Hub Operations Mathematical Framework — the operational baseline.

The Twilight sort inducts 90,000–120,000 packages each weekday across ~17 work areas and twelve primary-destination (PD) belts, on three nested timescales — each with its own data product and decision horizon. The framework binds them so a single mathematical object can serve all three:

Two structural results anchor the surface. The sorter is modeled as a discrete memoryless channel with right-stochastic transition matrix $M_{ij}=\Pr[\text{answer}=b_j \mid \text{correct}=b_i]$; accuracy is the diagonal, missorts are the off-diagonal, and per-sorter estimates carry a Wilson score interval rather than a point estimate. The Sort Quality Score (SQS) is a convex combination of four components — PPH, fidelity, staffing adherence, missort quality — each clipped to $[0,1]$ before weighting so that $\mathrm{SQS}\in[0,1]$ with interpretable bounds. v2.0 documents in place where the live-tracker code diverges from this specification rather than silently reconciling the two.

Every named structural constant is pinned to metric_contract.json — the ratified single source of truth — so the body cannot drift from the contract. Each is quoted with its basis and uncertainty; none is asserted.

The $\kappa_{9\text{-}12}$ SEAS-basis day-of-week vector is Mon 0.413 · Tue 0.395 · Wed 0.385 · Thu 0.380 · Fri 0.322 (full-year SEAS mean ≈ 0.368 ± 0.014). These are small-sample estimates (≈ 6 sorts per day of week): the Tue/Wed/Thu values are not statistically distinguishable from one another, and only the Monday-versus-Friday contrast — a ~9-percentage-point swing — is plausibly resolvable at this sample size. The Mon→Fri vector is the canonical object; a flat $\kappa = 0.311$ mis-calibrates Monday by ~0.10 (≈ 30% error) on the SEAS basis. PPH is always defined on the iGate-net ÷ SOR-hours basis; the SOR managed "PPH actual" field is not used, because it is inflated by district-level volume redistribution.

When a jam piles packages on a sort aisle, jam-breaker employees clear it — but during clearing they (a) work without scanning, contributing zero to iGate, and (b) may be clocked under overhead codes rather than production codes, so their hours never enter the SOR denominator. This produces two distortion layers on observed PPH that push in opposite directions: Layer 1 (hiding jam-breaker hours $H_\text{jam}$) compresses the denominator and inflates the rate, while Layer 2 (the unscanned scan-gap volume $V_\text{gap}$) deflates it.

The prior drafts asserted observed PPH is always biased upward. That claim is algebraically false. The exact distortion ratio $\Delta = \mathrm{PPH}_\text{obs}/\mathrm{PPH}_\text{L2}$ and its sign condition are:

The bias sign is therefore indeterminate when both $H_\text{jam}>0$ and $V_\text{gap}>0$: observed PPH is biased upward only if the denominator-compression effect $H_\text{jam}\cdot V$ exceeds the missing-volume effect $H_\text{prod}\cdot V_\text{gap}$, and downward otherwise. A worked counterexample — $H_\text{prod}=100$, $H_\text{jam}=10$, $V=1000$, $V_\text{gap}=200$ — gives $\Delta = \tfrac{110\cdot1000}{100\cdot1200} = 0.917 < 1$: a jam that biases the rate down, directly refuting the retracted statement. Accordingly v2.0 demotes this from a "Theorem (always upward)" to a Proposition (conditional sign), and the tracker reports the measured direction of the bias from the $(H_\text{jam}, V_\text{gap})$ pair rather than assuming inflation.

The routing function $f : Z \to B$ maps five-digit ZIP codes to belts. Define the longest-common-prefix length $\ell(z_1,z_2)$ and the prefix metric $d(z_1,z_2) = 10^{\,5-\ell}$ for $z_1\ne z_2$ (and $0$ otherwise). This is the one structural theorem of the framework that is both proved and externally verified:

The proof follows from $\ell_{13} \ge \min(\ell_{12}, \ell_{23})$ and the fact that $x \mapsto 10^{5-x}$ is decreasing. The operational consequence is prefix-based local constancy: ZIPs sharing a long prefix cluster together under $d$, and $f$ is constant on those clusters except where a SLIC-level rule cuts across a prefix. This is the load-bearing justification for prefix-based question selection in LTC and prefix-based aggregation in the tracker's per-zone summaries. Coordinator-side routing adjustments — a swap, an ad-hoc redirect, a one-day overflow override — are modeled as overlays: a finite partial function $o$ applied pointwise, $f \triangleleft o$, forming an idempotent monoid of partial overrides (v2.0 softened the prior unverified left-regular-band claim to this weaker, demonstrated one).

v2.0 reconciles three prior Branch A drafts (v1.27, v1.27_kappa-conformance, v2.0-ds) into one authoritative head and applies a publishability audit. What changed: Theorem 7.1 (Jam-Breaker) corrected to a conditional Proposition with the indeterminate sign and worked counterexamples; the ~40% self-certifying autoresearch appendix excised (its $N$=2–3 ADMM / Kan-extension / MDL "constants" removed); constants pinned to metric_contract.json (γ corrected from the stale 0.938→0.958 lineage to the full-corpus 0.982; ρ corrected to a 0.509 volume share, not a Pearson correlation; Π_SOR≈120.3 with $c=1/\Pi$ named). Statistical hygiene was applied throughout: the CURE × PPH correlations now report $n$, 95% CIs, and two-sided $p$ and are explicitly labeled not statistically distinguishable from zero (every $|r|<0.30$, every CI spans 0), with the "bottleneck-tail mechanism" demoted from a theorem to a hypothesis. The borrow/loan optimality condition was corrected to its true first-difference form; SQS components are clipped before weighting and the spec/implementation divergence is flagged in place; the decorative "Phase Markov chain" label was dropped for "phase partition."

Honestly disclosed limits: per-day-of-week sample sizes are small (≈ 6 sorts) and the κ DOW-vector and iGate-basis number are quoted from the contract / audit, not re-fit here; the 166→188.6→155 iGate avgPPH series is not reconstructable from any loader-derivable PPH metric and is carried as a historical UI artifact (deferred to Branch C), not a result; κ_Z1 and κ_Z2 remain uncomputed (a labeled data gap); the multi-period staffing optimizer is an engineering objective with no claimed approximation guarantee; and three SOR-exceeds-Hub sorts (2026-03-31, 04-02, 04-23) remain held back pending iGate/TMS disambiguation. OJS/GEMS and operator-confirmed facts are internal-source citations.