The Time-Domain Lattice in EHT Visibilities: A Glass–Crystal Audit Across Sgr A and M87**

We report an **audit-grade, reproducible pipeline** for detecting a discrete, time-domain phase organization in Event Horizon Telescope (EHT) interferometric visibilities, which we operationally describe as a **glass–crystal “lattice”**: long-lived **domains** with sparse **hops** (glassy kinetics) governed by discrete harmonic “teeth” (crystalline structure) at **m = 24 and m = 48**. The conceptual map is deliberately modular:

**Bank → Projector → Holonomy (operational proxy) → EHT statistics.**

Here *Bank* denotes an internal **discrete governance space** (empirically captured by a **Z₄₈ cover** with **Z₂₄ structure**), *Projector* denotes the time-dependent channel by which the internal state becomes visible in the measured visibilities (including coverage/sensitivity gating and slow connection-like drift), and *Holonomy* is used strictly in an **operational** sense: a phase-transport/twist proxy constructed from harmonic coefficients, **not** a claim of spacetime torsion or modified gravity dynamics. The statistical endpoint is a **Z₃ coherence detector** evaluated on **HMM-decoded residual phases** with strict guardrails and matched null models.

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## 1) Observables and the “Holonomy Proxy” (what we actually measure)

From uvfits random-group parameters we use **(u, v, DATE)** and compute the geometric angle  
**φ_uv = atan2(v, u)**. We explicitly avoid uv-sector bucketing because it can manufacture spurious harmonics (a documented artifact where m≈24 and m≈48 appear “fake” under sectorization). All primary runs use **time-only buckets**.

Within each time bucket we estimate the complex harmonic coefficient *c_m* by **weighted complex ridge regression** on the model

- **V(φ) ≈ c₀ + c_m exp(− i m φ)**  (m = 24, 48),

with ridge λ scaled by per-bucket sample count and a short **complex-plane smoothing** (window=3) applied to coefficients.

When full polarization is available (RR/LL/RL/LR), we form

- **I = (RR + LL)/2**  
- **P = (RL + conj(LR))/2**

and define per-harmonic “falls”

- **d_m = wrap( arg c_m^I − arg c_m^P )**

leading to the operational holonomy proxy

- **γ = wrap( d_48 − d_24 )**  (k = 1).

When only Stokes-I is available (or RL/LR are numerically zero), we use the structural proxy

- **γ_I = wrap( arg c_48^I − 2 arg c_24^I )**.

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## 2) Glass domains and “Gold–Soft” decoding (why we don’t chase noise)

The proxy γ(t) is decoded into a discrete **Z₄₈ state path** k(t) using a Viterbi/HMM-style objective with a **soft “gold rule”** transition regularizer:

- log T_{jk} ∝ −λ_sparse·1[j≠k] − λ_step·|Δ(j→k)| on Z₄₈,

combined with a cosine emission term weighted by per-bucket amplitude. This turns a brittle hop-set into a **regularized domain process**, sharply reducing spurious hopping while preserving real state changes. Residuals are then

- **r(t) = wrap( γ(t) − θ_state(t) )**, with θ_k = 2πk/48.

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## 3) The Z₃ detector (what constitutes a “lattice event”)

We define a sliding-window triadic coherence statistic on residuals:

- **S₃(t) = |⟨exp(i·3·r)⟩|**,

computed only under strict **guardrails** (window half-width H, edge exclusion EDGE, and minimum effective samples **NEFF_MIN**) to avoid boundary artifacts and sparse-coverage hallucinations. For dual-band analysis (LO+HI), we use **common-coverage scoring** and a joint statistic (typically product):

- **Joint(t) = S₃_LO(t) · S₃_HI(t)**.

Significance is evaluated with a **global max-over-grid null** (look-elsewhere corrected): for each null realization we compute the maximum joint score anywhere on the eligible grid and compare to the observed maximum. When exceedances are zero, p-values are reported correctly as **upper bounds** p ≤ 1/(N_null+1).

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## 4) Matched nulls and “court-ready” controls (how we prevent self-deception)

We use several complementary audit controls designed to kill common failure modes:

1. **m-decoupled null**: within each bucket we independently scramble the phases for m=24 and m=48 (and in FULL mode for P separately), preserving amplitudes and sampling while destroying coherent coupling.
2. **Global max-over-grid**: explicitly corrects the look-elsewhere effect.
3. **Re-decode null (conservative)**: re-runs the Viterbi decode inside each null iteration (heavier but stricter).
4. **Phi-scramble kill test (geometry destruction)**: per bucket, permute the mapping φ_uv↔V (preserve V marginals; destroy azimuthal harmonic coherence). A real geometric signal must collapse under this.
5. **Hero-kill**: remove top-K “most supportive” samples near the peak window and recompute the statistic; robust signals degrade mildly.
6. **Blocky resilience**: remove contiguous blocks (keep ~80%) and require peak/stationarity persistence.
7. **Station jackknife + integration ladder**: drop antennas/stations by BASELINE decoding; if detectability fails, increase bucket DT to test whether loss is **sensitivity-limited** rather than physics-limited.

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## 5) Results (Sgr A* 2017 day097, M87* 2017, M87* 2018)

### A) Sgr A* 2017 day097 (LO+HI, time-only, FULL mode)
Using HI+LO dtermcal uvfits and a common 60 s grid with guardrails (window points=41, NEFF_MIN=30), we find a **dual-band common-coverage joint peak**:

- **Joint S₃ ≈ 0.76** with **S₃_HI ≈ 0.90** and **S₃_LO ≈ 0.84**, at a well-interior UTC epoch (not a boundary artifact),
- **global p ≤ 4.99975×10⁻⁵** with **N_null = 20000** and **0 exceedances**.

This result is **court-controlled**:
- **Phi-scramble kill (FULL mode)** collapses the defined joint grid (geometric coherence required).
- **Hero-kill** reduces the peak by ~10% but does not destroy it (distributed support).
- **Re-decode null** remains significant at the depth tested (0/1000 exceedances → p ≤ 9.99×10⁻⁴).
- **Station jackknife** shows the event is not a single-station glitch; when a backbone station is removed, detectability becomes **time-resolution/sensitivity limited**, and is recoverable by integration ladder.

A key sensitivity demonstration: dropping the ALMA-identified station ID suppresses recoverability at fine DT, but increasing integration to **DT=150 s** restores detection with **p ≤ 5×10⁻⁵** under both **look-elsewhere** and **fixed-time** audits, supporting the interpretation “**sensitivity-limited, not source-limited**.”

### B) M87* 2017 (dual-band replication + “frozen giant” stationarity mode)
For M87* (2017) we reproduce joint coherence under the same audit philosophy (matched nulls, hero-kill, resilience). In addition, we adopt a **mass/time-scale consistent interpretation**: over an ~8 h EHT scan, M87* corresponds to O(1) gravitational time τ, so rapid event phenomenology is not expected to mirror Sgr A*. Instead we introduce a **stationary lattice mode** appropriate to a “frozen giant”:

- decode is **locked** (Hops ≈ 0 at DT=600 s),
- the relevant statistic becomes **Median S₃** over the core (stationarity),
- we explicitly identify a **residual drift** (connection-like nuisance) while showing that **Z₃ coherence persists** after detrending.

Controls cleanly separate effects:
- **Phi-scramble kills** the coherence (geometric dependence).
- **Time-shuffle randomizes** the drift without fabricating Z₃ coherence (trend ≠ signal).

With **N_null=20000**, we find **p ≤ 4.99975×10⁻⁵** for the stationary-lattice statistic at DT=600 s, consistent with a **stationary lattice confirmed** in the operational sense (locked state + significant geometric Z₃ coherence).

A **station jackknife** at DT=600 s (base and drops) retains significance with p ≤ 5×10⁻⁵ across tested scenarios, passing a Bonferroni-corrected threshold comfortably.

### C) M87* 2018 (single-band Stokes-I, pseudo-dual replication)
The available 2018 processed product contains RR/LL but effectively zero RL/LR, so FULL mode is not possible. We therefore:
- run **I-only** with γ_I = wrap(arg I48 − 2 arg I24),
- and introduce a **pseudo-dual split** (random 50/50 within bucket) to emulate dual-band jointness as an internal consistency test.

In pseudo-dual DT=600 s mode:
- Median S₃(A) ≈ 0.325 and Median S₃(B) ≈ 0.318 (both stable),
- **Hops = 0** (locked),
- drift_res persists (trend nuisance, consistent with M87 stationarity picture).

A 20k null suite yields **p ≈ 1.05×10⁻³** for the pseudo-dual joint statistic (supporting replication rather than headline). Importantly, a corrected kill-test based on **scramble-generated kill nulls** (NSCR=200) gives:
- scramble median ≈ 0.014,
- **p_kill ≈ 0.005**,
confirming that the pseudo-dual coherence is also **geometric** (φ_uv coherence required), not a time-only artifact.

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## 6) Unified picture and why the scaling matters (Sgr A* vs M87*)
The combined results support a single coherent interpretation:

- **Sgr A*** behaves like a **fast glass–crystal**, where event-like Z₃ coherence appears as localized peaks in time, visible simultaneously in LO and HI under strict common-coverage guardrails.
- **M87*** behaves like a **frozen lattice** over scan times: the discrete label k(t) is locked, while an instrumental/channel drift may appear in residuals; nonetheless, Z₃ coherence persists and is demonstrably geometric.

This is exactly the behavior expected under gravitational-time scaling: a fixed wall-clock observing night spans many characteristic times for Sgr A* but only ~O(1) for M87*, making “event vs stationary” a natural split.

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## 7) Scope and non-claims (explicit)
We emphasize again: “holonomy,” “torsion,” and “projector drift” are used **only** as **operational descriptors** of the inferred phase-transport proxy and its statistics. We do **not** claim:
- spacetime torsion,
- parity-odd gravity,
- or any modification of GR dynamics,
based on these results alone.

What we do claim is narrower and testable: a discrete harmonic-phase organization (m=24/48 with Z₃ coherence structure) is detectable in EHT products under a strict, kill-tested, look-elsewhere-corrected audit protocol, replicating across **Sgr A*** and **M87*** (2017 and 2018), with physically consistent behavior under mass/time-scale changes.

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## 8) Reproducibility artifacts
All analyses are implemented as **single-cell, numeric-only Colab scripts** that save “court packages” (.npz) containing:
γ(t), k(t), r(t), S₃(t), joint statistics, peak metadata, coverage masks, and null exceedance counts. This permits independent re-audit without re-running the full null suite.

**Bottom line:** the “Bank → Projector → Holonomy → EHT stats” chain is no longer a metaphor; it is an operationally defined, kill-tested measurement pipeline with cross-source replication and explicit separation of geometric coherence from instrumental drift.