> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# Projection Possibility Space · Paper 01 — Reconstructed Edition
## 3D Projection Possibility Space: Three-Dimensional Sources and Planar Configuration Families

**Version:** reconstructed-v1 · 2026-10-08  
**Nature:** At Neo.K's direction, a reconstruction of the foundation from references to the preceding Paper 01 in PPS Papers 02–06. The manuscript and original conversation have not been recovered; the definitions below are compatible editorial proposals, not a claim that the original author used exactly these notations.

## 1. Problem and source body

Given a planar research domain \(\mathcal C\) and task functional \(Q:\mathcal C\to\mathbb R\), ordinary methods enumerate planar shapes directly. PPS's foundational proposal instead uses a projectable three-dimensional source body \(M\subset\mathbb R^3\), together with pose and projection parameters, to generate a configuration family.

With transformation \(T_\theta\) and projection \(\Pi_\eta\), let

\[
G(\theta,\eta)=\Pi_\eta(T_\theta M),\qquad
\mathcal P(M)=\{G(\theta,\eta):(\theta,\eta)\in\Theta\times H\}.
\]

This defines a searchable family; it does not prove \(\mathcal P(M)=\mathcal C\). A global conclusion based on this family requires an additional coverage or reduction proposition.

## 2. Minimal three-layer object

| Layer | Retained information |
|---|---|
| 3D source | Source body, transformations, projection direction, shape provenance and parameter domain |
| 2.5D intermediate | Depth, layer order, occlusion/visibility and provenance labels for each projection patch |
| 2D projection | Research shape, boundary, support function, contacts and task functional |

2.5D is not another abstract space dimension: it is an intermediate record retaining some provenance information. Projection is generally non-injective, so a planar image alone need not suffice to recover the source body or occlusion events.

## 3. Basic stability proposition

If \(\Pi\) has Lipschitz constant L and K, K′ are nonempty compact sets, then

\[
d_H(\Pi K,\Pi K')\le Ld_H(K,K').
\]

Proof: For every \(x\in K\), choose y with distance arbitrarily close to the infimum of the distances from x to K′, and apply \(\|\Pi x-\Pi y\|\le L\|x-y\|\). Taking the supremum in both directions gives the result. For orthogonal projection one may take L=1.

This is a local tool for propagating geometric error; it does not yet bound error in the task functional Q. The latter requires a separate continuity/Lipschitz contract for Q.

## 4. Boundary provenance needs separate tracking

In general, \(\Pi(\partial K)\) cannot simply be identified with \(\partial(\Pi K)\). For example, the sphere bounding the three-dimensional unit ball projects onto an entire planar disc, while only the circle is the projected disc's boundary. A boundary-event ledger therefore needs to record which source patches actually generate the visible outer boundary.

This information loss is precisely what the subsequent subject/object/boundary layering in PPS-03 and hidden geometry in PPS-05 address.

## 5. Minimal replayable example

Take the three-dimensional unit segment

\[
M=\{(s,0,0):-1/2\le s\le1/2\},
\]

rotate it through θ about the y-axis, and project onto the xy-plane. The projected segment has length \(|\cos\theta|\). At θ=π/2 the projection collapses and re-expands on either side.

This illustrates how finitely many frames can miss a degeneracy event, while a continuous family and its event equation explicitly identify \(\cos\theta=0\). It is a teaching example, not a certificate for the Lebesgue or worm problem.

## 6. Minimum Projection Certificate fields

```text
source_id / source_hash
mother_geometry / parameter_domain
transform / projection / coordinate_conventions
intermediate_state / visible_boundary_sources
target_shape / task_functional / assumptions
critical_events / coverage_status / evidence
```

Certificates must distinguish coverage of the research domain, coverage of the entire configuration space, and numerical sampling. Use null for unrecovered original data; do not fill gaps with fabricated source hashes.

## 7. Connections to the subsequent series

- Paper 02: parameterize the path by time, yielding \(G_t=\Pi_t(T_tM)\).
- Paper 03: distinguish subject, object and boundary provenance.
- Paper 04: extend the framework to non-rigid perturbed objects.
- Paper 05: track projection-induced hidden geometry.
- Paper 06: test the scope through adversarial re-derivation, counterexamples and closure ledgers.

This edition retains the separation between definitions, contracts and closed items, together with the subsequent series' authority ladder. It does not impose a site-wide judgment on ladder orderings that remain unreconciled between Paper 06 and other cases.

## 8. Remaining responsibilities

An actual application to a conjecture still requires selection of the source body, coverage of the target domain, complete treatment of degeneracies and occlusion, task-functional stability, finite certificates, and a reduction of global quantifiers. This edition supplies readable prerequisite objects for the series; it does not claim these responsibilities are complete.

===END OF RECONSTRUCTED EDITION===