Himalaya Peak in Sgr A* Audit-Grade Time-Domain Glass-Crystal Dynamics from Gold-Regularized Z₄₈ HMM and a Joint Z₃ Residual Event

We present an audit-grade, falsifiable pipeline for detecting and characterizing a time-domain “glass-like crystal” structure in **EHT Sgr A\*** 2017 uvfits visibilities (ER6, day 097) across **LO/HI** bands at 230 GHz. The core conceptual stance is deliberately “metaphorical but operational”: the system behaves like a *glass that resembles a crystal*—exhibiting strong **local order** (high-coherence plateaus) while remaining **globally disordered** via metastable **hops** among multiple quasi-stable configurations. We formalize this using a latent **Z₄₈** ring state space with a visible **Z₂₄** channel, and define a cross-harmonic “fall” variable per time bucket, \[d^{(m)}_b=\mathrm{wrap}\!\big(\arg c^{I}_m(b)-\arg c^{P}_m(b)\big),\qquad \gamma_b=\mathrm{wrap}\!\big(d^{(48)}_b-k\,d^{(24)}_b\big),\]where \(c^I_m,c^P_m\) are per-bucket harmonic coefficients estimated from polarized visibilities with a ridge-regularized fit, \(P=(RL+\overline{LR})/2\), and \(k=1\). Critical controls reveal that time×uv-sector bucketing can create pathological degeneracies (e.g., \(m=24\approx m=48\) and \(\hat\gamma\approx0\)); we therefore pre-register and use **time-only buckets** (\(\sim 620\) bins) with large within-bucket \(\phi_{uv}\) dispersion (median circular std \(\sim 77^\circ\)), preventing “geometry locking” artifacts. To validate statistical significance, we distinguish naïve shuffles from the **correct nulls**. In particular, fall-coherence must be tested against an **m-decoupled null** that independently scrambles the polarization channel for \(m=24\) and \(m=48\) before refitting, breaking spurious shared-P coupling that survives simpler time or phase scrambles. Under this null, the LO band exhibits strong separation (e.g., \(R_{\mathrm{fall}}\approx0.626\) versus null \(0.091\pm0.047\), with empirical \(p\le2\times10^{-4}\) at 5000 reps), indicating nontrivial cross-harmonic structure that is not explained by estimator coupling alone. However, multiple teflon controls indicate the signal is **not purely geometric**: \(\phi\)-scrambles do not universally collapse \(R_{\mathrm{fall}}\), while time-domain perturbations do, motivating a time-domain (domain/hops) prior rather than a purely \(\phi_{uv}\)-driven lattice model. We then implement the “gold rule” *in its intended soft form*—not as a hard hop set (e.g., ±19 steps), but as a **regularizer** that penalizes excessive hopping and large step sizes on the Z₄₈ ring. We fit a **gold-regularized HMM** to \(\gamma(t)\) with von-Mises-like emissions and transition cost \(-\lambda_{\mathrm{sparse}}\mathbf{1}[j\neq k]-\lambda_{\mathrm{step}}|\Delta(j\!\to\!k)|\). This dramatically increases evidence and yields physically interpretable glass dynamics: compared to a uniform-transition baseline, the gold-soft model increases forward log-evidence by \(\Delta\log Z\approx+340\) in both LO and HI, while reducing Viterbi hop counts from \(\sim 350\) to \(\sim 50\) and lowering mean step size from \(\sim3\) to \(\sim1\). Independent time-domain segmentation of \(\gamma_b(t)\) exhibits heavy-tailed dwell times (e.g., LO \(q_{90}/q_{50}\approx4.54\), HI \(q_{90}/q_{50}\approx3.23\)), matching the “glass-like crystal” signature (few long plateaus, many short ones). On top of the HMM, we introduce a **Z₃ triadic residual event detector**—explicitly as a *secondary* diagnostic rather than a foundational axiom. We define residuals \(r(t)=\mathrm{wrap}(\gamma(t)-\theta_{\hat s(t)})\) and compute a guarded sliding statistic \(S_3(t)=|\langle e^{i3r}\rangle|\) with strict interior-window constraints and effective sample size \(N_{\mathrm{eff}}\ge30\). Because independent LO/HI maxima can differ under unequal coverage, we report the **common-coverage** joint event by maximizing the product \(S_{3,\mathrm{LO}}(t)S_{3,\mathrm{HI}}(t)\) only where both bands pass guardrails. This yields an audit-grade detection and timestamp: a **joint HMM-residual Z₃ peak** at **2017-04-07T07:10:59.997 UTC**, with \(S_{3,\mathrm{LO}}=0.797\), \(S_{3,\mathrm{HI}}=0.897\), joint score \(0.716\), and a **look-elsewhere-controlled empirical p-value** \(p_{\mathrm{joint}}\le4.99975\times10^{-5}\) (0/20000 exceeds). The resulting picture is consistent with a time-domain, metastable, domain-structured process that exhibits triadic residual coherence and benefits strongly from a gold-soft sparsity/small-step prior. We emphasize that this framework is designed for replication and falsification: the same pipeline, null generators, and guardrails can be applied to additional uvfits scans/days and to other EHT sources (e.g., M87, 3C279) to test universality versus source-specific structure. ## Abstract (Version 2) We report an audit-grade, time-domain replication of the “glass–crystal / lattice” phenomenology originally developed for EHT Sgr A* (2017) by applying the same analysis protocol to M87* in two campaigns: the EHT 2017 April campaign (M87* 2017) and the EHT 2018 campaign (M87* 2018). Our working definition of “lattice” is operational (not geometric): a discrete-phase, time-domain organization consistent with a Z₄₈/Z₂₄ governance picture that produces localized, robust Z₃ coherence episodes after regularizing a heavy-tailed “glass” time series. We emphasize that we are not claiming a discovery threshold; the goal is reproducibility under matched nulls, guardrails, and robustness audits. Methodologically, we process uvfits visibilities in time-only buckets (cadence matched to the data, typically 10 s) and avoid uv-sector bucketing to prevent known sector-induced m=24/48 artifacts. For each bucket we estimate complex harmonic coefficients via ridge-regularized regression, fitting V(φ_uv) ≈ c₀ + c_m e^{-imφ_uv} for m∈{24,48}, followed by short complex-plane smoothing of c_m(t). In the full polarization mode (2017 products with non-zero cross-hands) we form I=(RR+LL)/2 and P=(RL+conj(LR))/2 and define a “fall” series from phase differences d_m(t)=wrap(arg c_m^I − arg c_m^P), then γ(t)=wrap(d₄₈ − d₂₄). To regularize glass-like hopping in γ(t), we implement a Gold-Soft Hidden Markov Model over Z₄₈ states with an emission score proportional to cos(γ−θ_k) and a transition prior penalizing both state changes and large step sizes. The detector operates on HMM residuals r(t)=wrap(γ−θ_state) using a sliding Z₃ statistic S₃(t)=|⟨exp(i3r)⟩| with strict guardrails: edge exclusion, minimum effective sample size (NEFF≥30), and a common-coverage joint score when both bands are present. All windows are specified in the dimensionless time τ=t/t_g with t_g=GM/c³ to support cross-source scaling. To make the result court-ready, we use (i) a global max-over-grid null test (look-elsewhere control) where each null repeats the same peak search and records the maximum score, (ii) high-N permutations with N_null=20,000 (yielding p-resolution ≈5×10⁻⁵ when no exceedances occur), (iii) blocky subsampling resilience (removing contiguous time blocks to preserve autocorrelation structure), and (iv) a hero-kill test that drops the top-K most influential buckets/baselines. Where multiple files/grids are scanned, we report explicit multiple-testing corrections (e.g., Bonferroni bounds). Results for M87* 2017 (LO/HI joint, full I/P mode) show a common-coverage peak at UTC 2017-04-11T02:53:26.733 (τ≈0.2649) with a joint-min score ≈0.877 and NEFF≈36.4 (LO) and ≈30.5 (HI). Under the global max-over-grid null with N_null=20,000 we observe zero exceedances, giving p_global ≤ 4.99975×10⁻⁵ (one-sided z ≥ 3.89). The event survives blocky resilience at ≈88% (100 trials) and remains distributed under hero-kill (drop top-5), retaining score ≈0.827 (≈5.7% drop). A conservative multiple-testing example (M=4) yields p_corr ≤ 2×10⁻⁴ (one-sided z ≥ 3.54). For M87* 2018, the released processed product is Stokes-I with RL/LR channels identically zero, so the I/P fall metric is undefined. We therefore run a strictly supporting replication using an I-only structural proxy γ^(I)(t)=wrap(arg c₄₈^I − 2 arg c₂₄^I) with the same HMM regularization, Z₃ detector, and audit stack. We find a peak at UTC 2018-04-21T04:18:49.448 (τ≈0.5904) with score ≈0.9714 and NEFF≈33.5. Again, the global max-over-grid null (N_null=20,000) yields zero exceedances, giving p_global ≤ 4.99975×10⁻⁵ (z ≥ 3.89), with strong blocky resilience ≈98% and minimal hero-kill sensitivity (score ≈0.963; ≈0.9% drop). Taken together, M87* 2017 (full polarization fall metric) and M87* 2018 (I-only structural proxy) demonstrate that the lattice-like Z₃ event phenomenology is not unique to the 2017 EHT campaign and persists under strict look-elsewhere nulls and robustness audits. These results support interpreting the original Sgr A* “lattice” as a time-domain glass–crystal governed by discrete Z₄₈/Z₂₄ structure stabilized by Gold-Soft regularization, while remaining below discovery-level (5σ) claims and motivating further tests with additional polarization-rich releases.