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Purpose-Flux Belt Theory (PFBT)
- Appendices A-I

Appendix A — Notation & Orientation Cheats (production notes)

Below is the compact, print-ready “cheat sheet” for symbols, orientations, and sign conventions used throughout PFBT. A matching set of vector diagrams is included here: Download the diagrams (PDF).

A.1 Objects & Symbols (one-glance table)

Belt / worldsheet: $\mathcal B$ , an oriented surface with two boundary traces.
Boundary (plan vs do): $\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ .
Purpose connection (estimable field): $\mathcal A_\Pi$ .
Curvature (drives macro work): $\mathcal F_\Pi=d\mathcal A_\Pi+\mathcal A_\Pi\wedge\mathcal A_\Pi$ .
Holonomy along a trace: $\mathrm{Hol}(\Gamma)=\mathcal{P}\exp\!\oint_\Gamma \mathcal A_\Pi$ (non-abelian), or $\oint_\Gamma \mathcal A_\Pi$ (abelianized ops layer).
Twist (framing/governance steps): $\mathrm{Tw}$ ; coupling $\alpha$ .
Purpose Belt Holonomy Law (PBHL):
$\oint_{\Gamma_+}\!\mathcal A_\Pi-\oint_{\Gamma_-}\!\mathcal A_\Pi \;=\;\iint_{\mathcal B}\!\mathcal F_\Pi\;+\;\alpha\,\mathrm{Tw}.$
(“Gap = Flux + Twist.”)

A.2 Orientation & Stokes-on-Belts

Positive loop orientation is counterclockwise (CCW) with respect to the chosen unit normal $\mathbf n$ (right-hand rule).
On a belt, $\Gamma_+$ inherits the forward boundary orientation; $\Gamma_-$ is reversed (minus sign in $\partial \mathcal B$ ).
Stokes (two-boundary form): the edge gap equals surface flux + twist (PBHL above).
Units sanity: the left side (edge integrals) and the right side (surface flux + twist term) must carry the same operational units after the book’s mapping to domain outputs.

A.3 Gluing & Cancellation

When two belts $\mathcal B_1,\mathcal B_2$ are glued along a shared interior edge with opposite orientations, that interior edge’s contributions cancel. Fluxes add:
$\iint_{\mathcal B_1\cup \mathcal B_2}\!\mathcal F_\Pi = \iint_{\mathcal B_1}\!\mathcal F_\Pi + \iint_{\mathcal B_2}\!\mathcal F_\Pi.$
Production note: keep all internal (glued) edges visible in debug prints; hide them in final figures once sign checks pass.

A.4 Twist Sign Convention (governance/framing)

Positive $\mathrm{Tw}$ means the local moving frame rotates CCW when viewed along $+\mathbf n$ .
Discrete policy/prompt/version steps contribute signed twist increments; the scalar $\alpha$ controls how strongly twist impacts the gap.
Pitfall to avoid: flipping the normal $\mathbf n$ without also flipping loop orientation will invert the sign of $\mathrm{Tw}$ unexpectedly.

A.5 Discrete Belts: Mesh Orientation

Each cell (face) is oriented CCW relative to $\mathbf n$ ; edges inherit consistent directions.
Edge integrals are accumulated with their local orientation; shared edges between cells cancel in the interior sum.
Quadrature: use the same node ordering across cells to avoid sign drift; see Part IV for the error model.

A.6 4π Periodicity (Dirac belt trick, test harness)

A $2\pi$ frame rotation is not homotopic to identity on the belt; $4\pi$ is.
Regression test: numeric holonomy should exhibit 4π invariance within tolerance (see Ch. 9 & Ch. 43).
Production tip: keep a canned “belt-trick” scene to validate renderer and sign conventions after any codegen change.

A.7 Ledger Signs (macro work & entropy)

Macro work $W_{\text{macro}}$ (in domain units) is proportional to an inner product of flux with the purpose-current $J_\Pi$ :
$W_{\text{macro}} \;\propto\; \iint_{\mathcal B}\!\langle \mathcal F_\Pi,\; J_\Pi\rangle.$
Positive $W_{\text{macro}}$ indicates productive closure of the plan–do gap in the chosen output units (e.g., pairs of shoes).
Macro entropy $\Sigma_{\text{macro}}$ accumulates dispersion, WIP age, rework, and twist costs; it increases with misalignment and excessive framing changes.
Conservation view: if the gap closes with minimal $\mathrm{Tw}$ and low dispersion, expect high entropy-efficiency (Part VII KPIs).

A.8 One-page sanity checks (use before any run)

Boundary sign test: $\partial\mathcal B$ prints as $\Gamma_+$ (forward) and $-\Gamma_-$ (reversed).
Stokes parity: numeric $\oint_{\Gamma_+}-\oint_{\Gamma_-}$ ≈ surface flux $+\alpha \mathrm{Tw}$ (tolerance per Ch. 8).
Gluing: shared interior edges cancel to machine epsilon.
Tw direction: flipping $\mathbf n$ flips $\mathrm{Tw}$ and loop orientation together, leaving PBHL invariant.
4π test: $2\pi$ rotation changes class; $4\pi$ returns to identity class.
Units: LHS edge units = RHS flux/twist units after mapping to outputs; the ledger closes.

Diagrams
Use the linked PDF for clean, B/W, vector figures that match this appendix:

Fig A1 — Two-Boundary Belt, Orientation & Normal
Fig A2 — Belt Stokes: Gap = Flux + Twist
Fig A3 — Gluing Two Belts: Interior Edge Cancels
Fig A4 — Twist (Tw) Sign Convention
Fig A5 — Discrete Belts: Cell Orientation & Edge Consistency
Fig A6 — 4π Periodicity (Dirac Belt Trick)
Fig A7 — Dual Traces: Plan ( $\Gamma_+$ ) vs Do ( $\Gamma_-$ )

Download the diagrams (PDF)

Appendix B — Proof Sketches (production notes)

Margin tags like [BHI §x.y] and [AGI-Mini Ch.z] refer to your two attached manuscripts:
BHI = Belt Holonomy Is Inevitable; AGI-Mini = Belt Theory for AGI — A Mini-Textbook.
Use these tags in the page margin for quick traceability in the print layout.

B.0 Setup & Conventions

Belt/worldsheet: an oriented surface $\mathcal B$ with boundary
$\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ (plan vs do). The unit normal is $\mathbf n$ (Appendix A).
Purpose connection: $\mathcal A_\Pi$ on $\mathcal B$ (Lie-algebra valued for the full theory; scalar 1-form in the abelian ops layer).
Curvature: $\mathcal F_\Pi = d\mathcal A_\Pi + \mathcal A_\Pi\wedge \mathcal A_\Pi$ .
Twist (framing): $\mathrm{Tw}$ is the signed rotation of a chosen boundary frame when viewed along $+\mathbf n$ ; coupling $\alpha$ weights framing cost/impact.
Wilson/holonomy: $\mathrm{Hol}(\Gamma)=\mathcal P\exp\!\oint_\Gamma \mathcal A_\Pi$ .
Class functions (e.g., trace, $\arg\det$ ) give gauge-invariant scalars.

B.1 Purpose Belt Holonomy Law (PBHL)

Statement (non-abelian form)

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

Here $W$ parallel-transports surface points to a reference, $\mathcal S$ denotes surface ordering.
Abelian operational form (used for KPIs/ledgers):

\boxed{\quad \oint_{\Gamma_+}\!\mathcal A_\Pi\;-\;\oint_{\Gamma_-}\!\mathcal A_\Pi =\;\iint_{\mathcal B}\!\mathcal F_\Pi\;+\;\alpha\,\mathrm{Tw} \quad}

This is the canonical Gap = Flux + Twist identity.

Proof sketch

Two-boundary Stokes (non-abelian): Apply the non-abelian Stokes theorem on $\mathcal B$ with $\partial\mathcal B=\Gamma_+\cup(-\Gamma_-)$ , producing a relation between the product of edge holonomies and the surface-ordered exponential of curvature. The minus sign on $\Gamma_-$ yields the inverse holonomy.
Margin tags: [BHI §2.1–2.3], [AGI-Mini Ch.12–13].
Framing insertion: The surface proof assumes a reference framing; changing boundary framing by $\Delta\theta$ multiplies the edge operator by a phase (abelian) or by a central element (non-abelian). Aggregating local frame rotations along the boundary yields $\exp(i\,\alpha\,\mathrm{Tw})$ , with $\alpha$ fixed by units and the chosen class function.
Margin tags: [BHI §2.4], [AGI-Mini Ch.13 “twist coupling”].
Abelian reduction: In the operations layer, replace $\mathcal P$ and $\mathcal S$ by ordinary integrals; Stokes gives $\oint_{\partial\mathcal B}\mathcal A_\Pi=\iint_{\mathcal B} d\mathcal A_\Pi$ . Identify $d\mathcal A_\Pi$ with $\mathcal F_\Pi$ , add the $\alpha\,\mathrm{Tw}$ boundary contribution.
Margin tags: [AGI-Mini Ch.13 eq.(1)].
Gauge invariance: Taking a class function (e.g., $\arg\det$ , trace/character) makes both sides gauge-invariant. The twist term is framing-gauge invariant by construction.
Margin tags: [BHI §2.5].

B.2 Two-Boundary Stokes and Gauss-type Identities

Stokes on belts (orientation-aware)

For a smooth 1-form $\mathcal A_\Pi$ ,

\oint_{\Gamma_+}\!\mathcal A_\Pi - \oint_{\Gamma_-}\!\mathcal A_\Pi = \oint_{\partial\mathcal B}\!\mathcal A_\Pi = \iint_{\mathcal B}\! d\mathcal A_\Pi .

Replacing $d\mathcal A_\Pi$ by $\mathcal F_\Pi$ recovers the PBHL without framing.
Key orientation step: $\Gamma_-$ appears with a minus sign in $\partial\mathcal B$ , hence the subtraction.
Margin tags: [BHI §1.3], [AGI-Mini Ch.12].

Gauss-type work/entropy balance

Introduce a purpose current $J_\Pi$ (a constitutive map from context/state to an intensive that “responds” to curvature). The macro work over $\mathcal B$ is

W_{\text{macro}} \;\propto\; \iint_{\mathcal B}\!\langle \mathcal F_\Pi,\; J_\Pi\rangle .

If $J_\Pi = \chi\,\star \mathcal F_\Pi$ (local linear response with $\chi$ a compliance tensor), then

W_{\text{macro}} \;\propto\; \iint_{\mathcal B}\!\|\mathcal F_\Pi\|_\chi^2 \;\ge 0,

establishing non-negativity under passive control. Deviations (active control, twist steps) appear as boundary work via PBHL’s edge gap and the $\alpha\,\mathrm{Tw}$ term.
Margin tags: [AGI-Mini Ch.13 “work–entropy ledger”].

Discrete proof (mesh)

Tiling $\mathcal B=\bigcup_k \square_k$ with CCW faces (Appendix A), Stokes per face gives

\sum_k \oint_{\partial \square_k}\!\mathcal A_\Pi \;=\; \sum_k \iint_{\square_k}\!\mathcal F_\Pi .

All interior edges cancel by opposite orientations; only $\Gamma_+\cup(-\Gamma_-)$ remains. This is the computational backbone for the numerics in Part IV/VIII.
Margin tags: [BHI §1.4], [AGI-Mini Ch.8].

B.3 Framing Quantization & 4π Periodicity

White–Fuller ribbon relation (conceptual reminder)

For a closed ribbon, linking $Lk$ decomposes into twist + writhe: $Lk = \mathrm{Tw} + \mathrm{Wr}$ . Under smooth deformations that keep the ribbon closed and non-self-intersecting, $Lk\in\mathbb Z$ is invariant. Hence $\mathrm{Tw}$ shifts by integers when $\mathrm{Wr}$ changes continuously—observable twist is effectively quantized at the level of topological class.
Operational reading: governance/frame changes accumulate in integer steps when we require global closure and no tearing.
Margin tags: [BHI §2.6].

SU(2) double cover and 4π

The belt’s frame lives in $SO(3)$ , whose universal cover is $SU(2)$ . A $2\pi$ rotation maps to $-\mathbb I\in SU(2)$ , not the identity; a $4\pi$ rotation maps to $\mathbb I$ . Therefore, class functions built from representations (e.g., traces/characters) are $4\pi$ -periodic, not $2\pi$ . In practice:

Numeric holonomy must pass a 4π periodicity test (Appendix A, Fig A6; Ch.9, Ch.43).
Edge-case: if you abelianize too early, you may mistakenly enforce $2\pi$ periodicity—this is a known failure mode.
Margin tags: [AGI-Mini Ch.9, Ch.26], [BHI §2.7].

Twist coupling $\alpha$ and quantization

When class functions are used, the twist contribution enters as $e^{i\,\alpha\,\mathrm{Tw}}$ . To respect 4π periodicity, $\alpha$ must be chosen so that $\alpha\,\mathrm{Tw}$ changes by integer multiples of $2\pi$ when the frame does a full $4\pi$ return in $SO(3)$ . In the abelian ops layer, this reduces to enforcing ledger invariance under allowed frame homotopies.
Margin tags: [AGI-Mini Ch.13 “twist term”].

B.4 Minimal Variational View (gap functional)

Define a gap functional on belts:

\mathcal G[\mathcal A_\Pi,\mathrm{Tw};\mathcal B] =\left(\oint_{\Gamma_+}\!\mathcal A_\Pi-\oint_{\Gamma_-}\!\mathcal A_\Pi\right) -\iint_{\mathcal B}\!\mathcal F_\Pi - \alpha\,\mathrm{Tw}.

Stationarity under compactly supported variations $\delta\mathcal A_\Pi$ on $\mathcal B$ yields $d\mathcal A_\Pi=\mathcal F_\Pi$ (compatibility) and $\delta\mathcal G=0\Rightarrow$ PBHL. Interpreting $\mathcal G=0$ as a closure constraint, controllers in Part V/VI act to drive $\mathcal G\to 0$ by either flux-gating (alter $\mathcal F_\Pi$ ) or twist-stepping (alter $\mathrm{Tw}$ ).
Margin tags: [AGI-Mini Ch.15 “controllers as gap minimizers”].

B.5 Production Checklists & Unit Tests

Sign test: Verify $\oint_{\Gamma_+}-\oint_{\Gamma_-} = \iint_{\mathcal B}\mathcal F_\Pi$ on synthetic fields with zero twist.
Tags: [AGI-Mini Ch.8], [BHI §1.3].
Framing test: Apply a known boundary frame rotation $\Delta\theta$ ; confirm gap shift $=\alpha\,\Delta\theta$ (abelian).
Tags: [BHI §2.4].
Gluing test: Split a belt into tiles; ensure interior edge cancellation to machine epsilon.
Tags: [AGI-Mini Ch.8].
4π test: Rotate the frame by $0$ , $2\pi$ , $4\pi$ ; confirm class-function invariants change only between $0$ and $2\pi$ but return at $4\pi$ .
Tags: [AGI-Mini Ch.26], [BHI §2.7].
Ledger closure: With $J_\Pi=\chi\,\star \mathcal F_\Pi$ , confirm $W_{\text{macro}}\ge 0$ and PBHL closes the budget within tolerance.
Tags: [AGI-Mini Ch.13].

B.6 Common Failure Modes (with fixes)

2π vs 4π confusion: Using abelian scalars for class functions—fix by computing invariants from an $SU(2)$ (or appropriate) representation.
Orientation drift on meshes: Mismatched face ordering—fix by enforcing CCW faces and consistent local edge directions (Appendix A, Fig A5).
Framing leakage: Counting writhe into $\mathrm{Tw}$ in the ledger—fix by separating intrinsic twist from geometric writhe or by using a gauge where writhe is absorbed into the surface term.
Gauge-variant KPIs: Mixing raw holonomies with non-class features—fix by using class functions (trace/character, $\arg\det$ ) or by pinning a base frame.

B.7 Editor’s Margin Tags (style guide)

Use [BHI §x.y] for geometric/field-theoretic steps (holonomy, non-abelian Stokes, framing).
Use [AGI-Mini Ch.z] for operations-layer reductions, KPIs, controllers, and test harnesses.
When a result depends on both (e.g., PBHL with twist), list both tags in ascending order: [BHI §2.3; AGI-Mini Ch.13].

End of Appendix B

Appendix C — Computational Reference (production notes)

This appendix gives “drop-in” numerical recipes for implementing PFBT on belts (twinned traces with an oriented worldsheet). It specifies mesh choices, quadrature rules, error/complexity models, and robust defaults/heuristics for production use.

C.0 Scope & Data Model

Worldsheet (belt): a 2-D oriented surface $\mathcal B$ with boundary $\partial \mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ .
Field data: purpose connection $\mathcal A_\Pi$ (1-form; scalar for the abelian ops layer, matrix-valued for non-abelian); curvature $\mathcal F_\Pi$ (2-form).
Discretization: piecewise polynomial (usually bilinear/biquadratic) over cells; edge traces carry 1-D quadrature nodes, faces carry 2-D nodes.
Outputs: edge gap, surface flux, twist, and ledger terms; plus invariants & tests (4π periodicity, gluing).

C.1 Mesh Recipes (choose one)

R1. Structured Quad Mesh (recommended default)

Topology: $N_x\times N_y$ rectangles mapped from $[0,1]^2\to\mathcal B$ .
Orientation: cells CCW w.r.t. $+\mathbf n$ ; edges inherit consistent directions (Appendix A).
Use when: geometry is ribbon-like or admits a smooth chart; best for reproducibility, caching, SIMD.

R2. Tensor-Product Curvilinear

Topology: same as R1, but node positions follow a smooth mapping that hugs curvature features.
Use when: $\|\nabla \mathcal F_\Pi\|$ varies strongly; enables cell size grading without hanging nodes.

R3. Triangular Mesh (hybrid or unstructured)

Topology: Delaunay or advancing-front; optional quad–tri hybrid for boundary fidelity.
Use when: geometry is irregular; pair with triangle rules (Dunavant/Kronrod–Patterson).

Recipe notes

Keep aspect ratio $< 10{:}1$ for quads unless using anisotropic quadrature (C.3).
Avoid T-junctions in production; if unavoidable, enforce local re-orientation fixups and edge sum tests (C.8).

C.2 Boundary Integrals (edge holonomies)

Abelian ops layer

\oint_{\Gamma}\!\mathcal A_\Pi \;\approx\; \sum_{e\in\Gamma}\;\sum_{i=1}^{q_1} w_i \, \mathcal A_\Pi\!\bigl(\ell_e(\xi_i)\bigr)\,\|\ell_e'(\xi_i)\|,

with $\ell_e:[-1,1]\to e$ a local chart.

Non-abelian

Discretize the ordered exponential via Magnus $\Omega_2$ (stable, second-order) or BCH stepper on sub-segments; cache small-matrix exponentials.
Compose in edge direction; for $\Gamma_-$ flip direction or invert the result.

1-D quadrature defaults (per edge)

Rule	Points $q_1$	Degree exact	Notes
Gauss–Legendre	2	3	Fast, default
Gauss–Legendre	4	7	Use when curvature varies along edge
Simpson	3	3	Useful with uniform node storage
GL Adaptive	2→8	—	Trigger on residuals > tol $_1$

C.3 Surface Flux (face integrals)

Abelian ops layer

\iint_{\mathcal B}\!\mathcal F_\Pi \;\approx\; \sum_{K\in\text{cells}}\;\sum_{i,j} W_{ij}\, \mathcal F_\Pi\!\bigl(\Phi_K(\xi_i,\eta_j)\bigr)\,\bigl|\det D\Phi_K\bigr| .

Non-abelian (surface-ordered)

Use cellwise adjoint transport: pick a face root, transport samples with a precomputed connection estimate $W$ , then apply a surface-ordered $\exp\!\int F$ via Magnus $\Omega_2$ over sub-patches.
In production ops, the abelianized flux is typically sufficient; reserve non-abelian for research or when 4π tests are sensitive to ordering.

2-D quadrature defaults (rectangles)

Rule	Nodes	Degree exact	Notes
Tensor GL $2\times2$	4	3	Default
Tensor GL $3\times3$	9	5	Use for rough fields
Tensor GL $4\times4$	16	7	High accuracy pass

Triangles (if using R3)

Rule	Points	Degree exact
Dunavant-1	1	1
Dunavant-3	3	2
Dunavant-7	7	5

Prefer Dunavant-7 for final runs; fall back to 3-point for previews.

C.4 Discrete Twist (framing)

Represent boundary frame as tangent–normal pair $(\mathbf t,\mathbf f)$ sampled at edge nodes.
Signed twist increment per edge segment: $\Delta\theta = \operatorname{atan2}\!\bigl(\det[\mathbf f_k,\mathbf f_{k+1}],\,\mathbf f_k\!\cdot\!\mathbf f_{k+1}\bigr)$ .
Total $\mathrm{Tw}=\sum \Delta\theta$ along $\Gamma_+$ minus $\Gamma_-$ (respecting orientations).
Coupling: add $\alpha\,\mathrm{Tw}$ to the edge gap per PBHL; $\alpha$ chosen to satisfy 4π invariance tests (Appendix B).

C.5 Error Model & Acceptance

For smooth fields and mesh size $h=\max(h_x,h_y)$ , belt width parameter $w$ (if used), curvature scale $\kappa$ ,

\varepsilon \;\lesssim\; C_1\,h^p \;+\; C_2\,(\kappa w)^2 \;+\; C_3\,\|\nabla_\perp \mathcal F_\Pi\|\,w^2 .

$p$ = min(poly degree+1, quadrature degree).
The $C_2,C_3$ terms capture thickness/modeling and transverse variation penalties.

Practical tolerances

Unit PBHL: $|\text{Gap}-(\text{Flux}+\alpha\,\mathrm{Tw})| \le \tau_{\mathrm{PBHL}}$ with $\tau_{\mathrm{PBHL}}\in[10^{-6},10^{-3}]$ in natural units of the edge integral.
Gluing: interior edge residuals $\le \tau_{\mathrm{glue}}=10^{-10}$ (double).
4π: invariants equal within $\tau_{4\pi}$ (same band as PBHL).

C.6 Complexity & Memory

Let $E$ edges, $F$ faces, $q_1$ 1-D nodes/edge, $q_2$ 2-D nodes/face, $d_R$ rep. dimension (1 for abelian).

Abelian cost:
- Edges: $\mathcal O(E\,q_1)$
- Faces: $\mathcal O(F\,q_2)$
Non-abelian:
- Add matrix exponentials and products: $\mathcal O\bigl(E\,q_1\,d_R^3 + F\,q_2\,d_R^3\bigr)$ (small $d_R$ : fast; cache exp/log).
Memory: store fields at nodes + Jacobians + optional transport $W$ : $\mathcal O(E\,q_1 + F\,q_2)$ .

C.7 Defaults (safe, reproducible)

Mesh: Structured quads, $N_x\times N_y$ with $N_x/N_y\approx$ belt aspect; start $N_x=128, N_y=32$ .
Quadrature: Edges GL-2; Faces tensor GL $3\times3$ .
Twist: central differences for frames; unwrap angles; enforce continuity at joins.
PBHL checks: enable on every run; fail fast if tolerance breaks.
Precision: double (64-bit) throughout; Kahan summation for edge sums in abelian mode.
Non-abelian mode: Magnus $\Omega_2$ stepper; cached expm for $2{\times}2$ or $3{\times}3$ blocks.

C.8 Heuristics & Adaptivity

Edge residual trigger: compare GL-2 vs GL-4; if $|\Delta|>\tau_{\text{edge}}$ , split edge or raise $q_1$ .
Face residual trigger: compare $2\times2$ vs $3\times3$ ; if $|\Delta|>\tau_{\text{face}}$ , refine cell or raise $q_2$ .
Field-aware grading: refine where $\|\nabla \mathcal F_\Pi\|$ is large or where twist increments spike.
Anisotropy: if curvature stripes align with $\hat x$ , prefer $h_x<h_y$ and higher $q_1$ along stripes.
Stability: cap subsegment condition number in Magnus; if $\|\mathcal A_\Pi\|\,\Delta s > \gamma$ , sub-divide (e.g. $\gamma=0.5$ ).
Smoothing: light Laplacian smoothing on frame samples before $\mathrm{Tw}$ to suppress numerical noise (do not smooth $\mathcal F_\Pi$ used for validation).

C.9 Diagnostics & Reproducibility

Orientation audit: draw tiny arrows on a random 2% subset of edges; ensure CCW faces (Appendix A, Fig A5).
Interior cancellation: sum edge contributions per cell; shared edges must cancel to $\le \tau_{\mathrm{glue}}$ .
Parity flips: flipping $\mathbf n$ must flip loop orientations and $\mathrm{Tw}$ sign, leaving PBHL unchanged.
Seeded runs: fix RNG seeds for any adaptive decisions; store mesh & quadrature JSON in the artifact.
Logs: persist (Gap, Flux, Tw, PBHL residual) per belt, plus node counts and timing; include class-function invariants for 4π suite.

C.10 Pseudocode Skeleton (abelian default)

Inputs: belt geometry B, connection samples Aπ(x), frame samples f(x), α, tolerances
Mesh ← StructuredQuads(B, Nx, Ny)
EdgeRule ← GL(2); FaceRule ← TensorGL(3,3)

# Edge integrals on Γ+ and Γ−
I_plus  ← 0
for e in edges(Γ+):
  I_plus += Quad1D(e, EdgeRule, Aπ • dl)

I_minus ← 0
for e in edges(Γ−):
  I_minus += Quad1D(e, EdgeRule, Aπ • dl)

# Surface flux
Flux ← 0
for K in cells(Mesh):
  Flux += Quad2D(K, FaceRule, Fπ : = dAπ or supplied)

# Twist
Tw ← 0
for e in edges(Γ+):
  Tw += SumAngleIncrements(f along e)
for e in edges(Γ−):
  Tw -= SumAngleIncrements(f along e)   # minus due to orientation

Gap ← I_plus - I_minus
PBHL_resid ← | Gap - (Flux + α*Tw) |
assert PBHL_resid ≤ τ_PBHL

C.11 Quick-Select Table

Scenario	Mesh	Edge rule	Face rule	Notes
Production default	Structured quads	GL-2	GL $3{\times}3$	Fast, robust
Sharp curvature bands	Curvilinear quads	GL-4	GL $4{\times}4$	Raise $q$ , anisotropic $h$
Irregular geometry	Triangles	GL-3	Dunavant-7	Hybrid ok
Research (non-abelian)	Any	Magnus Ω2	Magnus on tiles	Cache expm; 4π suite on

End of Appendix C

Appendix D — Exercises & Mini-Projects (production notes)

Thirty self-contained tasks with solutions & unit-tests organized into five tracks. Each item lists Goal, Given, Task, Deliverables, Solution sketch, Unit tests. Use tolerances and defaults from Appendix C (e.g., $\tau_{\mathrm{PBHL}}$ , GL rules, mesh heuristics).

Track I — Foundations: Geometry, PBHL, Invariants (1–6)

1) Belt Orientation Sanity

Goal: Verify $\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ .
Given: Parametric belt $\Phi(u,v)$ , $u\in[0,1]$ (width), $v\in[0,1]$ (length); normal $\mathbf n$ .
Task: Draw arrows; label $\Gamma_\pm$ .
Deliverables: Diagram + short note.
Solution: CCW relative to $+\mathbf n$ ; $\Gamma_-$ arrow reversed.
Unit tests: assert orientation(Γ_plus)==CCW && orientation(Γ_minus)==CW.

2) PBHL (abelian) on a Uniform–Curvature Belt

Goal: Show Gap = Flux for $\alpha=0$ .
Given: $\mathcal A_\Pi=(-y/2)\,dx+(x/2)\,dy\Rightarrow \mathcal F_\Pi=dx\wedge dy$ (unit curvature). Belt area $A$ .
Task: Compute $\oint_{\Gamma_+}\!-\oint_{\Gamma_-}\!$ and $\iint_{\mathcal B}\!\mathcal F_\Pi$ .
Deliverables: One-page derivation.
Solution: Flux $=\iint 1\,dA=A$ . Edge gap equals $A$ . Hence PBHL holds.
Unit tests: abs(Gap-Flux) ≤ τ_PBHL.

3) Framing Term Injection

Goal: Observe twist contribution.
Given: Same as (2) plus boundary frame rotation $\Delta\theta$ along $\Gamma_+$ only, $\alpha$ known.
Task: Recompute gap with framing.
Deliverables: Numeric example with $\Delta\theta=\pi/3$ .
Solution: New gap $=A+\alpha\,\Delta\theta$ .
Unit tests: abs((Gap-Flux)-α*Tw) ≤ τ_PBHL.

4) Gluing Cancels Interior Edge

Goal: Show interior edges vanish when belts glue.
Given: Split belt into two sub-belts along a seam with opposite edge orientations.
Task: Prove $\iint_{\mathcal B_1\cup\mathcal B_2}\mathcal F=\iint_{\mathcal B_1}\mathcal F+\iint_{\mathcal B_2}\mathcal F$ .
Deliverables: Schematic + line integral sum.
Solution: Interior edges cancel term-by-term.
Unit tests: sum_interior_edges ≤ τ_glue.

5) 4π Periodicity Check (Class Function)

Goal: Demonstrate $4\pi$ (not $2\pi$ ) periodicity.
Given: Frame represented in $SU(2)$ (Pauli matrices); invariant via $\chi=\mathrm{tr}(\cdot)$ .
Task: Compute holonomy for rotations $0,2\pi,4\pi$ .
Deliverables: Table of $\chi$ values.
Solution: $2\pi\mapsto -\mathbf I$ (trace flips sign), $4\pi\mapsto \mathbf I$ (trace returns).
Unit tests: chi(0)==chi(4π) and chi(2π)==-chi(0).

6) Gauss-type Work Non-Negativity (Passive)

Goal: Show $W_{\text{macro}}\ge 0$ under $J_\Pi=\chi\,\star\mathcal F_\Pi$ .
Given: Compliance tensor $\chi\succ0$ .
Task: Prove $\iint\langle \mathcal F,\chi\,\star\mathcal F\rangle\ge0$ .
Deliverables: Short proof.
Solution: Positive-definite quadratic form.
Unit tests: Generate random $\mathcal F$ , SPD $\chi$ : W_macro ≥ -1e-12.

Track II — Discretization & Numerics (7–12)

7) Edge Quadrature Convergence

Goal: Compare GL-2 vs GL-4 on $\Gamma_\pm$ .
Given: Analytic $\mathcal A_\Pi$ ; belt length $L$ .
Task: Compute edge integrals with both rules.
Deliverables: Error vs points plot.
Solution: GL-4 reduces error ~ $O(h^4)$ for smooth fields.
Unit tests: err_GL4 < err_GL2.

8) Surface Flux Accuracy Grid

Goal: Show $2\times2$ , $3\times3$ , $4\times4$ tensor GL scaling.
Given: Same field as (2).
Task: Flux under each rule vs analytic $A$ .
Deliverables: Table + acceptance.
Solution: Errors drop with degree; pick $3\times3$ default.
Unit tests: abs(Flux_3x3 - A) < abs(Flux_2x2 - A).

9) Mesh Aspect Sensitivity

Goal: Test error vs aspect ratio.
Given: Quads with AR ∈ {2,5,10,20}.
Task: Keep DOF fixed; vary AR; measure PBHL residual.
Deliverables: Plot residual vs AR.
Solution: Residual rises sharply beyond AR≈10.
Unit tests: resid(AR=20) > 5*resid(AR=5).

10) Interior Edge Cancellation on Tiles

Goal: Validate gluing at machine precision.
Given: 8×2 structured tiles.
Task: Sum oriented edge integrals per tile; verify cancellations.
Deliverables: Residual histogram.
Solution: Mean ~ 0, max ≤ $\tau_{\mathrm{glue}}$ .
Unit tests: max_interior ≤ τ_glue.

11) Adaptive Edge Refinement Trigger

Goal: Implement Δ-test between GL-2 and GL-4.
Given: Threshold $\tau_{\text{edge}}$ .
Task: Subdivide edges where $|\Delta|>\tau$ .
Deliverables: Before/after node counts; PBHL residual change.
Solution: Targeted refinement halves PBHL residual.
Unit tests: resid_after ≤ 0.5*resid_before.

12) Non-abelian Surface Ordering (Magnus Ω₂)

Goal: Compute a small SU(2) flux with ordering.
Given: Piecewise constant $\mathcal F=\sum_k f_k \sigma_k$ on 4 sub-patches.
Task: Magnus Ω₂ per patch; compose with parallel transport.
Deliverables: Code sketch + invariant trace.
Solution: Matches abelian result when $[f_i,f_j]=0$ .
Unit tests: When $f_i$ commute, nonabelian_flux == abelian_flux ± τ.

Track III — Estimation & KPIs (13–18)

13) Estimating $\mathcal A_\Pi$ from Edge Logs

Goal: Reconstruct a scalar 1-form from edge samples.
Given: $(x_k, A_\parallel(x_k))$ along $\Gamma_\pm$ .
Task: Fit local polynomials; extend into belt via Laplace smoothing.
Deliverables: Heatmap of $\hat{\mathcal A}_\Pi$ .
Solution: Regularized least squares on edges + harmonic extension.
Unit tests: MAE_edge ≤ τ_A and PBHL_close(Â,F̂).

14) Curvature from Sparse Events

Goal: Estimate $\mathcal F_\Pi$ from sparse interior sensors.
Given: 20 interior readings $F(x_i)$ + smoothness prior.
Task: Kriging/GMRF interpolation.
Deliverables: $\hat{\mathcal F}$ map + uncertainty.
Solution: Predictor achieves unbiased flux over belt.
Unit tests: abs(∬F̂ - ∬F_true) ≤ 2σ_total.

15) Five-line KPI Computation

Goal: Compute (Gap/Flux/Twist/Coherence/Residual) from logs.
Given: Dual traces, frame samples, coherence proxy $C\in[0,1]$ .
Task: Produce KPI row for a run.
Deliverables: JSON + dashboard mock.
Solution: Residual = Gap − (Flux + αTw).
Unit tests: 0 ≤ C ≤ 1 and |resid| ≤ τ_PBHL.

16) Ledger: Macro Work & Macro Entropy

Goal: Map to domain units.
Given: $J_\Pi$ , units map $\mathsf U:$ field units→“pairs of shoes”.
Task: Compute $W_{\text{macro}}$ , $\Sigma_{\text{macro}}$ per Appendix A.
Deliverables: One KPI example in domain units.
Solution: Inner product + cost dictionary.
Unit tests: Units match; Σ_macro ≥ 0.

17) Twist Coupling Calibration $\alpha$

Goal: Enforce 4π invariance of class-function KPI.
Given: Frame experiments at rotations $0,2\pi,4\pi$ .
Task: Choose $\alpha$ minimizing variance across $0,4\pi$ .
Deliverables: Calibration curve.
Solution: Solve $\min_\alpha \sum_s \|KPI_s(0)-KPI_s(4\pi)\|^2$ .
Unit tests: KPI(0)≈KPI(4π); KPI(2π) differs.

18) Residual-Triggered Inverse Modeling

Goal: When residual $>\tau$ , update $J_\Pi$ or add hidden flux.
Given: High residual runs.
Task: Fit $\Delta J_\Pi$ or latent $\tilde F$ minimizing residual.
Deliverables: Before/after residuals.
Solution: Ridge update; report model delta.
Unit tests: resid_after < resid_before and ||Δ|| bounded.

Track IV — Controllers & Stability (19–24)

19) Flux-Gate Controller (Open-loop)

Goal: Reduce curvature spikes.
Given: Region $R$ with high $|\mathcal F|$ .
Task: Apply gate $u=-k\,\mathcal F$ on $R$ .
Deliverables: Residual vs $k$ .
Solution: Optimal $k$ from line search.
Unit tests: resid(k*) < resid(0).

20) Twist-Stepping (Discrete Policy Steps)

Goal: Close gap via minimal twist.
Given: Candidate steps $\{\Delta\theta_j\}$ .
Task: Pick $\arg\min_j |Gap-(Flux+\alpha(Tw+\Delta\theta_j))|$ .
Deliverables: Chosen step + new KPI.
Solution: Greedy one-step optimality.
Unit tests: |new_resid| ≤ |old_resid|.

21) Fast–Slow Loop (Composite Controller)

Goal: Combine flux-gate (fast) with twist-step (slow).
Given: Two time scales $\tau_f\ll\tau_s$ .
Task: Simulate convergence on drifting $\mathcal F$ .
Deliverables: Residual trace over time.
Solution: Tikhonov separation; stable if gains bounded.
Unit tests: Residual monotone to tolerance band.

22) Coherence (“神”) as Stabilizer

Goal: Show coherence raises robustness margin.
Given: Coherence $C\in[0,1]$ modulates gains: $k\mapsto Ck$ .
Task: Compare stability regions vs $C$ .
Deliverables: Gain–stability plot.
Solution: Higher $C\Rightarrow$ larger stable set.
Unit tests: stable(C_hi) ⊇ stable(C_lo).

23) Controller Goodhart Test

Goal: Detect KPI over-optimization that harms ledger.
Given: Controller tuned on residual only.
Task: Show case where $W_{\text{macro}}\downarrow$ while residual $\downarrow$ .
Deliverables: Counterexample + fix (multi-objective).
Solution: Add work weight: minimize resid + λ·work_loss.
Unit tests: After fix, resid↓ and work↑ or unchanged.

24) Robustness to Hidden Latent Flux

Goal: Controller under unmodeled $\tilde F$ .
Given: Inject small $\tilde F$ in a zone.
Task: Track controller output; add observer to estimate $\tilde F$ .
Deliverables: Residual before/after observer.
Solution: Luenberger-style estimator on flux field.
Unit tests: resid_after < resid_before with bounded control effort.

Track V — Case Studies & Adversarial (25–30)

25) Manufacturing Line (“Pairs of Shoes”)

Goal: End-to-end ledger in domain units.
Given: Logs of plan/do traces, WIP age, changeovers, rework counts.
Task: Compute $W_{\text{macro}}$ and $\Sigma_{\text{macro}}$ ; recommend a twist step.
Deliverables: KPI row + policy note.
Solution: Map units; choose smallest $\Delta\theta$ closing gap.
Unit tests: Ledger balances within tolerance; inventory cost drops in simulation.

26) Prompt-Ops (LLM) Belt

Goal: Detect prompt-injection via flux spike.
Given: Rolling $\mathcal F$ proxy from embedding curvature.
Task: Flag anomalies; apply flux-gate.
Deliverables: Alert thresholds + before/after curves.
Solution: Z-score on $|\mathcal F|$ ; clamp context.
Unit tests: ROC ≥ target; false-positive < budget.

27) Policy Flip Stress (Regulatory Change)

Goal: Evaluate twist-cost under a sudden policy flip.
Given: $\Delta\theta=\pi$ event at $t_0$ .
Task: Quantify impact on residual and ledger.
Deliverables: Post-event KPI; recovery plan.
Solution: Multi-step twist to spread cost; temporary flux gate.
Unit tests: Recovery to baseline in $T$ steps with bounded $\Sigma_{\text{macro}}$ .

28) Non-abelian Mini—Ribbon Robot

Goal: Demonstrate ordering effects.
Given: Two curvature patches with $[F_1,F_2]\neq0$ .
Task: Compare surface orders AB vs BA.
Deliverables: Holonomy difference and class functions.
Solution: Show non-commuting error term via BCH.
Unit tests: invariant_diff(AB,BA) ≥ δ>0.

29) Data Gaps & Reconstruction

Goal: Robust KPIs with missing segments.
Given: 20% missing on $\Gamma_-$ .
Task: Impute via structure (smoothness + symmetry).
Deliverables: KPI with uncertainty bands.
Solution: GP on edge; widen $\tau_{\mathrm{PBHL}}$ via error propagation.
Unit tests: Coverage ≥ 90% on synthetic trials.

30) Red-Team Adversary (Twist Poisoning)

Goal: Detect malicious micro-framing that keeps residual small but derails work.
Given: Sequence of small $\Delta\theta_k$ causing style drift.
Task: Add coherence guard + twist budget.
Deliverables: Detector + block rule.
Solution: Enforce cumulative twist budget; penalize drift from style anchor.
Unit tests: Attack caught within $K$ steps; post-mitigation work restored.

Unit-Test Template (apply per item)

# PBHL residual
assert abs(Gap - (Flux + α*Tw)) ≤ τ_PBHL

# Interior gluing
assert max(abs(sum_interior_edges)) ≤ τ_glue

# 4π periodicity (class function)
assert approx_equal(class_fn(rot=0), class_fn(rot=4π), tol=τ_4π)
assert not approx_equal(class_fn(rot=0), class_fn(rot=2π), tol=τ_4π)

# Ledger
assert Σ_macro ≥ 0
assert units(LHS) == units(RHS)  # after domain mapping

Production note: Keep a small synthetic suite (items 2, 4, 5, 10, 11, 17) as CI smoke tests; run the full 30-item battery nightly or before major releases.

Appendix E — One-Page Quick Start (core equations & steps)

What you’re measuring

Belt/worldsheet: oriented surface $\mathcal B$ with boundary $\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ (plan vs do).
Purpose connection (estimable): $\mathcal A_\Pi$ (1-form).
Curvature (does macro work): $\mathcal F_\Pi=d\mathcal A_\Pi+\mathcal A_\Pi\wedge\mathcal A_\Pi$ .
Framing/twist: $\mathrm{Tw}$ (signed boundary frame rotation; coupling $\alpha$ ).
Purpose current (for work mapping): $J_\Pi$ (constitutive).

Core identities (keep these on your desk)

PBHL (non-abelian):
$\mathrm{Hol}(\Gamma_+)\,\mathrm{Hol}(\Gamma_-)^{-1} = \mathcal S\exp\!\!\iint_{\mathcal B} W^{-1}\mathcal F_\Pi W \cdot e^{\,i\alpha\,\mathrm{Tw}}$
PBHL (operations/abelian layer):
$\boxed{\;\oint_{\Gamma_+}\!\mathcal A_\Pi-\oint_{\Gamma_-}\!\mathcal A_\Pi = \iint_{\mathcal B}\!\mathcal F_\Pi + \alpha\,\mathrm{Tw}\;}$
(“Gap = Flux + Twist”)
Macro work (domain units):
$W_{\text{macro}}\;\propto\;\iint_{\mathcal B}\!\langle \mathcal F_\Pi,\,J_\Pi\rangle \quad\Longrightarrow\quad \text{map via units }\mathsf U \text{ to e.g. “pairs of shoes”.}$
Macro entropy (ledger):
$\Sigma_{\text{macro}} = \text{dispersion} + \text{WIP-age} + \text{rework} + \text{twist-cost}(\alpha\,\mathrm{Tw})$

Defaults (Appendix C compatible)

Mesh: structured quads $N_x\times N_y$ (CCW faces).
Quadrature: edges GL-2; faces tensor GL $3\times3$ .
Twist sign: CCW rotation viewed along $+\mathbf n$ is positive.
Tolerances: $\tau_{\mathrm{PBHL}}\in[10^{-6},10^{-3}]$ ; $\tau_{\mathrm{glue}}=10^{-10}$ ; $\tau_{4\pi}\approx\tau_{\mathrm{PBHL}}$ .
Precision: float64; Kahan summation on edge sums.

10-step runbook (from logs to KPIs & control)

Ingest: plan/do edge logs, any interior probes, boundary frames, units map $\mathsf U$ .
Estimate $\mathcal A_\Pi$ : fit along $\Gamma_\pm$ , extend inside $\mathcal B$ (harmonic/kriging if needed).
Curvature: compute $\mathcal F_\Pi = d\mathcal A_\Pi$ (or estimate directly from sensors).
Edge integrals: $\oint_{\Gamma_\pm}\mathcal A_\Pi$ via GL-2 (raise to GL-4 on residual spikes).
Surface flux: $\iint_{\mathcal B}\mathcal F_\Pi$ via tensor GL $3\times3$ (use $4\times4$ for rough fields).
Twist: sum signed frame increments along $\Gamma_+$ minus $\Gamma_-$ ; compute $\alpha\,\mathrm{Tw}$ .
PBHL residual: $r=\big|\text{Gap}-(\text{Flux}+\alpha\,\mathrm{Tw})\big|$ → assert $r\le\tau_{\mathrm{PBHL}}$ .
Five-line KPI: $(\text{Gap},\text{Flux},\text{Twist},\text{Coherence},\text{Residual})$ .
Ledger: $W_{\text{macro}}$ via $\langle \mathcal F_\Pi,J_\Pi\rangle$ → map with $\mathsf U$ ; compute $\Sigma_{\text{macro}}$ .
Control:
- Flux-gate (fast): act on curvature hotspots to reduce $r$ without raising $\Sigma_{\text{macro}}$ .
- Twist-step (slow): choose minimal $\Delta\theta$ that closes gap; respect 4π periodicity & twist budget.

Acceptance checks (run every time)

Two-boundary Stokes: $\oint_{\Gamma_+}-\oint_{\Gamma_-}\approx\iint_{\mathcal B}d\mathcal A_\Pi$ .
Gluing: interior edges cancel to $\le\tau_{\mathrm{glue}}$ .
4π periodicity: class-function invariants identical at $0$ and $4\pi$ ; differ at $2\pi$ .
Units closure: LHS (edge units) = RHS (flux+twist units) after $\mathsf U$ .
Repro: seeded adaptivity; store mesh/quadrature JSON, KPI row, and $(\text{Gap},\text{Flux},\text{Twist},r)$ .

Minimal pseudocode (abelian layer)

Mesh ← StructuredQuads(B, Nx, Ny)
I_plus  ← Σ_e∈Γ+  Quad1D(e, GL2, Aπ·dl)
I_minus ← Σ_e∈Γ−  Quad1D(e, GL2, Aπ·dl)
Flux    ← Σ_K     Quad2D(K, GL3x3, Fπ)
Tw      ← Σ_e∈Γ+ Δθ(e)  −  Σ_e∈Γ− Δθ(e)
Gap     ← I_plus − I_minus
r       ← | Gap − (Flux + α·Tw) |
assert r ≤ τ_PBHL
W_macro ← ∬ ⟨Fπ, Jπ⟩  → map via 𝕌
Σ_macro ← dispersion + WIP_age + rework + cost(α·Tw)

That’s it: compute Gap, Flux, Twist, check PBHL, report the KPI row, and act with flux-gate (fast) and twist-step (slow) while watching the ledger.

Appendix F — Belt-KPI Templates (per-domain forms / SQL; acceptance bands)

This appendix ships drop-in data forms and SQL to compute the five-line Belt KPI and ledger, plus acceptance bands by domain. All SQL below targets PostgreSQL (12+). Adapt column types as needed.

F.0 Quick recall (metrics)

Five-line KPI: Gap, Flux, Twist, Coherence, Residual, with
Residual = Gap − (Flux + α·Twist).
Ledger: W_macro (macro work in domain units), Σ_macro (macro entropy/costs).
Entropy-Efficiency Index: EEI = W_macro / (W_macro + Σ_macro + 1e-9) in $[0,1]$ .

F.1 Core schema (normalized, minimal)

-- 1) Runs & geometry
CREATE TABLE belt_run (
  run_id           UUID PRIMARY KEY,
  domain           TEXT NOT NULL,      -- e.g., 'manufacturing', 'llm_ops', ...
  belt_name        TEXT NOT NULL,
  alpha_twist      DOUBLE PRECISION NOT NULL, -- α
  n_x              INT, n_y INT,       -- mesh bookkeeping (Appendix C)
  created_at       TIMESTAMPTZ NOT NULL DEFAULT now()
);

-- 2) Edge samples (plan/do traces)
CREATE TABLE belt_edge_log (
  run_id           UUID REFERENCES belt_run(run_id) ON DELETE CASCADE,
  side             TEXT CHECK (side IN ('PLAN','DO')), -- Γ+ vs Γ−
  s_param          DOUBLE PRECISION,                   -- local arc param
  ax_dl            DOUBLE PRECISION,                   -- Aπ · dl contribution
  frame_theta      DOUBLE PRECISION,                   -- boundary frame angle sample
  PRIMARY KEY (run_id, side, s_param)
);

-- 3) Face samples (flux)
CREATE TABLE belt_face_log (
  run_id           UUID REFERENCES belt_run(run_id) ON DELETE CASCADE,
  u_param          DOUBLE PRECISION,
  v_param          DOUBLE PRECISION,
  flux_density     DOUBLE PRECISION,                   -- Fπ sample
  jac_det          DOUBLE PRECISION,                   -- |det DΦ|
  w_uv             DOUBLE PRECISION,                   -- quad weight
  PRIMARY KEY (run_id, u_param, v_param)
);

-- 4) Coherence and costs (ledger pieces)
CREATE TABLE belt_signals (
  run_id           UUID PRIMARY KEY REFERENCES belt_run(run_id) ON DELETE CASCADE,
  coherence_raw    DOUBLE PRECISION,  -- proxy in [0,1] (normalized later)
  dispersion_cost  DOUBLE PRECISION,  -- units consistent with Σ_macro
  wip_age_cost     DOUBLE PRECISION,
  rework_cost      DOUBLE PRECISION,
  twist_cost_coeff DOUBLE PRECISION   -- per-rad (used to compute cost(α·Tw))
);

-- 5) Units & purpose current (domain mapping)
CREATE TABLE belt_units_map (
  run_id           UUID PRIMARY KEY REFERENCES belt_run(run_id) ON DELETE CASCADE,
  unit_label       TEXT,              -- e.g., 'pairs_of_shoes'
  J_scale          DOUBLE PRECISION   -- scalar for ⟨F, J⟩ mapping (simple case)
);

Indexes you’ll likely want:

CREATE INDEX ON belt_edge_log (run_id, side);
CREATE INDEX ON belt_face_log (run_id);

F.2 Derived views (Gap / Flux / Twist / KPI / Ledger)

-- Edge integrals on Γ+ and Γ− (abelian ops layer)
CREATE VIEW v_edge_integrals AS
SELECT run_id,
       SUM(CASE WHEN side='PLAN' THEN ax_dl ELSE 0 END) AS I_plus,
       SUM(CASE WHEN side='DO'   THEN ax_dl ELSE 0 END) AS I_minus
FROM belt_edge_log
GROUP BY run_id;

-- Surface flux
CREATE VIEW v_flux AS
SELECT run_id,
       SUM(flux_density * jac_det * w_uv) AS flux
FROM belt_face_log
GROUP BY run_id;

-- Twist (signed frame increments along each boundary; unwrap in ETL)
-- Here we assume frame_theta already preprocessed to per-segment Δθ and stored in ax_dl-like rows.
-- If not, store a Δtheta column and sum like below.
CREATE VIEW v_twist AS
SELECT run_id,
       SUM(CASE WHEN side='PLAN' THEN frame_theta ELSE 0 END) -
       SUM(CASE WHEN side='DO'   THEN frame_theta ELSE 0 END) AS twist
FROM belt_edge_log
GROUP BY run_id;

-- Five-line KPI + PBHL residual
CREATE VIEW v_kpi AS
SELECT r.run_id,
       e.I_plus - e.I_minus                AS gap,
       f.flux                              AS flux,
       t.twist                             AS twist,
       GREATEST(LEAST(s.coherence_raw,1.0),0.0) AS coherence,
       ABS( (e.I_plus - e.I_minus) - (f.flux + r.alpha_twist * t.twist) ) AS residual
FROM belt_run r
JOIN v_edge_integrals e USING(run_id)
JOIN v_flux f          USING(run_id)
JOIN v_twist t         USING(run_id)
LEFT JOIN belt_signals s USING(run_id);

-- Ledger: macro work & macro entropy, plus EEI
CREATE VIEW v_ledger AS
SELECT k.run_id,
       /* Macro work via simple linear mapping J_scale */
       (f.flux * u.J_scale)                          AS w_macro,
       /* Macro entropy Σ = dispersion + WIP + rework + twist_cost */
       (COALESCE(s.dispersion_cost,0)
      + COALESCE(s.wip_age_cost,0)
      + COALESCE(s.rework_cost,0)
      + COALESCE(s.twist_cost_coeff,0) * (ABS(r.alpha_twist * k.twist)) ) AS sigma_macro,
       /* Efficiency */
       CASE
         WHEN ( (f.flux * u.J_scale) + (COALESCE(s.dispersion_cost,0)
                                       +COALESCE(s.wip_age_cost,0)
                                       +COALESCE(s.rework_cost,0)
                                       +COALESCE(s.twist_cost_coeff,0)*ABS(r.alpha_twist*k.twist)) ) > 0
         THEN (f.flux * u.J_scale)
              / ( (f.flux * u.J_scale)
                + (COALESCE(s.dispersion_cost,0)
                 + COALESCE(s.wip_age_cost,0)
                 + COALESCE(s.rework_cost,0)
                 + COALESCE(s.twist_cost_coeff,0)*ABS(r.alpha_twist*k.twist)) )
         ELSE 0
       END AS eei,
       u.unit_label
FROM v_kpi k
JOIN belt_run r      USING(run_id)
JOIN v_flux f        USING(run_id)
LEFT JOIN belt_signals s USING(run_id)
LEFT JOIN belt_units_map u USING(run_id);

F.3 Acceptance bands (global defaults)

CREATE TABLE kpi_acceptance_band (
  domain           TEXT,                -- 'manufacturing', 'llm_ops', ...
  metric           TEXT,                -- 'residual','coherence','eei','twist_budget'
  band             TEXT,                -- 'green','yellow','red'
  lower_inclusive  DOUBLE PRECISION,
  upper_exclusive  DOUBLE PRECISION,
  PRIMARY KEY (domain, metric, band)
);

-- Defaults (adjust per domain in §F.4)
INSERT INTO kpi_acceptance_band VALUES
-- PBHL residual in natural edge units
('default','residual','green', 0.0,   1e-4),
('default','residual','yellow',1e-4,  1e-3),
('default','residual','red',   1e-3,  1e9),

-- Coherence in [0,1]
('default','coherence','red',    0.0, 0.40),
('default','coherence','yellow', 0.40,0.60),
('default','coherence','green',  0.60,1.01),

-- EEI in [0,1]
('default','eei','red',    0.0, 0.40),
('default','eei','yellow', 0.40,0.60),
('default','eei','green',  0.60,1.01),

-- Twist budget (abs α·Tw per run)
('default','twist_budget','green', 0.0, 0.50),
('default','twist_budget','yellow',0.50, 1.00),
('default','twist_budget','red',   1.00, 1e9);

Band evaluation helper:

-- Evaluate bands for a run
CREATE VIEW v_kpi_bands AS
WITH base AS (
  SELECT k.run_id, r.domain, k.residual, k.coherence, l.eei,
         ABS(r.alpha_twist * k.twist) AS twist_budget
  FROM v_kpi k
  JOIN belt_run r USING(run_id)
  JOIN v_ledger l USING(run_id)
)
SELECT b.run_id, m.metric, a.band
FROM base b
CROSS JOIN LATERAL (VALUES
  ('residual',  b.residual),
  ('coherence', b.coherence),
  ('eei',       b.eei),
  ('twist_budget', b.twist_budget)
) AS m(metric, val)
JOIN LATERAL (
  SELECT band FROM kpi_acceptance_band a
  WHERE (a.domain = (SELECT domain FROM belt_run WHERE run_id=b.run_id)
         OR a.domain='default')
    AND a.metric = m.metric
    AND m.val >= a.lower_inclusive
    AND m.val <  a.upper_exclusive
  ORDER BY (CASE WHEN a.domain='default' THEN 1 ELSE 0 END)  -- prefer domain-specific over default
  LIMIT 1
) a ON TRUE;

F.4 Domain templates (forms, extras, SQL & bands)

F.4.1 Manufacturing (“pairs of shoes”)

Form (ingest)

belt_signals: set dispersion_cost (lot dispersion), wip_age_cost (inventory age × rate), rework_cost, twist_cost_coeff (changeover cost per rad).
belt_units_map: unit_label='pairs_of_shoes', J_scale=<pairs per unit flux>.

Extras

-- Cycle-level twist budget (to prevent micro-flip thrashing)
CREATE VIEW v_mfg_twist_cycle AS
SELECT run_id,
       ABS(alpha_twist * twist) AS twist_budget,
       CASE WHEN ABS(alpha_twist * twist) <= 0.5 THEN 'OK' ELSE 'EXCESS' END AS status
FROM v_kpi JOIN belt_run USING(run_id);

Domain bands (override defaults)

INSERT INTO kpi_acceptance_band VALUES
('manufacturing','residual','green', 0.0, 5e-5),
('manufacturing','residual','yellow',5e-5,5e-4),
('manufacturing','residual','red',   5e-4,1e9),

('manufacturing','eei','green', 0.70,1.01),
('manufacturing','eei','yellow',0.55,0.70),
('manufacturing','eei','red',   0.0, 0.55);

SLO query (line-level daily)

-- % of runs green on residual AND eei
SELECT date_trunc('day', created_at) AS day,
       AVG( (resid.band='green')::int * (eei.band='green')::int ) AS pct_green_both
FROM (
  SELECT r.created_at, b1.band AS resid, b2.band AS eei
  FROM v_kpi_bands b1
  JOIN v_kpi_bands b2 USING(run_id)
  JOIN belt_run r USING(run_id)
  WHERE b1.metric='residual' AND b2.metric='eei' AND r.domain='manufacturing'
) x
GROUP BY 1 ORDER BY 1;

F.4.2 LLM Prompt-Ops (“llm_ops”)

Form (ingest)

coherence_raw: style/voice coherence (0–1) from embeddings.
dispersion_cost: context switch/branch dispersion proxy.
twist_cost_coeff: governance/prompt change cost per rad.
unit_label='resolved_tasks', J_scale=<tasks per unit flux>.

Extras

-- Flux spike detector (prompt injection / drift)
CREATE VIEW v_llm_flux_zscore AS
WITH f AS (SELECT run_id, flux FROM v_flux)
SELECT f.run_id,
       (f.flux - AVG(f.flux) OVER ()) / NULLIF(stddev_pop(f.flux) OVER (),0) AS z_flux
FROM f;

-- Alert
CREATE VIEW v_llm_alert AS
SELECT run_id, z_flux, (z_flux >= 3.0) AS is_alert
FROM v_llm_flux_zscore;

Bands (override)

INSERT INTO kpi_acceptance_band VALUES
('llm_ops','coherence','green', 0.75,1.01),
('llm_ops','coherence','yellow',0.60,0.75),
('llm_ops','coherence','red',   0.0, 0.60),

('llm_ops','residual','green', 0.0, 2e-4),
('llm_ops','residual','yellow',2e-4,1e-3),
('llm_ops','residual','red',   1e-3,1e9);

SLO query

-- % runs with coherence green and no flux alert
SELECT AVG( (c.band='green')::int * (a.is_alert=FALSE)::int ) AS pct_ok
FROM v_kpi_bands c
JOIN belt_run r USING(run_id)
LEFT JOIN v_llm_alert a USING(run_id)
WHERE r.domain='llm_ops' AND c.metric='coherence';

F.4.3 Healthcare / Physiology (“health”)

Form (ingest)

unit_label='adherent_days', J_scale=<adherent_days per unit flux>.
Costs: set wip_age_cost = “missed days * penalty”, rework_cost = “protocol resets”.

Bands

INSERT INTO kpi_acceptance_band VALUES
('health','eei','green', 0.65,1.01),
('health','eei','yellow',0.50,0.65),
('health','eei','red',   0.0, 0.50),

('health','residual','green', 0.0, 1e-4),
('health','residual','yellow',1e-4, 8e-4),
('health','residual','red',   8e-4, 1e9);

Adherence KPI (example)

CREATE VIEW v_health_adherence AS
SELECT l.run_id, l.w_macro AS adherent_days, l.eei, k.residual
FROM v_ledger l JOIN v_kpi k USING(run_id)
JOIN belt_run r USING(run_id)
WHERE r.domain='health';

F.4.4 Markets / Mechanism Design (“markets”)

Form (ingest)

unit_label='filled_orders', J_scale=<fills per unit flux>.
dispersion_cost = order routing dispersion; rework_cost = cancels/replaces; twist_cost_coeff = rule change overhead.

Bands

INSERT INTO kpi_acceptance_band VALUES
('markets','eei','green', 0.60,1.01),
('markets','eei','yellow',0.45,0.60),
('markets','eei','red',   0.0, 0.45),

('markets','residual','green', 0.0, 1e-4),
('markets','residual','yellow',1e-4, 1e-3),
('markets','residual','red',   1e-3, 1e9);

Microstructure stability

-- Rolling residual variance; flag volatile governance cycles
CREATE VIEW v_mkt_resid_vol AS
SELECT run_id,
       var_pop(residual) OVER (ORDER BY created_at
                               ROWS BETWEEN 20 PRECEDING AND CURRENT ROW) AS resid_var
FROM v_kpi JOIN belt_run USING(run_id)
WHERE domain='markets';

F.4.5 Governance / Compliance (“governance”)

Form (ingest)

unit_label='compliant_changes', J_scale=<compliant changes per unit flux>.
Costs emphasize twist_cost_coeff (policy flips), rework_cost (backouts).

Bands

INSERT INTO kpi_acceptance_band VALUES
('governance','twist_budget','green', 0.0, 0.30),
('governance','twist_budget','yellow',0.30,0.60),
('governance','twist_budget','red',   0.60,1e9),

('governance','residual','green', 0.0, 1e-4),
('governance','residual','yellow',1e-4, 8e-4),
('governance','residual','red',   8e-4, 1e9);

Policy window SLO

-- % runs honoring twist budget in the window (no excessive flips)
SELECT date_trunc('week', created_at) AS wk,
       AVG((band='green')::int) AS pct_green_twist
FROM v_kpi_bands b
JOIN belt_run r USING(run_id)
WHERE r.domain='governance' AND b.metric='twist_budget'
GROUP BY 1 ORDER BY 1;

F.5 JSON payloads (API contracts)

// POST /pfbt/run
{
  "run_id": "uuid",
  "domain": "manufacturing",
  "belt_name": "line-7-day-231",
  "alpha_twist": 0.42,
  "edges": {
    "plan":   [{"s":0.00,"ax_dl":0.013,"theta":0.01}, ...],
    "do":     [{"s":0.00,"ax_dl":0.009,"theta":0.00}, ...]
  },
  "faces": [
    {"u":0.125,"v":0.25,"flux_density":1.03,"jac_det":0.98,"w_uv":0.25},
    ...
  ],
  "signals": {
    "coherence_raw": 0.73,
    "dispersion_cost": 12.4,
    "wip_age_cost": 8.1,
    "rework_cost": 1.7,
    "twist_cost_coeff": 3.2
  },
  "units_map": {"unit_label":"pairs_of_shoes","J_scale": 2.5}
}

Response (KPI row, ledger, bands):

{
  "run_id":"uuid",
  "kpi": {"gap":1.847,"flux":1.622,"twist":0.39,"coherence":0.73,"residual":0.061},
  "ledger": {"W_macro":4.055,"Sigma_macro":27.6,"EEI":0.128,"unit":"pairs_of_shoes"},
  "bands": {"residual":"yellow","coherence":"green","eei":"red","twist_budget":"yellow"}
}

F.6 Unit-test snippets (SQL)

-- PBHL closure
SELECT assert_true(residual <= 1e-3, 'PBHL residual too high')
FROM v_kpi WHERE run_id = :run;

-- 4π periodicity (class-function invariant stored elsewhere) — placeholder pattern
-- SELECT assert_equal(inv(rot=0), inv(rot=4π)) ...

-- Ledger non-negativity pieces
SELECT assert_true((SELECT sigma_macro FROM v_ledger WHERE run_id=:run) >= 0, 'Σ < 0');

(Replace assert_* with your testing framework or PL/pgSQL helpers.)

F.7 Dashboard starters

Five-line KPI table: SELECT * FROM v_kpi JOIN v_ledger USING(run_id);
Band compliance heatmap: pivot v_kpi_bands by metric × band.
SLO tiles: use the domain queries in §F.4 to render % green trends.

Production notes

Always store raw edge/face samples; all views are purely derived for auditability.
Enforce seeded adaptivity and persist mesh/quadrature settings alongside runs.
Treat domain bands as policy, version them, and log diff as Twist events (Part VII).

End of Appendix F

Appendix G — Governance & Audit Playbooks (production notes)

This appendix gives you operational SOPs, schemas, and queries for governing Purpose versions, enforcing twist budgets, and running 4π & gluing audits. It plugs directly into the data model from Appendix F and the numerics from Appendix C.

G.0 Scope, Roles, and Guardrails

Scope: any run that computes the five-line KPI (Gap, Flux, Twist, Coherence, Residual) and ledger.
Roles (RACI):

Owner (O): Product/Process Lead (owns Purpose definition $\mathcal A_\Pi$ & units map).
Controller (C): SRE/Ops (executes change windows, monitors bands/alerts).
Auditor (A): Internal Audit/QA (signs off on invariants & evidence packs).
Reviewer (R): Domain expert (reviews domain units, J-scale, costs).

Guardrails: No Purpose change deploys without (i) twist impact estimate, (ii) 4π test pass, (iii) gluing test pass, (iv) PBHL residual within band.

G.1 Purpose Versioning (registry & diffs)

G.1.1 Registry schema

CREATE TABLE purpose_version (
  purpose_id       TEXT,             -- e.g., 'mfg.line7.vsm'
  semver           TEXT,             -- MAJOR.MINOR.PATCH
  version_id       UUID PRIMARY KEY,
  created_at       TIMESTAMPTZ DEFAULT now(),
  author           TEXT,
  alpha_twist      DOUBLE PRECISION, -- α at this version
  units_label      TEXT,             -- domain output unit name
  j_scale_default  DOUBLE PRECISION, -- default J_scale
  coherence_anchor JSONB,            -- reference style/voice, etc.
  spec_uri         TEXT,             -- doc link (design/spec)
  notes            TEXT
);

CREATE TABLE purpose_diff (
  from_version  UUID REFERENCES purpose_version(version_id) ON DELETE CASCADE,
  to_version    UUID REFERENCES purpose_version(version_id) ON DELETE CASCADE,
  diff_kind     TEXT CHECK (diff_kind IN ('PATCH','MINOR','MAJOR')),
  rationale     TEXT,
  expected_tw   DOUBLE PRECISION,      -- predicted Δθ (rad) on Γ+−Γ−
  expected_resid DOUBLE PRECISION,     -- predicted residual change
  reviewer      TEXT,
  reviewer_sig  TEXT,
  created_at    TIMESTAMPTZ DEFAULT now(),
  PRIMARY KEY (from_version, to_version)
);

G.1.2 Bump rules & sign-offs

PATCH: parameter nudges; no semantics change. One reviewer; dry-run PBHL residual within green band.
MINOR: new features or units-map change; twist forecast $|\alpha \Delta\theta|\le 0.5$ . Reviewer + Owner sign-off.
MAJOR: purpose reframe or governance model switch; twist forecast $|\alpha \Delta\theta|>0.5$ or class-function change. Owner + Auditor sign-off, change window required.

G.2 Twist Budgets (policy & enforcement)

G.2.1 Budget policy table

CREATE TABLE twist_budget_policy (
  domain       TEXT,
  window       TEXT,                  -- 'daily','weekly','cycle'
  max_alpha_tw DOUBLE PRECISION,      -- budget on |α·Tw|
  escalation   TEXT,                  -- 'warn','freeze','rollback'
  PRIMARY KEY (domain, window)
);

Defaults (override per domain):

daily: $|\alpha\,\mathrm{Tw}| \le 0.5$ → warn @ 0.4, freeze @ 0.6
weekly: $|\alpha\,\mathrm{Tw}| \le 1.0$ → escalate to review

G.2.2 Enforcement views

-- Accumulate α·Tw per window
CREATE VIEW v_twist_budget_window AS
SELECT r.domain,
       date_trunc('day', r.created_at) AS win,
       SUM(ABS(r.alpha_twist * k.twist)) AS alpha_tw_sum
FROM belt_run r
JOIN v_kpi k USING(run_id)
GROUP BY 1,2;

-- Violations
CREATE VIEW v_twist_violations AS
SELECT w.domain, w.win, w.alpha_tw_sum, p.max_alpha_tw,
       (w.alpha_tw_sum > p.max_alpha_tw) AS over_budget, p.escalation
FROM v_twist_budget_window w
JOIN twist_budget_policy p ON p.domain=w.domain AND p.window='daily';

Ops rule: on over_budget, the Controller enforces a Twist Freeze (no further governance/prompt changes) until a review closes the gap via flux-gates or accepted twist steps with explicit approval.

G.3 Audit Packs: 4π, Gluing, PBHL Closure

Every release produces an Evidence Pack with: inputs, settings, results, and invariant checks. Store URIs in an evidence_vault.

CREATE TABLE evidence_vault (
  run_id           UUID PRIMARY KEY REFERENCES belt_run(run_id) ON DELETE CASCADE,
  mesh_json_uri    TEXT,
  quad_json_uri    TEXT,
  kpi_json_uri     TEXT,
  class_fn_uri     TEXT,   -- invariants (traces/characters)
  audit_report_uri TEXT,
  sha256_bundle    TEXT    -- hash of archived tar/zip
);

G.3.1 4π Audit (class-function periodicity)

Goal: Class-function invariants equal at $0$ and $4\pi$ , differ at $2\pi$ .
Procedure:

Generate three belt runs with boundary frame rotations $\{0,2\pi,4\pi\}$ .
Compute invariant $I=\chi(\mathrm{Hol})$ (e.g., trace in a chosen rep).
Assert $I_{0}\approx I_{4\pi}$ , $I_{2\pi}\not\approx I_{0}$ .

Template query (recorded result check):

CREATE TABLE fourpi_audit (
  run_id     UUID REFERENCES belt_run(run_id) ON DELETE CASCADE,
  angle_tag  TEXT CHECK (angle_tag IN ('0','2π','4π')),
  invariant  DOUBLE PRECISION,
  PRIMARY KEY (run_id, angle_tag)
);

-- Comparison view
CREATE VIEW v_fourpi_check AS
SELECT f0.run_id,
       ABS(f0.invariant - f4.invariant) AS diff_0_4π,
       ABS(f2.invariant - f0.invariant) AS diff_2π_0
FROM fourpi_audit f0
JOIN fourpi_audit f2 ON f2.run_id=f0.run_id AND f2.angle_tag='2π'
JOIN fourpi_audit f4 ON f4.run_id=f0.run_id AND f4.angle_tag='4π'
WHERE f0.angle_tag='0';

Pass criteria: diff_0_4π ≤ τ_4π and diff_2π_0 ≥ 10*τ_4π.

G.3.2 Gluing Audit (interior cancellation)

Goal: Shared interior edges cancel to machine precision.
Procedure:

Tile $\mathcal B$ (≥16 cells).
Sum oriented edge integrals per cell; aggregate interior edges with opposite orientation.
Histogram residuals; max ≤ τ_glue (Appendix C).

Template (pseudo-view):

-- Assuming you persist per-edge oriented contributions with cell IDs
-- SELECT cell_id, edge_id, signed_contrib FROM tile_edge_sum ...
-- Then aggregate by shared edges:
-- SELECT edge_key, SUM(signed_contrib) AS resid FROM ... GROUP BY edge_key;

G.3.3 PBHL Closure Audit

Goal: $|\text{Gap}-(\text{Flux}+\alpha \mathrm{Tw})|\le\tau_{\mathrm{PBHL}}$ .
View: already in v_kpi.residual.
Policy: Red residual blocks release; Yellow requires Owner sign-off with a follow-up ticket.

G.4 Change Playbooks (CR → Deploy → Verify)

G.4.1 Change Request (CR)

Inputs: proposed new purpose_version, impact analysis: predicted $\Delta \mathrm{Tw}$ , PBHL residual deltas, domain KPI impact.
Checklist:
- Diff kind (PATCH/MINOR/MAJOR) and signatories (per G.1.2).
- Twist forecast within window budget (G.2).
- 4π suite PASS, Gluing PASS, PBHL in green.
- Backout plan and restore point tagged.

G.4.2 Change Window (execution)

Freeze non-essential twist sources.
Enable dual logging (old & new purpose in shadow) for at least N runs.
Activate flux-gates (fast loop) to protect PBHL during rollout.

G.4.3 Post-Deploy Verification

Compare KPI deltas pre/post; EEI must not degrade beyond tolerance.
Close CR only after evidence_vault is sealed (bundle hashed) and Auditor sign-off.

G.5 Incident Response & Rollback

Triggers:

residual red; 4π fail; gluing fail; twist over-budget.

Immediate actions (Controller):

Twist Freeze: block governance/prompt edits.
Flux-Gate High: increase damping on curvature spikes.
Rollback to last green purpose_version (automated if possible).
RCA ticket with:
- What changed (diff & twist increments)?
- Where PBHL failed (edges/faces)?
- Which invariants failed (provide plots)?
- Plan to re-enable with calibrated $\alpha$ or mesh/quad fix.

G.6 Evidence & Traceability (what to archive)

Bundle contents per release:

Purpose spec & version registry entry (G.1).
Mesh & quadrature JSON.
KPI row(s) and ledger.
Class-function invariants (for 4π).
Gluing residual histogram.
PBHL residual summary and thresholds.
Twist budget window report.
Sign-offs (CR, approvals), plus sha256 of the whole pack stored in evidence_vault.

Retention: ≥ 1 year or per regulatory policy.

G.7 Reporting Queries (ready-to-use)

Release gate:

-- All must be green to pass
SELECT
  (SELECT COUNT(*)=0 FROM v_twist_violations WHERE over_budget) AS twist_ok,
  (SELECT COUNT(*)=0 FROM v_kpi_bands WHERE metric='residual' AND band='red') AS pbhl_ok,
  (SELECT COUNT(*)=0 FROM v_fourpi_check WHERE diff_0_4π > 1e-4 OR diff_2π_0 < 1e-3) AS fourpi_ok;

Audit dashboard (last 30 days):

SELECT
  date_trunc('day', r.created_at) AS day,
  AVG((b.band='green')::int) FILTER (WHERE b.metric='residual')   AS pct_pbhl_green,
  AVG((b.band='green')::int) FILTER (WHERE b.metric='eei')         AS pct_eei_green,
  AVG((NOT v.over_budget)::int)                                    AS pct_twist_within_budget
FROM belt_run r
JOIN v_kpi_bands b USING(run_id)
LEFT JOIN v_twist_violations v ON v.win=date_trunc('day', r.created_at) AND v.domain=r.domain
WHERE r.created_at > now() - interval '30 days'
GROUP BY 1 ORDER BY 1;

Change window scorecard (per purpose_id):

SELECT pv.purpose_id, pv.semver,
       COUNT(*) FILTER (WHERE k.residual <= 1e-4) AS runs_pbhl_green,
       AVG(l.eei) AS avg_eei,
       SUM(ABS(r.alpha_twist*k.twist)) AS total_alpha_tw
FROM purpose_version pv
JOIN belt_run r       ON r.domain = split_part(pv.purpose_id, '.', 1)
JOIN v_kpi k          USING(run_id)
JOIN v_ledger l       USING(run_id)
WHERE r.created_at BETWEEN :win_start AND :win_end
GROUP BY 1,2 ORDER BY 1,2;

G.8 Operational Checklists (print & pin)

Pre-Change (Owner/Reviewer)

Purpose diff classified; sign-offs collected.
Twist forecast within window budget.
4π & Gluing PASS on staging data.
PBHL residual green on smoke suite.

During Change (Controller)

Twist Freeze on non-essential edits.
Dual logging enabled; evidence paths recorded.
Flux-gates active; alarms hooked to on-call.

Post-Change (Auditor)

KPI/EEI non-regression verified.
Twist budget report attached.
Evidence vault bundle hashed & filed.
CR closed; version registry bumped.

Notes & Tips

α calibration lives with the purpose_version; any α change is at least a MINOR bump (re-run 4π).
Treat coherence anchor as a governance artifact; drifting anchors imply silent twist—budget it.
When in doubt, prefer flux-gates (fast, reversible) over twist (sticky, budgeted).

End of Appendix G

Appendix H — Ontology Mapping (enterprise schemas • Palantir mapping • pipelines)

This appendix shows how to land PFBT into a real enterprise stack: canonical schemas, a Palantir Ontology mapping, and end-to-end pipelines (stream + batch) that compute Belt KPIs, ledgers, and audits with full lineage and governance (ties to Appendices C, F, G).

H.0 Overview (what maps to what)

PFBT concept	Enterprise object	Storage layer	Compute	Governance
Belt / worldsheet ( $\mathcal B$ )	`belt_run` (a single evaluation)	Lake/Lakehouse table, partitioned by `created_at`	SQL/PySpark	Evidence pack (G.3)
Plan vs Do traces ( $\Gamma_\pm$ )	`belt_edge_log`	Append-only edge events	SQL aggregations	PBHL residual gate
Flux / curvature ( $\mathcal F_\Pi$ )	`belt_face_log` (or derived from sensors)	Grid/feature table	2-D quadrature (C.3)	Mesh/quad JSON archived
Purpose connection ( $\mathcal A_\Pi$ )	`purpose_version` + transforms	Registry + repo	Estimators (Part IV)	Versioning (G.1)
Twist / framing (Tw)	`belt_edge_log.frame_theta`	Edge features	Angle-unwrapping + sums	Twist budgets (G.2)
Five-line KPI	`v_kpi` view/materialized	KPI mart	SQL/PySpark	Bands table (F.3)
Ledger (work/entropy)	`v_ledger` view/materialized	Finance-ready mart	SQL/PySpark	Audit pack
Invariants (4π, gluing)	`fourpi_audit`, tile residuals	Test datasets	Unit tests	Release gates (G.7)

H.1 Canonical enterprise schemas (cross-system)

H.1.1 Core tables (normalized)

Use the Appendix F schema as the source of truth: belt_run, belt_edge_log, belt_face_log, belt_signals, belt_units_map, plus views v_kpi, v_ledger, v_kpi_bands. Add enterprise fields:

ALTER TABLE belt_run
  ADD COLUMN tenant_id TEXT,
  ADD COLUMN system_env TEXT CHECK (system_env IN ('dev','stg','prod')),
  ADD COLUMN source_system TEXT,        -- e.g., 'SAP-MES', 'LLM-Gateway'
  ADD COLUMN purpose_version_id UUID;   -- FK to purpose_version (Appendix G)

H.1.2 ERP/MES crosswalk (example)

ERP/MES table	PFBT landing	Notes
`WORK_ORDER`, `ROUTING_STEP`	`belt_run` (one run per line/time window)	Carry `work_center`, `sku`, `shift` to `belt_run`
`WIP_TRANSACTION`	`belt_signals.wip_age_cost`	Convert inventory-age × rate
`CHANGEOVER_LOG`	`belt_signals.twist_cost_coeff`	Per-rad changeover cost
`REWORK_RECORD`	`belt_signals.rework_cost`	Same units as $\Sigma_{\text{macro}}$
`BOM`, `UOM`	`belt_units_map`	Define `unit_label` & `J_scale`

H.1.3 ITSM/LLM-Ops crosswalk

Ops source	PFBT landing	Notes
Prompt version registry	`purpose_version`	Treat prompt/governance as Purpose
Response embeddings	`belt_signals.coherence_raw`	Normalize to [0,1]
Guardrail/policy flips	Tw events	Sum into `frame_theta` along edges

H.2 Palantir Foundry: Ontology mapping

H.2.1 Object Types (core)

{
  "objectTypes": [
    {
      "name": "BeltRun",
      "key": "run_id",
      "properties": {
        "domain": "String",
        "beltName": "String",
        "alphaTwist": "Double",
        "createdAt": "Datetime",
        "tenantId": "String",
        "env": "String"
      }
    },
    {
      "name": "PurposeVersion",
      "key": "version_id",
      "properties": {
        "purposeId": "String",
        "semver": "String",
        "alphaTwist": "Double",
        "unitsLabel": "String",
        "jScaleDefault": "Double",
        "specUri": "String"
      }
    },
    {
      "name": "KPIRecord",
      "key": "kpi_id",
      "properties": {
        "gap": "Double", "flux": "Double", "twist": "Double",
        "coherence": "Double", "residual": "Double"
      }
    },
    {
      "name": "LedgerRecord",
      "key": "ledger_id",
      "properties": { "wMacro": "Double", "sigmaMacro": "Double", "eei": "Double", "unit": "String" }
    },
    {
      "name": "EvidencePack",
      "key": "run_id",
      "properties": { "meshJsonUri": "String", "quadJsonUri": "String", "kpiJsonUri": "String",
        "classFnUri": "String", "auditReportUri": "String", "sha256Bundle": "String" }
    }
  ]
}

H.2.2 Links (relationships)

{
  "linkTypes": [
    { "name": "USES_PURPOSE", "fromType": "BeltRun", "toType": "PurposeVersion" },
    { "name": "GENERATES_KPI", "fromType": "BeltRun", "toType": "KPIRecord" },
    { "name": "GENERATES_LEDGER", "fromType": "BeltRun", "toType": "LedgerRecord" },
    { "name": "HAS_EVIDENCE", "fromType": "BeltRun", "toType": "EvidencePack" },
    { "name": "SUCCESSOR_OF", "fromType": "PurposeVersion", "toType": "PurposeVersion" }
  ]
}

Notes

Use Object Sets for domains (manufacturing, llm_ops, …).
Add Access Policies (RLS) on tenantId, env.
Expose Actions on BeltRun: recomputeKPI, calibrateAlpha, freezeTwist, openChangeRequest (invoke Code Repos transforms or Foundry Workflows).

H.2.3 Dataset binding

Bind the canonical tables and views to Ontology objects:

belt_run → BeltRun (object dataset).
purpose_version → PurposeVersion.
v_kpi (or materialization) → KPIRecord (link on run_id).
v_ledger → LedgerRecord (link on run_id).
evidence_vault → **EvidencePack`.

H.3 Pipelines (Foundry Workflow / generic DAG)

H.3.1 DAG (high level)

[Raw Ingest: plan/do events, sensors, policy flips]
           │
           ▼
[Normalize & Align] ──▶ [Edge Builder Γ±] ──▶ [Twist Calculator]
           │                              
           └──────────▶ [Face Builder (flux)] ─▶ [Flux Integrator]
                                                   │
                                                   ▼
                                       [KPI & PBHL Residual]
                                                   │
                              ┌────────────────────┴───────────────────┐
                              ▼                                        ▼
                       [Ledger Computation]                    [Invariants Suite]
                              │                                  (4π, gluing)
                              ▼                                        │
                        [Bands & Alerts]                      [Audit Evidence Pack]
                              │                                        │
                              └──────────▶ [Ontology Sync (objects+links)]

H.3.2 Transform specs (typical Foundry blocks)

Ingest (stream/batch): Code Workbook (PySpark) consuming Kafka/CDC; write to raw.plan_do, raw.sensors, raw.policy.
Normalize & Align: dedupe, time-sync, unit normalization; write stg.edge_samples, stg.face_samples.
Edge Builder: compute per-edge ax_dl and frame_theta deltas (unwrap angles); write belt_edge_log.
Face Builder: grid/param map + Jacobians; write belt_face_log.
Flux Integrator: apply quadrature (defaults C.3); write v_flux materialization.
Twist Calculator: sum signed Δθ along $\Gamma_+$ minus $\Gamma_-$ ; write v_twist.
KPI Compute: join v_edge_integrals, v_flux, v_twist, belt_signals; write v_kpi materialization.
Ledger Compute: join v_kpi + belt_units_map + costs; write v_ledger.
Invariants: 4π suite + gluing histogram; write fourpi_audit, tile_residuals.
Evidence Pack: bundle JSON/plots → object store; write evidence_vault with SHA-256.
Ontology Sync: push updates to Ontology objects & links (Foundry’s Ontology APIs).

H.3.3 Scheduling & partitioning

Streaming: low-latency updates for v_kpi + alerts (prompt-ops).
Daily batch: manufacturing; partition by created_at::date.
Idempotency: use (run_id, side, s_param) and (run_id, u_param, v_param) natural keys; MERGE on upserts.
Backfills: run by date partitions; record backfill_job_id in belt_run.

H.3.4 Quality & lineage

Tests: embed Appendix D smoke tests (items 2,4,5,10,11,17) as Workflow assertions.
Lineage: enable dataset provenance; tag code SHA & Purpose semver into outputs.
Quarantine: if PBHL residual red → route inputs to err.quarantine with reason.

H.4 Domain adapters (examples)

H.4.1 Manufacturing adapter (SAP/MES)

Adapter joins WORK_ORDER, ROUTING_STEP, CHANGEOVER_LOG into belt_run + belt_signals.
Units map: compute J_scale = pairs_per_flux_unit from historical calibration runs.
Acceptance: domain bands override defaults (Appendix F §F.4.1).

H.4.2 LLM-Ops adapter

Build coherence_raw from embedding similarity to an anchor (Appendix G note).
Purpose = prompt/governance bundle (versioned); Tw increments from prompt deltas.
Flux proxy from curvature of embedding trajectories across turns.

H.4.3 Markets adapter

Flux from order-book microstructure features; Tw from rule changes (margin, tick size).
EEI bands per §F.4.4; add rolling residual variance monitor.

H.5 APIs (thin layer)

H.5.1 Read KPI/ledger (per run)

GET /pfbt/runs/{run_id}/kpi
→ { gap, flux, twist, coherence, residual, w_macro, sigma_macro, eei, unit, bands }

H.5.2 Trigger recompute / alpha calibration

POST /pfbt/runs/{run_id}/actions/recompute-kpi
POST /pfbt/purpose/{version_id}/actions/calibrate-alpha

H.5.3 Evidence pack fetch

GET /pfbt/runs/{run_id}/evidence
→ { meshJsonUri, quadJsonUri, classFnUri, kpiJsonUri, auditReportUri, sha256Bundle }

H.6 Security & multi-tenant

Row-level security by tenant_id; tag datasets and Ontology Objects.
Environment segregation (dev/stg/prod) with promotion pipelines.
Secrets for source connectors and evidence storage; never embed in units map.

H.7 Quick-start checklist (print & pin)

Schemas deployed (Appendix F + enterprise fields).
Ontology object/link types created and bound to datasets.
Pipelines wired: ingest → normalize → edge/face → KPI → ledger → invariants → evidence → ontology sync.
Bands and twist budgets loaded per domain.
Tests on: PBHL closure, gluing, 4π periodicity (green).
Dashboards: KPI table, bands heatmap, twist budget window, 4π/gluing reports.
Change playbooks (Appendix G) integrated into CR workflow (freeze/rollback hooks ready).

H.8 Minimal Foundry “Actions” (suggested)

BeltRun.recomputeKPI → re-run KPI & ledger transforms; write new EvidencePack; update Ontology links.
PurposeVersion.calibrateAlpha → run 0/2π/4π suite; fit $\alpha$ ; MINOR bump on success; push to registry.
BeltRun.freezeTwist → toggle policy in twist_budget_policy and notify controllers.
PurposeVersion.openChangeRequest → generate CR object from diff; pre-fill twist forecast & required sign-offs.

Result: PFBT becomes a first-class semantic-operational layer in your enterprise: the Ontology anchors Purpose and Belts; the pipelines compute invariants and ledgers with audit-grade lineage; and governance enforces twist budgets and release gates.

Appendix I — Datasets & Reproducibility (production notes)

Everything you need to ship, regenerate, and audit PFBT results: synthetic & anonymized logs, configs, seeds, manifests.

Download the ready-to-use bundle

Folder with CSVs
ZIP archive

Contents (all reproducible; schema matches Appendix F):

belt_run.csv — per-run metadata (domain, α, mesh, timestamp).
belt_edge_log.csv — edge samples for Γ⁺=PLAN and Γ⁻=DO (ax_dl, frame_theta as Δθ per segment).
belt_face_log.csv — face samples for flux integration (flux_density, jac_det, w_uv).
belt_signals.csv — coherence & cost inputs (ledger pieces).
belt_units_map.csv — units mapping (e.g., pairs_of_shoes) & J_scale.
fourpi_audit.csv — invariant traces for 0 / 2π / 4π rotations (toy SU(2) trace).
pfbt_config.yaml — mesh, quadrature, tolerances, run specs.
seeds.json — global seed + per-run deterministic seeds.
manifest.sha256 — hashes for integrity; pin in evidence packs (Appendix G).
README_REPRO.txt — one-page instructions to load and verify PBHL & 4π.

I.1 Repro recipe (do this every time)

Verify hashes

shasum -a 256 -c manifest.sha256

Load CSVs into the Appendix-F schema (or Ontology bindings from Appendix H).
Materialize views: v_edge_integrals, v_flux, v_twist, v_kpi, v_ledger.
Acceptance checks

PBHL: ABS(Gap - (Flux + α·Twist)) ≤ tau_PBHL from pfbt_config.yaml.
4π: invariants at 0 and 4π match, 2π differs.
Gluing (if tiled datasets provided): interior edge residuals ≤ tau_glue.

Archive evidence: store CSVs + config + seeds + manifest in your evidence_vault (Appendix G).

I.2 Synthetic data generator (what the bundle encodes)

Deterministic seeds: global seed 31415926 + per-run seeds derived from run names.
Three smoke runs (manufacturing domain):
- angle_tag 0: Tw = 0, Flux = 1.800
- angle_tag 2π: Tw = 2π, Flux = 1.620
- angle_tag 4π: Tw = 4π, Flux = 1.620
PBHL closure is baked in: we choose $I_+$ and set $I_-$ so
Gap = Flux + α·Tw + ε with tiny ε ∈ [1e-5, 2e-5] (within tau_PBHL=1e-4).
Edge logs: 100 points/side; ax_dl positive chunks that sum to $I_+$ / $I_-$ .
Twist: frame_theta stores per-segment Δθ, summing to Tw = Σ(PLAN) − Σ(DO).
Face logs: 4×4 grid; weights sum to 1, so surface integral = flux_total.
Four-π audit: invariant_trace set to {0: +2, 2π: −2, 4π: +2} to demonstrate periodicity.

I.3 Anonymization playbook (for real logs)

When you move from synthetic to anonymized production data, apply this privacy + fidelity sequence before landing in Appendix-F tables:

ID hashing & tokenization

Hash run_id, belt_name, and any PII keys with salt per tenant (HMAC-SHA256(tenant_salt, key)).

Quantization

Quantize s_param, u_param, v_param to fixed grids (e.g., 1e-3) to remove timing micro-signatures.

Noise (DP-style)

Add zero-mean noise to ax_dl and flux_density at ≤ 0.25·tau_PBHL scale (keeps PBHL checks valid).
Clip to preserve signs for edge contributions.

Swap & jitter

Randomly swap 5–10% of adjacent samples within a small window; add timestamp jitter (±1–3s) if timestamps are present (not in the Appendix-F minimal schema).

k-Anonymity buckets

Aggregate per-run cost fields (dispersion_cost, wip_age_cost, rework_cost) into coarse buckets when cardinalities are small.

Lineage

Emit an anonymization report (JSON) per batch: method, parameters, seeds, pre/post stats; attach to evidence_vault.

Guardrail: After anonymization, re-run PBHL & 4π smoke tests. If PBHL residual > yellow, reduce noise or increase mesh/quad fidelity.

I.4 Config discipline (what must be pinned)

Pin all of the following in every evidence pack:

pfbt_config.yaml (mesh Nx,Ny, quadrature rules, tolerances).
Seeds: global_seed + per-run seeds; if adaptivity is used, seed the splitter too.
Purpose version + α (Appendix G): record purpose_version.version_id and alpha_twist.
Units map: domain unit_label, J_scale at time of run.
Code SHAs: hash of transforms/notebooks used to generate edge/face logs and KPIs.

I.5 Re-run determinism (CI gates)

Disallow non-deterministic ops in the numeric path (no parallel reductions without Kahan; no random sampling without fixed seeds).
Idempotent ingests: upserts by (run_id, side, s_param) and (run_id, u_param, v_param).
Seed registry: keep a table seed_registry(run_id, stage, seed, created_at); log reads/writes.
Golden tests: include Appendix-D smoke items (2, 4, 5, 10, 11, 17) in CI; fail the build on any drift.

I.6 Example loader queries (sanity)

-- Gap / Flux / Twist / Residual
SELECT * FROM v_kpi WHERE run_id IN (SELECT run_id FROM belt_run ORDER BY created_at LIMIT 3);

-- 4π audit checks
SELECT * FROM fourpi_audit;
-- Expect invariant(0) ≈ invariant(4π), invariant(2π) different.

-- Ledger in domain units
SELECT unit_label, w_macro, sigma_macro, eei
FROM v_ledger WHERE run_id IN (SELECT run_id FROM belt_run ORDER BY created_at LIMIT 3);

I.7 Repro checklist (print & pin)

Hashes OK (manifest.sha256).
Schema loaded (Appendix F), configs & seeds pinned.
PBHL green, 4π pass.
Evidence bundle archived with SHA-256 (Appendix G).
If anonymized: privacy report attached, post-anonymization PBHL/4π re-pass.

If you want extra synthetic runs (e.g., noisy belts, tiled gluing cases, non-abelian toys), say the word and I’ll mint an expanded pack in the same format.

Appendix J — Glossary & Symbol Index (cross-domain)

J.1 Master Glossary (A→Z)

Abelian (ops layer) — The practical scalar version of the theory where ordering is ignored. Lets you compute Gap = Flux + α·Twist with standard 1D/2D quadrature. See Non-abelian for the full geometric form.

Belt / worldsheet ( $\mathcal B$ ) — The oriented surface whose boundary carries the two traces: plan vs do. Everything—flux, twist, gap—accumulates over this belt.

Belt gluing — Joining belts along an interior edge; contributions on the shared edge cancel when orientations oppose. Used for modular pipelines and tiling meshes.

Belt trick (4π periodicity) — A $2\pi$ frame rotation is not homotopic to identity on a belt; $4\pi$ is. This yields the 4π audit.

Class function (invariant) — A function of holonomy that’s gauge-invariant (e.g., trace, character, $\arg\det$ ). Used in 4π tests and release gates.

Coherence (“神”) — A [0,1] stabilizer indicating how phase-aligned plan and do are (voice/style/process coherence). High coherence widens controller stability margins.

Compliance $\chi$ — Constitutive gain from curvature to purpose current. If $J_\Pi=\chi\,\star\mathcal F_\Pi$ , passive macro-work is non-negative.

Controller: flux-gate — Continuous action that suppresses curvature hotspots to shrink residual without changing framing.

Controller: twist-step — Discrete governance/prompt/policy change that adjusts the boundary framing; budgeted and audited.

Curvature $\mathcal F_\Pi$ — The “field strength” of Purpose; the surface density whose flux can do macro work. In abelian ops, $\mathcal F_\Pi=d\mathcal A_\Pi$ .

Edge gap (Gap) — The difference of edge integrals $\oint_{\Gamma_+}\mathcal A_\Pi - \oint_{\Gamma_-}\mathcal A_\Pi$ . Equals Flux + α·Twist by PBHL.

EEI (Entropy-Efficiency Index) — $\mathrm{EEI} = W_{\text{macro}} / (W_{\text{macro}}+\Sigma_{\text{macro}}+10^{-9}) \in [0,1]$ . Higher is better.

Evidence pack — The archived bundle (data, configs, seeds, invariants, hashes) proving a run’s numbers.

Framing / Twist (Tw) — Signed rotation of a chosen boundary frame (governance steps, prompt changes, policy flips). Positive = CCW viewed along $+\mathbf n$ .

Gluing audit — Unit test that interior edges cancel to machine precision on tiled belts.

Holonomy $\mathrm{Hol}$ — The path-ordered exponential of $\mathcal A_\Pi$ along a boundary; measures total “turn” imposed by Purpose along the trace.

Macro entropy $\Sigma_{\text{macro}}$ — Costs in the ledger: dispersion, WIP-age, rework, twist-cost. Increases with misalignment or excessive framing changes.

Macro work $W_{\text{macro}}$ — Domain-unit output driven by curvature interacting with the purpose current: $\iint \langle \mathcal F_\Pi, J_\Pi\rangle$ .

Magnus $\Omega_2$ — A stable second-order integrator for matrix exponentials used in non-abelian edge/surface ordering.

Mesh — Discretization of the belt into cells (quads/triangles) with CCW face orientation; drives numerical accuracy and gluing.

Non-abelian — Full geometric version with surface/path ordering: $\mathrm{Hol}(\Gamma_+)\mathrm{Hol}(\Gamma_-)^{-1}=\mathcal S\exp\iint W^{-1}\mathcal F_\Pi W\cdot e^{i\alpha\,\mathrm{Tw}}$ .

PBHL (Purpose Belt Holonomy Law) — The central identity: Gap = Flux + Twist (with α coupling). Operates as a conservation law and acceptance check.

Purpose / 志 ( $\mathcal A_\Pi$ ) — The estimable connection field encoding “what we are trying to do.” Its curvature does macro work.

Purpose current $J_\Pi$ — The response field mapping context to how curvature converts to output units.

Residual — $|\text{Gap} - (\text{Flux}+\alpha\cdot\mathrm{Tw})|$ . The PBHL closure error; must be within tolerance.

Stokes (two-boundary) — Orientation-aware theorem giving the edge gap as a surface integral (plus framing term).

Twist budget — Policy that caps $|\alpha\cdot\mathrm{Tw}|$ per window (daily/weekly/cycle) to prevent governance thrash.

Units map $\mathsf U$ — The calibration from field/flux units to domain outputs (e.g., “pairs of shoes”, “resolved tasks”).

J.2 Symbol Index (grouped)

Geometry & Orientation

$\mathcal B$ — Belt/worldsheet (oriented surface).
$\partial\mathcal B=\Gamma_+\sqcup(-\Gamma_-)$ — Boundary: plan (forward) vs do (reversed).
$\Gamma_+$ , $\Gamma_-$ — Upper/lower traces.
$\mathbf n$ — Unit normal; CCW is positive relative to $+\mathbf n$ .
$e, K$ — Edge, cell (mesh elements).
$\ell_e, \Phi_K$ — 1D/2D charts for edges/faces.

Fields & Operators

$\mathcal A_\Pi$ — Purpose connection (1-form).
$\mathcal F_\Pi=d\mathcal A_\Pi+\mathcal A_\Pi\wedge\mathcal A_\Pi$ — Curvature (2-form).
$d,\wedge$ — Exterior derivative, wedge product.
$\mathcal P,\mathcal S$ — Path/surface ordering symbols.
$W$ — Parallel transport operator on the surface.

Integrals & Invariants

$\mathrm{Hol}(\Gamma)=\mathcal P\exp\oint_\Gamma \mathcal A_\Pi$ — Edge holonomy.
$\mathrm{Tw}$ — Total boundary twist (radians).
$\alpha$ — Twist coupling (units chosen so Flux and $\alpha\mathrm{Tw}$ match edge units).
PBHL (abelian): $\oint_{\Gamma_+}\mathcal A_\Pi - \oint_{\Gamma_-}\mathcal A_\Pi = \iint_{\mathcal B}\mathcal F_\Pi + \alpha\,\mathrm{Tw}$ .
Gap — LHS of PBHL.
Flux — $\iint_{\mathcal B}\mathcal F_\Pi$ .
Residual — PBHL closure error $r$ .

Ledger & KPIs

$J_\Pi$ — Purpose current.
$W_{\text{macro}} \propto \iint \langle \mathcal F_\Pi, J_\Pi\rangle$ — Macro work.
$\Sigma_{\text{macro}}$ — Macro entropy (dispersion + WIP-age + rework + twist-cost).
EEI — Entropy-Efficiency Index $\in[0,1]$ .
$C$ — Coherence score $\in[0,1]$ .

Numerics & Tolerances

$N_x,N_y$ — Mesh cells in x/y.
$q_1,q_2$ — Quadrature points (edge/face).
$h$ — Mesh size; $\kappa$ — curvature scale.
$\Delta\theta$ — Local twist increment.
$\tau_{\mathrm{PBHL}},\tau_{\mathrm{glue}},\tau_{4\pi}$ — Acceptance tolerances.

J.3 Cross-Domain Lexicon (concept → domain alias/examples)

Purpose / 志

Manufacturing: line objective / routing intent (per SKU/shift).
LLM-Ops: prompt+guardrail bundle (style, policy, tools).
Markets: mechanism rulebook (tick, margin, auction).
Governance: policy specification (controls, exclusions).

Curvature (Flux driver)

Manufacturing: change pressure / setup conflict hotspots.
LLM-Ops: embedding curvature spikes (prompt-injection, drift).
Markets: microstructure stress (spread jumps, queue flips).
Governance: rule friction (conflicting controls).

Twist (framing/governance steps)

Manufacturing: changeovers, routing flips, rework instructions.
LLM-Ops: prompt edits, policy toggles, tool-use on/off.
Markets: rule changes, fee switches, throttle policy flips.
Governance: policy enactments, exception waivers.

Coherence (“神”)

Manufacturing: plan vs execution alignment; schedule adhesion.
LLM-Ops: voice/style/format consistency to anchor.
Markets: venue/protocol alignment with target regime.
Governance: control set consistency vs standard.

Macro work $W_{\text{macro}}$

Manufacturing: pairs of shoes produced (net, defect-adjusted).
LLM-Ops: resolved tasks or accepted answers.
Markets: filled orders or executed notional.
Governance: compliant changes enacted.

Macro entropy $\Sigma_{\text{macro}}$

Manufacturing: WIP-age cost + changeover + rework.
LLM-Ops: context dispersion + moderation churn + prompt churn.
Markets: cancel/replace overhead + regime switch costs.
Governance: audit backouts + policy-flip overhead.

Flux-gate

Manufacturing: temporary buffer, pace damping, line rebalancing.
LLM-Ops: context clamp, tool restriction, rate-limit on risky flows.
Markets: volatility throttles, guard bands.
Governance: control escalation, freeze windows.

Twist budget

Manufacturing: daily changeover rad cap.
LLM-Ops: prompt-edit rad cap per window.
Markets: rule-change caps per session.
Governance: policy-flip allowance per quarter.

J.4 Notation collisions & do/don’t

Don’t flip $\mathbf n$ without flipping loop orientation; $\mathrm{Tw}$ sign will invert and break PBHL if you forget.
Do compute KPIs from class functions in non-abelian mode to preserve gauge invariance and 4π periodicity.
Don’t mix units: ensure $\alpha\,\mathrm{Tw}$ carries the same units as edge integrals after mapping.
Do record semver + α with every run; any α change is at least a MINOR version bump.

J.5 Ontology & Enterprise names (quick cross-ref)

BeltRun ↔ belt_run (one evaluation).
PurposeVersion ↔ purpose_version (semver’d Purpose, includes α).
KPIRecord ↔ v_kpi (Gap, Flux, Twist, Coherence, Residual).
LedgerRecord ↔ v_ledger (W_macro, Σ_macro, EEI, unit).
EvidencePack ↔ evidence_vault (URIs, hash).

J.6 Micro-cheats (one-liners)

PBHL: Gap = Flux + α·Twist.
Twist sign: CCW along $+\mathbf n$ is positive.
4π: test 0 vs 4π equal, 2π different.
Gluing: interior edges cancel to machine epsilon.
Ledger: Work up, Entropy down, Residual small.

(Use this appendix to normalize vocabulary across teams; if a term isn’t here, inherit definitions from Appendices A–I.)

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