# An anchored proper algebra and a continuous calibrated family

Section 11Z of *K-theory of the leaf space* proves the actual groupoid vertex geometry, the proper continuous nearest-face algebra, its continuous shifted exterior family, the whole-base ordinary positive product and the original-anchor deformation interfaces. Exercises 164–168 give complete solutions for 64 points. Section 11AA now proves the separately stated equivariant lifting condition. The general graph construction remains a distinct obligation.

Use Python 3.13.9 and the pinned package versions in requirements.txt with the bundled unchanged regular and bold fonts. Run `python -B draw_anchored_universal.py --output-dir OUTPUT` to regenerate PNG, SVG and the exact mathematical data, and `python -B self_check.py --output OUTPUT/checks.json` to run the finite model. The diagram's SVG date, hash salt and model seed are fixed. Compare regenerated bytes with the included PNG/SVG and figures/anchored-nearest-face-data.json.

The finite model is the action groupoid of C2×C2 on two units: the first factor exchanges units and the second is actual isotropy. It retains eight arrows, two integer labels per arrow, the range-fibre metric and the exterior permutation signs. It checks 2,040 simplex covariance identities and7,936 radius-one face-support samples, connected-origin agreement, Clifford compression, the even deficient line and actual source/range composition. These checks supplement the full proofs; finite samples do not establish infinite KK identities.

Figure 11Z.1 shows left translation `(h,n)→(kh,n)` with `r(h)=s(k)`, preserving distinct isotropy arrows. Its middle panel is an orthogonal isometry on the affine plane through e0,e1,e2. For v=(3.2,−2.2,0) and w=(3,−2.12,.12), the squared distance is.0608 and both nearest projections equal e0. The green disk is the plane section of the exact radius-one Bott-defect ball. The bottom panel is the whole-base ordinary identity, with both algebras balanced over the original unit space T. Proof locators: AG.2–AG.6, AF.6–AF.13, OC.1–OC.6 and DI.6–DI.8.

The nearest-face mechanism follows Jean-Louis Tu, *The Gamma Element for Groups which Admit a Uniform Embedding into Hilbert Space* (2004), printed pp. 275–281, available in the [free primary chapter scan](https://www.yumpu.com/en/document/view/50923940/the-gamma-element-for-groups-which-admit-a-uniform-embedding-). The analytic background follows Tu, *La conjecture de Baum–Connes pour les feuilletages moyennables*, K-Theory 17 (1999),215–264, Sections 7–9, available in the [free original journal article](https://citeseerx.ist.psu.edu/document?doi=486238991671dba031f9cb078eeba25443307a0a&repid=rep1&type=pdf). Sections 11W–11X contain the complete finite Bott, completed deformation and oscillator proofs used here.