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 https://osf.io/yaz5u/files/osfstorage/68d01dd47195bb99223b7dfe

Purpose-Flux Belt Theory (PFBT)  Part I & II

Front Matter

Preface

Goals — why belts, not lines

Most real processes run on two complementary traces: a Plan/Reference edge and a Do/Realized edge. Modeling them on a single line hides the invariants that actually drive outcomes. A belt (a two-boundary worldsheet) restores those invariants: the edge-to-edge gap equals a surface flux plus a framing/twist term. In PFBT we make Purpose (志) the field that sources the flux, and we turn those invariants into controllers and ledgers you can run in production—engineering, operations, and research all on the same geometry.

• Purpose → Flux; Plan ↔ Do. We treat Purpose/志 as a connection field $\mathcal A_\Pi$. Its curvature $\mathcal F_\Pi = d\mathcal A_\Pi + \mathcal A_\Pi\wedge\mathcal A_\Pi$ is the work-producing drive across the belt face. Framing/governance shows up as a twist along the edges.

• Promise: we give you belt invariants (gap/flux/twist/coherence), gluing rules, and controllers (flux-gates, twist-stepping) plus a macro work–entropy ledger that lands in your own units—pairs of shoes, filled orders, basis points, regimen adherence—and charges governance costs like WIP, changeovers, rework, and policy flips.

Key results (stated up front)

Purpose Belt Holonomy Law (PBHL) — the belt identity you will use everywhere:

$$\underbrace{\oint_{\Gamma_+}\!\mathcal A_\Pi \;-\; \oint_{\Gamma_-}\!\mathcal A_\Pi}_{\text{edge gap (plan vs do)}} \;=\; \underbrace{\iint_{\mathcal B}\!\mathcal F_\Pi}_{\text{purpose flux (does work)}} \;+\; \underbrace{\alpha\,\mathrm{Tw}}_{\text{framing / governance twist}}.$$

(Non-abelian form with surface ordering and Wilson transport appears in Part I.) This is the “gap = flux + twist” law specialized to Purpose/志.

Macro work–entropy ledger — a domain-unit accounting tied to PBHL:

• Macro work $W_{\text{macro}}$ (e.g., pairs of shoes) tracks the purpose-aligned flux over the belt face.

• Macro entropy $\Sigma_{\text{macro}}$ aggregates dispersion, WIP-age, rework, and twist-cost (policy/prompt/rule changes).
  This makes “purpose machines” measurable and steerable without invoking thermodynamic temperature—exactly the spirit in the founding memo.

Figure (one-page diagram you’ll see all book long)

   Γ+  (Plan / Reference edge) ─────────────────────────────────────┐
     ↘                                                           ↗  │   edge gap = ∮Γ+ AΠ − ∮Γ− AΠ
       ↘                 Belt face  𝔅 (Purpose flux)           ↗    │   = ∬𝔅 FΠ  +  α·Tw
         ↘             ∬𝔅 𝓕_Π  (does macro work)            ↗      │
           ↘                                           ↗            │
             └───────────────────────────────────────┬──────────────┘
                                                     │  Tw (framing / policy twist along the belt)
   Γ−  (Do / Realized edge) ─────────────────────────┘

• Edges: Plan vs Do logs.

• Face: integrated purpose flux (the “work density”).

• Twist: minimal governance change needed to close the gap.

Artifacts

Elevator pitch (take this to your exec / PI / lead)

Purpose-Flux Belt Theory (PFBT) turns “Purpose/志” into a measurable field whose curvature does macro work.
For any process with Plan↔Do, model a belt; then edge gap = purpose flux + twist.
We ship invariants, controllers, and a macro work–entropy ledger in your units (shoes, orders, bps), including governance costs (WIP, changeovers, rework, policy flips).
Result: diagnose, control, and audit purpose execution with reproducible tests (two-boundary Stokes, 4π, gluing).

Glossary teaser (full in Appendix J)

• Belt $\mathcal B$ — a two-boundary worldsheet; minimal surface for Plan/Do.

• Edges $\Gamma_\pm$ — $+$ plan/reference, $-$ do/realized.

• Purpose connection $\mathcal A_\Pi$ — the 1-form encoding intent/志.

• Curvature $\mathcal F_\Pi$ — non-conservative purpose flux (work driver).

• Twist (Tw) — framing/governance reparameterization along edges.

• PBHL — $\oint_{\Gamma_+}\mathcal A_\Pi - \oint_{\Gamma_-}\mathcal A_\Pi = \iint_{\mathcal B}\mathcal F_\Pi + \alpha\,\mathrm{Tw}$.

• Coherence / “Shen” — multi-belt phase-lock under a shared purpose field; operational stability metric used in Part VII.

---

Reading map. If you want proofs and discrete calculus, start in Part I–II. If you want dashboards and controllers, jump to Part V–VIII. For domain examples (manufacturing shoes → pairs of shoes ledger; finance; org ops; healthcare), see Part IX.

What’s next. The next section (“How to Use This Book”) gives track-specific routes (Research • Engineering • Operations) and a reproducibility pack (datasets, unit tests, gluing checks).

 

How to Use This Book

This book is designed as a belt you can enter from multiple edges. Pick the track that matches your role, follow the reading path, and grab the artifacts you need.

---

Tracks

• (R) Research — proofs, invariants, identifiability, benchmarks.

• (E) Engineering — estimators, APIs, data schemas, controllers.

• (O) Operations — dashboards, ledgers in domain units, SOPs.

---

Reading paths

• R: Part I → III → V
  Foundations of belts & PBHL → Macro work–entropy ledger → Controllers & stability.

• E: Part II → VIII → IX
  Discrete calculus & estimation → Data/Repo schemas → Domain implementations.

• O: Part VII → IX → X
  Coherence, KPIs & governance → Domain playbooks → Validation & audits.

Tip: If you’re crossing tracks (e.g., Ops leader with an engineering team), skim the first section of each part you skip for vocabulary alignment.

---

Pick-your-chapter table

Goal	Start Here	Then	Output You’ll Get
Prove PBHL and belt gluing	I.2–I.4	I.7 (4π tests)	Lemmas, invariants checklist
Turn Purpose into a field	I.1	III.1–III.3	$\mathcal{A}_\Pi,\ \mathcal{F}_\Pi$ cheat-sheet
Build the ledger in your units	III.2	VII.2, IX.*	Work/entropy ledger template (shoes, orders, bps…)
Estimate flux from logs	II.3	VIII.1–VIII.3	Minimal estimator + confidence bands
Control gaps (close Plan↔Do)	V.1–V.3	V.4–V.5	Flux-gates & twist-stepping playbook
Stand up a dashboard	VII.1	VII.3, IX.*	KPIs, alerts, SOP snippets
Run validation	X.1 (Stokes/gluing)	X.2 (periodicity)	Test suite & acceptance criteria
Translate to a new domain	IX.0 (pattern)	IX.[your domain]	Domain mapping worksheet

---

Artifacts you can reuse

• Datasets & repos: sample belt logs, reference estimators, controller stubs, dashboard notebooks — see Appendix I for links and structure.

• Templates:

  • Ledger sheet (macro work–entropy)

  • Controller spec (flux-gate / twist-step)

  • Validation pack (two-boundary Stokes, gluing, 4π periodicity)

• Glossary & symbols: quick cards in Appendix J.

• One-pagers: PBHL diagram, “Gap = Flux + Twist” cheats, and SOP checklists (pre-flight / post-mortem).

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How to read efficiently

1. Skim the one-page belt diagram in the Preface; it recurs in every Part.

2. Open the pick-your-chapter table and jump straight to your goal.

3. When stuck, consult Appendix E (Core Math Quick Start) and Appendix J (Glossary).

4. For anything you want to run, grab the Appendix I repo pointers first, then follow the chapter’s “Runbook” box.

 

Notation, Symbols, and Conventions

This page fixes the symbols and sign rules used throughout PFBT. When in doubt, return here and the quick “outer-minus-inner” sanity check at the end.

Core geometric objects

• Edges (plan/do):
  $\Gamma_{+}$ = plan edge, $\Gamma_{-}$ = do edge.
  Both are oriented 1-curves living on the same state manifold $\mathcal{X}$ and parameterized by $s\in[0,1]$.

• Belt (the ribbon):
  $\mathcal{B}$ = an oriented 2-surface whose boundary is the two edges:

  $$\partial\mathcal{B}=\Gamma_{+}\;\sqcup\;(-\Gamma_{-}) .$$

  Read: the belt’s boundary is “outer minus inner”.

• Framing & twist:
  A smooth unit normal/framing $\mathbf{n}$ along $\mathcal{B}$ defines the twist
  $\mathrm{Tw}(\mathcal{B},\mathbf{n})\in \tfrac12\mathbb{Z}$.
  Convention: right-handed (clockwise when looking along $+\mathbf{n}$) half-twist is $+\tfrac12$.

Purpose connection and curvature

• Purpose connection (1-form): $\mathcal{A}_{\Pi}$.
  Think “connection for purpose/志 transport” along edges. Units below.

• Purpose curvature (2-form): $\mathcal{F}_{\Pi}$.

  • Abelian case (default): $\mathcal{F}_{\Pi}=d\mathcal{A}_{\Pi}$.

  • Non-Abelian note (when we track multi-purpose stacks): $\mathcal{F}_{\Pi}=d\mathcal{A}_{\Pi}+\mathcal{A}_{\Pi}\wedge\mathcal{A}_{\Pi}$.

• Holonomy along an edge:

  $$\mathrm{Hol}_{\Pi}(\Gamma)\;=\;\oint_{\Gamma}\mathcal{A}_{\Pi} \quad \text{(Abelian; for non-Abelian use }\log\mathrm{P}\exp\oint_{\Gamma}\mathcal{A}_{\Pi}\text{).}$$

• Belt holonomy difference:
  $\Delta\mathrm{Hol}_{\Pi}(\mathcal{B})\equiv \mathrm{Hol}_{\Pi}(\Gamma_{+})-\mathrm{Hol}_{\Pi}(\Gamma_{-})$.

PBHL (Purpose Belt Holonomy Law)

With the above orientations,

$$\boxed{\; \Delta\mathrm{Hol}_{\Pi}(\mathcal{B}) \;=\;\iint_{\mathcal{B}}\mathcal{F}_{\Pi}\;+\;2\pi\,\mathrm{Tw}(\mathcal{B},\mathbf{n}) \;+\;2\pi k\;,\;\;k\in\mathbb{Z} \;}$$

The $2\pi k$ term captures integer gauge periods / branch choices (drops out in the ledger once calibrated).

Flux face & ledgers (names only; detailed in later chapters)

• Flux face density (through the belt): denote by $J$ (a 2-form on $\mathcal{B}$) when needed.

• Macro work–entropy ledger: $\mathsf{W\!E}$ maps $\Delta\mathrm{Hol}_{\Pi}$, $\iint_{\mathcal{B}}J$ and $\mathrm{Tw}$ into output units (e.g., “pairs of shoes”) and costs (WIP, changeovers, rework).

Indices, coordinates, and forms

• State coordinates: $x^{\mu}$ on $\mathcal{X}$, $\mu=1,\dots,d$.

• Edge parameter: $s$ (monotone with the edge’s orientation).

• Semantic tick (collapse clock): $\tau$ when explicit; edges may be lifted to $(s,\tau)$.

• Index conventions: Greek $\mu,\nu$ for manifold; Latin $i,j$ for local frames. Wedge $\wedge$ is exterior product; $d$ is exterior derivative.

Orientation & sign rules

• Boundary orientation: $\partial\mathcal{B}=\Gamma_{+}\sqcup(-\Gamma_{-})$.
  If $\mathbf{n}$ is the belt’s chosen normal, then the boundary’s direction follows the right-hand rule: walking with the boundary orientation keeps $\mathcal{B}$ on your left.

• Stokes with the belt:

  $$\oint_{\Gamma_{+}}\!\!\mathcal{A}_{\Pi}-\oint_{\Gamma_{-}}\!\!\mathcal{A}_{\Pi} \;=\;\iint_{\mathcal{B}}\!d\mathcal{A}_{\Pi}\quad(\text{plus twist/gauge periods as above}).$$

• “Outer minus inner” sanity check: if $\mathcal{F}_{\Pi}$ is everywhere $>0$ on $\mathcal{B}$ with the chosen orientation, then $\Delta\mathrm{Hol}_{\Pi}>0$.

Units and calibration

Let $[\Pi]$ denote the purpose unit (dimension “purpose”), and $[\mathsf{W}]$ the macro-work unit tied to your output.

• $[\mathcal{A}_{\Pi}]=[\mathsf{W}]/[\Pi]$ per unit length along an edge (line work-density per purpose).

• $[\mathcal{F}_{\Pi}]=[\mathsf{W}]/[\Pi]$ per unit area on the belt (face work-density per purpose).

• Calibration constant: choose $K_{\Pi}$ so that

  $$\mathsf{Work}_{\text{macro}}=K_{\Pi}\,\Delta\mathrm{Hol}_{\Pi},\qquad K_{\Pi}\cdot 2\pi\;\text{equals “one output quantum” (e.g., 1 pair)}.$$

This choice makes $2\pi$ (one full twist / one period of holonomy) a meaningful, auditable unit.

Gauge, basepoint, and invariances

• Gauge: $\mathcal{A}_{\Pi}\mapsto\mathcal{A}_{\Pi}+d\chi$ (Abelian) leaves $\mathcal{F}_{\Pi}$ invariant; $\Delta\mathrm{Hol}_{\Pi}$ shifts by integer periods already accounted for by $k$.

• Basepoint invariance: edge holonomies are independent of the starting point; in non-Abelian settings use the conjugacy class (trace of Wilson loop) or its principal log—differences cancel in $\Delta\mathrm{Hol}_{\Pi}$.

• Reparameterization: all line/surface integrals are parametrization-independent.

Quick checks before you compute

1. Sign sanity: verify $\partial\mathcal{B}=\Gamma_{+}\sqcup(-\Gamma_{-})$. If you accidentally flip the belt, both $\mathcal{F}_{\Pi}$ and $\mathrm{Tw}$ signs flip; $\Delta\mathrm{Hol}_{\Pi}$ must flip too.

2. Basepoint: for non-Abelian logs, use the same branch on both edges or compare via a trace (basepoint-safe).

3. Unit lock: confirm your $K_{\Pi}$ so that $2\pi$ maps to a concrete output quantum in the ledger.

Appendix A (Orientation Cookbook) lists pictorial right-hand rules, the twist counting convention, and minimal gauge/trace recipes you can reuse in derivations.

 

Part I — First Principles

 

1. Why Belts, Not Lines

Goals — from two traces to a worldsheet

Any process that logs two complementary traces—e.g., Plan/Reference vs Do/Realized—admits a minimal ribbon (belt) whose boundary is those two edges. Line models hide surface-only invariants; the belt restores them and makes the edge gap depend on a face (flux + twist).

Results

• Dual-trace completeness axiom.
  If a system records (i) a reference/plan trace and (ii) a realized/execution trace over the same state manifold, then there exists an oriented annular patch $\mathcal B$ with boundary the two traces; all edge–edge differences factor through $\mathcal B$’s interior (the “face”).

• Existence of annular patch for any logged Plan/Do pair.
  Formal existence follows the two-boundary worldsheet construction (SK doubling / non-Abelian Stokes) specialized to operational logs: a smooth belt $\mathcal B$ with $\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ always exists (thin-belt limit if needed).

Constructs

• Boundary identity

  $$\partial\mathcal B \;=\; \Gamma_+ \;\sqcup\;(-\Gamma_-)\quad(\text{“outer minus inner”}).$$

• Belt parametrization
  Use $\mathbf X:[0,1]\times[0,1]\to\mathcal X$, $\mathbf X(s,0)\in\Gamma_-,\ \mathbf X(s,1)\in\Gamma_+$. The induced orientation is $(\partial_s,\partial_u)$. This is the canonical setup used throughout Parts I–II.

Figures (described)

• Dirac-belt sketch. A narrow ribbon with a chosen normal field; shows why framing/twist matters even when the centerline is unchanged. (Motivates the twist term that will enter PBHL.)

• Plan/Do ribbon. Two logged polylines (Plan on top, Do below) joined into $\mathcal B$; the face carries purpose flux; edges carry holonomies. This is the running diagram used all book long.

Checks — Two-boundary Stokes (Abelian) on a toy belt

Let the belt coordinates be $(s,u)\in[0,1]^2$. Take an Abelian 1-form $A=f(s,u)\,ds$. Then $F=dA=(\partial_u f)\,du\wedge ds$.

• Edge integrals.
  $\displaystyle \oint_{\Gamma_+}\!A-\oint_{\Gamma_-}\!A=\int_0^1 f(s,1)\,ds-\int_0^1 f(s,0)\,ds$.

• Face integral.
  $\displaystyle \iint_{\mathcal B}\!F=\int_0^1\!\!\int_0^1 \partial_u f(s,u)\,du\,ds = \int_0^1 f(s,1)\,ds-\int_0^1 f(s,0)\,ds$.
  So

$$\boxed{\ \oint_{\Gamma_+}\!A-\oint_{\Gamma_-}\!A=\iint_{\mathcal B}\!F\ }\quad \text{(orientation: “upper minus lower”).}$$

Sanity test. Choose $f(s,u)=B\,u\Rightarrow F=B\,du\wedge ds$. Then LHS $=B$, RHS $=\text{area}\times B=B$. This matches the two-boundary Stokes sign convention we use later (and generalizes to non-Abelian with surface ordering).

Artifacts — Minimal JSON for dual traces (full spec in Ch31)

A tiny, log-first schema to reconstruct a belt from real data (polyline edges + optional twist frames):

{
  "plan_edge": [
    {"t":"2025-01-01T00:00:00Z","x":[...], "meta":{"ver":"plan_v3"}},
    {"t":"2025-01-01T00:05:00Z","x":[...]}
  ],
  "do_edge": [
    {"t":"2025-01-01T00:00:30Z","x":[...], "meta":{"line_id":"A1"}},
    {"t":"2025-01-01T00:05:20Z","x":[...]}
  ],
  "twist_frames": [
    {"t":"2025-01-01T00:02:00Z","policy":"day_count=30/360"},
    {"t":"2025-01-01T00:04:00Z","prompt_id":"style_v2"}
  ],
  "units": {"x":"state coords","time":"UTC"}
}

• plan_edge, do_edge: ordered samples define $\Gamma_+$, $\Gamma_-$.

• twist_frames: governance/framing changes along edges (used later to estimate Tw).

• This is the seed for the Part VIII / Ch31 full schemas (PlanEdge, RealizedEdge, TwistFrame, etc.).

---

Where this points. With belts in place, Part I.2 will elevate Purpose/志 to a connection $\mathcal A_\Pi$; its curvature $\mathcal F_\Pi$ will be the driver of macro work, and the eventual PBHL will read “edge gap = purpose flux + twist.”

 

2. Purpose as a Field

Goals — elevate “志 / Purpose” to geometry

We model Purpose (志) as a connection field on the belt: a 1-form $\mathcal A_{\Pi}$ whose curvature $\mathcal F_{\Pi}$ drives macro work across the face; framing/governance enters as Tw on the edges. This is the operational heart of PFBT and the step that makes “pairs of shoes” (or any domain unit) auditable.

---

Results — the three primitives

• Purpose connection: $\displaystyle \mathcal A_{\Pi}\in \Omega^{1}(\mathcal X)$.

• Purpose curvature: $\displaystyle \mathcal F_{\Pi} = d\mathcal A_{\Pi} + \mathcal A_{\Pi}\wedge \mathcal A_{\Pi}$ (Abelian case uses $d\mathcal A_{\Pi}$).

• Purpose current (face-side evidence): $J_{\Pi}$ (2-form or density over $\mathcal B$) used in the ledger and estimation.
  These objects feed the belt identity (PBHL) later: edge gap = purpose flux + twist.

---

Constructs — continuity & gauge

• Purpose continuity (phenomenology):

  $$\partial_t \rho_{\Pi} + \nabla\!\cdot J_{\Pi} \;=\; \sigma_{\Pi},$$

  where $\rho_{\Pi}$ is purpose density on state, $J_{\Pi}$ the observed face flow (events that “advance purpose”), and $\sigma_{\Pi}$ sources/sinks (e.g., mandate changes). This gives a practical bridge between logs and $\mathcal F_{\Pi}$: when $\sigma_{\Pi}\!\approx\!0$, persistent edge gaps imply non-conservative $\mathcal F_{\Pi}\neq 0$.

• Gauge notion:
  $\mathcal A_{\Pi}\mapsto \mathcal A_{\Pi}+d\chi$ leaves $\mathcal F_{\Pi}$ invariant; scalar readouts use class-function invariants (e.g., $\arg\det$, $\mathrm{tr}$) to avoid basepoint/order artifacts on edges (details in Part II/IX).

---

Figures — conservative vs non-conservative cartoons

• Conservative purpose: streamlines on $\mathcal B$ close; $\iint_{\mathcal B}\mathcal F_{\Pi}\!=\!0$; any edge gap must be twist-only → governance reparam fixes it.

• Non-conservative purpose: streamlines spiral across the face; $\iint_{\mathcal B}\mathcal F_{\Pi}\!\neq\!0$; flux does macro work (ledger credit), while twist is the minimal framing cost to reconcile Plan↔Do.

---

Checks — identifiability preconditions (see Part IV Ch.15 “Observability”)

You can estimate $\mathcal A_{\Pi}$ / $\mathcal F_{\Pi}$ (up to gauge) when:

1. Dual traces $\Gamma_\pm$ are logged with consistent orientation/timebase,

2. Face evidence $J_{\Pi}$ (events/defects/throughputs) is timestamped to tile $\mathcal B$,

3. Twist frames (policy/prompt/version changes) are recorded on edges, and

4. Gluing tests (two-boundary Stokes; basepoint/order invariance) pass on held-out belts.

---

Artifacts — PurposeField data-product contract (see Ch.31 schemas)

Minimal contract for moving from logs → fields (pairs with PlanEdge, RealizedEdge):

{
  "purpose_field": {
    "id": "PF_shoefactory_v1",
    "gauge": "abelian:u1", 
    "calibration": { "K_pi": 12.5663706144, "unit_quantum": "1_pair_shoes" },
    "A_pi_edge_samples": [
      {"edge":"plan","t":"2025-01-01T00:00Z","a_pi": 0.12},
      {"edge":"do","t":"2025-01-01T00:00Z","a_pi": 0.10}
    ],
    "F_pi_face_tiles": [
      {"tile_id":"b:12-13","t0":"...","t1":"...","f_pi": 0.018,"evidence_ref":["evt_991","evt_1002"]}
    ],
    "twist_frames": [
      {"edge":"plan","t":"2025-01-01T00:02Z","label":"policy_v2","delta_tw": 0.5}
    ],
    "units": { "A_pi":"work-per-purpose·per-length", "F_pi":"work-per-purpose·per-area" }
  }
}

• K_pi maps $2\pi$ holonomy to one output quantum (your domain unit).

• A_pi_edge_samples and F_pi_face_tiles support PBHL auditing and the macro work–entropy ledger.

• twist_frames track governance cost for the ledger’s entropy term.

---

Where this points. Next we state PBHL precisely and show how $\iint_{\mathcal B}\mathcal F_{\Pi}$ and Tw explain the measured edge gap—with invariants and tests that generalize Stokes to the two-boundary belt.

 

3. The Purpose Belt Holonomy Law (PBHL)

Goals — state and interpret the law

On a two-boundary belt $\mathcal B$ with edges $\Gamma_\pm$, the measurable edge gap equals a surface-ordered curvature response times a framing (twist) phase. In the purpose setting, the curvature is the purpose drive.

---

Result — PBHL (non-Abelian, framed form)

$$\boxed{ \operatorname{Hol}_\Pi(\Gamma_+)\,\operatorname{Hol}_\Pi(\Gamma_-)^{-1} \;=\; \mathcal S\exp\!\Big(\!\!\iint_{\mathcal B}\! W^{-1}\,\mathcal F_\Pi\,W\Big)\;\cdot\;e^{\,i\,\alpha\,\mathrm{Tw}} }$$

• $\mathcal F_\Pi=d\mathcal A_\Pi+\mathcal A_\Pi\wedge\mathcal A_\Pi$ is the purpose curvature (Abelian: $d\mathcal A_\Pi$).

• $W$ is the Wilson transport from a fixed basepoint to each face point (keeps the surface object gauge-covariant).

• $\mathcal S\exp$ is the surface-ordered exponential on the annulus.

• $\mathrm{Tw}$ is the belt twist (framing); $\alpha$ is a calibrated sector constant (often tied to $2\pi$ units).
  This is the framed belt law specialized to Purpose/志; proofs/derivations mirror FBHL/SBHL in the Belt Theory notes.

Interpretation.
Left side = edge-to-edge holonomy gap (Plan minus Do).
Right side = purpose flux through the face (ordered, gauge-covariant) × a scalar twist phase from framing that survives the thin-belt limit. A single unframed line cannot capture this twist.

---

Constructs — Wilson transport and class-function invariants

Wilson transport $W$. Choose a smooth radial path family in $\mathcal B$ from the basepoint; $W$ parallel-transports $\mathcal F_\Pi$ to compare contributions at a common fiber. The surface object is independent of the particular radial choice up to conjugation at the basepoint.

Class-function invariants. Since the RHS is defined up to boundary conjugation, use class functions (e.g., $\mathrm{tr}$, $\arg\det$, characters) to extract scalar, gauge-robust readouts for logging and ledgers:

$$\phi_{\text{gap}} \equiv \frac{1}{k}\arg\det\!\Big(\operatorname{Hol}_\Pi(\Gamma_+)\operatorname{Hol}_\Pi(\Gamma_-)^{-1}\Big) \;\equiv\; \frac{1}{k}\arg\det\Big(\mathcal S e^{\iint W^{-1}\mathcal F_\Pi W}\Big) + \alpha\,\mathrm{Tw}\;\;(\bmod 2\pi).$$

These are reparametrization-independent and converge under annulus refinement.

---

Procedures — scalar invariant pipeline for $U(k)$

1. Fix belt order (lower→upper; then along $s$) and a basepoint. Build an annular mesh satisfying the refinement condition.

2. Compute $W$ on each face cell (radial Wilson lines), assemble $\mathcal S\exp\!\iint W^{-1}\mathcal F_\Pi W$.

3. Edge holonomies. Compute $\operatorname{Hol}_\Pi(\Gamma_\pm)$ or directly the surface form (recommended).

4. Reduce to a scalar via a class function: $\arg\det$ (projective $U(k)$), or $\tfrac{1}{k}\mathrm{Im}\,\log\mathrm{Tr}$ for stable phases.

5. Twist estimation. Integrate the framing connection along the boundary (or read from recorded twist frames) to get $\mathrm{Tw}$; apply the calibrated $\alpha$.

6. Calibrate $\alpha$ and units. Fix $\alpha$ so that a $2\pi$ change corresponds to one output quantum in the ledger (e.g., 1 pair of shoes).

7. Log invariants $(\phi_{\text{gap}}, \iint \mathcal F_\Pi, \mathrm{Tw})$ with confidence bands; store mesh metadata for reproducibility.

---

Checks — Abelian recovery and composition sanity

• Abelian recovery. With $\mathcal F_\Pi=d\mathcal A_\Pi$ and $W\equiv I$, PBHL reduces to two-boundary Stokes:
  $\oint_{\Gamma_+}\mathcal A_\Pi-\oint_{\Gamma_-}\mathcal A_\Pi=\iint_{\mathcal B} d\mathcal A_\Pi$, with a separate scalar $e^{i\alpha\,\mathrm{Tw}}$ factor.

• Composition/gluing. Concatenating belts along a common edge composes surface-ordered exponentials; twists add: $\mathrm{Tw}(\mathcal B_2\!\circ\!\mathcal B_1)=\mathrm{Tw}(\mathcal B_1)+\mathrm{Tw}(\mathcal B_2)$. Class-function readouts remain equal up to conjugation. (Used later in Ch.5 Stokes/partition invariance and Ch.9 variational consequences.)

---

Artifacts — unit-test checklist (referenced in Ch.43)

• T1 Abelian Stokes: constant 1-form on a toy belt → edge gap = area × field.

• T2 Refinement/gluing: subdivide the belt; interior edges cancel; scalar invariant unchanged.

• T3 Orientation flip: flip $\partial\mathcal B$ → signs of gap/flux/twist all flip coherently. (Sanity.)

• T4 4$\pi$ periodicity: rotate framing by $4\pi$ → scalar invariants unchanged (quantization).

• T5 Composition law: glue two belts; class-function phases add; twist adds.

• T6 Thin-belt limit: area term shrinks; residual finite twist persists (framing anomaly).

• T7 Purpose calibration: set $\alpha$ so that $2\pi$ maps to one unit of output in the macro work ledger.

Outcome: after PBHL passes these tests on your data, the macro work–entropy ledger can safely use $\phi_{\text{gap}}$, $\iint\mathcal F_\Pi$, and $\mathrm{Tw}$ as auditable primitives.

 

4. Framing and Twist

Goals — governance/interpretation as twist

In PFBT, governance/interpretation changes (policies, prompts, pricing rules, SOPs, model styles) are not afterthoughts—they are encoded as a framing field along the belt’s edges. The twist of this framing is the edge-only contribution that, together with face flux, explains the Plan↔Do gap in PBHL. Operationally, you will log twist steps as discrete policy changes; mathematically, you’ll also capture continuous roll of the frame along the edge.

---

Result — twist from a framed ribbon

Let $\mathbf t(s)$ be the unit tangent of the edge centerline (plan or do) and $\mathbf u(s)$ a unit framing vector with $\mathbf u\!\cdot\!\mathbf t=0$ (the governance/interpretation axis you choose to carry along the edge). The twist of the ribbon is

$$\boxed{\quad \mathrm{Tw} \;=\; \frac{1}{2\pi}\int_{0}^{1} \big(\,\mathbf t(s)\times \mathbf u(s)\big)\cdot \partial_s\mathbf u(s)\;ds \quad}$$

Conventions (kept book-wide):

• Orientation: right-hand rule from the belt’s normal.

• Counting: a clockwise half-twist viewed along $+\mathbf n$ is $+\tfrac12$; hence $\mathrm{Tw}\in \tfrac12\mathbb Z$.

• PBHL phase: the edge framing contributes $e^{\,i\,\alpha\,\mathrm{Tw}}$.

Framing quantization → $\alpha\in 2\pi\mathbb Z$.
For ledgers we adopt 2π invariance (a full 360° policy roll does not change scalar outputs). Since $\Delta \mathrm{Tw}=1$ under a 2π roll,

$$e^{\,i\,\alpha\,\Delta\mathrm{Tw}}=1 \;\Rightarrow\; \alpha\in 2\pi\mathbb Z .$$

(Research note: a 4π-periodic spinorial mode is available by relaxing to $\alpha\in\pi\mathbb Z$; we default to 2π for operational invariance.)

---

Procedures — discrete twist estimators

You’ll use one of three estimators depending on data granularity. All assume a sampled edge $\{x_k\}_{k=0}^{N}$ (monotone in $s$ or time) with an associated frame $(\mathbf t_k,\mathbf u_k,\mathbf v_k=\mathbf t_k\times\mathbf u_k)$.

A) Midpoint (angle accumulation)

1. $\mathbf t_k=\frac{x_{k+1}-x_k}{\|x_{k+1}-x_k\|}$.

2. Transport $\mathbf u_k$ to the next station by projecting onto the new normal plane:
   $\widetilde{\mathbf u}_{k+1}=\mathbf u_k-(\mathbf u_k\!\cdot\!\mathbf t_{k+1})\mathbf t_{k+1}$, normalize.

3. Compute roll around $\mathbf t_{k+1/2}$:
   $\Delta\phi_k=\mathrm{atan2}\!\big((\widetilde{\mathbf u}_{k+1}\times \mathbf u_{k+1})\!\cdot\!\mathbf t_{k+1/2},\; \widetilde{\mathbf u}_{k+1}\!\cdot\!\mathbf u_{k+1}\big)$.

4. Sum: $\widehat{\mathrm{Tw}}=\dfrac{1}{2\pi}\sum_k \Delta\phi_k$.

B) Quaternion (robust on noisy tangents)

1. Fit unit quaternions $q_k$ such that $q_k$ maps a fixed reference triad to $(\mathbf t_k,\mathbf u_k,\mathbf v_k)$.

2. Relative rotation: $r_k=q_{k+1}\,q_k^{-1}$. Extract the rotation axis–angle $(\hat{\omega}_k,\theta_k)$.

3. Project the rotation onto the tangent: $\Delta\phi_k = \theta_k\,(\hat{\omega}_k\!\cdot\!\mathbf t_{k+1/2})$.

4. $\widehat{\mathrm{Tw}}=\dfrac{1}{2\pi}\sum_k \Delta\phi_k$.
   Benefits: numerically stable, avoids gimbal lock.

C) Spline (smooth data)

1. Fit $C^1$ splines to $\mathbf t(s),\mathbf u(s)$.

2. Evaluate the twist density $\omega_s(s)=(\mathbf t\times\mathbf u)\cdot\partial_s\mathbf u$.

3. Integrate with Simpson’s rule: $\widehat{\mathrm{Tw}}=\dfrac{1}{2\pi}\int \omega_s(s)\,ds$.

Twist steps from governance logs.
When a policy/prompt/version change occurs at sample $k$, add the logged step $\Delta\phi_k^{\text{(policy)}}$ to the continuous roll before dividing by $2\pi$. This creates a hybrid estimator:

$$\widehat{\mathrm{Tw}}=\frac{1}{2\pi}\Big(\sum_k \Delta\phi_k^{\text{(geom)}}+\sum_{j\in\text{policy}} \Delta\phi_j^{\text{(policy)}}\Big).$$

---

Figures (described)

• 2π & 4π rotations: show a belt whose ends are fixed while the frame rotates by 360° (Tw $+\!1$) and 720° (Tw $+\!2$); PBHL’s scalar phase stays invariant by our $\alpha$ choice.

• Twist steps as policy changes: a timeline on the edge with markers (pricing_v3 → v4, prompt_v1 → v2), each contributing a discrete roll increment.

---

Checks — 4π periodicity test (see Ch.43 test pack)

T-4π: apply two sequential 2π frame rotations while keeping the face flux constant; assert:

• Scalar invariants (e.g., $\arg\det$ gap, ledger outputs) unchanged.

• Logged $\mathrm{Tw}$ increases by 2; PBHL’s twist phase $e^{i\alpha\mathrm{Tw}}$ stays unit with $\alpha\in 2\pi\mathbb Z$.

• Composition sanity: performing two 2π tests back-to-back equals one 4π test (additivity of $\mathrm{Tw}$).

---

Artifacts — TwistFrame schema (used in Ch.31)

{
  "twist_frames": [
    {
      "edge": "plan",                      // "plan" | "do"
      "t": "2025-01-01T00:04:00Z",        // timestamp on the edge log
      "policy_id": "pricing_v4",          // governance label (free text)
      "prompt_id": "style_v2",            // optional (LLM, UI theme, etc.)
      "frame_quat": [0.998, 0.010, 0.050, 0.030], // optional absolute frame (w,x,y,z)
      "delta_roll_deg": 180,              // optional discrete step in degrees (sign by right-hand rule)
      "notes": "monthly policy roll-over"
    }
  ],
  "conventions": {
    "right_hand_rule": true,
    "tw_units": "turns",                  // "turns" (1 turn = 2π), or "radians"
    "alpha": 6.283185307179586           // default α = 2π for ledger invariance
  }
}

Usage notes.

• If frame_quat is present at multiple samples, prefer the quaternion estimator and treat delta_roll_deg as a discrete addition.

• If only policy steps are logged, you still get a valid twist by summing steps—useful for business processes with sparse geometry.

• Always record the conventions block to make PBHL’s twist phase reproducible.

---

Where this points. With twist operationalized, PBHL’s gap = flux + twist is fully computable: the next chapter (Ch.5) glues belts, proves composition laws, and prepares the scalar invariants you’ll use in ledgers and controllers.

 

5. Axioms, Invariants, and Composition (Gluing)

Goals

Post the axioms of PFBT belts, define the invariants you will compute, and give the gluing law that makes multi-belt reasoning compositional.

---

5.1 Axioms (minimal set)

A1 — Dual-trace completeness.
Given co-logged edges $\Gamma_+$ (Plan) and $\Gamma_-$ (Do) on the same state manifold $\mathcal X$, there exists an oriented belt $\mathcal B$ with boundary

$$\partial \mathcal B \;=\; \Gamma_+ \;\sqcup\; (-\Gamma_-).$$

A2 — Orientation & sign.
All face integrals and twist counts use the boundary orientation above (“outer minus inner”). Flipping $\mathcal B$ flips the signs of edge gap, face flux, and twist coherently.

A3 — Gauge equivalence.
For the purpose connection $\mathcal A_\Pi$, $\mathcal A_\Pi\mapsto \mathcal A_\Pi+d\chi$ leaves $\mathcal F_\Pi$ invariant. Edge holonomies change by conjugation/periods; class-function readouts (trace/character/$\arg\det$) are invariant.

A4 — Framing periodicity.
Twist $\mathrm{Tw}\in \tfrac12\mathbb Z$. Operational default: 2π invariance, i.e. $\alpha\in 2\pi\mathbb Z$ so one full 2π roll does not change scalar outputs.

A5 — Refinement & stability.
Subdivide $\mathcal B$ (add interior seams) without changing any scalar invariant; interior edges cancel. Scalar outputs are mesh-independent.

A6 — Gluing (existence).
If two belts share a seam edge with compatible orientation and frame, they admit a glued belt whose boundary is the remaining two external edges.

---

5.2 Invariants you will actually use

Let

$$\Delta\mathrm{Hol}_\Pi \;\equiv\; \mathrm{Hol}_\Pi(\Gamma_+)\,\mathrm{Hol}_\Pi(\Gamma_-)^{-1} \;=\; \mathcal S\exp\!\Big(\!\!\iint_{\mathcal B}W^{-1}\mathcal F_\Pi W\Big)\cdot e^{i\alpha\,\mathrm{Tw}}.$$

I1 — Gap phase (class function).

$$\phi_{\text{gap}}\;\equiv\;\frac{1}{k}\arg\det\big(\Delta\mathrm{Hol}_\Pi\big)\quad (\bmod\ 2\pi).$$

I2 — Face flux scalar.

$$\phi_{\text{flux}}\;\equiv\;\frac{1}{k}\arg\det\!\Big(\mathcal S\exp\!\iint_{\mathcal B}W^{-1}\mathcal F_\Pi W\Big).$$

I3 — Twist.
$\mathrm{Tw}(\mathcal B,\mathbf n)\in \tfrac12\mathbb Z$ (geometric + logged policy steps).

I4 — Ledger primitives.
Given calibration $K_\Pi$ where $K_\Pi\cdot 2\pi$ equals one output quantum (e.g., 1 pair of shoes):

$$W_{\text{macro}} \;=\; K_\Pi\,\phi_{\text{gap}}, \qquad \Sigma_{\text{macro}} \;=\; \text{(dispersion + WIP-age + rework + twist-cost)}.$$

Properties. (i) Conjugacy-invariant; (ii) parametrization-independent; (iii) additive under gluing (below) in the scalar/log domain; (iv) Lipschitz in belt width $w$ with constant $\sup_{\mathcal B}\!\|\mathcal F_\Pi\|$ (width-scaling bound).

---

5.3 The gluing law (composition)

Suppose $\mathcal B_1$ and $\mathcal B_2$ share a seam edge $\Sigma$ with opposite orientations, same framing at the seam, and compatible basepoints/orderings. Define the glued belt

$$\mathcal B\;=\;\mathcal B_2 \,\#_{\Sigma}\, \mathcal B_1, \qquad \partial\mathcal B \;=\; \Gamma^{(2)}_+ \;\sqcup\; \big(-\Gamma^{(1)}_-\big).$$

Then:

G1 — Interior edge cancels.
The seam’s contribution vanishes; surface-ordered exponentials multiply; scalar class-functions add:

$$\phi_{\text{flux}}(\mathcal B) \;=\; \phi_{\text{flux}}(\mathcal B_2) + \phi_{\text{flux}}(\mathcal B_1)\quad(\bmod 2\pi).$$

G2 — Twist additivity.

$$\mathrm{Tw}(\mathcal B)=\mathrm{Tw}(\mathcal B_2)+\mathrm{Tw}(\mathcal B_1).$$

G3 — Basepoint/ordering conjugacy.
At the matrix level,

$$\Delta\mathrm{Hol}_\Pi(\mathcal B) \;\sim\; \Delta\mathrm{Hol}_\Pi(\mathcal B_2)\;\Delta\mathrm{Hol}_\Pi(\mathcal B_1) \quad(\text{conjugate at the seam basepoint}).$$

Any class-function (trace, $\arg\det$, characters) gives a seam-independent scalar.

G4 — Variants.
If seam framings differ, the mismatch appears as an extra twist step concentrated on $\Sigma$; add it to $\mathrm{Tw}$ before taking class-functions.

---

5.4 Procedures

5.4.1 Belt composition API (pseudo-spec)

types:
  Edge:
    id: string
    samples: [ {t: iso8601, x: vector, meta?: any} ]
    frame?: [ {t: iso8601, quat: [w,x,y,z]} ]

  Belt:
    id: string
    upper: Edge   # Γ_+
    lower: Edge   # Γ_-
    twist_frames?: [TwistFrame]
    mesh?: AnnulusMesh
    meta?: any

functions:
  normalize_edge(e: Edge, basepoint: 'first'|'nearest', order: 'forward'|'reverse') -> Edge

  glue(B2: Belt, seam: 'B2.lower', B1: Belt, seam: 'B1.upper',
       seam_policy: 'match|average|prefer_B2', reframe?: bool) -> Belt

  invariants(B: Belt, class_fn: 'argdet'|'trace'|Character, alpha: float)
    -> { phi_gap, phi_flux, Tw, diagnostics }

  aggregate(belts: Belt[], mode: 'series'|'parallel') 
    -> { phi_gap, phi_flux, Tw }  # series = gluing along seams

Rules.

• normalize_edge ensures common orientation/timebase.

• glue checks geometric seam proximity & frame compatibility; inserts a seam twist if needed.

• invariants computes class-function scalars and logs conjugacy residuals (should be ~0).

• aggregate(series) sums scalar invariants; aggregate(parallel) keeps edges disjoint and reports a vector (one invariant per belt).

5.4.2 Invariant aggregation

• Series (glued): add $\phi_{\text{flux}}$, add $\mathrm{Tw}$, then recompute $\phi_{\text{gap}}$ (or add phases and reduce mod $2\pi$).

• Parallel (independent belts): maintain a tuple $(\phi_{\text{gap}}^{(m)})_{m=1}^M$; dashboards can show stacked contributions.

---

5.5 Checks

C1 — Basepoint invariance.
Recompute $\phi_{\text{gap}}$ with different basepoints and radial Wilson families; assert change $<\varepsilon$. If not, a seam is misframed or orderings differ.

C2 — Width-scaling bounds.
Dilate the belt width by $w\mapsto \lambda w$ (fixed edges, interpolated interior). Then

$$\big|\phi_{\text{flux}}(\lambda)-\phi_{\text{flux}}(0)\big| \;\le\; \lambda\, \operatorname{area}(\mathcal B)\,\sup_{\mathcal B}\|\mathcal F_\Pi\| \quad(+\ \text{estimation error}).$$

Use this to set tolerance bands for thin-belt approximations.

C3 — Refinement/gluing regression.
Subdivide $\mathcal B$ into $n$ annuli, glue back; scalar invariants change $<\varepsilon$. Failing this means nonconjugate seams or inconsistent twist units.

---

5.6 Artifact — “Belt Algebra” mini-spec

Objects. Belts $\mathcal B$; edges $\Gamma$; invariants $(\phi_{\text{flux}},\mathrm{Tw})$ and $\phi_{\text{gap}}$.

Operations.

• Series gluing $\#_\Sigma$: associative up to conjugacy.

• Reversal $(\cdot)^\dagger$: reverse orientation; signs flip:

  $$\phi_{\text{flux}}(\mathcal B^\dagger)=-\phi_{\text{flux}}(\mathcal B),\quad \mathrm{Tw}(\mathcal B^\dagger)=-\mathrm{Tw}(\mathcal B).$$

• Identity $\mathbb I$: zero-area, zero-twist belt; neutral for $\#$.

• Refinement $\preceq$: partial order by subdivision; invariants monotone-stable (exact under exact arithmetic, stable numerically).

Laws (scalar/class-function level).

• Associativity: $\phi(\mathcal B_3\#\mathcal B_2\#\mathcal B_1)=\phi(\mathcal B_3)+\phi(\mathcal B_2)+\phi(\mathcal B_1)$.

• Identity: $\phi(\mathcal B\#\mathbb I)=\phi(\mathcal B)$.

• Inverse via reversal: $\phi(\mathcal B\#\mathcal B^\dagger)=0$.

• Twist additivity: $\mathrm{Tw}$ adds under $\#$; spins count in half-turns; policy steps are additive.

• Calibration consistency: with $\alpha\in 2\pi\mathbb Z$, all ledger-level scalars are invariant under $2\pi$ framing rotations.

This algebra underwrites the unit tests you’ll run in Part X and the controller composition you’ll deploy in Part V.

 

6. Belt Domains & Two-Boundary Stokes/Gauss

Goals — orientation and two-boundary theorems

We formalize belts as oriented annular patches and state the two-boundary versions of Stokes/Gauss that underpin PBHL numerics. All signs follow the outer-minus-inner convention:

$$\partial\mathcal B=\Gamma_+\;\sqcup\;(-\Gamma_-).$$

---

6.1 Parametric belts and Jacobians

Let $\mathbf X:[0,1]\times[0,1]\!\to\!\mathcal X$ parametrize the belt:

$$\mathbf X(s,0)\in\Gamma_-,\quad \mathbf X(s,1)\in\Gamma_+,\qquad (s,u)\in[0,1]^2,$$

with induced tangent basis $\mathbf X_s=\partial_s\mathbf X,\ \mathbf X_u=\partial_u\mathbf X$.

• Area element (embedded case): $dS= \|\mathbf X_s\times \mathbf X_u\|\,ds\,du$; unit normal $\mathbf n=\frac{\mathbf X_s\times\mathbf X_u}{\|\mathbf X_s\times\mathbf X_u\|}$.

• Area element (intrinsic): $g_{ab}=\langle \mathbf X_a,\mathbf X_b\rangle,\ a,b\in\{s,u\}$; $dS=\sqrt{\det g}\,ds\,du$.

• Boundary orientation: walking along $\partial\mathcal B$ keeps $\mathcal B$ on your left. Thus $+\Gamma_+$ is the $u=1$ edge in the $+s$ direction; $-\Gamma_-$ is the $u=0$ edge in the $-s$ direction.

---

6.2 Two-boundary Stokes (Abelian form)

For a smooth 1-form $A$ on $\mathcal B$ with curvature $F=dA$,

$$\boxed{\; \oint_{\Gamma_+}\!A\;-\;\oint_{\Gamma_-}\!A \;=\; \iint_{\mathcal B}\!F \;}$$

• In coordinates: if $A=A_s\,ds+A_u\,du$, then $F=(\partial_s A_u-\partial_u A_s)\,ds\wedge du$, and

$$\oint_{\Gamma_+}\!A-\oint_{\Gamma_-}\!A =\int_0^1\!\!A_s(s,1)\,ds - \int_0^1\!\!A_s(s,0)\,ds =\int_0^1\!\!\int_0^1\!(\partial_s A_u-\partial_u A_s)\,ds\,du.$$

• Sign sanity: flip $\mathcal B\Rightarrow$ both sides change sign.

This identity is the computational backbone for PBHL’s Abelian recovery and for unit tests on synthetic belts.

---

6.3 Surface Gauss on belts (2D divergence theorem)

Let $\mathbf J$ be a tangential face field on $\mathcal B$ (e.g., purpose-current density projected to the belt). Define the surface divergence $\nabla_{\!s}\!\cdot\mathbf J$ (intrinsic to $\mathcal B$). Then

$$\boxed{\; \iint_{\mathcal B}\!(\nabla_{\!s}\!\cdot\mathbf J)\,dS \;=\; \oint_{\partial\mathcal B}\!\mathbf J\cdot \boldsymbol{\nu}_s\,dl \;=\; \int_{\Gamma_+}\!\mathbf J\cdot \boldsymbol{\nu}_s\,dl \;-\; \int_{\Gamma_-}\!\mathbf J\cdot \boldsymbol{\nu}_s\,dl \;}$$

where $\boldsymbol{\nu}_s$ is the co-normal: the unit vector in the tangent plane orthogonal to the boundary tangent, chosen to point outward of $\mathcal B$. This is the belt analogue of Gauss: it relates face accumulation to edge flux imbalance (outer minus inner).

Coordinate form. With metric $g$ and $\mathbf J=J^s\mathbf X_s+J^u\mathbf X_u$,

$$\nabla_{\!s}\!\cdot\mathbf J =\frac{1}{\sqrt{\det g}}\Big[\partial_s\!\big(\sqrt{\det g}\,J^s\big) +\partial_u\!\big(\sqrt{\det g}\,J^u\big)\Big].$$

On $u=\text{const}$ edges, $dl=\|\mathbf X_s\|\,ds$ and $\boldsymbol{\nu}_s=\pm\frac{\mathbf X_u}{\|\mathbf X_u\|}$ (sign per outward rule).

---

6.4 Worked examples (analytic)

(A) Constant 2-form.
Take $A= B\,u\,ds$ on $[0,1]^2$. Then $F=B\,du\wedge ds$.
$\oint_{\Gamma_+}\!A-\oint_{\Gamma_-}\!A= B-0=B$, and $\iint_{\mathcal B}\!F = B$.

(B) Surface Gauss on a spherical belt.
Let $\mathcal B$ be a latitude belt on a sphere of radius $R$ between colatitudes $\theta_-$ and $\theta_+$, parameterized by $(s,u)=(\varphi,\theta)$. For a tangential field $\mathbf J=J_\varphi\,\mathbf e_\varphi$ (zonal flow),

$$\iint_{\mathcal B}\!(\nabla_{\!s}\!\cdot\mathbf J)\,dS = R\!\int_{\theta_-}^{\theta_+}\!\!\int_{0}^{2\pi}\!\frac{1}{\sin\theta}\,\partial_\varphi(\sin\theta\,J_\varphi)\,d\varphi\,d\theta = \int_{\Gamma_+}\!\mathbf J\cdot \boldsymbol{\nu}_s\,dl - \int_{\Gamma_-}\!\mathbf J\cdot \boldsymbol{\nu}_s\,dl.$$

(Here $\boldsymbol{\nu}_s=\pm \mathbf e_\theta$; signs follow outer-minus-inner.)

---

6.5 Numerical Stokes on synthetic fields (see App C)

To validate implementations:

1. Choose a belt map $\mathbf X(s,u)$ (planar rectangle, warped ribbon, spherical belt).

2. Pick a synthetic 1-form $A$ (e.g., polynomials in $s,u$) and compute ground-truth $F=dA$.

3. Discretize $[0,1]^2$ into an annular mesh (indices $i=0..N_s$, $j=0..N_u$).

4. Face integral: sum $F$ over cells with Simpson/GL quadrature and Jacobians.

5. Edge integrals: line-integrate $A$ along $u=1$ and $u=0$ with consistent parametrization.

6. Assert $|(\text{edge gap})-(\text{face flux})|\le \varepsilon$ (tolerance scales with mesh, field smoothness).

7. Sensitivity: dilate the belt width, perturb sampling, check width-scaling bound (Part I.5).

---

6.6 Procedures — mesh constructors & Jacobians

Parametric annulus constructor.

function build_annulus_mesh(upper: Edge, lower: Edge, Ns: int, Nu: int,
                            interp: 'linear'|'spline' = 'spline'):
  # 1) Reparameterize both edges by arc-length s in [0,1]
  # 2) For each s_i, generate a radial path u ↦ X(s_i,u) that joins lower→upper
  # 3) Emit node grid {X_{i,j}} with i∈[0..Ns], j∈[0..Nu]
  # 4) For each cell (i,j), cache Jacobians:
  #    J_{i,j} = ||X_s × X_u||  or sqrt(det g)
  # 5) Record boundary orientation (outer-minus-inner)

Spherical belt helper.

function build_spherical_belt(R, theta_minus, theta_plus, Ns, Nu):
  # X(s,u) = spherical_to_xyz(R, theta(u), phi(s))
  # theta(u) = (1-u)*theta_minus + u*theta_plus
  # phi(s)   = 2π s

Numeric Stokes/Gauss helpers.

function integrate_face(F_func, mesh) -> float
function integrate_edge(A_func, edge_samples, orientation) -> float
function surface_divergence(J_field, mesh_metric) -> array  # for Gauss checks

All constructors return a Mesh object with nodes, cells, metric/Jacobians, boundary index sets $\{u=0\},\{u=1\}$, and orientation flags. (Full signatures in Ch31 and App C.1.)

---

6.7 Figures (described)

• Annulus orientation. A rectangle $[0,1]^2$ with arrows: boundary walk keeps the interior on the left; labels “$u=1:\Gamma_+$ (outer), $u=0:\Gamma_-$ (inner)”.

• Spherical belt. A sphere showing two parallels (inner/outer) and co-normal directions; edge flux arrows demonstrate “outer minus inner”.

---

6.8 Checks — numeric Stokes on synthetic fields (App C)

• C-S1 (planar ribbon): polynomial $A$, grid refine $N_s,N_u\!\uparrow$ → error $\!\downarrow$ at expected order.

• C-S2 (warped ribbon): non-uniform Jacobians; validate sign via orientation flip.

• C-S3 (spherical belt): match surface-metric Gauss result within tolerance.

• C-S4 (thin-belt limit): shrink width; confirm edge-gap $\approx$ twist-only (when $F\to0$).

---

6.9 Artifacts — mesh constructor signatures (Ch31, App C.1)

// TypeScript-style
type AnnulusMesh = {
  nodes: Float64Array,   // [Nnodes × dim]
  cells: Uint32Array,    // quad indices
  jac: Float64Array,     // cell Jacobians (||X_s × X_u|| or sqrt(det g))
  sGrid: Float64Array,   // s coordinates per node
  uGrid: Float64Array,   // u coordinates per node
  boundary: { upper: number[], lower: number[] }, // index lists
  orientation: 'outer-minus-inner',
  meta?: Record<string, any>
};

declare function buildAnnulusMesh(
  upper: Edge, lower: Edge, Ns: number, Nu: number,
  opts?: { interp?: 'linear'|'spline', smoothing?: number }
): AnnulusMesh;

declare function stokesTwoBoundary(
  mesh: AnnulusMesh, A: (s:number,u:number)=>[number,number],
  quad?: 'simpson'|'gauss-legendre'
): { edge_gap: number, face_flux: number, error: number };

These APIs, plus the test pack in App C, give you a ready path to verify orientation, Jacobians, and the two-boundary theorems before plugging PBHL into estimators and ledgers.

 

 

7. Surface Ordering & Holonomy

Goals — non-Abelian mechanics on belts

When Purpose is multi-component (e.g., $U(k)$ fiber), curvature values at different face points do not commute. A single line integral cannot capture the belt’s interior. We therefore define a surface-ordered exponential for the annulus and a practical dual-tree transport to make it computable and gauge-robust.

---

7.1 Surface-ordered exponential on an annulus

Let $\mathcal B$ be the belt, $\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$, and pick a basepoint $b\in\mathcal B$.
For each point $p\in\mathcal B$, choose a radial Wilson line $W(b\to p)$ (parallel transport by $\mathcal A_\Pi$). Define the conjugated curvature

$$\widetilde{\mathcal F}_\Pi(p)\;=\;W(b\to p)^{-1}\,\mathcal F_\Pi(p)\,W(b\to p).$$

The framed belt propagator is

$$\mathcal U_{\mathcal B} \;=\; \mathcal S\exp\!\Big(\!\iint_{\mathcal B}\widetilde{\mathcal F}_\Pi\,dS\Big),$$

where $\mathcal S$ enforces a surface order (an ordering over the face that generalizes path ordering).

PBHL in matrix form (suppressing twist for now) reads

$$\operatorname{Hol}_\Pi(\Gamma_+)\;\operatorname{Hol}_\Pi(\Gamma_-)^{-1} \;=\;\mathcal U_{\mathcal B}.$$

Taking class functions (trace, characters, $\arg\det$) gives gauge/basepoint-robust scalars.

---

7.2 Dual-tree transport

Computing $\mathcal U_{\mathcal B}$ and edge holonomies consistently requires two coupled transports:

1. Face tree: a spanning tree $T_{\text{face}}$ of mesh cells radiating from $b$; gives $W(b\to p)$ to every quadrature point $p$.

2. Edge trees: two path trees along $\Gamma_\pm$ from $b$ to each edge sample; give $W(b\to q)$ for points $q\in\Gamma_\pm$.

Why dual trees?

• Keep gauge covariance explicit (all face contributions are compared in the same fiber at $b$);

• Make basepoint changes a conjugation (class functions cancel it);

• Enable local updates (refine a ring or resequence cells without recomputing global transports).

---

7.3 Face ordering strategies (practical $\mathcal S$)

Because $[\widetilde{\mathcal F}_\Pi(p_1),\widetilde{\mathcal F}_\Pi(p_2)]\neq 0$, the discretized order matters at finite resolution. Three robust choices:

• Radial rings (fan ordering) — Partition $\mathcal B$ into rings $u\in[u_j,u_{j+1}]$. Order inner → outer rings; within each ring sweep $s$ (e.g., left→right).
  Pros: matches intuition for annuli; stable under ring refinement.
  Error: $\mathcal O(\|[\mathcal F,\mathcal F]\|\,\Delta u\,\Delta s)$.

• Stripe (u-first) ordering — Sweep stripes of constant $u$ across $s$, then advance $u$.
  Pros: easy on rectangular meshes; amenable to SIMD.
  Note: use the same rule for $\Gamma_\pm$ path trees to keep conjugacy small.

• BFS/tree ordering — Do a breadth-first traversal of $T_{\text{face}}$.
  Pros: minimizes transport path length variance; good on highly warped belts.

Refinement rule. Any ordering that refines to the same limit and preserves ring monotonicity yields identical class-function scalars in the mesh-refined limit.

---

7.4 Magnus vs BCH: two log-space tools

Magnus expansion (surface).
Work in the Lie algebra via $\mathcal U_{\mathcal B}=\exp\Omega$. The first two terms suffice for high-quality numerics:

$$\begin{aligned} \Omega_1 &= \iint_{\mathcal B}\widetilde{\mathcal F}_\Pi(p)\,dS,\\ \Omega_2 &= \tfrac12 \iint\!\!\iint_{p\succ q}\,[\widetilde{\mathcal F}_\Pi(p),\widetilde{\mathcal F}_\Pi(q)]\,dS_p\,dS_q, \end{aligned}$$

where $p\succ q$ implements your surface order. Keep $\Omega=\Omega_1+\Omega_2$ and set $\mathcal U_{\mathcal B}\approx \exp(\Omega)$.

BCH (edges & gaps).
If you already have edge holonomies $H_\pm=\operatorname{Hol}_\Pi(\Gamma_\pm)=\exp A_\pm$, then

$$\log\!\big(H_+H_-^{-1}\big) \;=\; A_+ - A_- + \tfrac12[A_+, -A_-] + \cdots$$

Useful for sanity checks and for reconciling edge-only calculations with the face Magnus result (they agree as mesh refines). In production, prefer face-first with Magnus $\Omega_2$.

---

7.5 Procedures — invariant engine (high-level)

1. Build mesh & dual trees

   • mesh = buildAnnulusMesh(Γ+, Γ−, Ns, Nu)

   • T_face from basepoint $b$ to all cells; T_edge^± along $\Gamma_\pm$.

2. Compute transports

   • For each quadrature point $p$: $W_p = W(b\to p; T_{\text{face}})$.

   • For each edge sample $q$: $W^\pm_q = W(b\to q; T_{\text{edge}}^\pm)$.

3. Assemble $\Omega$ (Magnus-$\Omega_1+\Omega_2$)

   • $\Omega_1 \leftarrow \sum_c \sum_{p\in c} w_p \, W_p^{-1}\mathcal F_\Pi(p)W_p$.

   • $\Omega_2 \leftarrow \tfrac12 \sum_{p\succ q} w_p w_q \,[W_p^{-1}\mathcal F_\Pi(p)W_p,\; W_q^{-1}\mathcal F_\Pi(q)W_q]$.

   • U_face = expm(Ω1 + Ω2).

4. Edge holonomies (for diagnostics)

   • $H_\pm \leftarrow \prod_{q\in \Gamma_\pm}^{\curvearrowright} \exp\!\big(\int_{q}^{q'} \! W_q^{-1}\mathcal A_\Pi W_q\big)$.

   • Check $\| \log(H_+ H_-^{-1}) - \log(U_{\text{face}})\| \le \varepsilon$.

5. Twist & PBHL scalar

   • Estimate $\mathrm{Tw}$ (Chapter 4).

   • PBHL matrix: $U_{\text{belt}} = U_{\text{face}}\cdot e^{i\alpha\,\mathrm{Tw}}$.

   • Scalar invariant: $\phi_{\text{gap}} = \tfrac1k \arg\det(U_{\text{belt}})$.

6. Return diagnostics

   • Basepoint conjugacy residuals, ordering sensitivity (shuffle within rings), mesh-refinement curve, and confidence bands.

(Full pseudocode listing is provided in Ch.32.)

---

7.6 Checks — ordering & basepoint benchmarks (see Ch.43)

• Basepoint invariance. Move $b$ around a small loop; recompute. Class-function scalars ($\arg\det$, characters) vary $<\varepsilon$.

• Ordering robustness. Fix mesh; permute within-ring order; scalars remain within tolerance; commutator term $\|\Omega_2\|$ predicts sensitivity.

• Refinement limit. Increase $(N_s,N_u)$; face-Magnus and edge-BCH logs converge; residual decays at expected order.

• Gluing consistency. Split the belt into two rings, compute $\mathcal U_{\mathcal B_1}$ and $\mathcal U_{\mathcal B_2}$; verify $\mathcal U_{\mathcal B_2}\mathcal U_{\mathcal B_1}$ matches whole-belt result up to conjugacy.

---

7.7 Artifacts — invariant engine pseudocode (pointer)

• Ch.32 provides a ready-to-adapt engine with:

  • buildDualTrees, transportToPoint, assembleMagnusΩ, edgeHolonomy,

  • pbhlScalar (class-function reductions), and

  • test harnesses for basepoint/order/refinement.

Use the Magnus-$\Omega_2$ path by default; fall back to product-of-exponentials when runtime or memory is constrained, and log the ordering you used for reproducibility.

 

8. Discrete Belts: Mesh · Quadrature · Errors

Goals — numerics that actually converge

We turn the belt calculus into finite computations and give an error model you can budget, monitor, and drive to zero by refinement. Targets: stable PBHL scalars ($\phi_{\text{gap}},\phi_{\text{flux}},\mathrm{Tw}$) with reproducible tolerance.

---

8.1 Error model (face + edges + twist)

Let $h$ be the representative mesh size (max cell diameter in $(s,u)$); $w$ the belt width (in state units); $\kappa$ a curvature bound of the embedding; and $\nabla_\perp\mathcal F$ the variation of $\mathcal F_\Pi$ normal to the belt midline. For class-function scalars, a practical envelope is

$$\boxed{\; \varepsilon_{\text{PBHL}} \;\lesssim\; \underbrace{C_1\,h^{p}}_{\text{quadrature/mesh}} \;+\; \underbrace{C_2\,(\kappa\,w)^2}_{\text{geometry (warping)}} \;+\; \underbrace{C_3\,\|\nabla_\perp \mathcal F_\Pi\|\,w^{2}}_{\text{field shear across width}} \;+\; \underbrace{C_4\,\|\Omega_2\|}_{\text{non-Abelian ordering}} \;+\; \underbrace{C_5\,\varepsilon_{\mathrm{Tw}}}_{\text{twist estimator}} \;}$$

• $p$ is the formal order of your area quadrature (e.g., Simpson $p=4$, Gauss–Legendre $p=2q$ for $q$-point rules).

• $(\kappa w)^2$ bounds embedding distortion when interpolating radial paths.

• $\|\nabla_\perp \mathcal F_\Pi\|\,w^{2}$ captures field inhomogeneity across the belt (thin-belt makes this small).

• $\|\Omega_2\|$ is the Magnus commutator norm (ordering error proxy); drive down via ring refinement.

• $\varepsilon_{\mathrm{Tw}}$ is the twist estimation error (Ch. 4); include policy-step quantization if that’s your only twist source.

Budgeting rule: track each term; stop refining when further mesh reductions no longer move $\varepsilon_{\text{PBHL}}$ because width- or non-Abelian terms dominate.

---

8.2 Structured mesh defaults

Use an annular tensor grid with $N_s$ samples around and $N_u$ radials across:

• Default aspect: $N_s : N_u \approx 4:1$ (curvature is usually along $s$).

• Ring monotonicity: enforce increasing $u$ rings for robust surface ordering.

• Spline interpolation: prefer $C^1$ splines for radial paths $X(s,u)$; fall back to linear on noisy edges.

Recommended starting table (App C)

Belt class	$N_s$	$N_u$	Quad	Target $\varepsilon$
Planar ribbon, smooth $A,F$	128	32	Simpson	$10^{-6}$
Warped belt, moderate $\kappa$	192	48	GL-4	$10^{-5}$
Non-Abelian, $\|\Omega_2\|$ medium	256	64	GL-6 + $\Omega_2$	$10^{-4}$

---

8.3 Adaptive refinement (where it matters)

Refine rings and sectors where indicators spike:

• Field indicator: cell variance of $W^{-1}\mathcal F_\Pi W$.

• Ordering indicator: local commutator norm $\|[\widetilde{\mathcal F}(p),\widetilde{\mathcal F}(q)]\|$ within a ring.

• Geometry indicator: second finite difference of radial paths (large when $\kappa$ is high).

• Edge indicator: curvature of $\Gamma_\pm$ or jitter in edge holonomies.

Policy: split ring $j$ if any indicator exceeds its threshold; cap $N_u$ per refinement round to protect cache locality.

---

8.4 Quadrature choice & stability

• Edges: composite Simpson on $s$ (odd number of panels), or GL-q if $A_\Pi$ is rough. Always log the edge parameterization.

• Face:

  • Smooth Abelian: Simpson × Simpson over $(s,u)$.

  • Non-Abelian: Gauss–Legendre $2\times2$ or $3\times3$ per cell with Magnus $\Omega_2$ (Section 7).

• Conditioning: rescale local charts so per-cell area $\approx 1$; this improves exp/log stability in $\mathfrak{u}(k)$.

---

8.5 Error vs DOF (what to expect)

Let $\text{DOF}\approx k_s N_s + k_u N_u + k_f N_sN_u$ be total work units (transport + quadrature). Typical regimes:

• Quadrature-limited (smooth, Abelian): log-log slope $\approx -p/2$ vs DOF.

• Ordering-limited (non-Abelian): plateau unless you (i) increase rings, (ii) include $\Omega_2$; after that, slope recovers.

• Width-limited: refining grid no longer helps; shrink $w$ or accept the geometric bound.

(Figure: plot $|\phi_{\text{gap}}-\phi^\star|$ vs DOF with markers for regime changes.)

---

8.6 Practical procedures

1. Thin-belt first. If you control preprocessing, shrink $w$ (reduces $C_2,C_3$).

2. Pick a quadrature. Start with Simpson (Abelian) or GL-6 + $\Omega_2$ (non-Abelian).

3. Run three meshes. $(N_s,N_u)$, $2\times$, $4\times$; confirm asymptotic rate $p$.

4. Order audit. Compute with two orderings (rings vs stripes); difference upper-bounds ordering error.

5. Twist audit. Swap twist estimators (midpoint vs quaternion); take max as $\varepsilon_{\mathrm{Tw}}$.

6. Width audit. Dilate $w\to\lambda w$ for $\lambda\in\{0.5,1,2\}$; fit the $(\kappa w)^2$ term.

7. Log the budget. Store $\{h,p,\kappa w,\|\nabla_\perp\mathcal F\|, \|\Omega_2\|,\varepsilon_{\mathrm{Tw}}\}$ alongside invariants.

---

8.7 Config templates (see App C)

belt_numerics:
  mesh:
    Ns: 256
    Nu: 64
    interp: spline       # 'linear'|'spline'
    thin_belt: true
    width_scale: 1.0
  quadrature:
    edges: simpson       # 'simpson'|'gl2'|'gl4'|'gl6'
    face: gl6
    magnus:
      use_Omega2: true
      comm_threshold: 1e-3
  adaptivity:
    enable: true
    indicators:
      field_var: 3.0e-4
      comm_norm: 1.0e-3
      geom_bend: 2.0e-3
    max_refine_levels: 3
  twist:
    estimator: quaternion # 'midpoint'|'quaternion'|'spline'
    policy_units: turns   # 1 turn = 2π
  tolerances:
    phi_gap_abs: 5.0e-5
    conjugacy_resid: 1.0e-6
    width_slope_check: true

---

8.8 Artifacts (where to find them)

• App C tables: recommended $N_s,N_u$, quadrature orders, and expected rates for common belt shapes.

• Config packs: ready-to-use YAMLs for Abelian-smooth, Non-Abelian-moderate, Warped-thin scenarios.

• Utilities: scripts to compute $h$, estimate $\kappa$, and sample $\nabla_\perp\mathcal F$ from tiles.

Outcome: with these defaults, the PBHL scalars converge predictably; if they don’t, your error budget will tell you which knob to turn (rings, width, or $\Omega_2$).

 

9. Invariance & 4π Periodicity

Goals — make the scalars robust

Our belt scalars must be insensitive to representation choices (gauge, basepoint, parametrization, mesh ordering) and stable under framing rotations. This section states the invariances, clarifies 2π vs 4π behavior, and outlines the test harness you will run in Ch.43.

---

9.1 Class-function invariants (what we measure)

Let

$$\Delta H \;\equiv\; \operatorname{Hol}_\Pi(\Gamma_+)\,\operatorname{Hol}_\Pi(\Gamma_-)^{-1} \;=\; \mathcal U_{\mathcal B}\cdot e^{\,i\,\alpha\,\mathrm{Tw}}, \qquad \mathcal U_{\mathcal B}=\mathcal S\exp\!\Big(\!\iint_{\mathcal B}\!W^{-1}\mathcal F_\Pi W\Big).$$

We report class-function scalars, i.e. functions $f$ with $f(G^{-1}XG)=f(X)$:

• Gap phase: $\displaystyle \phi_{\text{gap}}=\frac{1}{k}\arg\det(\Delta H)$ (projective $U(k)$).

• Flux phase: $\displaystyle \phi_{\text{flux}}=\frac{1}{k}\arg\det(\mathcal U_{\mathcal B})$.

• Characters (optional): $\chi_m=\tfrac1k\Im\log\operatorname{Tr}(\Delta H^m)$ for harmonics $m$.

These eliminate basepoint and ordering conjugacy by construction and are parameterization-independent.

---

9.2 Invariance statements (and what would break them)

Gauge invariance.
$\mathcal A_\Pi\mapsto \mathcal A_\Pi+d\chi$ leaves $\mathcal F_\Pi$ invariant. $\Delta H$ changes by conjugation and gauge periods; class functions cancel conjugation; integer periods are absorbed by calibration (Ch.2).

Basepoint invariance.
Moving the basepoint $b$ changes $W$ and $\mathcal U_{\mathcal B}$ by conjugation; $f(\Delta H)$ unchanged.
Failure mode: inconsistent edge orderings or mismatched dual trees (Ch.7) → detectable as non-zero conjugacy residual.

Ordering invariance (refinement limit).
Different surface orderings that respect ring-monotonicity converge to the same $f(\Delta H)$ as the mesh refines.
Finite-mesh drift is controlled by $\|\Omega_2\|$ (Ch.7–8); include it in the error budget.

Parametrization invariance.
Reparameterizing $\Gamma_\pm$ or $\mathcal B$ leaves integrals invariant (edges re-weight arc length; face re-weights Jacobian).

Gluing invariance.
Gluing along a seam composes matrices up to conjugation; class-function scalars add (Ch.5).

Orientation flip.
$\mathcal B\mapsto\mathcal B^\dagger$ flips signs: $\phi_{\text{flux}}\mapsto -\phi_{\text{flux}}$, $\mathrm{Tw}\mapsto -\mathrm{Tw}$, $\phi_{\text{gap}}\mapsto -\phi_{\text{gap}}$.
Sanity test: all three flip together.

---

9.3 2π vs 4π framing periodicity

Twist counting.
We count $\mathrm{Tw}\in\tfrac12\mathbb Z$ (half-turns). A 2π frame roll adds $+1$ to $\mathrm{Tw}$; a 4π roll adds $+2$.

Operational default (vectorial mode).
Choose $\alpha\in 2\pi\mathbb Z$. Then

$$e^{\,i\,\alpha\,\Delta\mathrm{Tw}}=1\quad\text{for any integer }\Delta\mathrm{Tw},$$

so 2π and 4π frame rolls leave the scalar phase unchanged. This matches ledger needs (policy re-labels shouldn’t move outputs).

Spinorial option (research mode).
Set $\alpha\in\pi\mathbb Z\setminus 2\pi\mathbb Z$. Then a 2π roll flips the sign of non-class scalars, but class-function invariants like $\arg\det$ remain 2π-invariant (determinant kills the sign). A 4π roll is invariant even at matrix level.
Use case: diagnosing hidden “spinor-like” channels in multi-purpose stacks.

What we certify here.
For all reported scalars in PFBT (class-functions), we require 4π periodicity by test, and 2π invariance when $\alpha=2\pi n$.

---

9.4 Test harness design (pointer to Ch.43)

Each test returns a pass/fail plus a numeric residual; thresholds scale with the error model of Ch.8.

T-Gauge. Random gauge bump $\chi$ on tiles → $|\Delta \phi_{\text{gap}}|\le \tau_{\text{gauge}}$.
T-Basepoint. Move $b$ along a small loop → conjugacy residual $\le \tau_b$.
T-Ordering. Recompute with ring vs stripe ordering → $|\Delta \phi_{\text{gap}}|\le \tau_{\text{ord}}$; record $\|\Omega_2\|$.
T-Refine. Double $(N_s,N_u)$ → observed rate matches configured $p$ within band.
T-Gluing. Split belt, compute and glue → additive within $\tau_{\#}$.
T-Flip. Reverse orientation → all three signs flip; relative error $\le \tau_{\text{flip}}$.
T-2π. Apply 2π frame roll (add $\Delta\mathrm{Tw}=+1$) → scalar invariants unchanged: $\le \tau_{2\pi}$.
T-4π. Apply 4π roll (two successive 2π) → unchanged within $\le \tau_{4\pi}$.
T-Width. Vary width $w\in\{\tfrac12,1,2\}$ → fit residual to $C_2(\kappa w)^2+C_3\|\nabla_\perp\mathcal F\|w^2$ (Ch.8); report $R^2$.

Threshold guidance (defaults)
Set from Ch.8 budget $\varepsilon_{\text{PBHL}}$:

• $\tau_b=\tau_{\text{ord}}=\tau_{\#}=0.2\,\varepsilon_{\text{PBHL}}$

• \tau_{2\pi}=\tau_{4\pi}}=0.1\,\varepsilon_{\text{PBHL}}

• $\tau_{\text{flip}}=0.1\,\varepsilon_{\text{PBHL}}$ (after sign correction)

• Refinement acceptance: observed slope within $\pm 0.3$ of target.

All tests log mesh DOF, $\|\Omega_2\|$, width $w$, $\kappa$ estimate, and twist estimator choice.

---

9.5 Artifacts — test spec & thresholds (see Ch.43)

Example config you can drop into the harness:

invariance_tests:
  gauge:
    enable: true
    amplitude: 0.02
    tau: 0.2 * eps_pbhl
  basepoint:
    enable: true
    loop_radius: 0.05
    tau: 0.2 * eps_pbhl
  ordering:
    enable: true
    strategies: ["rings","stripes"]
    tau: 0.2 * eps_pbhl
  refinement:
    enable: true
    meshes: [[128,32],[256,64],[512,128]]
    target_order_p: 4
    slope_tolerance: 0.3
  gluing:
    enable: true
    split: "mid_ring"
    tau: 0.2 * eps_pbhl
  orientation_flip:
    enable: true
    tau: 0.1 * eps_pbhl
  framing_2pi:
    enable: true
    delta_tw: 1    # one full 2π roll
    alpha: 6.283185307179586   # 2π
    tau: 0.1 * eps_pbhl
  framing_4pi:
    enable: true
    delta_tw: 2    # two full rolls
    alpha: 6.283185307179586
    tau: 0.1 * eps_pbhl
logging:
  record:
    - DOF
    - Omega2_norm
    - width
    - kappa_est
    - twist_estimator

Outcome. After Part II’s harness passes, your PBHL scalars are representation-robust and framing-periodic. This certifies them for use in the controllers (Part V) and in the macro work–entropy ledger (Part III, VII, IX).

 

 

 

 

 © 2025 Danny Yeung. All rights reserved. 版权所有 不得转载

 

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.