Cross-Branch Synthesis — consolidating four lines of theory into one consistent reference.

Branches A, B, and C each developed along a single line of theory. Branch A formalizes the mathematical structure; Branch B builds the digital shadow / state model; Branch C specifies the purposeful-systems objective. Each is individually usable, and none, on its own, closes the loop from raw data architecture — what the coordinator's screen actually shows — through theory, to a concrete tool specification for the tracker.

Branch D juxtaposes the three, checks them for mutual consistency, and names the integration gaps. It does not formally unify them. There is no proof here that A's constraints imply B's state model, that B's state feeds C's objective, or that C's objective determines operational choices. The prior heads called this work a "unification"; v5.0 honestly relabels it a synthesis by juxtaposition, in keeping with how the data actually supports it.

The contribution is therefore readability, traceability, and constants coherence: one coherent narrative replacing four sequential patches (v4.0 → v4.1 → v4.1.1 → v4.1.2), with every Session 19 correction applied inline and every named constant tied to a single source of truth. The data-bearing content is unchanged from the v4.1.2 endpoint.

The Loading Bottleneck Regime (Branch B) describes a condition in which the rate-limiting step is dock-side, not floor-side: above some utilization, increasing scan rate fails to increase throughput because trailer loading creates back-pressure. Branch D makes this computable by joining CURE destination utilization to the LTC ZIP→belt truth table, producing a per-zone dock-pressure score. The belts are partitioned into coordinator zones — Zone 1 (PD-01…04), Zone 2 (PD-05…08), Zone 3 (PD-09…12), plus Smalls Sort and Air.

where $B(d)$ is the CURE-to-belt function, $U_d$ destination $d$'s utilization, and $V_d$ its loaded volume. v5.0 states the construction as a definition plus a falsifiable empirical hypothesis, not a theorem: when $P_{Z_i}$ exceeds the Twilight cap $U_T^\text{cap} = 0.85$, the bottleneck for $Z_i$ is hypothesized to be dock-side, so the correct response is dock-side intervention (trailer sequencing, door availability, early release of full trailers). Prior heads stated this as a theorem whose "proof" merely restated the definition and then asserted the conclusion; the causal claim was never derived or tested, so it is demoted.

A worked operator correction: L.L. Bean (SLIC 0432) showed CURE utilization of 88–98% and was earlier placed in Zone 3. Session 19 ground truth re-anchors it to Zone 2 (PD-05, Bottom Red, bay 116), reached via Smalls Sort → RNC bags → the two Smalls Sort Primary Feed belts → sort aisle. The 88–98% is bag-weighted cube utilization, which CURE weights differently than loose cube. The coupling mechanism still applies — but the zone is $Z_2$, mediated by smalls-sort bag throughput.

Quality propagates through three measurement layers, each measuring a different manifestation of the same upstream cause. Level 1 — Routing Knowledge (LTC): the Label Training Certification quiz measures whether an employee can correctly assign a ZIP to a belt; let $\bar{\rho}$ be the workforce routing accuracy. Level 2 — Load Accuracy (SEAS): wrong routing puts a package on the wrong trailer — a misload regardless of loader scan discipline. Level 3 — Service Failure (SFR): HUB DERIVED LIB per 10,000 PD packages.

where $\varepsilon_\text{struct}$ is a structural misload floor independent of routing knowledge (damage, label obscurement, outages), empirically $\approx 0.01$–$0.02$ at CHEMA (asserted, not yet sourced to a counted table). This was stated as a hard-bound theorem in prior heads. v5.0 demotes it to Claim 5.1, a structural hypothesis: the prior "proof" never justified the product form (no step warrants multiplying the two loss channels rather than subtracting), and it mixed two distinct employee populations — sorters'/pick-offs' quiz accuracy and loaders' FidelityScore — with no stated linking assumption. No new proof is offered; the missing independence/uniformity assumptions and the role-population mismatch are flagged as the reason it is not a theorem.

The operational implication survives the demotion: monitoring LIB and misload frequency without LTC accuracy is reading Level 3 without Level 1. When SFR is elevated, the diagnostic question is not "was the sort rushed?" but "do the sorters on this belt know their routes?" — high quiz accuracy with elevated SFR is a staffing or physical signal; low quiz accuracy with elevated SFR is a training signal.

A small set of quantities anchors pre-sort planning and live diagnostics. The 30-sort + 100-snapshot pass and the contract reconciliation revised them: $\gamma$ recalibrated at full-corpus scale; $\varepsilon_\text{schema}$ retired from constant status to a small empirical measurement; $\rho_\text{PD/Hub}$ corrected by 22%; $\kappa_{9\text{-}12}$ established as a DOW-indexed vector with a named basis. All values match metric_contract.json.

The canonical $\kappa_{Z3}$ on the SEAS basis is Mon 0.413 · Tue 0.395 · Wed 0.385 · Thu 0.380 · Fri 0.322, full-year mean 0.368 ± 0.014 (16 SEAS Hub Monthly files, Jan 2025 → Apr 2026). The iGate basis sits ~0.04–0.06 lower at every phase (≈ 0.31–0.33; the historical "0.311" is the iGate-basis summary). The two differ by the CCHIL → PD-09 attribution rule — SEAS bundles all CCHIL volume under the single PD-09 row, while iGate distributes it by physical scanner — not by phase scope. The tracker exposes both with a cchil_distribution: SEAS|iGate toggle. The other anchors: $U_T^\text{cap} = 0.85$ (a Twilight-scope cap derived from a mean of per-sort maxima, with a stated caveat that it is not a measured onset point); $\bar{\varepsilon}_\text{schema} \approx 0.04$ DOW-flat (an empirical measurement, not a structural invariant).

v5.0 reconciles three prior documents — v4.17, v4.17_kappa-conformance, and cross_branch_synthesis_v5.0-ds — into one authoritative Branch D reference, and applies the Branch-D rigor audit (whose verdict on the prior head was Reject / blocker). What changed: the primary blocker — the internal $\gamma$ self-contradiction, where the header banner said 0.982 while §6.2, the summary table, and the glossary said 0.958 — is resolved, with $\gamma = 0.982$ everywhere (0.958/0.938 retained only as explicitly-labeled historical lineage). $\kappa$ is corrected to the contract definition (a DOW-indexed vector on the SEAS basis) rather than the backwards-labeled flat scalars. The document is relabeled "synthesis" rather than "unification." adj_factor = 9200 is demoted to a fitted scaling factor. The tautology-theorems are demoted (CURE×PD → definition + falsifiable hypothesis; Fidelity Ceiling → structural claim; phase-aware $\kappa_z$ → stratification observation), conservatively, with no new proofs. The self-certifying autoresearch appendix is removed in full. The external citation is corrected and demoted: Yang et al. (2025), Step Back to Leap Forward: Self-Backtracking for Boosting Reasoning of Language Models (arXiv:2502.04404), is treated as a borrowed term/analogy only, not a load-bearing methodology.

Honestly disclosed open items (carried, not resolved here): the v3.6 avgPPH 166→188.6→155 trajectory is still not reproducible from any directly-derivable PPH metric in the corpus, so SOR PPH (Actual, 115–128) is canonical for the live tracker and the 166–188 series carries a "definition pending" badge; $\alpha_\text{CURE}$ remains a single-snapshot tilt with no validated variance band; the $U_T^\text{cap} = 0.85$ threshold is a summary statistic (mean of per-sort maxima), not a measured saturation/bottleneck-onset point, and should be re-grounded; and the formal cross-branch unification — how A constrains B, B feeds C, and C directs operations — is not demonstrated and remains an open research direction.