# Geometry-First Teacher: Three-Pass Audit

**Reviewed:** 2026-09-05  
**Artifact:** `understand-geometry-deep.html`  
**Scope:** 39 chapters, 456 stable teaching moments, 456 exact scene-step bindings  
**Status:** sequence, visual correspondence, mathematical wording, source boundaries, and responsive layout checked

## Why this is a separate edition

The geometry-first Teacher is a working copy. It does not overwrite the earlier `understand.html` narration. The two remain available side by side so every restructuring can be compared and recovered.

The number of teaching moments is a result of the dependency sequence, not a quota. Stable semantic IDs—not display ordinals—hold links together when a missing bridge is inserted.

## Pass 1 — conceptual dependency

The narration was read in order from the first distinction through recursive Mind. The pass repaired these boundaries:

1. a co-produced triad is not narrated as three consecutive causes;
2. recursion through occurrences is distinct from recursion through depth;
3. delay retains causal relation in transit, but reusable memory requires an addressable trace;
4. contact begins an observer-cycle but is not itself the completed observation;
5. Mind-structure, active Mind, observerhood, and local cognitive Mind capability are different claims;
6. layer seven is a turning boundary, not a small observer or correctness oracle;
7. observation, retained observation, return, modelling, self-reference, and metacognitive correction are earned in order;
8. Pi, golden inheritance, seven-depth addressing, wave, spiral, helix, and toroidal closure keep separate derivations and conditions.

The resulting operational distinction is:

```text
Mind-structure = retained recursive topology and available relational routes
active Mind    = bounded history-bearing circulation through that organization
cognitive Mind = model-guided returned redirection of later operation
```

## Pass 2 — narration-to-visual correspondence

Every teaching moment has one explicit `chapter + scene + step` visual address. No nearest-slide visual inheritance is allowed.

The audit replaced several misleading pictures:

- three isolated decorative circles became overlapping and shared recurrent routes;
- resonance now shows a compatible mode gaining influence;
- coupled loops now show directed, delayed, weighted contributions and their return;
- interference now uses actual aligned, opposed, and non-collinear vector relations;
- the observer aperture became a state-space projection rather than a literal window;
- the observer became the distributed `present whole → retained whole → comparison → retained observation → return` operation rather than a central dot;
- the observed component and presently unintegrated remainder are revealed on separate teaching moments;
- layer seven visibly turns consequence toward whole-level integration without becoming the observer.

All 456 visual routes resolve. Portrait labels were shortened and separated where three vector examples share one narrow canvas.

## Pass 3 — executable and claim-boundary audit

The engine and narration were compared with the source trace and the current numerical implementations.

The new flow sequence treats electrical or electrochemical signalling as one biological carrier example, never as the universal definition of Mind. Gain, loss, phase shift, intrinsic recurrence rate, and propagation delay remain separate response quantities unless a typed domain equation relates them.

The phase-two aperture is now executable in `observer-aperture-engine.js` as a standalone candidate. For a general finite operator `W`, the implementation selects the dominant right-singular participation mode of `W†W`; it does not silently pretend every `W` satisfies the source candidate's eigenmode equation. Projection preserves both components:

```text
z_obs  = v* v*† z
z_perp = (I - v* v*†) z
z      = z_obs + z_perp
```

This aperture is not yet wired into the governed seven-depth observer runtime. Aperture alone does not establish observerhood, correctness, canonical proof, or phenomenal experience.

An unintegrated local change can influence a later observer-cycle only when an addressable consequence remains and later reconnects through coupling, delay, attention, or a changed collective mode. No retained relation means no later causal contribution.

## Responsive review

The final layout was checked at:

- 360 × 740 portrait;
- 390 × 844 portrait;
- 740 × 360 short landscape;
- the desktop split view.

Portrait keeps the geometry above a scrollable narration. Short landscape switches to a side-by-side stage and reader, while preserving separate brand, canvas, and footer rows so captions and controls do not cover the visual.

## Automated evidence

- Full Node project suite: **498/498 passing**.
- Focused geometry, source-registry, and phase-two aperture suite: **50/50 passing**.
- Slide and chapter count: **39 / 456**.
- Missing or duplicate visual bindings: **0**.
- Source registry: exact file and inclusive-span SHA-256 verification passes.

Passing tests establish implementation and narration conformance only. They do not turn a candidate operator, visual analogy, metaphysical statement, or phenomenal interpretation into external empirical proof.