= ln(L/L0). Search for a cosine at β; check that detrending (choice of B) does not absorb periodic structure (vary smoothness/spline tension as a control). Null reporting: Quote 95% CL limits a1(β) over the scanned range; provide sensitivity from injection–recovery. 7.3 (ii) Lamb-Shift Compilations Dataset: Hydrogenic level shifts across principal quantum numbers with modern QED fits.[8– 10] Baseline: State-of-the-art QED predictions B(n) including radiative/recoil/vacuum terms. Analysis: Construct residuals vs. x= ln n(or ln(∆E/∆E0)). Search for shallow periodicity with βconsistent with (i). Null reporting: Limits on a1(β); show stability under alternative compilations and error inflation tests. 7.4 (iii) CMB Power Spectrum Residuals Dataset: Planck 2018 temperature/polarization spectra and ground-based ACT/SPT high-ℓ data.[11–13] Baseline: Best-fit ΛCDM transfer functions; residuals Rℓmapped to R(ln k) via standard ℓ↔k 7 approximations. Apply beam/foreground marginalizations as in official likelihoods. Analysis: Periodogram/likelihood search for βin R(ln k); account for multipole windowing and correlated errors (use published covariance). Control the large trials space (broad krange, multiple spectra TT/TE/EE) with global p-values.[18–20,23] Null reporting: Tabulate a1(β) limits for TT, TE, EE and combined; confirm that phasescrambles erase any apparent peaks. 7.5 (iv) SGWB Posteriors vs. ln f Dataset: PTA common-spectrum posteriors (e.g., NANOGrav 15-year) and LIGO/Virgo/KAGRA isotropic limits.[14,15] Baseline: Power-law or broken-power-law strain spectra B(f) used by the collaborations. Analysis: Build R(f) against ln f; include band-limited covariances and window functions. Search for βconsistent with (iii). For PTAs, respect Hellings–Downs spatial correlations as handled in released posteriors; for LIGO, use the stochastic cross-correlation pipelines’ covariances. Null reporting: Frequency-dependent limits a1(β) and combined CMB×SGWB joint constraints; publish analysis notebooks for replication. 7.6 Upper Limits and Global Significance For a scan over β∈[βmin, βmax], report: a95% 1(β) : the smallest amplitude excluded at 95% CL from the likelihood profile, (3) pglobal : the global false-alarm probability after trials correction (Gross–Vitells or MC), (4) ∆AIC,∆BIC : model-selection penalties when adding the cosine term(s). (5) A claim requires a globally significant peak and cross-domain consistency of βwithin quoted uncertainties; otherwise, results should be framed as upper limits with complete robustness checks. 8 Statistical Methods 8.1 Baseline Fitting and Detrending The choice of baseline B(X) is critical, as over-flexible models can absorb the very residual structure of interest. In practice, B(X) may be taken as a power-law, spline, or parametric form motivated by the physics of the system (e.g. QED corrections for Casimir/Lamb data; ΛCDM transfer functions for CMB).[11,28] Robustness checks include varying the smoothing scale or spline tension, and verifying that candidate periodicities persist. Apodization and binning choices can introduce leakage into the residual spectrum. We therefore recommend multiple windowing functions (Hann, Tukey, Kaiser) and bin-width scans to test for stability. This is standard practice in spectral analysis of physical data [29]. 8.2 Periodogram and Likelihood For irregularly spaced xvalues (e.g. CMB multipoles, Casimir separations), we employ Lomb– Scargle periodograms [18,19]. For uniform x, fast Fourier transforms can be used for efficiency. A complementary likelihood-based approach uses the Whittle likelihood for stationary residuals: −2 ln L(θ) = r−rθ⊤C−1r−rθ+ const,(6) where θ={β, am, ϕm},rare the observed residuals, rθthe model prediction, and Cthe covariance matrix estimated from noise modeling or split-sample methods.[20] 8 Uncertainties on βcan be obtained from the Fisher information matrix or from Monte Carlo simulations of synthetic datasets. Bootstrap resampling provides a further check against nonGaussian noise. The overall analysis pipeline, from baseline fitting to joint-consistency tests, is illustrated in Fig. 4. 8.3 Model Selection and Trials Model comparison employs information criteria such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), which penalize the addition of cosine terms cos(βx) relative to smooth baselines.[21,22] Because βcan span a wide range and multiple harmonics mmay be tested, trials factors (the “look-elsewhere” effect) must be included when quoting significances. This can be addressed either via Monte Carlo simulations or using analytic approximations such as the Gross–Vitells method [23]. For multiple datasets, false-discovery-rate (FDR) control can be applied to maintain overall error rates [24]. A detection claim therefore requires: (i) global p-values below conventional thresholds after trials correction, (ii) improvement in AIC/BIC over baselines, and (iii) consistency of βacross independent domains. 9 Results The framework was validated through simulated datasets and application to representative analysis pipelines. Results are presented in two categories: archetypal recovery tests using injected signals, and null or exclusion scenarios that define falsifiability. 9.1 Archetypal Signal Recovery Synthetic residuals with injected log-periodic modulations were generated to test the sensitivity of the analysis pipeline. Figure 5illustrates recovery of an injected cosine with amplitude a1= 0.02 at β= 4.5 in the presence of Gaussian noise. A prominent peak is recovered in the Lomb–Scargle periodogram at the injected frequency, with significance surviving global trials correction. These examples confirm that the pipeline correctly identifies shallow modulations when present, and quantifies their global statistical significance. 9.2 Null and Exclusion Outcomes In the absence of an injected signal, the pipeline produces no spurious peaks above the 95% global confidence threshold. Phase-scrambled and jackknife resampling confirm the stability of null outcomes, ensuring that significant peaks cannot arise from standard baseline fluctuations or statistical artifacts. Such null tests form the basis of publishing upper limits on a1(β) in real datasets. 9.3 Falsifiability Conditions To guide interpretation, a concise falsifiability checklist was constructed. The template is ruled out if: (i) no globally significant βpeak is found in any domain after proper trials correction; (ii) inconsistent βvalues emerge across independent domains; or (iii) apparent signals fail under splits, jackknifes, or surrogate null tests. This structure ensures that results, whether discovery or null, are decisive and reproducible. 10 Results Archetypes (Simulated Examples) To illustrate the expected outcomes of the proposed framework, we present simulated examples that demonstrate how the analysis pipeline behaves under different scenarios. These examples 9 Aspect Photons (EOM/AOM gating) Electrons (pulsed/RF gating) Source Narrowband diode/DPSS; SPDC/singlephoton optional Thermionic/photoemission gun; monoenergetic beam Gate EOM/AOM, ps–ns response RF deflectors/choppers, ns–µs Coherence time τcτc≈1/∆ν(laser linewidth) Energy spread ∆E⇒τc∼ℏ/∆E Count rate High (SPAD saturation caution) Lower (MCP/pixel detector efficiency) Alignment Standard optics (mirrors, irises) Beamline (lenses, apertures, stigmators) Environmental sensitivity Air currents, vibration, temperature Magnetic/electric fields, vacuum stability Complexity Moderate; off-the-shelf parts High; vacuum/beam diagnostics required Table 1: High-level trade-offs for photon vs. electron implementations. so a laser of ∆ν= 10 MHz yields τc∼100 ns and Lc∼30 m. For electrons with mean kinetic energy Eand spread ∆E, the temporal coherence is set by τc∼ℏ ∆E, λdB =h √2meE,(8) implying that strong energy monochromation (or narrow photoemission bandwidth) is essential for large ∆t. In practice, choose a logarithmic sweep of ∆t: ∆ti= ∆t0ρi, ρ ∈[1.05,1.2], i = 0, . . . , N, covering at least 2–3 decades while keeping ∆tmax ≲τc. As summarized in Table 1, photon and electron implementations offer complementary strengths and challenges. Photon setups benefit from off-the-shelf optics and high count rates, while electron variants require stricter vacuum and field control but probe different coherence regimes. B.3 Optical Layout and Alignment (Photon Variant) 1. Source and mode-cleaning: Launch a narrowband laser through a single-mode fiber (optional) to stabilize spatial mode; pick off a small fraction for reference power monitoring. 2. Temporal gates: Cascade two EOMs/AOMs to define temporal apertures. Drive with synchronized pulses (from an AWG or pulser) with programmable separation ∆tand duty cycle ≪1. 3. Interferometer geometry: Use a fixed-path Mach–Zehnder/Michelson so that only the temporal degree of freedom is varied. Equalize arms to within ≪Lc. 4. Detection: Image the far-field to a SPAD or sCMOS. Calibrate linearity (avoid SPAD dead-time saturation) and record timestamps if available. 5. Reference point: Define ∆t0where visibility V0is high and stable; monitor slow drifts by re-acquiring V0periodically. B.4 Beamline and Alignment (Electron Variant) 1. Source: Use a stable DC/photoemission gun; set energy Eand minimize ∆Evia monochromator/apertures. 16 2. Temporal gating: RF deflector or chopper to pass two short packets separated by ∆t; verify timing jitter <0.1 ∆tat smallest separations. 3. Imaging: MCP/phosphor or pixelated detector at far-field; calibrate point-spread function (PSF) and gain. 4. Field control: Magnetic shielding and active cancellation; residual fields can wash out visibility before temporal effects appear. B.5 Calibration, Baselines, and Error Budget Visibility extraction. Fit the fringe profile I(θ) at each setting to I(θ)=I0[1+Vcos(θ+ϕ)] , and record V=V(∆t) with uncertainty from fit covariance. Normalize to V0=V(∆t0) and form residuals R(∆t)=V(∆t)/V0−1. Instrumental baselines. Characterize: (i) gate impulse responses (rise/fall times, ringing), (ii) spectral bandwidth at the interferometer (OSA for photons; retarding-field analyzer for electrons), (iii) detector nonlinearity/dead-time, (iv) mechanical/vibrational noise (accelerometer/FFT). Systematic budget. Track contributions to σRfrom counting noise, drift, background subtraction, bandwidth/line-shape uncertainty, and timing jitter. Quote both statistical and systematic components; propagate to the periodogram/likelihood fits. B.6 Diagnostic Plots and Health Checks •Gate metrology: Measured temporal transmission vs. time for each gate; convolution model vs. measured pulse pair. •Coherence validation: Visibility Vvs. spectral bandwidth; extrapolate to narrow bandwidth to confirm expected monotonic trend. •Stability: V(∆t0) vs. run index; Allan deviation to detect drifts. •Controls: Single-gate (single-slit) run should eliminate fringes and any periodic residuals; reversed gate order should not change results. •Binning/apodization: Show persistence (or disappearance) of any βpeak across Hann/Tukey/Kaiser windows and multiple bin widths in ln ∆t. B.7 Recommended Operating Points Photons: laser linewidth ∆ν≲10 MHz, EOM rise/fall ≲200 ps, ∆trange 0.1 ps–100 ns, acquisition per point sufficient for SNR>50 on V. Electrons: energy spread ∆E≲0.1 eV, timing jitter <100 ps, vacuum ≲10−7mbar, magnetic field residuals <1 mG. B.8 Distance-Dependent Complement (Consistency Check) As an independent probe, a spatial double-slit with a translatable detection plane scans slit– screen distance Lwhile measuring V(L). Standard optics predicts monotonic decay of visibility (after accounting for beam divergence and detector PSF). Any reproducible non-monotonic revivals periodic in ln Lwould mirror the time-domain search and must pass the same robustness tests (single-slit null, wavelength variation, binning/apodization stability). 17 Narrowband laser Gate 1 (EOM/AOM) Gate 2 (EOM/AOM) 50/50 BS Recombiner SPAD / sCMOS AWG / Pulse Gen. OSA (bandwidth) Gate impulse response Detector linearity / dead-time ∆tprogrammable Two short windows create temporal “slits” Arms path-matched: ≪Lc Measure V(∆t) at each setting t ∆t ln(∆t/∆t0) R βscan (a) Source & gating (b) Interference & detection (c) Residual analysis Figure 3: Time-domain double-slit (photon implementation). (a) Narrowband laser passes two fast temporal gates (EOM/AOM) driven by an AWG to create two short transmission windows separated by a programmable delay ∆t; interferometer arms are path-matched (≪Lc) so that only the temporal degree of freedom is varied. (b) Far-field interference is recorded on a SPAD/sCMOS detector and fringe visibility V(∆t) is extracted at each setting. (c) Residual analysis is performed on normalized visibility R(∆t) = V(∆t)/V0−1 as a function of x= ln(∆t/∆t0) using Lomb–Scargle and likelihood scans in β(with trials correction). Calibration blocks (OSA for bandwidth, gate impulse-response, detector linearity) monitor systematics throughout. B.9 Failure Modes and Mitigations •Apparent periodicity from electronics: Check for harmonics in gate drivers/AWG (spectrum analyzer). Repeat with independent drivers. •Timing jitter masquerading as structure: Measure jitter spectrum; convolve with model; require that inferred βis stable under jitter deconvolution. •Spectral breathing: Bandwidth drift can modulate V; interleave ∆tpoints to decorrelate from slow drifts; monitor OSA/RFA continuously. •Detector systematics: Verify linearity; vary count rate; use different detector segments; require consistent βacross splits. 18 Raw data Y() Baseline B() fit Residuals R()=Y/B −1 Log-grid = ln(/0) Spectral search(Lomb–Scargle, Whittle) Candidate Robustness tests(jackknife, surrogates) Joint-consistencyacross domains Input Output: claim or upper limits Figure 4: Analysis pipeline for residual-template searches. Raw spectrum Y(X) is divided by the accepted baseline B(X) to form normalized residuals R(X). Residuals are reparameterized onto a logarithmic grid x= ln(X/X0). 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