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Purpose-Flux Belt Theory (PFBT) - Part V - VII

Part V — Control: Geometric Belt Control (GBC)

19. Flux Gates

Purpose. Flux Gates are real-time controllers that watch curvature in the purpose field and take decisive, auditable actions when it “spikes.” In PFBT, purpose is a connection $\mathcal A_{\Pi}$ ; its curvature $\mathcal F_{\Pi}$ is the driver of macro work. Flux Gates act on $\mathcal F_{\Pi}$ (and its time variation) to prevent destructive surges and to harvest constructive ones—without thrashing. This chapter specifies signals, thresholds, hysteresis, cooldowns, and the operational playbook (re-plan/reroute/failover) together with SLOs and a deployable ruleset format.

19.1 Signals: What a Flux Gate Watches

Let $\mathcal B_t$ be the belt surface active over a control window $[t-\Delta, t]$ with boundary $\partial \mathcal B_t=\Gamma_+(t)\sqcup(-\Gamma_-(t))$ (plan vs do).

Primary spike metric (work-aligned):

M_t \;\;=\;\; \left\langle \iint_{\mathcal B_t}\! \widehat{\mathcal F}_{\Pi}(\xi),\; \widehat{J}_{\Pi}(\xi)\right\rangle,

where $\widehat{\mathcal F}_{\Pi}$ is the estimated curvature (Ch. 16) and $\widehat{J}_{\Pi}$ is the current purpose-flow (plan/do allocation, resource routing). This aligns detection with domain units (e.g., “pairs of shoes per hour”).

Auxiliary features (all derived from belt-minimal logs; cf. Ch. 15):

Amplitude: $A_t=\|\iint_{\mathcal B_t}\widehat{\mathcal F}_{\Pi}\|$ .
Impulse: $I_t=\int_{t-\Delta}^{t} A_{\tau}\,d\tau$ .
Slope: $S_t=dA_t/dt$ .
Localized heatmap: $\phi(x,t)=\|\widehat{\mathcal F}_{\Pi}(x,t)\|$ to pinpoint cells for reroute.
Uncertainty: $\sigma_t=\mathrm{Var}[\widehat{\mathcal F}_{\Pi}]$ (from estimator posteriors).
Twist proximity: budget headroom vs coupling $\alpha$ (Ch. 17).

All gate decisions run on a filtered stream (median or biweight + EWMA) to suppress chatter:

\widetilde{M}_t=\mathrm{EWMA}(\mathrm{MedianFilter}(M_t;k),\;\lambda).

19.2 Gate Types

Hard Gate (binary open/close).
Enforces immediate policy steps (pause, freeze, circuit-break) when $\widetilde{M}_t$ crosses an “on” threshold; releases only after crossing a lower “off” threshold and cooldown.
Soft Gate (graded).
Outputs a control level $u_t\in[0,1]$ to scale levers (rate caps, WIP throttles, changeover pacing). Useful when throughput must not drop to zero.
Zonal Gate (spatial).
Applies hard/soft logic to sub-regions of $\mathcal B_t$ found by $\phi(x,t)$ clustering; enables reroute instead of halt.

19.3 Thresholding with Hysteresis

Let $\theta_{\text{on}}>\theta_{\text{off}}>0$ .

Binary logic

\text{state}_t=\begin{cases} \text{OPEN} & \text{if } \widetilde{M}_t\ge \theta_{\text{on}}\ \text{and not in cooldown},\\[2pt] \text{CLOSED} & \text{if } \widetilde{M}_t\le \theta_{\text{off}}\ \text{and cooldown elapsed},\\[2pt] \text{hold previous} & \text{otherwise.} \end{cases}

Graded control

u_t =\mathrm{clip}\!\left(\frac{\widetilde{M}_t-\theta_{\text{off}}}{\theta_{\text{on}}-\theta_{\text{off}}},\,0,\,1\right)^{\gamma},

with $\gamma\ge 1$ to be conservative near the off threshold.

Directional sensitivity. When sign is meaningful (e.g., cost-raising vs value-creating curvature),
use $\theta_{\text{on}}^{+},\theta_{\text{off}}^{+}$ for $M_t>0$ and $\theta_{\text{on}}^{-},\theta_{\text{off}}^{-}$ for $M_t<0$ . For example: open a harvest gate for beneficial spikes but cap for harmful ones.

Confidence gating. Require $p_t=\mathbb P[\widetilde{M}_t>\theta]\ge p_{\min}$ under the estimator’s uncertainty $\sigma_t$ ; otherwise degrade to soft gate with low $u_t$ .

19.4 Cooldowns, Rate Limits, and Anti-Thrash

Cooldown window $T_{\mathrm{cd}}$ . After an OPEN action, enforce a no-reopen or no-flip period. Typical: minutes for software deploys, hours for factory changeovers.
Rate limit. Max $K$ OPENs per $H$ hours; exceeding triggers failover tier (Sec. 19.6).
Deadband. Expand $[\theta_{\text{off}}, \theta_{\text{on}}]$ dynamically when volatility $\mathrm{Std}(M)$ rises.
Action half-life. Decay $u_t$ slowly even if $\widetilde{M}_t$ drops—protects against ping-ponging.

19.5 Gate Actions: Re-Plan, Reroute, Re-time

When a gate opens, the controller must choose a minimal-twist action that reduces curvature while preserving macro work. Preferred order:

Reroute within the belt (zonal): swap tasks, redistribute labor/machines/traffic along low- $\phi(x)$ cells; keep $\mathrm{Tw}$ small.
Re-time (pace/queue): stagger starts, stretch batch size, or insert micro-buffers to flatten $S_t$ .
Re-plan (edge updates): adjust $\Gamma_{+}$ commitments (plans) or $\Gamma_{-}$ execution modes. Use smallest twist step that closes the gap (cf. Ch. 17 quantized steps).
Circuit-break: stop a sub-flow and invoke failover (next section) when risk of macro-entropy surge exceeds budget.

Choice principle (micro-MPC): Select action $a\in\mathcal A$ to minimize a short-horizon objective

\min_{a}\ \ \underbrace{\mathbb E\!\left[\iint_{\mathcal B_{t:t+\tau}}\!\!\langle \mathcal F_{\Pi}^{(a)},J_{\Pi}^{(a)}\rangle\right]}_{\text{future harmful curvature}} \;-\;\lambda\,\underbrace{W_{\text{macro}}^{(a)}}_{\text{throughput}} \;+\;\mu\,\underbrace{\Sigma_{\text{macro}}^{(a)}}_{\text{entropy cost}} \;+\;\nu\,\underbrace{\Delta\mathrm{Tw}^{(a)}}_{\text{governance friction}},

subject to SLOs and current budgets. Implement with 1–2 step look-ahead using fast surrogates (no heavy solve in the loop).

19.6 Failover Tiers

Define progressive, pre-approved responses with escalation conditions:

Tier 0 — Local reroute. No plan change; keep $\mathrm{Tw}\approx 0$ .
Tier 1 — Bypass pipeline. Temporary alternate path; small $\Delta\mathrm{Tw}$ .
Tier 2 — Partial freeze & roll-forward. Halt the hotspot, open an alternate capacity pool; medium $\Delta\mathrm{Tw}$ .
Tier 3 — Global stop & rollback. Full circuit breaker; large $\Delta\mathrm{Tw}$ , invoked only when $\Sigma_{\text{macro}}$ explosion or safety risk is imminent.

Each tier has: (i) entry rule (thresholds + rate limit), (ii) exit rule (off-threshold + cooldown elapsed + residual $R$ back under cap), (iii) audit payload (Sec. 19.9).

19.7 SLOs and Budgets

Tie the controller to observable outcomes:

Detection latency: $L_{95}\le L_{\max}$ from spike onset to gate decision.
False open rate: $\le \epsilon_{\text{FO}}$ per day (validate on backtests with counterfactuals).
Throughput SLO: $W_{\text{macro}}$ within $[W_{\min}, W_{\max}]$ weekly.
Entropy SLO: $\Sigma_{\text{macro}}$ (WIP-age + rework + twist-cost) $\le \Sigma_{\text{cap}}$ .
Twist budget: weekly $\sum \Delta\mathrm{Tw}\le B_{\mathrm{Tw}}$ .
Residual closure: post-event residual $R$ (Ch. 18) decays below $R_{\text{target}}$ within $T_{\text{heal}}$ .
Escalation cap: $\le N_{\text{Tier2}}$ and $\le N_{\text{Tier3}}$ per quarter.

19.8 Reference Logic (Pseudocode)

# Inputs (streaming every Δt):
#   M_t: curvature-work metric; sigma_t: uncertainty; phi_map: spatial heatmap
#   budgets: {twist_headroom, entropy_headroom, escalation_counters}
#   thresholds: {theta_on+, theta_off+, theta_on-, theta_off-}
#   config: {cooldown, rate_limit, deadband_coeff, gamma}

state = "CLOSED"
last_action_time = -inf
opens_in_window = 0

def flux_gate_step(M_t, sigma_t, phi_map, now):
    global state, last_action_time, opens_in_window

    M_t_f = filter_stream(M_t)                 # robust median + EWMA
    conf_ok = prob_exceeds(M_t_f, sigma_t) >= p_min

    # hysteresis thresholds (possibly directional)
    th_on, th_off = select_thresholds(M_t_f)

    cooldown_active = (now - last_action_time) < config.cooldown
    rate_limited = (opens_in_window >= rate_limit(now))

    if state == "CLOSED":
        if (M_t_f >= th_on) and conf_ok and (not cooldown_active) and (not rate_limited):
            a = choose_action(phi_map, budgets)  # reroute->retime->replan->break
            execute(a)
            log_gate_event(now, "OPEN", a, M_t_f, sigma_t)
            state = "OPEN"
            last_action_time = now
            opens_in_window += 1
    else:  # state == "OPEN"
        if (M_t_f <= th_off) and (not cooldown_active) and healed_residual():
            a = choose_release_action(budgets)
            execute(a)
            log_gate_event(now, "CLOSE", a, M_t_f, sigma_t)
            state = "CLOSED"

    # graded control output (even while CLOSED)
    u_t = clip(((M_t_f - th_off) / (th_on - th_off)) ** config.gamma, 0, 1)
    apply_soft_controls(u_t, phi_map)

Action chooser (sketch).

Prefer zonal reroutes minimizing $\int\phi(x)dx$ and $\Delta\mathrm{Tw}$ .
If entropy headroom low: re-time.
If repeated spikes in same zone: re-plan.
If Tier 1–2 quota exhausted or safety risk: escalate to next tier.

19.9 Audit, Telemetry, and Safety

Every OPEN/CLOSE must emit a Gate Event to the belt registry:

Header: belt id, zone(s), version/basepoint, actor (auto/manual), tier.
Signals: $\widetilde{M}_t, A_t, S_t, I_t, \sigma_t$ ; thresholds applied.
Decision: action class (reroute/re-time/re-plan/failover), expected $\Delta R$ , expected $\Delta\Sigma$ , $\Delta\mathrm{Tw}$ .
Outcome (T+τ): realized $W_{\text{macro}}, \Sigma_{\text{macro}}, R$ and recovery time.
Safety: whether any SLO breached; if yes, attach post-mortem link.

This payload powers Ch. 26 benchmarks and Ch. 18 residual analytics.

19.10 Configuration Artifacts

Gate Ruleset (YAML)

gate_id: "factory-shoes/main-assembly"
window: {seconds: 300}
thresholds:
  positive:
    on:  8.0     # M units (pairs-of-shoes aligned)
    off: 4.0
  negative:
    on:  6.0
    off: 3.0
confidence_min: 0.8
cooldown: {minutes: 20}
rate_limit:
  opens_per: {hours: 6}
  max_opens: 3
deadband_coeff: 1.5
twist_budget_weekly: 12
tiers:
  - name: T0-reroute
    max_retries_zone: 2
  - name: T1-bypass
    alt_line: "aux-assembly-02"
  - name: T2-partial-freeze
    freeze_percentage: 60
  - name: T3-global-stop
    approvals: ["plant-head","safety-officer"]
slo:
  detection_p95_ms: 2000
  false_open_per_day: 0.2
  throughput_min_per_week: 42000    # pairs
  entropy_cap_index: 1.15

19.11 Domain Patterns (Illustrative)

Manufacturing (shoes). A leather cutting hotspot raises $\phi(x)$ . T0 reroute shifts stitching to line B, T1 bypass pre-assembles soles, keeping $\Delta\mathrm{Tw}$ small; entropy SLO honored as WIP-age stays below cap.
Software delivery. Curvature aligns with failure-correlated change clusters; soft gate caps deploy rate $u_t$ ; Tier 2 freezes a microservice slice while rerouting traffic; rollback (Tier 3) only when error-integral exceeds impulse cap.
Customer ops. Spikes occur on refund policy flips; gate holds the new script (re-time) and routes high-risk tickets to senior agents (zonal reroute) until residual $R$ clears.

19.12 Validation & Tuning

Backtest: Replay weeks of logs; compute ROC of gates vs incidents (entropy surges, residual spikes).
A/B belts: Run with/without Flux Gate; compare $W_{\text{macro}}$ , $\Sigma_{\text{macro}}$ , recovery $T_{\text{heal}}$ .
4π robustness: Verify invariants under framing changes; thresholds should be stable to basepoint shifts.
Sensitivity: Sweep $\theta_{\text{on/off}}$ , $T_{\mathrm{cd}}$ , $\gamma$ ; pick Pareto points (low false-open, high protection).

19.13 What to Hand Over

Gate Rulesets (per belt and per zone) with versioned thresholds and budgets.
Runbook: escalation tiers, approvals, and message templates.
Dashboards: live $\widetilde{M}_t$ , action state, budgets, SLO status, and post-event outcomes.
Test fixtures: curated spike traces for regression.

Takeaway

Flux Gates turn the identity gap = flux + twist into a controller: they sense work-relevant curvature, act with hysteresis and cooldown to avoid oscillation, and choose the smallest-twist intervention that preserves throughput while capping macro entropy. They are the first line of Geometric Belt Control and the foundation for the stabilization methods in the next chapters.

20. Twist Stepping

Purpose. Twist Stepping is the governance-side actuator of PFBT. When Flux Gates (Ch. 19) detect persistent, work-relevant curvature, Twist Stepping adjusts the system’s framing/policies—encoded by a parameter vector $\theta$ —to close the gap with the smallest possible governance change, i.e., minimal twist. Operationally, we choose a discrete policy move from a parameter ladder that achieves the target gap while respecting guardrails and enabling fast, safe rollback.

20.1 Objects & Notation

Purpose connection / curvature: $\mathcal A_{\Pi},\ \mathcal F_{\Pi}$ .
Belt & edges: $\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ (plan vs do).
Gap (edge-form, operational):
$\mathrm{gap}\;\equiv\;\oint_{\Gamma_+}\!\mathcal A_{\Pi}-\oint_{\Gamma_-}\!\mathcal A_{\Pi} \;-\;\iint_{\mathcal B}\!\widehat{\mathcal F}_{\Pi}.$
(Positive means “plan outruns execution” after flux adjustment.)
Twist (framing/governance): $\mathrm{Tw}=\mathrm{Tw}(\theta)$ .
PBHL: $\mathrm{gap}=\alpha\,\mathrm{Tw}+\text{residual back-reaction}$ .
Governance dial vector: $\theta\in\mathbb R^d$ (discrete in practice). Examples: release cadence, batch sizes, WIP caps, staffing rosters, escalation tiers, pricing nudges, routing rules, SLA clauses, safety gates, model/prompt versions.

We write the post-step predicted gap under a change $\Delta\theta$ as

\widehat{\mathrm{gap}}^{\,\text{post}} \;\approx\; \mathrm{gap}^{\,\text{now}} +\underbrace{\alpha\,\Delta\mathrm{Tw}(\Delta\theta)}_{\text{direct twist}} +\underbrace{\sum_{i=1}^{d}G_i\,\Delta\theta_i}_{\text{flux back-reaction}}

where $G_i$ are local gap sensitivities estimated from logs/experiments (Ch. 16–18).

20.2 Problem Statement (Minimal-Churn Step)

Given a target $g_\star\ge 0$ (often zero or a tight band), choose a discrete policy step $\Delta\theta$ from an allowed set $\mathcal L$ (the parameter ladder) that solves

\boxed{ \Delta\theta^\*=\arg\min_{\Delta\theta\in\mathcal L}\; \mathcal C_{\text{gov}}(\Delta\theta) \quad\text{s.t.}\quad \widehat{\mathrm{gap}}^{\,\text{post}}(\Delta\theta)\;\le\; g_\star,\; \text{Guardrails}. }

A simple, interpretable governance cost is

\mathcal C_{\text{gov}}(\Delta\theta) \;=\; w_T\,|\Delta\mathrm{Tw}| \;+\; w_0\,\|\Delta\theta\|_0 \;+\; w_A\,\text{approvals} \;+\; w_X\,\text{cross-team touches},

penalizing twist magnitude, number of knobs touched, required approvals, and blast radius.

Rule-of-thumb step size. If back-reaction is weak ( $G\approx 0$ ),

\Delta\mathrm{Tw}\;\approx\;-\frac{\mathrm{gap}^{\,\text{now}}-g_\star}{\alpha}.

Pick the smallest ladder move whose $\Delta\mathrm{Tw}$ meets the above, adjusted for $G$ .

20.3 Parameter Ladders (Discrete, Auditable Moves)

A ladder gives pre-approved rungs—small, composed policy moves with known deltas and risk profiles.

Examples (manufacturing):

WIP cap $-5\%$ , batch size $-10\%$ , changeover window $+15$ min, overtime $+1$ hr, cross-train pool $+1$ cell, QC sampling $+20\%$ .

Examples (software):

Canary % $+2/5/10$ , deploy freeze $+2$ h, autoscale floor $+1$ , circuit-break threshold $-15\%$ , retry backoff $+100$ ms, error budget mode ON.

Examples (customer ops):

Triage rule shift (A→B) for queue $X$ , script v. $k\to k+1$ , senior-review for risk class $R$ , callback window $+1$ day, refund cap $-5\%$ for 2 weeks.

Rung metadata (required):

Expected $\Delta\mathrm{Tw}$ , $G$ -vector, confidence, risk class, dependencies, approvals, rollout scope, blast radius, rollback recipe, SLO probes.

20.4 Guardrails (Pre-Flight Checks)

Before stepping, enforce:

Budgets: $|\Delta\mathrm{Tw}|\le B_{\mathrm{Tw}}^{\text{step}}$ , weekly $\sum|\Delta\mathrm{Tw}|\le B_{\mathrm{Tw}}^{\text{week}}$ .
Safety SLOs: no step may violate safety/ethics constraints even if it closes the gap.
4π robustness: invariants unchanged under framing/basepoint moves; rungs that break invariants are forbidden.
Decoupling sanity: if $\|G\|$ is large/uncertain, require canary + shorter probe window.
Blast radius cap: max affected volume/tenure; default to zonal rollout.
Interference with Flux Gates: if a gate is OPEN (Ch. 19), ensure step does not defeat cooldown/hysteresis; optionally pause soft controls during probe.

20.5 Rollout & Probe Windows

Each rung defines a probe window $[t, t+\tau_{\mathrm{probe}}]$ with:

Primary checks: gap ≤ $g_\star$ ; residual $R$ trending down; entropy $\Sigma_{\text{macro}}$ within cap.
Secondary checks: throughput $W_{\text{macro}}$ within SLO band; no spike in incidents; $\Delta\mathrm{Tw}$ within budget.
Decision: promote (expand scope), hold (extend probe), or rollback.

Typical $\tau_{\mathrm{probe}}$ : hours (software), days (ops), 1–2 cycles (factory).

20.6 Rollback (Fast, Loss-Averse)

Rollback reverts to the previous rung (or a safe baseline) via the inverse twist step:

\mathrm{Tw}\;\leftarrow\;\mathrm{Tw}-\Delta\mathrm{Tw}_{\text{applied}}, \quad \theta\;\leftarrow\;\theta-\Delta\theta.

Trigger conditions (any):

Gap fails to meet $g_\star$ by time $t+\tau_{\mathrm{probe}}$ with low probability of recovery.
Entropy $\Sigma_{\text{macro}}$ breaches cap or safety near-miss detected.
Adverse residual trend: $R$ increasing for $k$ consecutive intervals.
Canary harm beyond guardrail.

Post-rollback cooldown prevents immediate re-attempt; log a Twist Incident with diagnostics.

20.7 Choosing the Minimal Step (Algorithms)

A. Ladder Search (coarse→fine, recommended)

Rank rungs by $\mathcal C_{\text{gov}}$ (low→high).
Predict $\widehat{\mathrm{gap}}^{\,\text{post}}$ using $\alpha, G$ and rung deltas.
Pick the first rung meeting $g_\star$ + guardrails.
If none qualify, consider two-rung compositions with small cumulative $|\Delta\mathrm{Tw}|$ and disjoint blast radii.

B. Sensitivity-Aware One-Shot

Solve a small 0–1 program over rungs $\{r_j\}$ with binary $z_j$ :

\min_{z\in\{0,1\}^m}\ \sum_j c_j z_j \quad\text{s.t.}\quad \mathrm{gap}^{\,\text{now}} +\alpha\sum_j \Delta\mathrm{Tw}_j z_j +\sum_j \langle G,\Delta\theta_j\rangle z_j \le g_\star,

plus budgets and incompatibilities. Use a short time limit; fall back to A if infeasible.

C. Greedy with Backtracking

Iteratively add the lowest-cost rung that most reduces predicted gap; backtrack if guardrails trip in simulation.

20.8 Reference Logic (Pseudocode)

# Inputs:
#  gap_now, alpha, G: current estimates
#  ladder: list of rungs with {id, dTw, dtheta, cost, risk, guardrails, probe}
#  budgets, slo, invariants, cooldowns

def pick_twist_step(gap_now, alpha, G, ladder, target):
    # 1) filter rungs by hard guardrails
    R = [r for r in ladder if respects_guardrails(r, budgets, invariants)]

    # 2) predict post-gap for each rung
    for r in R:
        r.gap_post = gap_now + alpha * r.dTw + dot(G, r.dtheta)
        r.feasible = (r.gap_post <= target)

    # 3) choose minimal-cost feasible rung
    feasible = [r for r in R if r.feasible]
    if feasible:
        return argmin(feasible, key=lambda r: r.cost)

    # 4) try small compositions (two-rung combos, disjoint scope)
    combos = compose_small(R, max_len=2, disjoint_scope=True)
    for c in combos:
        c.gap_post = gap_now + alpha * sum(r.dTw for r in c) + dot(G, sum(r.dtheta for r in c))
        c.cost = sum(r.cost for r in c) + coupling_penalty(c)
    feasible_c = [c for c in combos if c.gap_post <= target and respects_guardrails(c, budgets, invariants)]
    if feasible_c:
        return min(feasible_c, key=lambda c: c.cost)

    return None  # escalate or re-estimate sensitivities

def execute_twist_step(step):
    apply_policy(step)                 # change θ according to rung(s)
    open_probe_window(step.probe)      # start monitoring plan
    log_twist_event("APPLY", step)

def evaluate_probe(step, metrics):
    ok = (metrics.gap <= step.target and
          metrics.Sigma <= slo.entropy_cap and
          metrics.safety_ok)
    if ok:
        promote_or_hold(step)
        log_twist_event("PROMOTE/HOLD", step, metrics)
    else:
        rollback(step)
        log_twist_event("ROLLBACK", step, metrics)

20.9 SLOs for Twist Stepping

Gap closure: $\Pr[\mathrm{gap}\le g_\star \text{ within } T_{\text{close}}]\ge p_{\min}$ .
Governance churn cap: weekly $\sum|\Delta\mathrm{Tw}|\le B_{\mathrm{Tw}}^{\text{week}}$ ; $\mathbb E[\|\Delta\theta\|_0]\le K$ .
Safety/ethics: zero critical breaches; near-miss rate $\le \epsilon$ .
Rollback discipline: median rollback time $\le T_{\text{rb}}$ ; rollback success rate $\ge q_{\min}$ .
Stability: no >1 oscillatory flip on the same belt within $H$ hours (pair with Ch. 19 cooldowns).

20.10 Domain Walk-Throughs

Manufacturing (shoes). Persistent planning overhang on stitching cell. Ladder yields two candidates: reduce WIP cap 5% ( $\Delta\mathrm{Tw}=0.6$ , small cost) vs add 1 hr overtime ( $\Delta\mathrm{Tw}=1.1$ , higher cost). Sensitivities $G$ show WIP affects gap twice as effectively; pick WIP rung, probe 2 shifts, promote if residual $R$ decays.

Software delivery. Canary incidents keep tripping a flux gate. Twist step: raise canary from 2%→5% with SLO guard, enable error-budget mode, or deploy freeze 2 h. Cost ranks favor error-budget mode (small blast radius). Probe 6 h, rollback if incident integral exceeds impulse cap.

Customer ops. New refund script inflates entropy. Twist step: route high-risk to seniors (zonal, $|\Delta\mathrm{Tw}|$ small) vs tighten refund cap 5% (larger churn). Choose zonal route; gap closes without backlash; promote across regions.

20.11 Policy Template Library (Artifact)

Schema (YAML)

template_id: "ops/refund-triage-v3"
domain: "customer-ops"
rung:
  description: "Route risk-class R≥0.7 to senior queue; keep others on v2."
  dtheta:
    triage_rule: {from: "v2", to: "v3", scope: "risk>=0.7"}
  dTw: 0.45
  sensitivities:
    gap: {G: {"triage_rule": -0.8}, confidence: 0.75}
  expected_effects:
    gap_post_delta: -0.6
    entropy_delta: -0.2
    throughput_delta: -0.05
  risk_class: "Low"
  approvals: ["ops-lead"]
  blast_radius: "zonal: high-risk only"
  guardrails:
    safety: true
    invariants_4pi: true
    budgets: {tw_step_max: 1.0}
  probe:
    duration: "48h"
    monitors: ["gap","R","Sigma","incidents"]
    promote_criteria:
      gap_leq: 0.0
      R_trend: "down"
      Sigma_leq_index: 1.10
    rollback_criteria:
      incidents_gt: 3
      Sigma_spike_gt: 0.15
  rollback:
    recipe: "revert triage_rule to v2"
    cooldown: "24h"
  notes: "Pairs with flux-gate FG-ops-02; do not co-roll with pricing."

Maintain a library of such templates per belt/domain with versions, owners, and test fixtures. Link each to past outcomes to refine $(\alpha,G)$ priors.

20.12 Validation & Tuning

Counterfactual replay: apply historical spikes; compare gap/entropy under different rungs.
Pareto sweep: trade off $\mathcal C_{\text{gov}}$ vs closure probability; pick robust knee points.
Stability tests: ensure no oscillations with Flux Gate cooldowns; verify 4π invariance.
Learning loop: update rung posteriors for $\Delta\mathrm{Tw}$ and $G$ from probe outcomes (Bayesian shrinkage toward recent belts).

Takeaway

Twist Stepping is governance control in a belt world: choose the smallest policy/frame move that closes the gap, prove it with a probe window, and keep your hand on a rollback lever. Together with Flux Gates, it turns “gap = flux + α·twist” from an identity into a repeatable, low-churn operating doctrine.

21. Multi-Objective Arbitration Belts

Purpose. Many belts run at once—throughput, quality, safety, cost, fairness, carbon, etc.—and they compete for the same governance attention (twist budget) and sometimes the same physical capacity. An Arbitration Belt is the meta-controller that decides which goals to pursue now, how hard, and in what order, while keeping the system polycentric (no single objective quietly captures the whole loop).

21.1 Objects & Notation

Index objectives by $k=1,\dots,K$ . Each has a purpose connection $\mathcal A^{(k)}_{\Pi}$ , curvature $\mathcal F^{(k)}$ , and (optional) work-aligned flow $J^{(k)}$ .
Over a control window with belt surface $\mathcal B$ , define each objective’s macro utility
$U_k \;=\; \iint_{\mathcal B}\!\langle \mathcal F^{(k)}, J^{(k)}\rangle.$
(If $J^{(k)}$ omitted, treat $U_k=\iint\|\mathcal F^{(k)}\|$ with signed convention.)
Twist actions come as discrete rungs $r \in \mathcal L$ (Ch. 20) with deltas:
$\Delta\mathrm{Tw}_r,\qquad \Delta U_{k,r},\qquad \text{risk}(r),\qquad \text{scope}(r).$
Budget: $\sum_{r \in S} |\Delta\mathrm{Tw}_r|\le \tau$ per arbitration window (weekly by default).
Fairness floors (optional): $U_k \ge \underline U_k$ or share quotas $\sum_{r\in S} \Delta U_{k,r}\ge q_k$ .

21.2 The Arbitration Program (Abelian Core)

Given weights $w_k\ge 0$ (Sec. 21.4), choose a set of rungs $S\subseteq \mathcal L$ :

\boxed{ \max_{S \subseteq \mathcal L}\;\; \sum_{k=1}^{K} w_k \Big( U_k + \sum_{r\in S}\Delta U_{k,r}\Big) \quad \text{s.t.}\quad \sum_{r\in S} |\Delta\mathrm{Tw}_r| \le \tau,\;\; \text{Guardrails}. }

Guardrails include safety, ethics, invariant checks (4π), blast-radius limits, inter-rung incompatibilities, and live SLO bounds on entropy/throughput (Ch. 19–20).

Interpretation. The arbiter spends a limited twist budget $\tau$ across competing goals to achieve the highest weighted macro utility improvement.

21.3 Non-Abelian Reality: Commutator-Aware Scheduling

Twist steps don’t always commute. Applying rung $a$ then $b$ may differ from $b$ then $a$ . Using a BCH-style approximation, the effective twist for an ordered sequence $\pi$ is

\mathrm{Tw}_{\text{eff}}(\pi)\;\approx\;\sum_{r\in \pi}\Delta\mathrm{Tw}_r \;+\;\tfrac12\!\!\sum_{i<j}\!\kappa_{ij}\,[r_i,r_j] \;+\;\cdots

where $\kappa_{ij}$ is an interaction coefficient learned from logs/experiments and $[r_i,r_j]$ is a signed interference term (e.g., extra churn, surprise entropy, or undoing benefits).

Scheduling problem. Choose $S$ and an order $\pi$ to

\max_{\pi \text{ over } S}\;\; \sum_{k} w_k \sum_{r\in \pi}\Delta U_{k,r} \;-\;\lambda\!\left(\sum_{r\in \pi}\!|\Delta\mathrm{Tw}_r| +\tfrac12\!\!\sum_{i<j}\!\kappa_{ij}\,[r_i,r_j]\right),

subject to budgets/guardrails. This is a weighted sequencing problem with pairwise penalties.

Heuristics that work:

Cluster-then-order. Partition rungs into near-commuting clusters (low $|\kappa_{ij}|$ ); freely permute inside clusters; order clusters by utility-per-twist ratio.
Max-separation for antagonists. If $a,b$ strongly anti-commute ( $\kappa_{ab}\ll 0$ ), separate them by a cooldown epoch to let residuals $R$ (Ch. 18) decay.
Zonal first. Apply zonal/local rungs before global ones; they typically commute better and preserve headroom.
Exploit positive commutators. Some pairs have $\kappa_{ij}[r_i,r_j]>0$ : apply in the energy-harvesting order to reduce twist cost and boost $U$ .

21.4 Healthy Polycentric Weights

Weights determine whose voice counts right now. To keep the system healthy (no capture, no starvation):

Constitutional priors. Fix $w_k^{\text{base}}$ for hard principles (safety, legality, equity).
State-contingent boosts. Additive terms for incident heat, risk posture, or seasonality:
$w_k = w_k^{\text{base}} + \beta_1\,\text{risk}_k + \beta_2\,\text{backlog}_k + \beta_3\,\text{season}_k.$
Fairness governor. If $U_k$ falls below a rolling floor, multiply its weight by $\eta>1$ until recovered.
Regret tuning. Adjust weights to minimize historical regret:
$\min_{w \in \mathcal W}\ \sum_{t}\Big[\max_{k} \hat U_{k,t} - \sum_{k} w_k \hat U_{k,t}\Big] \quad\text{s.t.}\ \|w-w^{\text{base}}\|_2\le \rho.$
Human council. A small polycentric committee approves weight changes over thresholds, with public rationale (anti-capture transparency).

21.5 Procedures

A. Weight Tuning

Compute base $w_k$ from the governance charter.
Apply state-contingent boosts (risk/backlog/season).
Run Pareto sweep off-line; publish a frontier card showing trade-offs (ΔU vs twist).

B. Candidate Harvest

Each belt/controller proposes rungs with $(\Delta\mathrm{Tw}, \Delta U_k, \text{risk}, \kappa_{ij})$ .
Validate with guardrails and attach probe windows (Ch. 20).

C. Arbitration Solve

Stage 1 (selection): solve the abelian program to pick $S$ under $\tau$ .
Stage 2 (ordering): build a DAG using negative penalties on edges with large $\kappa_{ij}$ ; topologically sort to minimize expected commutator cost; insert cooldowns where needed.

D. Execute & Observe

Execute $\pi$ with probe windows; Flux Gates remain active (Ch. 19).
Update $\kappa_{ij}$ and $\Delta U_{k}$ posteriors from outcomes; feed back into the next arbitration.

21.6 Reference Logic (Pseudocode)

def arbitrate(rungs, weights, tau, guardrails):
    # Filter by hard guardrails
    R = [r for r in rungs if respects_guardrails(r, guardrails)]

    # Stage 1: selection under L1 twist budget (knapSack-like)
    # value = sum_k w_k * dU[k]; cost = |dTw|
    for r in R:
        r.value = sum(weights[k] * r.dU[k] for k in r.dU)
        r.cost  = abs(r.dTw)
    S = budgeted_max_sum(R, budget=tau)  # greedy+DP hybrid

    # Stage 2: commutator-aware ordering
    G = build_interaction_graph(S)  # edges weighted by kappa_ij * [ri,rj]
    pi = cluster_then_order(G)      # cluster commuting, order clusters by value/cost, insert cooldowns

    return pi  # sequence of rungs with timing

21.7 SLOs for Arbitration

Budget adherence: $\sum |\Delta\mathrm{Tw}| \le \tau$ per window, p99.
No starvation: every protected objective $k$ gets $U_k\ge \underline U_k$ over rolling horizon $H$ .
Regret bound: moving-window regret $\le \mathcal R_{\max}$ .
Commutator losses: realized extra twist from interactions $\le \epsilon_{\text{comm}}$ of budget.
Stability: at most one global oscillation per $H$ ; arbitration latency $L_{95}\le L_{\max}$ .

21.8 Domain Patterns

Factory poly-KPIs. Quality and throughput both want twist. Arbiter spends $\tau$ on zonal QC uplift (commutes with staffing) before global WIP cap changes; observed $\kappa_{ij}\approx 0$ inside zone → low extra twist.
SRE portfolio. Reliability vs velocity: select “error-budget mode ON” + “canary 2→5%” first; schedule “retry backoff +100 ms” after traffic calms to avoid antagonistic commutators with autoscaling.
Customer fairness. Refund fairness and cost sharing: enforce fairness floor $U_{\text{fair}}$ , then allocate residual $\tau$ to cost efficiency; schedule pricing nudges after triage policy rollout.

21.9 Arbitration Service Config (Artifact → Ch. 33)

arbiter_id: "global-arbiter/v1"
window: {days: 7}
twist_budget: 24.0          # weekly |ΔTw| budget
objectives:
  - id: safety
    weight_base: 1.0
    floor: {type: hard, U_min: 0.0}
  - id: quality
    weight_base: 0.6
    floor: {type: rolling, horizon_days: 28, share_min: 0.2}
  - id: throughput
    weight_base: 0.6
    floor: {type: soft}
  - id: cost
    weight_base: 0.3
    floor: {type: soft}
weight_updates:
  boosts:
    risk_coeff: 0.5
    backlog_coeff: 0.3
    season_coeff: 0.2
  fairness_governor: {eta: 1.5, trigger: "below_floor"}
selection:
  solver: "budgeted-max-sum"
  guardrails:
    invariants_4pi: true
    safety: true
    blast_radius_max: "zonal:80%, global:1 per window"
scheduling:
  commutator_matrix_ref: "s3://…/kappa_ij.parquet"
  strategy: "cluster-then-order"
  cooldown_default_hours: 24
  zonal_first: true
slo:
  budget_p99: 1.0
  regret_max: 0.15
  commutator_loss_max: 0.1
audit:
  council_emails: ["safety@org","ethics@org","ops@org"]
  publish_frontier_card: true

Takeaway

Arbitration Belts spend a limited twist budget across multiple goals, carefully ordering moves to respect non-commutativity and preserve system health. With principled weight tuning, commutator-aware scheduling, and explicit fairness floors, polycentric control becomes both effective and legitimate.

22. Fast–Slow Separation & Stability

Purpose. Flux Gates (fast) and Twist Stepping (slow) must cooperate without oscillation. This chapter gives a two-time-scale controller and a Lyapunov proof sketch that guarantees decay of the gap error with explicit rules for gain scheduling, sampling, lag compensation, and audit. The artifact is a Stability Checklist you can run before enabling GBC in production.

22.1 Signals, States, and Errors

Fast loop (F): real-time Flux Gate controls (rate caps, WIP throttles, zonal reroutes).
State $x_f$ summarizes short-horizon queues/loads; input $u_f$ .
Slow loop (S): Twist Step governance (policy/frame parameters $\theta$ ).
State $x_s \equiv \theta$ (piecewise constant with discrete jumps); input $u_s$ = ladder rungs.

Let the gap target be $g_\star$ (often 0). Define the error

e(t)\;=\;\mathrm{gap}(t)-g_\star.

We model (after linearization around an operating belt and time-scale separation)

\begin{aligned} \dot x_f &= f(x_f,x_s) + B_f u_f + d_f,\\ \dot x_s &= \varepsilon\, g(x_f,x_s) \qquad (\varepsilon\ll 1),\\ e &= C_f x_f + C_s x_s + d_e, \end{aligned}

with disturbances $d_\bullet$ (estimation noise, workload shocks). Twist steps induce jumps $x_s^{+}=x_s+\Delta\theta$ at decision times.

22.2 Controller Architecture (Two Time Scales)

Fast (F) controller $\mathcal C_f$ (Flux Gate):
$u_f \;=\; \mathcal K_f(\widetilde{M}_t,\ \phi(x,t),\ \text{budgets}) \quad\text{(hysteresis + cooldown + graded output, Ch. 19)}.$
Goal: damp $\dot e$ and keep $e$ within a small band quickly.
Slow (S) controller $\mathcal C_s$ (Twist Step):
choose discrete $\Delta\theta$ from ladder $\mathcal L$ only after a quiet-belt condition holds:
$\max_{\tau\in[t-T_q,t]}|e(\tau)| \le \epsilon_q,\quad \text{gate state}=\text{CLOSED}.$
Goal: shift the operating point so the steady gap meets $g_\star$ with minimal twist (Ch. 20).
Arbitration (optional third loop, weekly): allocates twist budget $\tau$ across belts (Ch. 21).

Design principle: the slow loop acts between fast transients; the fast loop never fights moving targets.

22.3 Lyapunov Template and Decay Condition

Pick positive definite $P_f,P_s$ and define

V(x_f,x_s)=x_f^\top P_f x_f + \eta\,x_s^\top P_s x_s,\qquad \eta>0.

Continuous evolution (no twist jump)

Assume the fast loop renders the fast subsystem ISS in $x_s$ :

\dot V_f \le -c_f\|x_f\|^2 + \kappa_f \|x_s\|^2 + \rho_f\|d_f\|^2.

The slow drift (with $\varepsilon\ll 1$ ):

\dot V_s \le \varepsilon\left(-c_s\|x_s\|^2 + \kappa_s \|x_f\|^2\right).

Then

\dot V \le -\Big(c_f - \eta\varepsilon \kappa_s\Big)\|x_f\|^2 \;-\;\eta\varepsilon c_s\|x_s\|^2\;+\;\kappa_f\|x_s\|^2 + \rho_f\|d_f\|^2.

Pick $\eta,\varepsilon$ so that

c_f - \eta\varepsilon \kappa_s \ge \tfrac{c_f}{2},\qquad \eta\varepsilon c_s - \kappa_f \ge c_s',

for some $c_s'>0$ . Then

\dot V \;\le\; -\tfrac{c_f}{2}\|x_f\|^2 - c_s' \|x_s\|^2 + \rho_f\|d_f\|^2 \;\le\; -c\|e\|^2 + \rho\|d\|^2,

establishing practical stability with dissipation rate $c>0$ .

Jumps (twist steps)

At a twist application time $t_k$ :

V^+ - V \;=\; \eta \left((x_s+\Delta\theta)^\top P_s (x_s+\Delta\theta) - x_s^\top P_s x_s\right).

Require step-wise decay

V^+ - V \le \underbrace{-\mu \|e\|^2}_{\text{net benefit}} + \underbrace{\sigma\|\Delta\theta\|^2}_{\text{governance churn}},

and enforce a dwell time $\Delta t \ge T_{\min}$ between steps so the continuous decay overtakes any jump-induced increase. This is guaranteed by the probe window and cooldowns in Ch. 20.

Conclusion. With (i) separation $\varepsilon$ small, (ii) fast ISS gains, (iii) guarded jump sizes with dwell time, we obtain a hybrid Lyapunov inequality

\dot V \le -c\|e\|^2 \quad\text{(flows)},\qquad V^+ - V \le -\mu\|e\|^2 \quad\text{(jumps)},

ensuring monotone decrease of $V$ and no oscillations.

22.4 Sampling & Timing (Emulation Rule)

Loop rates. Choose periods $(T_f,\ T_e,\ T_s)$ for fast control, estimator, and slow governance:
$T_f \le \tfrac{1}{5}\,T_e,\qquad T_s \ge 10\,T_f^{\max},\qquad T_q \in [3,10]\,T_f^{\max}.$
Sampled-data bound. For a Lipschitz fast model with constant $L$ , pick $T_f$ so that
$V[k{+}1]-V[k] \le -\alpha\|e[k]\|^2 + \beta L^2 T_f^2 \|e[k]\|^2$
yields $\alpha-\beta L^2 T_f^2>0$ . This is the emulation condition: make digital $T_f$ small enough that the discrete loop inherits the continuous stability.
Quiet-belt detector. Require $\max_{[t-T_q,t]}|e|\le\epsilon_q$ and gate=CLOSED before evaluating any twist.

22.5 Lag Compensation

Delays come from estimation (curvature windowing), actuation (mechanical/organizational), and communications.

Lead/Smith compensation. If total delay $L$ is known, predict
$\widehat{M}_{t+L} = M_t + L\,\dot M_t,$
and apply thresholds to $\widehat{M}_{t+L}$ (Flux Gate lead). For slow loop, use a Smith predictor: fit an internal model of belt response and compare predicted vs observed to avoid over-stepping.
Anti-alias filters. Median $+$ EWMA before gating; set cutoff $\omega_c \approx \frac{\pi}{2T_f}$ .
Dead-time margin. Keep $L \le 0.3\,T_q$ . If violated, widen hysteresis ( $\theta_{\text{on}}-\theta_{\text{off}}$ ) and reduce gate gain $\gamma$ .

22.6 Gain Scheduling

Schedule gains/thresholds against state and uncertainty:

Volatility-aware thresholds.
$\theta_{\text{on/off}} \gets \theta_{\text{on/off}}\times(1+\kappa_{\sigma}\,\widehat{\mathrm{Std}}(M))$ .
Headroom-aware softness.
Gate exponent $\gamma \uparrow$ when twist headroom is low; $\gamma \downarrow$ when high.
Slope-aware cooldown.
$T_{\mathrm{cd}} \gets T_{\mathrm{cd}} + \kappa_S\,\max(0, S_t-S_\text{cap})$ .
Zonal priority.
Prefer controls that reduce $\int \phi(x)\,dx$ with minimal $\Delta\mathrm{Tw}$ .

All schedules must monotone-bound the closed-loop gain so that the ISS constants $c_f,c_s'$ remain above configured floors.

22.7 Inter-Loop Etiquette (No Fighting)

Single owner of the plant at a time. Slow loop acts only when fast loop is CLOSED and the belt is quiet.
Dwell after twist. Enforce $T_{\min}$ (e.g., hours/days) before re-evaluating another twist.
Arbitration cooldowns. When Ch. 21 schedules multiple rungs, insert cooldowns to let residual $R$ (Ch. 18) decay; avoid back-to-back antagonistic moves.

22.8 Reference Scheduler (Pseudocode)

# Periods: Tf (fast), Te (estimator), Ts (slow); Tq quiet window
last_twist_time = -inf
state_gate = "CLOSED"

every Tf:
    M_f = filtered_metric()                   # median + EWMA
    state_gate = flux_gate_step(M_f, ...)     # Ch. 19 logic
    log_fast(state_gate, M_f)

every Ts:
    if state_gate == "CLOSED" and quiet_belt(Tq, eps_q) and time_since(last_twist_time) >= Tmin:
        step = pick_twist_step(gap_now(), alpha, G, ladder, target)  # Ch. 20
        if step:
            apply(step)
            last_twist_time = now()
            open_probe_window(step.probe)
            log_twist(step)

22.9 Stability Checklist (Artifact)

A. Separation & Timing

$T_s \ge 10\,T_f^{\max}$ (≥7 acceptable; target 10–15).
Quiet-belt detector configured: $T_q,\ \epsilon_q$ .
Dwell $T_{\min}$ after twists ≥ probe duration.

B. Fast Loop Margins

Hysteresis: $\theta_{\text{on}} > \theta_{\text{off}}$ with deadband ≥ $3\times$ noise std.
Cooldown $T_{\mathrm{cd}}$ ≥ actuation + estimation latency.
False-open rate ≤ $\epsilon_{\text{FO}}$ on backtests.

C. Slow Loop Discipline

Ladder rungs carry $\Delta\mathrm{Tw}$ , $G$ , probe windows, rollback recipes.
Weekly twist budget $B_{\mathrm{Tw}}$ set; no over-spend in last 4 windows.
Guardrails (safety/ethics/4π invariants) enabled.

D. Delay & Filters

Total delay $L \le 0.3\,T_q$ ; otherwise lead/Smith compensator on.
Anti-alias filtering configured (median $+$ EWMA); cutoff $\omega_c$ set.

E. Lyapunov/ISS Evidence

Linearized $A_f$ with feedback $K_f$ yields $P_f>0$ solving $A_f^\top P_f + P_f A_f - Q_f \prec 0$ .
Empirical $V$ decay on replay: $V_{t+\Delta}-V_t \le -c\|e\|^2$ .
Jump analysis: per-rung $V^+ - V \le -\mu\|e\|^2 + \sigma\|\Delta\theta\|^2$ .

F. SLOs

Recovery $T_{\text{heal}}$ met on last three incidents.
No >1 oscillation on same belt within horizon $H$ .

22.10 Domain Patterns

Manufacturing (shoes). $T_f=1$ min (line sensors), $T_e=5$ min (curvature estimation), $T_s=6$ h (policy). Quiet belt $T_q=30$ min. Flux Gates absorb rush orders; one small WIP cap rung per day keeps gap at target without ping-pong.
Software delivery. $T_f=30$ s (deploy pace), $T_e=2$ min (error/curvature), $T_s=4$ h (release policy). Smith predictor uses observed canary ramp to offset telemetry lag, preventing the classic “freeze/unfreeze” oscillation.
Customer ops. $T_f=5$ min (routing), $T_e=15$ min (curvature from ticket semantics), $T_s=1$ day (script/policy). Zonal reroutes first; only after 3 quiet hours apply triage rung; weekly arbitration allocates twist to fairness floor without destabilizing throughput.

22.11 What to Hand Over

Timing spec: $T_f,T_e,T_s,T_q,T_{\min}$ with rationale.
Lyapunov card: chosen $P_f,P_s$ , bounds $c,c_s'$ , and replay plots showing decay.
Lag plan: measured $L$ , compensator settings, widened hysteresis if needed.
Probe catalog: per-rung decay/rollback outcomes feeding back into $G,\alpha$ priors.

Takeaway

A belt system is stable when fast controls kill curvature spikes now, and slow twist steps shift policy only after the belt is quiet. With a composite Lyapunov and strict timing etiquette, you get

\dot V \le -c\|e\|^2

between steps and negative jumps at steps—no oscillations, no governance thrash, just steady closure of the purpose gap.

23. Coherence (“Shen”) as Stabilizer

Purpose. Flux Gates (fast) and Twist Stepping (slow) keep each belt stable on its own (Ch. 19–22). Coherence (“Shen”) keeps many belts phase-aligned enough to avoid cross-belt oscillations, policy thrash, and latent entropy spikes. We formalize “Shen” as phase-locking among belts, measure it with coherence coefficients $C_{ij}$ , and manage it with minimal-twist synchronization routines that preserve diversity.

23.1 Belt Phase: What Exactly Locks?

Each belt $i$ has a phase $\varphi_i(t)$ extracted from its purpose geometry over a control window $\mathcal B_t$ :

Let $\mathrm{Hol}_i(t)$ be an edge invariant (e.g., principal eigenphase or $\arg\det$ ) of the belt’s plan–do holonomy.
Define the belt phase as $\varphi_i(t)=\mathrm{Arg}(\mathrm{Hol}_i(t))\in\mathbb S^1$ , with unwrapped derivative $\dot{\varphi}_i$ the cadence (natural cycle rate).

Intuition:

$\varphi_i$ locks when plan/do cycles for belt $i$ recur with a stable relative timing and style framing.
Cross-belt phase-locking means these recurrences align enough to avoid destructive interference (e.g., a factory line “breathing” against QA audits; a deploy train stepping on customer-ops policy windows).

23.2 Coherence Metrics

Pairwise coherence $C_{ij}$

C_{ij} \;=\; \big|\ \mathbb E_{t\in W} \big[e^{\,i\big(\varphi_i(t)-\varphi_j(t)-\delta_{ij}^{\star}\big)}\big]\ \big| \ \in [0,1].

$W$ : rolling window; $\delta_{ij}^{\star}$ : learned phase offset (the “healthy lag” between $i$ and $j$ ).
$C_{ij}=1$ : perfect lock at the desired offset; $C_{ij}\approx 0$ : desync.

Global order parameter $R$

R \;=\; \left|\frac{1}{N}\sum_{i=1}^{N} e^{\,i(\varphi_i-\psi^\star_i)}\right|, \quad \psi^\star_i \text{ are target offsets (charter/cadence plan).}

$R$ near 1 means strong system-wide lock; too close to 1 for too long may imply brittleness (see 23.6).

Coherence heatmap

A matrix $[C_{ij}]$ with cluster structure reveals commuting cohorts vs antagonists; use it to decide who should synchronize.

23.3 “Shen” Indices

We operationalize “Shen” as phase-lock quality balanced against governance churn required to maintain it.

Phase-Lock Index (PLI)

\mathrm{PLI} \;=\; \frac{2}{N(N-1)} \sum_{i<j} \big( C_{ij} \wedge \mathbf 1\{|\,\hat\delta_{ij}-\delta^\star_{ij}\,|\le \Delta_{\text{tol}}\} \big),

rewarding both magnitude and correct offset.

Minimal-Twist Efficiency (MTE)

\mathrm{MTE} \;=\; \frac{\text{coherence gain per window}}{\text{twist spent per window}} \;=\;\frac{\Delta \mathrm{PLI}}{\sum |\Delta \mathrm{Tw}| + \epsilon}.

Shen Score

\mathrm{Shen} \;=\; \mathrm{HarmonicMean}\big(\mathrm{PLI},\,\mathrm{MTE}\big),

prioritizing high lock achieved with low churn.

Use Shen as a first-class SLO: keep $\mathrm{Shen}\ge S_{\min}$ on active weeks.

23.4 Fast–Slow–Shen Coupling

Augment Ch. 22 dynamics with a Kuramoto-style cross-belt term:

\dot{\varphi}_i \;=\; \omega_i \;+\;\sum_{j\neq i} K_{ij}\,\sin\!\big(\varphi_j-\varphi_i-\delta_{ij}^{\star}\big) \;-\;\beta_i\,u^{\text{gate}}_i \;-\;\gamma_i\,u^{\text{twist}}_i \;+\;\xi_i,

where $K_{ij}$ encodes structural coupling (shared resources, policy interlocks), $u^{\text{gate}}$ and $u^{\text{twist}}$ are the Ch. 19/20 actions projected into phase space, and $\xi_i$ is noise.

Positive $K_{ij}$ promotes lock; negative discourages simultaneity (e.g., deploy and pricing changes).
Set $\delta_{ij}^{\star}$ to encode desired lags (QA after build; policy after deploy; billing after policy).

Goal: Raise coherence within clusters with $K_{ij}>0$ and maintain healthy offsets where $K_{ij}<0$ .

23.5 Procedures

A. Cross-Belt Sync (when to lock)

Cadence calendar. Publish target offsets $\psi^\star_i$ and $\delta_{ij}^\star$ per belt cluster.
Basepoint alignment. Normalize phase extraction (same holonomy basepoint, same windowing) to make $C_{ij}$ comparable.
Soft pulls. Use soft gates (Ch. 19) to nudge cadences toward lock before trying any twist.
Minimal-twist nudge. Apply the smallest ladder rung that adjusts cadence (e.g., canary ramp timing, batch-start jitter, QC sample windows).

B. Desync Alarms (when to separate)

Trigger alarms if any:

$C_{ij}\downarrow$ crosses $\theta_C$ and $|\dot C_{ij}|$ large (rapid drift).
$|\hat\delta_{ij}-\delta_{ij}^{\star}|$ exceeds tolerance for $\ge T_{\text{hold}}$ .
Cross-entropy spike: $\Sigma_{\text{macro}}$ co-moves with coherence loss.

On alarm: prefer desync by soft gating (re-time, reroute) before governance steps; if antagonistic pairs $K_{ij}\ll 0$ , enforce cooldowns between their windows.

C. Entropy–Efficiency Trade (how tight to lock)

Tight lock $\Rightarrow$ less WIP jitter and fewer hand-offs → lower entropy, but risks systemic fragility (shared shocks).
Loose lock $\Rightarrow$ resilience via asynchrony, but more coordination waste.

Maintain a coherence band $R\in[R_{\min},R_{\max}]$ and cap weekly $\sum |\Delta\mathrm{Tw}|$ used purely for synchronization.

23.6 Reference Logic (Pseudocode)

# Inputs each window:
#   phases: {phi_i(t)}; targets: {psi_i*, delta_ij*}; twist_budget_sync
#   K_ij: coupling estimates; alarms cfg

C = pairwise_coherence(phases, delta_star)    # matrix C_ij
PLI = phase_lock_index(C, delta_star)
MTE = minimal_twist_efficiency(PLI, twist_spent_sync)
Shen = harmonic_mean(PLI, MTE)

# Cross-belt decisions
clusters = spectral_clusters(C, K)            # lock cohorts
for cl in clusters:
    if Shen < S_min and twist_headroom():
        # minimal-twist sync nudges
        rungs = cadence_rungs_for(cl)         # small timing rungs only
        step = pick_min_twist_to_raise_PLI(rungs, target_delta_PLI)
        if step:
            apply(step); log_shen_event("SYNC_STEP", step)

# Desync alarms
for (i,j) in antagonistic_pairs(K):
    if C[i,j] < theta_C and dCdt(i,j) < -theta_rate:
        open_soft_desync(i,j)  # re-time windows, insert cooldowns
        log_shen_event("DESYNC_SOFT", {"pair":[i,j]})

23.7 SLOs (Shen as a First-Class Objective)

Shen floor: $\mathrm{Shen}\ge S_{\min}$ on active weeks.
Lock integrity: $\Pr\big(|\hat\delta_{ij}-\delta^\star_{ij}|\le \Delta_{\text{tol}}\big)\ge p_{\min}$ for all $i,j$ in a cluster.
Minimal-twist discipline: $\mathrm{MTE}\ge E_{\min}$ ; sync-twist spend $\le \tau_{\text{sync}}$ per window.
Fragility guard: Global $R\le R_{\max}$ p95; no simultaneous OPEN states across antagonistic pairs more than $m$ times / window.
Recovery: After a desync alarm, $C_{ij}$ returns to band within $T_{\text{relock}}$ .

23.8 Domain Patterns

Factory (shoes). Lock stitching → QA → packaging with offsets $(0, +20\text{min}, +45\text{min})$ . Sync by micro-jitters in batch starts; desync from pricing updates on e-com (antagonistic). Entropy index drops 12% with no extra twist beyond cadence templates.
Software delivery. Lock build → canary → feature flag flips; keep pricing and refund policy out of phase by $\pi/2$ . A cheap “Shen nudge” is canary ramp cadence; avoid governance churn.
Customer ops. Lock triage script v $k$ rollout after deploy quiet window; desync marketing push by one day. Coherence cluster across regions improves SLA variance without global uniformity.

23.9 “Shen Report Pack” (Artifact → see Ch. 29)

Deliverable bundle for weekly ops/council:

Overview: Shen score, PLI, MTE, global $R$ , trend sparkline.
Matrices: $C_{ij}$ (heatmap), $\hat\delta_{ij}$ vs $\delta^\star_{ij}$ , coupling $K_{ij}$ .
Cohorts: discovered clusters, antagonistic pairs, proposed offsets.
Sync ledger: twist spent on synchronization, $\Delta \mathrm{PLI}$ , MTE per rung.
Alarms: desync incidents with root-cause notes and recovery times.
Playbook diffs: cadence calendar changes (adds/removals), next-week nudges.

Schema (YAML)

shen_pack_id: "shen-weekly/2025-W38"
summary:
  Shen: 0.73
  PLI: 0.81
  MTE: 0.62
  R_global: 0.77
coherence_matrix_ref: "s3://.../C_ij.parquet"
offsets:
  delta_star:
    - {pair: [build, qa],     offset_minutes: 20}
    - {pair: [qa, packaging], offset_minutes: 25}
clusters:
  - name: "factory-core"
    members: ["cut","stitch","qa","pack"]
  - name: "antagonists"
    members: [["deploy","pricing"],["policy","marketing"]]
sync_ledger:
  twist_spent_sync: 2.4
  delta_PLI: 0.08
  MTE: 0.033
alarms:
  - pair: ["deploy","pricing"]
    C_ij: 0.28
    action: "DESYNC_SOFT"
    eta_relock_hours: 12
next_actions:
  - "Apply canary cadence rung v5 in cluster factory-core"
  - "Hold pricing changes until post-deploy quiet window"
owners: ["ops-lead","sre-lead","finance-policy"]

Takeaway

Coherence (“Shen”) is the phase-lock stabilizer of poly-belt operations: quantify lock with $C_{ij}$ , govern it with minimal twist, and stay inside a coherence band that lowers entropy without creating fragility. With a weekly Shen report pack and light-touch cadence nudges, belts stay in step where it helps—and deliberately out of step where it’s safer.

24. Many Purposes, Many “Tops”

Purpose. Real organizations are polycentric: safety has a “top,” throughput has a “top,” compliance has a “top,” etc. Sometimes multiple highest layers (“tops”) are healthy—they check and balance one another; sometimes they are harmful—they thrash belts, drain twist budgets, and trap the plant in stalemates. This chapter tells you how to tell the difference, how to run governance patterns that keep many tops healthy, and how to fall back to localization/splitting when pathology appears. The artifact is a Playbook Matrix you can run operationally.

24.1 Objects & Notation

Belts $b\in\mathcal B$ with plan/do edges $\Gamma_+^b,\Gamma_-^b$ and curvature $\mathcal F_\Pi^b$ .
Tops $\tau\in\mathcal T$ (e.g., safety, quality, throughput, fairness, carbon).
Each $\tau$ owns a charter, a ladder $\mathcal L_\tau$ of twist rungs (Ch. 20), and a time-scale $T_\tau$ (how often it acts).
Each rung $r\in\mathcal L_\tau$ has twist $\Delta\mathrm{Tw}_r$ and impact vector $\Delta U_{k,r}$ on objectives $k$ .
Cross-top interaction. For rungs $a\in\mathcal L_{\tau_i}, b\in\mathcal L_{\tau_j}$ ,
$\kappa^{\top}_{ij} \,[a,b] \quad\text{(signed commutator cost; learned from logs)}.$
Large $|\kappa^{\top}_{ij}|$ = “these tops step on each other.”
Budgets. Per-top twist budgets $B_\tau$ and a shared cap $B_{\text{global}}$ per window.

24.2 When Many Tops Are Healthy

A polycentric system is healthy when the following hold (all are measurable):

Charter orthogonality (intent level).
The purpose statements for tops have low alignment overlap beyond declared treaties.
Metric: cosine between charter embeddings $< \theta_{\text{orth}}$ .
Arbitration in the loop (resource level).
Any rung targeting the same belt routes through an Arbitration Belt (Ch. 21).
Metric: fraction of cross-top belt touches that went via arbitration $> 0.9$ .
Time-scale separation (control level).
Tops that often conflict act on different cadences: $T_{\tau_i}\ge 3\,T_{\tau_j}$ or they use offsets $\delta_{ij}^{\star}$ (Ch. 23).
Metric: realized phase-offset adherence $p\big(|\hat\delta_{ij}-\delta_{ij}^\star|\le \Delta\big)\ge p_{\min}$ .
Budget fences (governance level).
Per-top twist spend $\sum|\Delta\mathrm{Tw}|\le B_\tau$ and cross-charges when a rung harms a protected objective.
Metric: on-time settlement of cross-charges; budget leakage $<\epsilon$ .
Commutator smallness (geometry level).
The spectral radius of the cross-top commutator matrix $(\kappa^\top_{ij})$ on the active set is $<1$ .
Metric: realized commutator loss (extra twist beyond abelian sum) $\le 10\%$ of budget.
Stability & residual decay (plant level).
After multi-top actions, per-belt residuals $R^b$ (Ch. 18) decay within the target window.
Metric: $R^b\to R_{\text{target}}$ by $T_{\text{heal}}$ on $>95\%$ of cases.
Coherence banding (network level).
Belts influenced by the same top maintain cluster coherence (Shen), antagonistic tops keep healthy offsets.
Metric: cluster PLI $\ge$ floor; global $R$ $\le R_{\max}$ (Ch. 23).

If you can demonstrate these seven, multiple tops are an asset: they create redundancy, limit capture, and raise system legitimacy.

24.3 Pathological Signatures (Red Flags)

Watch for these diagnostics that many tops have turned harmful:

Ping–pong: alternation of $\tau_i$ and $\tau_j$ rungs > $K$ times in a window on the same belt.
Symptom: repeated OPEN/CLOSE in Flux Gates coupled with alternating twist steps.
Budget thrash: net $\sum|\Delta\mathrm{Tw}|$ high but $\Delta U$ near zero; MTE (coherence gain per twist) collapses.
Approval cycles: directed cycle in the approvals DAG across tops.
Symptom: time-to-decision $>T_{\max}$ , escalations without resolution.
Commutator blow-up: realized extra twist $> \epsilon_{\text{comm}} B_{\text{global}}$ .
Symptom: two harmless rungs become harmful in sequence.
Residual stalling: $R^b$ plateaus after multi-top moves; no decay within $T_{\text{heal}}$ .
Coherence fracture: $C_{ij}\downarrow$ within a top’s cluster while antagonistic pairs converge (unwanted lock).
Starvation: a protected objective’s utility $U_k$ stays below floor for $H$ windows.

24.4 Governance Patterns (What Works)

A. Subsidiarity-first Ownership

One owner-of-plant per belt at a time. Other tops file advisory rungs which route via arbitration.
Flux Gates remain under the belt owner; Twist Stepping by others requires owner co-sign.

B. Treaty Edges & Neutral Interfaces

Define treaty belts between tops (e.g., Release–Policy Interface): a thin scope with fixed offsets, KPIs, and a small shared budget $B_{\text{treaty}}$ .
Only interface rungs may cross; everything else remains local.

C. Cadence Constitution

Publish a cadence calendar: which top acts on which weekdays/hours; enforce cooldowns between antagonists.
Shen monitor enforces offsets; arbitration enforces budget.

D. Cross-Charge & Floors

If a rung from $\tau_i$ harms a floor objective of $\tau_j$ , it must include a cross-charge budget or a compensating rung.
Floors are encoded as hard constraints in the arbiter (Ch. 21).

E. Incident vs Steady-State Modes

In incidents, elevate one top (e.g., safety) to temporary single-top on impacted belts; others suspend.
Exit criteria restore polycentric mode with a post-mortem.

24.5 Fallback: Localization & Splitting (How to Exit Pathology)

When red flags persist:

Localize (shrink the blast radius)
- Zonalize the belt; give local owners temporary autonomy with mini-budgets $B_{\text{local}}$ .
- Freeze cross-top global rungs; allow only zonal rungs until $R^b$ decays.
Split Tops by Domain/Time
- Partition scope (products, regions, time-of-day).
- Or split cadences: $\tau_A$ acts on even weeks, $\tau_B$ on odd weeks.
Quarantine Edges (air-gap)
- Insert a treaty belt with fail-closed defaults.
- Require two-key approval for any cross-edge rung for $N$ windows.
Shadow Mode & A/B Tops
- Let the contending top run shadow policy with zero plant actuation; compare residuals and KPIs; the winner keeps ownership.
Decompose Objectives
- Factor an overloaded top into two objectives with cleaner charters (reduce charter cosine; Sec. 24.2.1).

24.6 A Simple Health Test (Math Lens)

Let $V_b$ be per-belt Lyapunov cards (Ch. 22). Build a global potential

\mathcal V=\sum_{b} V_b \;+\; \sum_{i<j}\lambda_{ij}\,\Xi_{ij},

where $\Xi_{ij}$ penalizes harmful cross-top sequencing (commutator proxies, coherence violations).

Healthy-many-tops condition: There exist $\lambda_{ij}\ge 0$ such that during flows and allowed jumps,

\dot{\mathcal V} \le -c\sum_b \|e_b\|^2 \quad\text{and}\quad \Delta \mathcal V \le -\mu\sum_b\|e_b\|^2.

Operationally: demonstrate block-diagonal dominance—interactions are small enough that local stability persists globally.

24.7 Procedures

A. Weight Tuning Across Tops (Weekly)

Use the Arbitration Belt with constitutional weights for safety/ethics and state-contingent boosts (Ch. 21).
Publish a frontier card (trade-offs) and a regret bound for transparency.

B. Commutator-Aware Scheduling (Daily)

Cluster near-commuting rungs; order clusters to minimize $\sum \kappa^\top_{ij}[r_i,r_j]$ ; insert cooldowns.

C. Shen Sync/Desync (Daily)

Strengthen intra-top lock; maintain healthy offsets between antagonists (Ch. 23).
Cap sync-only twist spend at $\tau_{\text{sync}}$ .

D. Safety Nets (Always)

Circuit breakers for commutator blow-ups.
Escalation path to single-top mode during incidents.

24.8 SLOs for Many-Tops Health

Arbitration adherence: $\ge 90\%$ of cross-top belt touches go through arbiter.
Commutator losses: $\le 10\%$ of budget per window.
No ping–pong: $\le 1$ alternation sequence per belt per window.
Residual recovery: $R^b$ heals $< T_{\text{heal}}$ after multi-top actions, p95.
Shen band: cluster PLI $\ge$ floor; global $R \le R_{\max}$ .
Starvation-free: every protected objective at or above floor over horizon $H$ .

24.9 Reference Logic (Diagnosis Loop)

def many_tops_health(belts, tops, window):
    # 1) Gather events
    actions = fetch_rungs(window)               # who changed what, where, when
    comm = estimate_commutators(actions)        # kappa^top_ij, [a,b]
    routes = check_arbitration_routes(actions)  # routed via arbiter?
    shen = compute_shen_metrics(belts)          # PLI, C_ij, R

    # 2) Scores
    pingpong = detect_pingpong(actions)         # alternations per belt
    budget = twist_spend_by_top(actions)
    comm_loss = realized_commutator_loss(actions, comm)
    residuals = residual_decay_stats(belts)

    # 3) Decisions
    if comm_loss > eps_comm or pingpong:
        apply_cooldowns(); create_treaty_belt(); split_cadence()
    if starvation_detected():
        raise_weight_floor_in_arbiter()
    if shen.global_R > R_max or cluster_breaks():
        desync_antagonists(); sync_intra_clusters(min_twist=True)

    # 4) Fallbacks
    if residuals.stalled or approvals_cycle_found():
        localize_zones(); freeze_cross_edges(); start_shadow_mode()

24.10 Domain Patterns

Factory. Throughput-top and Quality-top conflict on the main assembly. Healthy mode uses treaty belt “QC-after-stitching (+20 min)”; pathology appeared when both tops acted hourly → split cadences (T_quality=4 h; T_throughput=1 h) and re-routed via arbiter.
Software. SRE-top and Product-top contended over deploy cadence. After commutator losses spiked, a Release–Policy treaty fixed offsets; canary cadence became the neutral interface; incident mode temporarily gave SRE single-top authority.
Customer Ops. Fairness-top floors starved by Cost-top nudges. Arbiter added a fairness floor, and cost rungs now carry cross-charges; desync with Marketing-top by one day to avoid multi-top pileups on the same queue.

24.11 Artifact — Many-Tops Playbook Matrix

Symptom	Measurement	Likely Cause	First Moves (Minimal Twist)	If Persisting (Fallback)	Owner
Ping–pong on a belt	>1 alternation / window	Competing tops at same cadence	Insert cooldowns; route via arbiter; set healthy offset $\delta^\star$	Localize zones; split cadences; temporary single-top	Belt owner + Arbiter
High commutator loss	Extra twist >10% of budget	Non-commuting rung pair	Reorder (cluster-then-order); apply zonal-first	Treaty belt; forbid pair until new rungs	Arbiter
Residuals don’t heal	$R^b\not\downarrow$ by $T_{\text{heal}}$	Overlapping rungs; no owner	Owner-of-plant rule; freeze cross-edges	Quarantine interface; shadow-mode A/B tops	Belt owner
Starvation of a floor	$U_k$ below floor H windows	Weight drift / capture	Raise weight floor; apply compensating rung	Split objectives; re-charter	Council
Coherence fracture	$C_{ij}\downarrow$ , unwanted global lock	Cluster/antagonist mix-up	Sync inside clusters; desync antagonists	Cap sync-twist; adjust cadence constitution	Shen lead
Approval cycle	Cycle in DAG	Ambiguous authority	Define treaty owner; two-key for cross-edge	Single-top incident mode	Governance

YAML sketch

playbook_id: "many-tops/v1"
thresholds:
  arbitration_route_min: 0.9
  commutator_loss_max: 0.10
  pingpong_max: 1
  residual_heal_hours: 48
  shen:
    PLI_floor: 0.75
    R_global_max: 0.85
responses:
  pingpong:
    minimal: ["add_cooldown","arbiter_route","set_phase_offset"]
    fallback: ["localize_zones","split_cadence","single_top_incident"]
  commutator_loss:
    minimal: ["cluster_then_order","zonal_first"]
    fallback: ["treaty_belt","forbid_pair"]
  starvation:
    minimal: ["raise_weight_floor","compensating_rung"]
    fallback: ["split_objective","recharter"]
owners: ["belt_owner","arbiter","shen_lead","council"]

Takeaway

Multiple tops are not a bug—they’re how real systems keep themselves honest. They’re healthy when charters are orthogonal, arbitration and cadence offsets are honored, commutators are small, budgets are fenced, residuals heal, and Shen keeps clusters in phase. When red flags appear, localize, split, and treaty your way back to stability—with the Playbook Matrix as your operational guide.

25. Belt Networks

Purpose. Real operations are not single belts but networks of belts that branch, merge, feed back, and cross-influence. This chapter gives the cobordism rules for composing belts at junctions, the conservation laws that must hold under gluing, how errors/uncertainty accumulate across the network, and the procedures for tiling and composing invariants. The artifact is a versioned Network Schema you can run in ops.

25.1 Network Model

A belt $\mathcal B_e$ (edge $e$ ) has boundary
$\partial\mathcal B_e=\Gamma_{+,e}\sqcup(-\Gamma_{-,e}),$
with plan/do edges (Ch. 19–22). Purpose is a connection $\mathcal A_\Pi$ ; curvature $\mathcal F_\Pi$ drives macro work.
A junction $v$ is a compact surface $\mathcal J_v$ where multiple boundary loops glue (series, merge, split). Networks are directed multigraphs of belts with cobordisms $\{\mathcal J_v\}$ between their edges.
Holonomy & gap on a belt.
$\Delta_e \;\equiv\; \oint_{\Gamma_{+,e}}\!\mathcal A_\Pi - \oint_{\Gamma_{-,e}}\!\mathcal A_\Pi, \qquad \Delta_e = \iint_{\mathcal B_e}\!\mathcal F_\Pi + \alpha\,\mathrm{Tw}_e.$
(Edge-form PBHL; “gap = flux + α·twist.”)

25.2 Cobordism Rules (Junction Types)

Let $\mathcal S_v$ be the cobordism surface that fills the small junction region and glues incident belt edges with consistent orientation.

Series (1→1). Glue $\Gamma_{-,B}$ to $\Gamma_{+,A}$ with opposite orientation.
- Interior loop cancels: no external boundary created.
- Network belt is $A$ followed by $B$ .
Merge (2→1). Glue $\Gamma_{+,A}$ and $\Gamma_{+,B}$ into the single $\Gamma_{-,C}$ via a “pair-of-pants” surface $\mathcal S_v$ .
Split (1→2). Reverse of merge.
Feedback (cycle). Glue an outgoing boundary to an incoming boundary of the same or downstream belt to form a closed loop.

Orientation law. Every glued interior edge appears with opposite sign exactly once:

\Gamma_{\text{glued}} \oplus (-\Gamma_{\text{glued}}) = \varnothing.

This is what enables conservation under Stokes.

25.3 Conservation Under Gluing (Kirchhoff–Stokes Laws)

Network domain. Let $\Omega$ be the union of all belts and junction cobordisms after gluing. Its external boundary $\partial\Omega$ consists only of those loops left unglued (true network inputs/outputs).

Global law.

\boxed{ \sum_{\Gamma\in\partial\Omega}\!\!\sigma(\Gamma)\!\oint_{\Gamma}\!\mathcal A_\Pi \;=\; \iint_{\Omega}\!\mathcal F_\Pi \;+\;\alpha\,\sum_{e}\mathrm{Tw}_e \;+\;\alpha\,\sum_{v}\mathrm{Tw}_v, }

where $\sigma(\Gamma)=+1$ for outgoing (plan) and $-1$ for incoming (do). Twist at junctions $\mathrm{Tw}_v$ accounts for governance/frame changes performed at the junction (handoff rules, contract edges, routers, releases).

Local node law (merge). For two inputs to one output:

\Delta_A+\Delta_B-\Delta_C \;=\; \iint_{\mathcal S_v}\!\mathcal F_\Pi \;+\;\alpha\,\mathrm{Tw}_v.

If $\mathcal S_v$ carries negligible curvature and no policy change at $v$ , the law reduces to gap additivity: $\Delta_C\approx \Delta_A+\Delta_B$ .

Series. $\Delta_{A\to B}=\Delta_A+\Delta_B$ (interior edge cancels exactly).

Cycles. For any fundamental cycle $\gamma$ of the network,

\oint_{\gamma}\!\mathcal A_\Pi = \iint_{\Sigma_\gamma}\!\mathcal F_\Pi +\alpha\,\mathrm{Tw}_\gamma,

with $\Sigma_\gamma$ any spanning cobordism of the cycle; class-function invariants (trace/character, $\arg\det$ ) are cycle invariants.

25.4 Non-Abelian Composition & Commutators

On a path $p=e_1\cdots e_m$ , the holonomy is the ordered product
$\mathrm{Hol}(p)=\mathcal P\exp\!\left(\int_{e_m}\!A\right)\cdots\mathcal P\exp\!\left(\int_{e_1}\!A\right).$
Class invariants (trace, principal eigenphase, $\arg\det$ ) are stable under basepoint changes and are the recommended network observables.
Commutator penalties. Non-commuting rungs and frames across junctions introduce second-order corrections (BCH):
$\mathrm{Hol}(ab)\approx \mathrm{Hol}(a)\,\mathrm{Hol}(b)\,\exp\!\Big(\tfrac12 [A_a,A_b]+\cdots\Big).$
We track them as commutator losses in twist/entropy budgets (Ch. 21, 24).

25.5 Error & Uncertainty Accumulation

Let each belt estimate carry bias $b_e$ , variance $\sigma_e^2$ , and a discretization error $\varepsilon_e \lesssim C_1 h_e^{p}+C_2(\kappa_e w_e)^2 + C_3\|\nabla_\perp \mathcal F\| w_e^2$ (mesh width $h_e$ , window $w_e$ ).

Series path $p$ .
$\widehat{\Delta}_p=\sum_e \widehat{\Delta}_e,\quad b_p=\sum_e b_e,\quad \sigma_p^2\approx \sum_e \sigma_e^2 + 2\!\!\sum_{e<f}\!\mathrm{Cov}_{ef}.$
Junction cobordism. Add $\varepsilon_{\mathcal S_v}$ and junction estimation noise; large $|\kappa^\top_{ij}|$ at $v$ raises interaction variance.
Cycle invariants. Use log-Euclidean aggregation for phase statistics to avoid wrap-around bias; quote p95 cones.

Bound. For a connected network with $E$ belts and $V$ junctions,

\|\widehat{\mathcal F}_\Omega-\mathcal F_\Omega\| \;\lesssim\; \sum_{e=1}^{E}\varepsilon_e \;+\;\sum_{v=1}^{V}\varepsilon_{\mathcal S_v}\;+\;\mathcal O(\text{commutators}).

Keep the commutator term below budget via cluster-then-order scheduling (Ch. 21, 24).

25.6 Procedures: Network Tiling & Composition

A. Tiling

Cut at natural interfaces: ownership changes, cadence boundaries, physical handoffs.
Refine where curvature varies: $\|\nabla \mathcal F\|$ high → smaller tiles.
Respect governance seams: put junction cobordisms exactly where policy/contract changes happen.

B. Basepoints & Frames

Choose a network basepoint per component; fix orientation conventions; store re-basing maps at each junction.

C. Compose Invariants

Pick a cycle basis of the network graph (fundamental cycles).
Form ordered products of edge holonomies; reduce with re-basing.
Compute class invariants (trace, $\arg\det$ , principal eigenphase) and 4π periodic tests; store in the network card.

D. Junction KPIs

Junction residual: $R_v\equiv(\sum_{\text{in}}\Delta)-(\sum_{\text{out}}\Delta)-\widehat{\Phi}_{\mathcal S_v}-\alpha\widehat{\mathrm{Tw}}_v$ .
SLOs: $|R_v|\le r_{\max}$ ; commutator loss at $v$ $\le\epsilon_v$ budget; recovery time $\le T_{\text{heal}}$ .

25.7 Reference Logic (Pseudocode)

def evaluate_belt_network(net):
    # net: {belts: E, junctions: V, gluings, cycles}
    # 1) per-belt estimates
    for e in net.belts:
        e.delta = gap_from_flux_twist(e)       # Ch. 16–20 estimators
        e.err   = error_model(e)               # bias/var/disc

    # 2) junction residuals (local laws)
    for v in net.junctions:
        inflow  = sum(e.delta for e in v.inputs)
        outflow = sum(e.delta for e in v.outputs)
        phi_J   = curvature_over_cobordism(v.surface)
        Rv      = inflow - outflow - phi_J - alpha * v.twist
        record_kpi(v.id, Rv)

    # 3) cycles and invariants
    for cyc in net.cycles:
        Hol = ordered_holonomy_product(cyc.edges)
        inv = class_invariants(Hol)            # trace, argdet, principal eigenphase
        record_cycle(cyc.id, inv)

    # 4) aggregate errors
    net_err = aggregate_errors(net)
    return net_err, net.kpis, net.cycles

25.8 Network SLOs

Junction closure: $|R_v|\le r_{\max}$ p95; no >1 breach per window.
Commutator loss: realized extra twist from non-commuting junctions $\le \epsilon_{\text{comm}}$ of twist budget.
Cycle invariants: drift within tolerance bands; 4π periodic checks pass.
Latency: end-to-end detection latency $L_{95}\le L_{\max}$ across longest path.
Stability: composed Lyapunov $\mathcal V$ (sum of belt cards + node penalties) decreases on replays.

25.9 Domain Patterns

Factory line. Belts: cut → stitch → QA → pack (series), merge two stitching lines into one QA line, split to two packaging lanes. Junction residuals flag that the merge cobordism carries policy twist (handoff rule), not just geometry; adding a small treaty at the merge fixes $R_v$ .
Software microservices. Belts per service with feedback cycles (retry loops). Merge at API gateway; split into canary/steady traffic. Cycle invariants on deploy trains detect latent non-abelian effects between feature-flag rungs and autoscale frames.
Customer ops. Merge web and phone intake belts into triage; split into regions. Junction SLOs keep script/policy twists local; network tiling isolates marketing-policy cobordisms to cap commutator loss.

25.10 Artifact — Network Schema (YAML)

network_id: "plant-shoes/v3"
owners: ["ops-lead","qa-lead","arbiter"]
belts:
  - id: cut
    edges: {plan: "Γ+_cut", do: "Γ-_cut"}
    params: {mesh_h: 0.5, window_min: 10}
  - id: stitch_A
    edges: {plan: "Γ+_stA", do: "Γ-_stA"}
  - id: stitch_B
    edges: {plan: "Γ+_stB", do: "Γ-_stB"}
  - id: qa
    edges: {plan: "Γ+_qa", do: "Γ-_qa"}
  - id: pack_1
  - id: pack_2
junctions:
  - id: merge_stitch_to_qa
    type: "merge"
    inputs:  ["stitch_A", "stitch_B"]
    outputs: ["qa"]
    surface: "pair_of_pants_v1"
    twist_budget: 2.0
    treaty: {owner: "qa-lead", rules: ["handoff-window:+20min","sample+10%"]}
  - id: split_qa_to_pack
    type: "split"
    inputs:  ["qa"]
    outputs: ["pack_1","pack_2"]
    surface: "Y_split_v2"
cycles:
  - id: rework_loop
    edges: ["qa","pack_1","qa"]   # feedback
    invariants: ["argdet","trace"]
constraints:
  invariants_4pi: true
  commutator_loss_max: 0.1
  junction_residual_max: 1.2
telemetry:
  sampling: {fast_sec: 60, est_min: 5}
  metrics: ["gap","flux","twist","Rv","cycle_invariants"]
slo:
  junction_closure_p95: 1.0
  latency_p95_sec: 5
  cycle_invariant_drift_max: 0.05

Takeaway

Belt Networks make “gap = flux + α·twist” a global statement: internal boundaries cancel, junction cobordisms account for merges/splits, and cycle invariants expose feedback structure. With clear cobordism rules, conservation laws, and explicit error aggregation, you can tile, glue, and govern complex plants without losing stability—or your twist budget.

26. Conflict Diagnostics

Purpose. Conflicts show up as wasted twist, oscillations, “ping–pong” between tops, and stubborn residuals. This chapter gives you a precise way to find and quantify those conflicts—via coherence heatmaps, commutator norms, and twist volatility—and then root-cause them with residual partitioning tied back to concrete rungs, belts, and junctions. Deliverable: a set of Diagnostic Notebooks you can run on logs.

26.1 What We Call a “Conflict”

A conflict is any interaction that raises entropy or burns twist without improving the intended utilities:

Cross-belt timing clashes: phase drift (low $C_{ij}$ ) causing rework, queues, or missed handoffs.
Non-commuting governance moves: order-dependent rungs creating extra churn (large commutator loss).
Budget thrash: high $\sum|\Delta\mathrm{Tw}|$ with flat $U_k$ and non-healing residuals $R$ (Ch. 18).
Alias/lag fights: fast Flux Gates acting against slow twist steps (violating Ch. 22 etiquette).
Junction leaks: merge/split cobordisms with persistent node residuals $R_v$ (Ch. 25).

26.2 Signals & Views

You’ll look at the plant through three lenses, computed on a rolling window $W$ .

A. Coherence Heatmaps

For belts $i,j$ with phases $\varphi_i,\varphi_j$ (Ch. 23),

C_{ij} \;=\; \big|\ \mathbb E_{t\in W}\big[e^{\,i(\varphi_i-\varphi_j-\delta_{ij}^\star)}\big]\ \big|\in[0,1].

Heatmap: matrix $[C_{ij}]$ with trend layer $\dot C_{ij}$ .
Flags: $C_{ij}\!\downarrow$ below a floor and $|\dot C_{ij}|$ large.

B. Commutator Norms

Let a window contain ordered rungs $\pi=(r_1,\dots,r_m)$ with pairwise interaction weights $\kappa_{pq}$ (learned) and signs $[r_p,r_q]\in\{-1,0,+1\}$ . Define the commutator matrix

\mathbf K_{pq}\;=\;\kappa_{pq}\,[r_p,r_q].

Norms: $\|\mathbf K\|_2$ (spectral), $\|\mathbf K\|_F$ (Frobenius).
Realized commutator loss

L_{\text{comm}} \;\equiv\; \Big|\mathrm{Tw}_{\text{eff}}(\pi)\Big| - \sum_{j=1}^{m}\big|\Delta\mathrm{Tw}_{r_j}\big| \ \approx\ \tfrac12 \sum_{p<q}\!\big|\mathbf K_{pq}\big|.

High $\|\mathbf K\|$ or $L_{\text{comm}}$ → order-sensitive thrash.

C. Twist Volatility

For each belt/top in $W$ ,

\mathrm{RV}_{\mathrm{Tw}}=\sum_{t\in W}(\Delta\mathrm{Tw}_t)^2, \qquad \mathrm{AV}_{\mathrm{Tw}}=\sum_{t\in W}\big|\Delta\mathrm{Tw}_t\big|, \qquad \lambda_{\text{jumps}}=\frac{\#\{\Delta\mathrm{Tw}\ne 0\}}{|W|}.

Spike in RV or jump intensity with flat utilities $U_k$ → budget burn.

26.3 Conflict Index (per Belt / Junction / Top)

Define a composite Conflict Index $\mathrm{CI}\in[0,\infty)$ that lights up when the three lenses agree:

\mathrm{CI} =\underbrace{\omega_C \sum_{j}(1-C_{ij})_{+}}_{\text{desync}} +\underbrace{\omega_K \frac{L_{\text{comm}}}{\text{budget}}}_{\text{non-commute}} +\underbrace{\omega_V \frac{\mathrm{RV}_{\mathrm{Tw}}}{\text{budget}^2}}_{\text{twist thrash}} +\underbrace{\omega_R \frac{\overline{R^+}}{\text{target}}}_{\text{residual stall}}.

Tune $\omega_\bullet$ by domain.
Track $\mathrm{CI}$ for nodes too using $R_v$ (Ch. 25) and local $L_{\text{comm}}(v)$ .

26.4 Residual Partitioning (Root-Cause)

We want to explain the residual $R$ (Ch. 18) in a window by who did what, where, and in what order.

A. Event-Kernel Attribution

Model incremental residual as a sum of rung-aligned impulse responses plus noise:

\Delta R_t \;=\; \sum_{r\in \pi}\!\big(\beta_r\,k(t-t_r)\big)\;+\;\sum_{p<q}\!\big(\gamma_{pq}\,k_\times(t-t_p,t-t_q)\big)\;+\;\varepsilon_t.

$k$ : short causal kernel (e.g., decaying ramp).
$k_\times$ : pairwise interaction kernel (only for nearby times).
Fit $\beta_r,\gamma_{pq}$ with lasso or group-lasso → sparse culprit set.

Attributions.

Direct: $A_r=\beta_r\!\int\!k$ .
Interaction: $A_{pq}=\gamma_{pq}\!\!\iint\!k_\times$ .
Normalize so $\sum_r A_r+\sum_{p<q}A_{pq}\approx \sum \Delta R_t$ .

B. Junction Partition

At node $v$ (merge/split), decompose (Ch. 25):

R_v \;=\; \underbrace{\sum_{\text{in}}\!b_e - \sum_{\text{out}}\!b_e}_{\text{estimation bias}} \;+\;\underbrace{\Phi_{\mathcal S_v}^{\text{miss}}}_{\text{unmodeled curvature}} \;+\;\underbrace{\alpha\,\Delta\mathrm{Tw}_v^{\text{unlogged}}}_{\text{governance leak}} \;+\;\underbrace{\tilde L_{\text{comm}}(v)}_{\text{non-commute at junction}}.

Use local probes to bound each term; if $\Delta\mathrm{Tw}_v^{\text{unlogged}}\neq 0$ , the treaty/interface is being bypassed.

C. Counterfactual Replay

Re-run the week without each candidate rung (or with swapped order for pairs) using the quick response surrogate (Ch. 19–21). The delta residual is the rung’s Shapley-lite contribution.

26.5 Diagnostic Procedures

Step 1 — Scan

Compute $C_{ij},\dot C_{ij}$ , $\mathbf K$ , $L_{\text{comm}}$ , twist volatility.
Rank belts/nodes/tops by $\mathrm{CI}$ ; open Conflict Cases for top-k.

Step 2 — Zoom

For each case, render:
- Phase panel: $\varphi_i$ , $C_{ij}$ , targets $\delta_{ij}^\star$ .
- Governance panel: rung timeline, $\Delta\mathrm{Tw}$ , commutator graph.
- Plant panel: $R, W_{\text{macro}}, \Sigma_{\text{macro}}$ (with alarms).

Step 3 — Attribute

Fit event-kernel model; partition $\Delta R$ into direct vs interaction attributions.
At junctions, decompose $R_v$ with node probes.
Run counterfactual replays for top suspects.

Step 4 — Decide & Record

If mostly desync → Chapter 23 remedies (sync inside cluster / desync antagonists).
If non-commute → Chapter 21/24 scheduling (cluster-then-order, treaty belts, cooldowns).
If thrash → Chapter 22 etiquette (increase deadband/cooldowns; slow loop dwell).
If governance leak → lock interface; enforce two-key treaty.

Log the root cause, the minimal-twist fix, and expected $\Delta R,\ \Delta \Sigma$ .

26.6 Reference Notebook Flows (Artifact)

Notebook A — Coherence & Phase

Load $\varphi_i$ , compute $C_{ij}, \dot C_{ij}$ , spectral clusters, antagonistic pairs.
Heatmaps with tool-tips: belts, owners, target offsets.
“What-if” sliders for cadence nudges; predicted $\Delta \mathrm{PLI}$ .

Notebook B — Commutators & Order

Build $\mathbf K$ from rung logs; show commutator graph; $\|\mathbf K\|$ , $L_{\text{comm}}$ .
Try alternative orders with cluster-then-order; estimate twist savings.

Notebook C — Twist Volatility

RV/AV/jump intensity; change-point detection; ping–pong finder.
Correlate volatility with utilities and residuals; flag “budget burn” zones.

Notebook D — Residual Partitioning

Fit event-kernel model; show contributions (bar chart) by rung/pair.
Junction sheet: $R_v$ terms with probes and treaty compliance.

Notebook E — Counterfactual Replay

One-click replays removing single rungs or swapping pairs; plot $R,W,\Sigma$ deltas.
Export “Minimal-Twist Fix Pack” (links to Ch. 19–21 actuators).

26.7 Pseudocode (Batch Diagnostic)

def run_conflict_diagnostics(window_logs):
    phases = extract_phases(window_logs)                  # φ_i(t)
    C, Cdot = coherence_matrix(phases)                    # C_ij, trends
    rungs = extract_rungs(window_logs)                    # sequence with times, ΔTw, owners
    K = build_commutator_matrix(rungs)                    # κ_pq [r_p, r_q]
    L_comm = realized_commutator_loss(rungs, K)
    tw_stats = twist_volatility(rungs)                    # RV, AV, jumps
    resid = residual_series(window_logs)                  # R(t), R_v(t)

    CI = conflict_index(C, Cdot, L_comm, tw_stats, resid)
    cases = select_topk(CI)

    for case in cases:
        timeline = slice_logs(window_logs, case)
        model = fit_event_kernel_attribution(timeline)
        attributions = model.attributions()
        cf = counterfactual_replay(timeline, suspects=top_suspects(attributions))
        recommendation = minimal_twist_fix(cf, scheduling_rules="cluster-then-order")

        emit_report(case, C, K, L_comm, tw_stats, resid, attributions, recommendation)

    return cases

26.8 SLOs for Diagnostics

Detection latency: conflict raised within $L_{95}$ of onset.
Attribution precision: p95 of counterfactual checks confirm sign of top-2 attributions.
Repair guidance: recommended minimal-twist plan reduces $R$ by $\ge \Delta R_{\min}$ on replay.
Noise robustness: false-positive rate $\le \epsilon$ on synthetic nulls.
Coverage: 100% of junctions have $R_v$ sheets; 100% of antagonistic pairs monitored.

26.9 Domain Patterns

Factory. Coherence heatmap shows stitching vs QA drift after a roster change; $L_{\text{comm}}$ low → it’s desync. A cadence nudge (+20 min QA window) fixes $R$ with $|\Delta\mathrm{Tw}|<1$ .
Software. High $L_{\text{comm}}$ between “retry-backoff +100 ms” and “autoscale floor +1” rungs; order swap (scale first) drops commutator loss 60% in replay.
Customer ops. Twist volatility spikes when marketing pushes overlap refund-policy steps; event-kernel points to the pair. Arbitration schedules a 1-day offset and caps sync-twist spend.

26.10 What to Hand Over

Diagnostic Notebooks (A–E) with sample datasets and regression tests.
Config presets (thresholds for $C$ , $\|\mathbf K\|$ , RV; kernel choices; windows).
Report template: “Conflict Case” packet—signals, root cause, minimal-twist recommendation, and expected KPI changes.
Ops runbook mapping each conflict class → the right chapter’s actuator (Ch. 19–24).

Takeaway

Conflicts leave fingerprints: belts fall out of phase, rungs stop commuting, twist starts to thrash, and residuals stall. By triangulating coherence, commutators, and volatility, then partitioning the residual back to concrete actions, you get fast, auditable diagnoses—and minimal-twist fixes that actually heal the plant.

27. Governance Patterns

Purpose. You now have working belts, polycentric tops, arbitration, Shen, and networks (Ch. 19–26). This chapter shows how to scale governance from a single plant to an ecosystem—multiple products, sites, partners, or even multi-org supply chains. We give three deployable patterns—Federated, Hub-and-Spoke, Marketplace—with precise roles, budget processes, cadence constitutions, and anti-thrash guarantees. The artifact is a Pattern Catalog (templates + configs) you can adopt and mix.

27.1 Building Blocks (recap → how they appear at ecosystem scale)

Belts: unit of control; plan/do edges with purpose connection $\mathcal A_\Pi$ , curvature $\mathcal F_\Pi$ .
Tops: objective owners (safety, throughput, fairness…); each has rungs (Ch. 20) and time-scale (Ch. 22).
Arbitration Belt: spends twist budget $\tau$ across goals (Ch. 21).
Shen: phase-lock across belts; target offsets $\delta^\star$ , coherence $C_{ij}$ (Ch. 23).
Treaty Belts: neutral interfaces between owners/top pairs with tiny dedicated budgets.
Network Cobordisms: gluing rules at junctions; node residuals $R_v$ as health tests (Ch. 25).
Diagnostics: coherence heatmaps, commutator norms, twist volatility, residual partitioning (Ch. 26).

Patterns differ mainly by who owns arbitration, how budgets are allocated, and how cadence/offsets are set.

27.2 Pattern Overview—when to use what

Pattern	Works best when	Risk it mitigates	Risk it introduces	First knobs
Federated	Many capable sub-orgs with clear boundaries	Central bottlenecks, political capture	Drift/divergence across sites	Charter orthogonality; treaty belts; weekly federation council
Hub-and-Spoke	Tight coupling, safety-critical or crisis	Cross-team ping-pong; slow incident response	Central overload; single point of failure	Hub SLOs; incident single-top mode; cadence constitution
Marketplace	Diverse proposals and measurable utilities	Budget opacity; low ROI twist spend	Gaming, short-termism if unchecked	Auction/clearing rules; ex-post audits; fairness floors

27.3 Pattern A — Federated Purpose

Decentralized control with a Federation Council that sets constitutional floors and arbitrates between autonomous nodes.

Roles

Federate (site/product/partner): owns local belts and tops; runs Ch. 19–26 locally.
Federation Arbiter: meta-arbitration across federates with soft weights and floors.
Treaty Owners: steward interfaces (release–policy, QA–ops, partner SLAs).

Budget program (weekly)

Let $f\in\mathcal F$ index federates. Each proposes rungs $r\in\mathcal L_f$ with $\Delta \mathrm{Tw}_r$ and objective deltas $\Delta U_{k,r}$ .

\max_{\{S_f\}}\ \sum_{f}\sum_{k} w_{k,f}\!\!\sum_{r\in S_f}\!\Delta U_{k,r} \quad\text{s.t.}\quad \sum_{f}\sum_{r\in S_f}\!|\Delta \mathrm{Tw}_r|\le \tau_{\text{global}}, \ \sum_{r\in S_f}\!|\Delta \mathrm{Tw}_r|\le \tau_f, \ \text{floors},

with floors (safety/equity) as hard constraints and commutator penalties if cross-federate rungs don’t commute at shared junctions.

Governance mechanics

Charter orthogonality check (cosine threshold) before admitting a new top.
Cadence constitution: publish $\delta^\star$ offsets between federates’ key windows; Shen monitors compliance.
Cross-charges: when a rung in $f_1$ harms a protected floor in $f_2$ , allocate budget or compensating rung.
Escalation: incidents flip impacted belts to single-top (safety) temporarily; council restores polycentric mode after post-mortem.

Federated template (YAML)

pattern: federated
council:
  window_days: 7
  members: ["safety","quality","ops","finance","partners"]
budgets:
  global_twist: 36.0
  per_federate: {site_A: 8.0, site_B: 6.0, partner_X: 4.0, default: 5.0}
floors:
  safety: {type: hard, U_min: 0.0}
  fairness: {type: rolling, horizon_days: 28, share_min: 0.2}
cadence:
  offsets:
    - {pair:["site_A.deploy","policy.global"], minutes: 60}
    - {pair:["partner_X.SLA","billing"], hours: 24}
treaties:
  - {id: "release-policy", budget: 2.0, owner: "release_office"}
audit:
  publish_frontier_card: true
  cross_charges_enabled: true

27.4 Pattern B — Hub-and-Spoke

A central Hub (control tower) runs arbitration and cadence; Spokes (teams/sites) execute with local autonomy inside bands. Best for high coupling or when safety/latency dominate.

Roles

Hub: owns arbitration, cadence calendar, incident response; curates ladder libraries.
Spokes: propose rungs, run Flux Gates, close local gaps; escalate via templates.

Operating rules

Band control: each spoke has throughput/entropy/Shen bands; falling outside triggers hub intervention.
Calendar: hub publishes a weekly cadence with protected quiet windows and antagonistic offsets.
Failover tiers (Ch. 19): hub can invoke T2/T3 globally; spokes keep T0/T1 local.

Hub program

\max_{S}\ \sum_k w_k \sum_{r\in S}\Delta U_{k,r} - \lambda \Big(\sum_{r\in S} |\Delta\mathrm{Tw}_r| + \tfrac12\!\!\sum_{i<j}\!\kappa_{ij}[r_i,r_j]\Big) \ \ \text{s.t. bands + floors.}

Hub-and-Spoke template

pattern: hub_and_spoke
hub:
  owner: "ops_control_tower"
  cadence_calendar: "weekly/v5"
  quiet_windows: ["Tue 14:00-18:00 deploy", "Fri 10:00-16:00 policy"]
spokes:
  - id: "factory_A"
    bands: {W: [40k, 52k], Sigma_max: 1.2, Shen_min: 0.7}
  - id: "svc_payments"
    bands: {SLO_err: 0.1, deploy_rate_max: 4/h}
failover:
  tiers: ["T0","T1","T2","T3"]
  global_escalation_rights: ["hub"]
interaction_costs:
  kappa_matrix_ref: "…/kappa_ij.parquet"

27.5 Pattern C — Marketplace (Twist Exchange)

Treat twist budget as a scarce resource and allocate it by auction. Rungs submit bids with expected utility uplift; the arbiter clears at a twist price $\lambda$ . Works when proposals are plentiful and utilities are measurable.

Mechanism

Each rung $r$ declares $(\Delta U_{k,r},\,|\Delta\mathrm{Tw}_r|,\,\text{risk},\,\text{scope})$ .
Arbiter solves a budgeted max-sum; the KKT multiplier becomes the clearing price $\lambda$ (twist per unit utility).
VCG-lite or ex-post penalty keeps truthfulness: realized uplift below declared → rebate clawback or reduced future quota.

Program

\max_S \sum_{r\in S}\Big(\sum_k w_k \Delta U_{k,r}\Big) \quad\text{s.t.}\quad \sum_{r\in S}|\Delta\mathrm{Tw}_r|\le \tau,\quad \text{floors},\quad \text{penalize commutators}.

Clearing price $\lambda^\*=\partial \mathcal L/\partial \tau$ . Accept $r$ iff value $/$ cost $\ge \lambda^\*$ and guardrails pass.

Anti-gaming rails

Probe windows (Ch. 20) and ex-post audits.
Fairness floors and per-actor caps to prevent capture.
Commutator fees: pairs with large $|\kappa_{ij}|$ pay order-penalties.

Marketplace template

pattern: marketplace
auction:
  window_days: 7
  objective_weights: {safety:1.0, quality:0.6, throughput:0.6, cost:0.3}
  floors: {safety:"hard", fairness:{type:"soft_floor", share_min:0.2}}
  commutator_fee_coeff: 0.5
  vcg_lite: {enabled:true, penalty_factor: 0.6}
budget:
  tau_weekly: 24.0
  caps_per_actor: 6.0
admission:
  min_probe_quality: 0.7
  max_blast_radius: "zonal:80%, global:1"
clearing:
  publish_price: true
  tie_break: "minimal_twist_then_smallest_scope"

27.6 Purpose Federation (common procedure across patterns)

Charter & Policy Grammar
- Write short purpose charters per top; check orthogonality.
- Normalize rung grammar (fields: $\Delta\mathrm{Tw}$ , $G$ , risk, scope, probe, rollback).
Interface Layout
- Identify natural seams; install treaty belts with owners and micro-budgets.
- Define cadence constitution: offsets $\delta^\star$ , quiet windows, incident handover.
Budgets & Floors
- Global $\tau$ , per-unit caps, protected floors (safety/fairness/ethics), and cross-charge rules.
Arbitration Setup
- Choose Federated, Hub, or Marketplace as the primary allocator; keep others as fallback modes.
Shen & Diagnostics
- Enable coherence monitors; register antagonistic pairs; wire Conflict Diagnostics notebooks.

27.7 Budget Negotiation (playbooks)

A. Cooperative bargaining (Federated)

Compute proposals $\{S_f\}$ , then solve:

\max_{\{x_f\}}\ \prod_f \left(U_f(x_f)-\underline U_f\right) \quad\text{s.t.}\ \sum_f x_f \le \tau,\ x_f\ge 0,

with $U_f$ the best utility each federate can realize with twist $x_f$ . This Nash bargaining yields fair splits; implement with bisection + knapsack oracles.

B. Priority bands (Hub)

Assign bands and reserve twist: critical tops hold a standing tranche; the rest compete for the residual via the arbiter.

C. Price discovery (Marketplace)

Iterate $\lambda$ until demand for twist equals $\tau$ ; accept rungs with value $/$ cost $\ge \lambda$ .

Negotiation loop (pseudocode)

def negotiate_budget(pattern, proposals, tau):
    if pattern == "federated":
        return nash_bargain(proposals, tau)      # cooperative split, oracle to each S_f
    if pattern == "hub_and_spoke":
        return hub_reserve_then_arbitrate(proposals, tau)
    if pattern == "marketplace":
        return price_discovery(proposals, tau)   # find λ*, accept value/cost ≥ λ*

27.8 Pattern Selection Matrix

Symptom / Need	Prefer	Why
High coupling, frequent incidents	Hub-and-Spoke	Single cadence & fast escalation
Many capable semi-autonomous units	Federated	Local learning, reduced central bottlenecks
Portfolio of diverse proposals, measurable uplifts	Marketplace	Best ROI on twist; transparent trade-offs
Trust deficit between units	Federated + Treaties	Neutral interfaces & cross-charges
Capture risk by a dominant top	Marketplace + Floors	Price + constitutional protections

27.9 SLOs & Anti-Failure Rails

Budget adherence: $\sum|\Delta\mathrm{Tw}|\le \tau$ p99; per-unit caps respected.
Fairness floors: protected objectives never below floor over horizon $H$ .
Commutator loss: $\le 10\%$ of twist spend.
Arbitration latency: $L_{95}\le L_{\max}$ for decision publish.
Transparency: publish frontier/clearing cards; log cross-charges; archive probe outcomes.
Anti-gaming: ex-post audits; probe falsification penalties; quota back-offs on repeated misses.
Resilience: incident mode tested (tabletop) quarterly; failback between patterns (e.g., marketplace → hub) under stress.

27.10 Pattern Catalog (Artifact)

Schema (YAML)

catalog_id: "governance-patterns/v1"
patterns:
  - include: "federated"
    template_ref: "s3://pfbt/patterns/federated.yml"
    defaults:
      global_twist: 36.0
      cadence_offsets: [...]
      floors: {safety:"hard", fairness:{type:"rolling_floor", share_min:0.2}}
  - include: "hub_and_spoke"
    template_ref: "s3://pfbt/patterns/hub.yml"
    defaults:
      quiet_windows: [...]
      failover_tiers: ["T0","T1","T2","T3"]
  - include: "marketplace"
    template_ref: "s3://pfbt/patterns/market.yml"
    defaults:
      auction_window_days: 7
      vcg_lite: {enabled:true, penalty_factor:0.6}
selection_matrix_ref: "s3://pfbt/patterns/matrix_v2.csv"
guardrails:
  invariants_4pi: true
  commutator_loss_max: 0.10
  starvation_guard: {horizon_weeks: 4}
ops_runbooks:
  - "budget_negotiation.md"
  - "cadence_constitution.md"
  - "incident_single_top.md"
audit:
  publish_frontier_cards: true
  cross_charges_policy: "v3"

27.11 Domain Walk-Throughs

Manufacturing supply web. Tier-2 factories run Federated locally; OEM runs Hub for global cadence; a light Marketplace allocates a weekly innovation twist pot. Treaty belts at handoff to OEM QA cap commutator loss; floors protect safety.
Cloud platform. Core SRE operates Hub; product teams are Spokes with bands; a Marketplace funds performance experiments. Incident mode elevates SRE to single-top; once stable, marketplace resumes.
Public services. Regions act as Federates; central regulator provides floors and cadence constitution; a Marketplace with fairness floors allocates twist for policy pilots, with Shen syncing rollouts across neighboring regions.

Takeaway

Governance patterns are how you scale PFBT: decide who spends twist, how cadence is set, and how conflicts are priced and resolved. Pick Federated for autonomy with treaties, Hub-and-Spoke for tightly coupled or safety-critical contexts, and Marketplace to maximize ROI under floors. All three live on the same spine—gap = flux + α·twist, minimal-twist doctrine, Shen bands, and commutator-aware scheduling—so you can switch modes without losing stability.

28. Belt-KPI Specification

Purpose. Define a compact, auditable five-line KPI for every belt: Gap / Flux / Twist / Coherence / Residual. Specify units, scaling, acceptance bands, and the streaming pipeline that produces them. Deliverables: Metric Contracts and SLI/SLO packs ready for ops.

28.1 Five-Line KPI (definitions & units)

Let a belt operate over window $W=[t-\Delta,t]$ with surface $\mathcal B_W$ and edges $\Gamma_+,\Gamma_-$ .

Gap

\mathrm{Gap}\equiv \oint_{\Gamma_+}\!\mathcal A_{\Pi} -\oint_{\Gamma_-}\!\mathcal A_{\Pi} -\iint_{\mathcal B_W}\!\widehat{\mathcal F}_{\Pi} \quad[\text{domain units}]

Interpretation: plan–do mismatch after accounting for estimated curvature (work driver).

Flux (work-aligned)

\mathrm{Flux}\equiv \iint_{\mathcal B_W}\!\langle \widehat{\mathcal F}_{\Pi},\widehat{J}_{\Pi}\rangle \quad[\text{domain units}/\Delta]

Examples: pairs-of-shoes/shift, tickets/day, deploy-value/hour.

Twist

\mathrm{Tw}\equiv \text{framing/ governance change in }W \quad[\text{twist units}];\quad \left|\mathrm{Tw}\right|=\sum|\Delta\mathrm{Tw}|

Minimal-twist doctrine measures magnitude and count of policy/frame steps.

Coherence ( $C\in[0,1]$ )

C\equiv \left|\frac{1}{|N|}\sum_{j\in N} e^{\,i(\varphi-\varphi_j-\delta_{\,j}^{\star})}\right|

Phase-lock vs neighbors $N$ at target offsets $\delta^\star$ (Ch. 23).

Residual

R\equiv \text{observed shortfall not explained by Flux or Twist} \quad[\text{domain units}]

Used for attribution/diagnostics (Ch. 18, 26).

Identity check (belt-form):

\mathrm{Gap}=\alpha\,\mathrm{Tw}+R \quad (\text{operational consistency})

28.2 Scaling & normalization

Domain units first. Gap, Flux, Residual in native output units (e.g., “pairs”, “tickets”).
Indexed views (dimensionless):
$\mathrm{FluxIdx}=\frac{\mathrm{Flux}}{\text{target}_\mathrm{Flux}},\quad \mathrm{TwIdx}=\frac{|\mathrm{Tw}|}{B_{\mathrm{Tw}}},\quad \mathrm{GapIdx}=\frac{|\mathrm{Gap}|}{\text{band}_\mathrm{Gap}},\quad \mathrm{RIdx}=\frac{|R|}{\text{band}_R}$
Robust smoothing: median-of-means over sub-windows + EWMA( $\lambda\!\in\![0.1,0.3]$ ).
Uncertainty bands: carry $p95$ cones from estimators: $\pm \sigma_{\text{Flux}},\pm \sigma_R$ .

28.3 Acceptance bands (green / amber / red)

Per belt, publish bands and weekly budgets:

Metric	Green (G)	Amber (A)	Red (R)
Gap	(	\mathrm{Gap}	\le \text{band}_\mathrm{Gap})
Flux	$\mathrm{Flux}\ge \text{target}_\mathrm{Flux}$	within $\pm10\%$	$<-10\%$
Twist	$\mathrm{TwIdx}\le 0.5$	$\le 0.85$	$>0.85$ or rate-limit hit
Coherence	$C\in[C_{\min},C_{\max}]$	near edges	outside band
Residual	(	R	\le \text{band}_R)

Dynamic rails. When volatility rises, widen Gap/Residual bands by factor $1+\kappa\,\widehat{\mathrm{Std}}(M)$ ; tighten Twist band as weekly budget depletes.

28.4 The “Five-Line” row (what prints on dashboards)

Example (shift window):

Belt: stitch_A | Gap: +420 ±90 pairs | Flux: 5,600/shift | Twist: 1.3 (of 6/wk) |
Coherence: 0.78 (target 0.75–0.85) | Residual: +25 ±40 pairs

Color each cell by its band; tooltips show uncertainty and last three windows.

28.5 Stream processing & aggregation

Inputs (Belt-Min logs; Ch. 15)

Dual edges (plan/do traces), event/rung log (ΔTw, owners), curvature estimates, purpose-flow $J_\Pi$ , phase extractor.

Windowing

Sliding windows: $\Delta_{\text{fast}}$ (1–5 min), $\Delta_{\text{ops}}$ (shift/day), $\Delta_{\text{govern}}$ (week).
Compute Five-Line at each cadence; store rollups (p50/p95, EWMA).

Pipeline (pseudocode)

def belt_kpi_stream(events, est, phases, now, Δ):
    B = surface_window(now, Δ)                 # belt surface for window
    Flux = integrate(dot(est.F_pi, est.J_pi), B)
    Gap  = line_int(A_pi, Γ_plus) - line_int(A_pi, Γ_minus) - integrate(est.F_pi, B)
    Tw   = sum_abs_twist(events.rungs in B.window)
    C    = coherence(phases.local, phases.neighbors, delta_star)
    R    = Gap - alpha * Tw

    # smoothing + bands
    Flux_s, Gap_s, Tw_s, C_s, R_s = robust_smooth(Flux, Gap, Tw, C, R)
    bands = adapt_bands(volatility(), budgets())
    return FiveLine(Flux_s, Gap_s, Tw_s, C_s, R_s, bands, uncertainty=est.sigmas)

Aggregation levels

Zone / Belt / Junction / Network.
Sum Gap, Flux, Residual; sum Twist; average Coherence with pairwise weights.
Cycles (network): track class invariants alongside Five-Line (Ch. 25).

Data hygiene

Late or missing slices → impute with last-valid + confidence downgrade.
Counter rollbacks: tag windows containing rollbacks to avoid double counting Twist.

28.6 Metric Contracts (artifact)

Formal spec each service must satisfy to emit Five-Line metrics.

contract_id: "belt-kpi/v3"
scope:
  belt_id: "factory.stitch_A"
  owner: "ops-lead"
signals:
  gap:
    unit: "pairs"
    method: "edges-minus-flux"
    smoothing: {type: "median+ewma", ewma_lambda: 0.2}
  flux:
    unit: "pairs/shift"
    integrand: "<F_pi·J_pi>"
  twist:
    unit: "twist"
    source: "rung-log"
    rate_limit: {opens_per: "6h", max_opens: 3}
  coherence:
    unit: "[0,1]"
    neighbors: ["cut","qa","pack"]
    target_band: {min: 0.75, max: 0.85}
  residual:
    unit: "pairs"
    identity_check: "gap = alpha*twist + residual"
uncertainty:
  report: ["p50","p95"]
  provenance: "estimator-v4.2"
storage:
  windowing: ["5m","shift","day","week"]
  retention_days: 365

28.7 SLIs / SLOs (artifact)

Service Level Indicators (SLIs)

SLI-Flux: median FluxIdx over ops windows.
SLI-Gap: p95 $|\mathrm{Gap}|$ vs band.
SLI-Twist: weekly $\sum|\Delta\mathrm{Tw}|$ / budget.
SLI-Coh: time in coherence band.
SLI-Res: p95 $|R|$ vs band; healing time to bring $|R|$ below band after events.

SLO pack (example)

slo_pack_id: "belt-kpi/slo-shoes-2025Q4"
targets:
  flux_idx_p50: {gte: 1.00}
  gap_p95_vs_band: {lte: 1.0}
  twist_budget_weekly: {lte: 0.85}
  coherence_time_in_band: {gte: 0.80}
  residual_heal_time_hours: {lte: 24}
error_budget:
  twist_overspend_quarter: 0.10
  red_windows_fraction: 0.05
actions:
  breach_gap_p95: ["enable_flux_gate","consider_twist_step"]
  breach_coh_band: ["shen_sync_or_desync"]
  breach_twist_budget: ["throttle_rungs","raise_commutator_fees"]
review:
  cadence: "weekly"
  owners: ["belt_owner","arbiter","shen_lead"]

28.8 Operational notes & edge cases

Incident mode: temporarily ignore Coh band; elevate safety-Flux and tighten Twist band.
Seasonality: maintain seasonal targets; compare FluxIdx to same-season baselines.
Antagonistic belts: report pairwise coherence explicitly alongside per-belt $C$ .
Governance leak: if $R$ consistently positive while Gap≈α·Tw, audit junctions (Ch. 25) and rung logs.

28.9 What to hand over

Metric Contracts per belt (YAML).
SLI/SLO pack for the plant.
Streaming configs (window sizes, smoothing, volatility adapters).
Dashboards: Five-Line row, trend sparklines, uncertainty cones, budget gauges.
Unit tests: identity check, conservation under gluing, band coloring, red/amber gating.

Takeaway

The Five-Line KPI makes PFBT operational: Gap, Flux, Twist, Coherence, Residual—each with units, bands, and uncertainty—computed continuously and rolled up cleanly. With contracts and SLOs in place, belts talk the same language from the cell to the network—so control, arbitration, and diagnostics snap into one coherent loop.

29. Dashboards & SLOs

Purpose. Give day-to-day ops visibility for belts and networks: a clean set of widgets, two composite indices (Entropy–Efficiency and Stability), alerting rules that tie back to Chapters 19–26, and drilldowns that land on the right runbooks. Artifact: Dashboard JSON + mockups you can drop into your tool of choice.

29.1 What ops must see (at a glance)

Five-Line KPI per belt (Ch. 28): Gap / Flux / Twist / Coherence / Residual with bands.
Two composite indices to read the room quickly:
- EEI (Entropy–Efficiency Index): “How much output per unit disorder?”
- SI (Stability Index): “How well are spikes damped and gaps healed?”
Budgets/SLOs: twist spend, floors, incident state.
Alarms: who’s red, why, and the smallest next action (Flux Gate / Twist Step / Arbitration / Shen nudge).

29.2 Indices

29.2.1 Entropy–Efficiency Index (EEI)

Let

$\mathrm{FluxIdx}=\mathrm{Flux}/\text{target}_\mathrm{Flux}$ ,
$\SigmaIdx=\Sigma_{\text{macro}}/\Sigma_{\text{cap}}$ (macro entropy: WIP-age + rework + twist-cost),
$\mathrm{TwIdx}=|\mathrm{Tw}|/B_{\mathrm{Tw}}$ (governance churn).

Define a penalized ratio (bounded in $[0,1]$ ):

\boxed{ \mathrm{EEI} \;=\; \frac{\mathrm{FluxIdx}} {1 + \lambda_\Sigma\,\SigmaIdx + \lambda_{\mathrm{Tw}}\,\mathrm{TwIdx}} \;\;\Bigg|\_{[0,1]} }

Default $\lambda_\Sigma=1,\ \lambda_{\mathrm{Tw}}=0.5$ . High EEI = strong output with low entropy & low churn.

29.2.2 Stability Index (SI)

Use a geometric mean of five normalized stabilizers over an ops window:

Gap compliance: $S_{\text{gap}}=\mathrm{clip}\big(1-\mathrm{GapIdx}_{p95},\,0,\,1\big)$ .
Residual healing: $S_{\text{heal}}=\mathrm{clip}\big(1-T_{\text{heal}}/T_{\text{heal,max}},\,0,\,1\big)$ .
Gate calmness: $S_{\text{gate}}=\mathrm{clip}\big(1-\lambda_{\text{flip}}\!/\lambda_{\max},\,0,\,1\big)$ (OPEN↔CLOSE flips per window).
Coherence in-band: $S_{\text{coh}}=\text{time\_in\_band}\in[0,1]$ .
Lyapunov slope: $S_{V}=\mathrm{clip}\big(\overline{-\dot V}/c_{\min},\,0,\,1\big)$ from replay card (Ch. 22).

\boxed{\mathrm{SI}=\big(S_{\text{gap}}\cdot S_{\text{heal}}\cdot S_{\text{gate}}\cdot S_{\text{coh}}\cdot S_{V}\big)^{1/5}}

SI near 1 = quiet, well-damped belt; near 0 = oscillatory, slow-healing, or out of band.

29.3 Layouts (overview → belt → drilldown)

A. Plant Overview (top)

Grid of belts with Five-Line row, EEI & SI mini-gauges, color by band.
Twist budget gauge (weekly), floors status (safety/fairness), incident banner.
Coherence heatmap (Ch. 23) with cluster overlays.

B. Network & Tops (middle)

Junction cards (node residual $R_v$ , treaty status).
Tops board (safety/quality/throughput…): per-top twist spend, commutator loss, share of arbitration.

C. Belt Detail (bottom/drilldown)

Time-series: Gap, Flux, Residual with uncertainty cones; Twist steps timeline.
Gate state (OPEN/CLOSED/graded) & actions (Ch. 19).
Twist ladder decisions & probe windows (Ch. 20).
Conflict panel: commutator norms, twist volatility, suggested reorder (Ch. 26).
Shen panel: $C_{ij}$ , offsets, sync/desync nudges (Ch. 23).

29.4 Alerts (rules → triage → runbook)

Severity levels

P1: Safety floor breach or $\mathrm{SI}<0.2$ for $>T_{\text{p1}}$ .
P2: $\mathrm{GapIdx}_{p95}>2$ or $\mathrm{EEI}<0.5$ and falling.
P3: Twist overspend risk (TwIdx>0.85), coherence drift ( $C\downarrow$ fast), junction $R_v$ > limit.

Grouping & dampening

Group by belt/junction; 10-min mute window to avoid page-storms.
Cascade: if three neighboring belts red, raise a network incident.

Next Action (always minimal-twist)

P2 Gap high → Flux Gate open (Ch. 19) or smallest ladder rung (Ch. 20) after quiet check.
Coherence drift → Shen sync/desync nudge (Ch. 23).
Commutator spike → cluster-then-order schedule (Ch. 21/24).

29.5 Drilldowns (one click = right chapter)

ALERT → Belt Detail with “Why” ribbon: desync / non-commute / thrash / leak (Ch. 26 taxonomy).
Twist step tag → rung card (ΔTw, probe plan, rollback).
Node residual → junction sheet (cobordism, treaty rules, local KPIs; Ch. 25).
Weight change → Arbitration record (objective weights, commutator fees; Ch. 21).

29.6 Dashboard JSON (artifact)

{
  "dashboard_id": "pfbt-ops/v1",
  "time_window_default": "shift",
  "layouts": [
    {
      "row": 1,
      "widgets": [
        {
          "type": "belts.grid",
          "id": "belts_overview",
          "columns": ["belt", "five_line", "EEI", "SI", "TwIdx", "status"],
          "query": "belts:*",
          "bands_ref": "s3://pfbt/bands/latest.json"
        },
        {
          "type": "gauge",
          "id": "twist_budget",
          "title": "Twist Budget (week)",
          "value": "sum_abs_twist(window='week')/budget_week",
          "thresholds": [0.5, 0.85, 1.0]
        },
        {
          "type": "heatmap",
          "id": "coherence_heatmap",
          "title": "Coherence C_ij",
          "matrix_query": "coherence_matrix(window='shift')",
          "cluster_overlay": true
        }
      ]
    },
    {
      "row": 2,
      "widgets": [
        {
          "type": "cards",
          "id": "junction_cards",
          "title": "Junction Residuals",
          "items_query": "junctions:*",
          "fields": ["Rv", "treaty", "commutator_loss", "owner"]
        },
        {
          "type": "table",
          "id": "tops_board",
          "title": "Tops & Arbitration",
          "columns": ["top", "twist_spend", "commutator_loss", "share_tau", "floors_ok"],
          "query": "tops:*"
        }
      ]
    },
    {
      "row": 3,
      "widgets": [
        {
          "type": "belt.detail",
          "id": "belt_detail",
          "route_param": "belt_id",
          "charts": [
            {"series": "Gap", "uncertainty": true},
            {"series": "Flux"},
            {"series": "Residual", "uncertainty": true},
            {"series": "TwistSteps", "style": "lollipop"},
            {"series": "GateState"}
          ],
          "panels": ["conflicts", "shen", "rungs", "arbitration"]
        }
      ]
    }
  ],
  "indices": {
    "EEI": "FluxIdx / (1 + lambda_Sigma*SigmaIdx + lambda_Tw*TwIdx)",
    "SI": "geom_mean(S_gap, S_heal, S_gate, S_coh, S_V)"
  },
  "alert_policies": [
    {
      "id": "p1_safety_or_si",
      "where": "floor_breach == true OR SI < 0.2 for 30m",
      "severity": "P1",
      "notify": ["oncall-safety", "ops-lead"],
      "runbook": "incident_single_top.md",
      "group_by": ["belt_cluster"],
      "mute_window": "10m"
    },
    {
      "id": "p2_gap_ei",
      "where": "GapIdx_p95 > 2 OR (EEI < 0.5 AND slope(EEI)<0)",
      "severity": "P2",
      "notify": ["oncall-ops"],
      "next_action": "flux_gate_or_smallest_twist",
      "runbook": "gap_repair.md"
    },
    {
      "id": "p3_comm_coh",
      "where": "commutator_loss > 0.1*budget OR dCdt < -0.1",
      "severity": "P3",
      "notify": ["arbiter", "shen-lead"],
      "next_action": "cluster_then_order_or_desync",
      "runbook": "scheduling_and_shen.md"
    }
  ],
  "drilldowns": {
    "conflicts": "open_notebook('Diagnostics-B', context=belt_id)",
    "shen": "open_notebook('Shen-Pack', context=cluster_id)",
    "rungs": "open_sheet('Twist-Library', filter=belt_id)",
    "arbitration": "open_card('Arbitration-Decision', id=arb_id)"
  }
}

29.7 Mockups (wireframe)

+──────────────────────────────────────────────────────────────────────────────+
|  PFBT Ops — Plant Overview                     [Week: W38]  [Shift: D1]     |
|  Twist Budget: [████████░░] 78%    Floors: SAFE ✓     Incidents: 0 (P1)     |
+─────────────+─────────────────────────────────────────+──────────────────────+
| Belts       | Five-Line KPI (Gap | Flux | Tw | Coh | R) | EEI |  SI | State |
|-------------+-------------------------------------------------+-----+-------|
| cut         | +80±20 | 3.2k | 0.3 | 0.79 | +12±15             | 0.78| 0.82 | G |
| stitch_A    | +420±90| 5.6k | 1.3 | 0.78 | +25±40             | 0.65| 0.71 | A |
| stitch_B    | +110±50| 5.3k | 0.9 | 0.83 |  -5±30             | 0.81| 0.86 | G |
| qa          |  -40±25| 6.1k | 0.4 | 0.80 | +10±20             | 0.89| 0.88 | G |
| pack_1      | +15±10 | 6.0k | 0.2 | 0.77 |  +3±8              | 0.92| 0.90 | G |
| pack_2      | +18±12 | 5.9k | 0.2 | 0.76 |  +4±9              | 0.90| 0.88 | G |
+──────────────────────────────────────────────────────────────────────────────+
| Coherence Heatmap (C_ij)   [clusters: {cut,stitch_A,qa,pack}]               |
|  █ 0.9  █ 0.8  █ 0.7  █ 0.6 ...                                              |
+───────────────────────+──────────────────────────────────────────────────────+
| Junctions (Rv)        | Tops Board                                           |
| merge_stitch→qa: 0.8  | safety: Tw 4.2, loss 0.02, share 0.31, floors ✓     |
| split_qa→pack: 0.3    | quality: Tw 3.1, loss 0.01, share 0.27, floors ✓    |
+──────────────────────────────────────────────────────────────────────────────+

Clicking stitch_A opens Belt Detail: time-series, gate state, rung probe, conflict panel.

29.8 SLOs (dashboard-level)

EEI floor: $\mathrm{EEI}_{p50}\ge 0.75$ per belt; plant median $\ge 0.80$ .
SI floor: $\mathrm{SI}_{p50}\ge 0.70$ ; no belt $<0.3$ for $>1$ ops window.
Twist spending: weekly $\sum|\Delta \mathrm{Tw}|\le 0.85\,B_{\mathrm{Tw}}$ .
Coherence band time: $\ge 80\%$ in cluster bands (Ch. 23).
Residual healing: $T_{\text{heal},p95}\le 24$ h from red event.

SLO JSON (excerpt)

{
  "slo_pack_id": "pfbt-dashboard/2025Q4",
  "targets": {
    "EEI_p50": {"gte": 0.75},
    "SI_p50": {"gte": 0.70},
    "TwIdx_weekly": {"lte": 0.85},
    "Coh_time_in_band": {"gte": 0.80},
    "T_heal_p95_hours": {"lte": 24}
  },
  "error_budget": {
    "red_windows_fraction": 0.05,
    "commutator_loss_ratio": 0.10
  },
  "owners": ["belt_owner", "arbiter", "shen_lead"]
}

29.9 Wiring alerts to runbooks (procedures)

Alert creation: rule → severity → owner(s) → Next Action (pre-filled minimal-twist move) → link to notebook (Ch. 26) or ladder card (Ch. 20).
Triage (≤15 min): confirm estimation health (Ch. 15/16), check cooldowns (Ch. 19), ensure quiet-belt before twist (Ch. 22).
Closure: attach counterfactual replay showing expected $\Delta R,\Delta \Sigma$ ; tag with rung ID and arbitration ticket if used.

29.10 What to hand over

Dashboard JSON (above) with your belt IDs, bands, and budgets.
Alert policies mapped to runbooks.
Widget library: Five-Line row, EEI/SI gauge, coherence heatmap, junction card, conflict panel.
Sample mockups to align with stakeholders.
Validation notebook: checks EEI/SI math, band coloring, alert fire/drain behavior on replay logs.

Takeaway

Dashboards translate PFBT’s math into operational sightlines: Five-Line KPIs for truth, EEI for cost-aware output, SI for calm, plus alerts that point to the smallest necessary move. With consistent JSON and drilldowns wired to Chapters 19–26, teams can see problems early, act with minimal twist, and prove stability with the numbers.

30. Audit & Compliance

Purpose. Make purpose governance auditable. This chapter specifies: (i) versioning for the purpose connection $\mathcal A_\Pi$ ; (ii) twist-budget ledgers & audits; (iii) 4π invariance checks; and (iv) operational SOPs for sign-offs, change windows, and evidence storage. Artifact: Audit SOPs (Appendix G) plus ready-to-run schemas and verifiers.

30.1 Audit Principles (what “good” looks like)

Reproducible math. Every KPI and decision is recomputable from stored inputs.
Gauge-aware evidence. We version $\mathcal A_\Pi$ up to framing/gauge; we store class invariants.
Minimal-twist accountability. Every governance move has ΔTw, probe, rollback, and smallest-step proof.
Conservation at scale. Junctions obey Kirchhoff–Stokes; cycles obey class-invariant checks.
Non-repudiation. Immutable, signed logs; monthly Merkle anchors.
Privacy & least data. Store only what’s needed to recompute KPIs, with role-based access.

30.2 Versioning the Purpose Field $\mathcal A_\Pi$

30.2.1 Purpose Bundle (what we version)

A Purpose Bundle is the minimal packet needed to reproduce belts’ geometry on a window:

\mathbf P = \big\langle \text{belt id},\; \text{basepoint},\; \mathcal A_\Pi,\; \mathcal E,\; \theta,\; \text{estimators},\; \text{frames}\big\rangle

$\mathcal A_\Pi$ : connection parameters (as coefficients or basis expansion).
$\mathcal E$ : event surface definition (windowing, mesh).
$\theta$ : policy vector (governance frame) at that version.
Estimators: model version/priors for $\widehat{\mathcal F}_\Pi, \widehat J_\Pi$ .
Frames/basepoint: to track gauge/framing choices explicitly.

30.2.2 Gauge-equivalent versions

Two bundles $\mathbf P,\mathbf P'$ are equivalent if there exists a smooth $g$ with

\mathcal A'_\Pi = g^{-1}\mathcal A_\Pi g + g^{-1}dg, \quad\text{and}\quad \text{frames}',\text{basepoint}'\ \text{updated accordingly.}

We store class invariants $I(\mathrm{Hol})$ (trace, $\arg\det$ , principal eigenphase) for all belt cycles and record the gauge map hash when known.

30.2.3 Registry & diff

ID: Aπ-v{major}.{minor}.{patch} (monotone, not time-based).
Semantic diff:
- math-diff (normed Δ on coefficients in a canonical gauge);
- effect-diff (Δ on Five-Line KPIs in replay, Sec. 30.8).
Signature: SHA-256 of serialized bundle; sign by owner + auditor.

Registry (YAML)

purpose_registry:
  - id: "Aπ-v4.2.1"
    belt: "factory/stitch_A"
    basepoint: "Γ0"
    hash: "sha256:…"
    invariants:
      cycles:
        - {cycle_id: "stitch→qa→stitch", argdet: 1.742, trace: 0.63}
    estimators: {F_pi: "v4.2", J_pi: "v3.1"}
    frames: {gauge: "Coulomb-like", twist_coupling_alpha: 0.92}
    owners: ["ops-lead"]
    signoffs: ["ops-lead@sig", "audit@sig"]

30.3 Twist-Budget Ledgers & Audits

A Twist Ledger is the journal of governance steps and their budgets.

Entry fields (minimal): time, belt, rung id, $\Delta\mathrm{Tw}$ , owner, approvals, probe window, rollback, expected $\Delta \mathrm{gap}$ , realized $R, \Sigma$ .

Budget rules.

Weekly spend: $\sum |\Delta\mathrm{Tw}|\le B_{\mathrm{Tw}}$ (per belt/top & global).
Cross-charges: if a rung harms a floor (safety/fairness), it carries a compensating budget or fee (Ch. 21/27).
Commutator loss accounted as spend: realized extra twist beyond abelian sum (Ch. 21).

Audit queries (examples):

Overspend: belts with $\sum|\Delta\mathrm{Tw}|>0.85\,B_{\mathrm{Tw}}$ .
ROI: rungs with $\Delta R$ not improving after probe → flag for library pruning.
Thrash: jump intensity $\lambda_{\text{jumps}}$ vs net utility flat → governance thrash report.

Ledger (JSON)

{
  "twist_ledger_id": "plant-W38",
  "entries": [
    {"ts":"2025-09-08T10:05Z","belt":"stitch_A","rung":"WIPcap-5%",
     "dTw":0.6,"owner":"ops-lead","approvals":["qa-lead"],"probe":"2shifts",
     "expected":{"gap_delta":-420},"realized":{"R":-410,"Sigma_idx":-0.08}}
  ],
  "budgets":{"weekly":6.0,"spent":4.8,"commutator_loss":0.32}
}

30.4 4π Checks & Invariance Tests

Why: Governance/frame changes shouldn’t alter physics; period-4π and class-invariant checks catch framing errors and silent drift.

30.4.1 4π periodicity (belt)

Apply a controlled twist loop of total $4\pi$ (in normalized units). Verify identity on belt invariants and Five-Line within tolerance.

I(\mathrm{Hol})\Big|_{\mathrm{Tw}\to\mathrm{Tw}+4\pi} \stackrel{?}{=} I(\mathrm{Hol}), \quad \mathrm{Gap}\stackrel{?}{=}\mathrm{Gap},\ \mathrm{Flux}\stackrel{?}{=}\mathrm{Flux}

30.4.2 Re-basing invariance

Change basepoint/framing per Sec. 30.2; invariants must be unchanged; KPI deviations must lie within estimator uncertainty cones.

30.4.3 Network conservation

At every junction $v$ :

R_v=\sum_{\text{in}}\Delta-\sum_{\text{out}}\Delta-\widehat\Phi_{\mathcal S_v}-\alpha\,\widehat{\mathrm{Tw}}_v

must satisfy $|R_v|\le r_{\max}$ p95 (Ch. 25). Cycle invariants drift $<\epsilon_{\text{cyc}}$ .

Verifier (pseudocode)

def four_pi_check(belt, window):
    inv0 = class_invariants(belt, window)
    kpi0 = five_line(belt, window)
    apply_virtual_twist(belt, total=4*math.pi)
    inv1 = class_invariants(belt, window)
    kpi1 = five_line(belt, window)
    assert close(inv0, inv1, tol=ε_inv)
    assert close(kpi0, kpi1, tol=ε_kpi)

30.5 Sign-offs & Separation of Duties

RACI per rung: Run (belt owner), Approve (top owner), Consult (safety/quality), Inform (arbiter).
Two-key treaties: any cross-edge rung requires two independent approvals (treaty owner + belt owner).
Risk class: L/M/H; H requires independent auditor review and a quiet-belt attestation (Ch. 22).
E-signatures: detached signatures over the rung card (hash of YAML below) + purpose bundle id.

Rung card (YAML)

rung_id: "ops/WIPcap-5%/v7"
belt: "stitch_A"
risk: "Low"
dTw: 0.6
sensitivities: {gap: -0.7}
probe: {duration: "2shifts", monitors: ["gap","R","Sigma"]}
rollback: {recipe: "revert WIPcap to v6", cooldown: "24h"}
owners: {run: "ops-lead", approve: ["qa-lead"], consult: ["safety"]}
signatures: ["ops-lead@sig","qa-lead@sig"]

30.6 Change Windows & Etiquette

Quiet-belt check: $\max_{[t-T_q,t]}|e|\le\epsilon_q$ and gate=CLOSED (Ch. 22).
Cadence constitution: obey target offsets $\delta^\star$ (Ch. 23); forbid antagonists in same window.
Dwell: post-twist cooldown $T_{\min}$ ≥ probe duration.
Incident override: single-top mode allowed with stamped incident ticket and post-mortem due date.

30.7 Evidence Storage & Retention

WORM store for ledgers, purpose registry, invariants, and Five-Line rollups.
Chain of custody: monthly Merkle root anchored (e.g., notary or internal KMS note).
PII minimization: store hashed IDs for personnel/customers; keep re-identification keys off the audit tier.
Retention: KPIs & ledgers 1 year (ops), 3–7 years (regulated) configurable.
Provenance fields: data source, estimator version, hash of raw slice, replay seed.

Evidence object (JSON)

{
  "evidence_id":"ev-2025W38-qa-merge",
  "kind":"junction_audit",
  "inputs":{"registry":"Aπ-v4.2.1","ledger":"plant-W38","raw_hash":"sha256:…"},
  "verifiers":{"four_pi":"pass","rebasing":"pass","junction_residual_p95":0.82},
  "signatures":["audit@sig"],
  "merkle_anchor":"mroot:…/2025-09"
}

30.8 Automated Replay & “Same-Numbers” Test

Every audit pack includes a replay of KPIs and outcomes from stored inputs:

Same-numbers: rebuild Five-Line and confirm equality within cones.
Counterfactuals: remove rungs or swap pairs to validate minimal-twist choices (Ch. 20/26).
Stability: show Lyapunov decay on flows & negative jump deltas on steps (Ch. 22).

Replay harness (pseudocode)

def audit_replay(window):
    P = load_purpose_bundle(window)
    L = load_twist_ledger(window)
    kpi_rebuilt = recompute_kpis(P, L)
    assert within_cones(kpi_rebuilt, stored_kpis)
    # spot CF
    for r in pick_sample_rungs(L):
        delta = counterfactual(window, remove=r)
        record_cf(r, delta)

30.9 Compliance Hooks (map to common controls)

Change management: rung cards + approvals + probe/rollback (SOC-style change control).
Config/version control: Purpose Registry + diffs + signatures (config integrity).
Ops metrics & SLOs: Five-Line + dashboard SLO pack (service commitments).
Incident handling: single-top override + post-mortems (incident policy).
Separation of duties: two-key at treaties; independent audit role.
(Use your org’s control catalog; the artifacts below slot directly into standard audits.)

30.10 Audit Verifiers (batch)

def run_audit_pack(week_id):
    reg  = load_purpose_registry(week_id)
    ledg = load_twist_ledger(week_id)
    kpis = load_kpi_rollups(week_id)
    net  = load_network_schema()

    # 1) Same-numbers
    assert replay_equals(kpis, reg, ledg)

    # 2) 4π & re-basing
    for belt in belts():
        assert four_pi_check(belt, week_id)
        assert rebasing_invariant(belt, week_id)

    # 3) Conservation at nodes & cycles
    for v in net.junctions:
        assert node_residual(v) <= r_max_p95
    for cyc in net.cycles:
        assert drift(cyc.invariants) <= eps_cyc

    # 4) Budgets & commutators
    assert twist_spend(ledg) <= weekly_budget*1.0
    assert commutator_loss(ledg) <= loss_cap

    # 5) Governance etiquette
    assert all(quiet_belt_before_twist(ledg))
    assert all(two_key_treaty(ledg))

30.11 Artifacts — Audit SOPs (Appendix G)

SOP-01: Purpose Version Control

Inputs: proposed $\mathbf P$ update; Outputs: signed registry entry.
Steps: math-diff → effect-diff → invariants → sign → publish.

SOP-02: Twist Budget Audit (weekly)

Reconcile spend, commutator loss, cross-charges; publish variances & causes.

SOP-03: 4π & Re-basing Checks (monthly)

Sample belts; run verifiers; file exceptions with fixes.

SOP-04: Change Window & Sign-off

Quiet-belt attestation → approvals → pre-flight checklist → post-probe report.

SOP-05: Junction/Treaty Compliance

Node residual sheets, unlogged twist leak checks, treaty two-key attestations.

SOP-06: Replay & Counterfactuals

Same-numbers test; top-k CFs for library pruning and training priors.

SOP-07: Incident Override & Restoration

Elevate single-top; evidence of cause; restore polycentric mode with post-mortem.

SOP-08: Dashboard/SLO Evidence Pack

Export EEI/SI, band times, breaches, and runbook links for the audit period.

SOP index (YAML)

audit_sops:
  - id: SOP-01-purpose-version
    cadence: "on_change"
    owners: ["ops-lead","audit"]
  - id: SOP-02-twist-budget
    cadence: "weekly"
    owners: ["arbiter","finance"]
  - id: SOP-03-4pi-rebasing
    cadence: "monthly"
    owners: ["audit"]
  - id: SOP-04-change-window
    cadence: "on_change"
    owners: ["belt_owner","top_owner"]
  - id: SOP-05-junction-treaty
    cadence: "weekly"
    owners: ["treaty_owner","audit"]
  - id: SOP-06-replay-cf"
    cadence: "weekly"
    owners: ["audit","ops-analytics"]
  - id: SOP-07-incident-override
    cadence: "on_incident"
    owners: ["safety","hub"]
  - id: SOP-08-dashboard-slo
    cadence: "weekly"
    owners: ["ops-analytics","audit"]

30.12 What to hand over

Purpose Registry with signed versions & invariants.
Twist Ledger (weekly) with budgets, commutator loss, cross-charges.
Audit Verifiers (scripts, configs) for same-numbers, 4π, conservation, etiquette.
Evidence store config (WORM, retention, Merkle anchors).
Audit SOPs (App G) with checklists & templates.

Takeaway

Auditable purpose governance means: you can rebuild the numbers, prove invariance (4π, re-basing, conservation), show where twist went (and why), and demonstrate discipline (quiet-belt, two-key, rollback). With a versioned purpose registry, twist ledgers, invariance verifiers, and SOPs, PFBT operations meet both engineering rigor and compliance-grade accountability.

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.

Sunday, September 21, 2025