Hub Ops Unified Opus — a synthesis of four research branches for one hub.

This monograph places four research branches — A (mathematical framework), B (digital shadow), C (purposeful systems), D (synthesis under common constants) — plus the Sort Intelligence (SI) and Label Training Certification (LTC) bodies side by side under one ratified set of constants, reconciles their contradictions, and cross-references their results. It is research-grade in the operational sense: every load-bearing claim is traceable to a measured empirical anchor, a contractual operational document (OJS GEMS #1760), or a clearly-flagged open problem, and carries a version-origin tag so a future reader can audit the trail.

What it explicitly does not claim is a single formal object that derives all four branches as special cases. Earlier versions advanced the §19 EKF-marginalization theorem as the formal bridge "unifying" the operational baseline (Part II) and the idealized digital shadow (Part III). That theorem is unproven and false in general — a marginal covariance is not "strictly larger" than the projected sub-block — and has been retracted to a conjecture. With its keystone gone, the honest description is juxtaposition: the parts are coherent and consistent, but the connective tissue is design intent and shared calibration, not a proven identity. The "self-referential / already-complete" framing of prior versions is removed.

A weaker, defensible sense of coherence does hold: every load-bearing claim is cross-referenced to its definition or theorem within the opus, so a coordinator can trace any tracker number back to the equation that produced it. The version lineage itself is described — as an analogy, not a derivation — using the backtrack-then-regenerate pattern of Yang et al. (2025), Step Back to Leap Forward (arXiv:2502.04404), offered as an organizing metaphor only.

Four structural constants govern the Twilight sort. Each is pinned to a single source of truth, metric_contract.json; where the body text reproduces an older derivation value it is labelled historical lineage and is not canonical. $\gamma$ takes exactly one value — 0.982 — everywhere in this document.

The $\kappa_{Z_3}$ vector on the SEAS basis runs Mon 0.413 · Tue 0.395 · Wed 0.385 · Thu 0.380 · Fri 0.322 (full-year mean $\approx 0.368 \pm 0.014$). The iGate basis sits $\approx 0.04$–$0.06$ lower at every phase ($\approx 0.31$–$0.33$; the historical "0.311"); the two bases differ by the CCHIL → PD-09 attribution rule — SEAS bundles all CCHIL volume / misload / LIB under PD-09 while iGate distributes by physical scanner location — not by phase scope.

Alongside the four structural constants, the canonical hub PPH is $\Pi_\text{SOR} \approx 120.3$ (the un-ratified six-week 124.1 is superseded), with the associated cost field $c = 1/\Pi$ — the per-package paid-labor cost the pre-sort plan minimizes when it converts a volume forecast into planned paid hours ($H = c \cdot V = V/\Pi$).

The single most consequential operational fact for the analytical framework is the role decomposition of the outbound operation. Three roles execute the package path; each has zero scanning overlap with the next; the iGate PPH metric attaches to exactly one of the three.

The implication: every iGate PPH figure is loader PPH, always. Sorter and pick-off performance is invisible to iGate — it surfaces only downstream, as missorts (L1), misloads (L2), and service failures. The contractual confirmation comes from a different angle than observation: the sorter-ojs (111 methods) and pickoff-ojs (87 methods) certification sheets contain zero scanning methods, while the loader sheet (168 methods) makes the bay-door ULD scan mandatory under Method #20. Any framework claim that treats sorter or pick-off behavior as a scanning-compliance problem is structurally wrong.

Each role carries a knowledge layer certified by the LTC quiz — sort-chart knowledge (L1), ZIP-exception routing (L2), scan discipline (L3) — with all three bound together by OJS GEMS Code #1760.

Part IV writes the observed PPH of a loader as the product of a physical-capacity floor and a residual split into two latent coefficients — commitment $c_j$ and alignment $\pi_j$:

This is a bookkeeping identity, not a falsifiable model. Because $c_j$ and $\pi_j$ are latent and back-solved — defined so the equation holds — any observed $\Pi_j$ is consistent with it by construction. The decomposition has no predictive content: making it predictive would require an independent estimator for $c_j$ or $\pi_j$ from observable data, which does not exist. Earlier versions stated this as a "theorem" (PPH as Purposeful Projection) and asserted independence of the multiplicative effects; both are removed. It is retained only as a vocabulary for talking about where unexplained PPH variance might sit — capacity vs commitment vs alignment — not as an estimator. The Marcus/Diana "worked examples" are now explicitly illustrative back-solves, not measurements.

One sharp caveat closes the section: the latent commitment multiplier $c_j$ here is distinct from the operational cost field $c = 1/\Pi$ of §02. The latter is a real per-package paid-labor cost used in forecasting; the former is a behavioral bookkeeping coefficient. They share a letter and nothing else.

v6.0 supersedes v5.7, v5.7_kappa-conformance, and v6.0-ds, reconciling all three into one authoritative head under a conservative rule: unproven flagship claims are retracted or demoted with a one-line reason, not re-proven. What changed: $\gamma$ resolved to one value (0.982) everywhere, the stale 0.958/0.938 figures kept as labelled lineage only; the §13.5-vs-§14.2 $\kappa$ contradiction resolved by demoting the "gradient-invariant zone share" theorem to a within-DOW conditional Proposition and naming the SEAS-vs-iGate denominator; $\Pi_\text{SOR}$ pinned to 120.3 with $c = 1/\Pi$ defined.

The keystone correction: the §19 marginalization theorem was retracted to a conjecture — it was the only claimed formal bridge unifying Parts II–III, so the "unified / self-referential" framing was dropped and the opus reframed as a synthesis-by-juxtaposition. Inherited theorem demotions from the cleaned branch heads followed: Jam-Breaker → conditional Proposition (the "Δ>1 whenever H_jam>0 OR V_gap>0" claim is algebraically false), Nash equilibrium → descriptive cost-ratio heuristic (there is no game; the supervisor has no payoff or strategy space), and the ~300-dim EKF migration-value RMSE numbers removed (the model was never built). Citations and statistics were fixed: Yang 2025 corrected to Step Back to Leap Forward and re-scoped as analogy; the Hackman "35–45% of variance" figure cut as unverifiable false precision; decorative mathematics (dependent type theory, category theory, Brouwer) removed — the validated ZIP ultrametric kept.

Honestly disclosed open items: the proposed EKF remains uninstantiated (the marginalization relation survives only as a conditional conjecture); $\kappa_{Z_1}$ and $\kappa_{Z_2}$ are still uncomputed, so the zone-share collapse is established for Z3 only; the full SEAS–iGate gap is "consistent with" the CCHIL rule, not yet "fully explained by" a matched-sort recomputation; and $\varepsilon_\text{schema}$ is retained as a "constant" with the caveat that $\sigma/\text{mean} \approx 0.77$.