Square-paper tray reference model (v0.10.0)

Scope, authorship and provenance

This release implements one recognizable object: an open square-paper tray with four upright walls and four inward double-layer triangular gussets. It is not a traditional locking masu box. It is an original implementation and independently derived analytic construction, not a claim of historical novelty. No external pattern, instructions, model file, photograph, diagram, software or source code was copied. No third-party pattern license is relied on. The implementation belongs to this private project; no new external software dependency is added.

Modelling references, checked 5 October 2026:

- He and Guest, *On Rigid Origami I: Piecewise-planar Paper with Straight-line Creases*, §2, Definitions 5–7: https://arxiv.org/html/1803.01430v3 . This primary research supplies the rigid-panel framework, continuity and the distinction between allowed contact and forbidden crossing. The tray coordinates and proof below are our own, not taken from this paper. Its arXiv license is not treated as permission to redistribute the paper or its illustrations.
- Abel et al., *Rigid Origami Vertices: Conditions and Forcing Sets*: https://erikdemaine.org/papers/RigidOrigami_JoCG/ . Primary mathematical context only; no algorithm, pattern or illustrations incorporated.
- FOLD format: https://github.com/edemaine/fold/blob/main/doc/spec.md . Export follows the format's flat frame plus inherited folded frame. We do not bundle the FOLD software library.
- Tachi's software license https://www.tsg.ne.jp/TT/software/ requires separate commercial permission. We do not use or redistribute that software or its model files; reading the research does not grant such rights.

This reference studio is deterministic and entirely local. The separate AI lab supports AI-guided fan parameters or tray sheet size and wall depth on the same validated construction. The tray topology and motion are reference geometry, not AI-invented or arbitrary-object synthesis. Using this reference page makes no API call or change to keys, model settings or budget.

Material topology

Start with square side S, wall width h, base half-width a=S/2−h. Parameters are 120≤S≤190 mm and 10≤100h/S≤16. Therefore 0<h<a/2 and a−h/√2>0.

The flat grid coordinates on each axis are −a−h, −a, a, a+h. Its central cell is the base. The four non-corner surrounding cells are rectangular walls. Each corner square is divided along the diagonal from its inner base corner to its outer sheet corner. All 13 faces retain fixed vertex identities: 1 base, 4 walls, 8 triangles. The disk has 16 material vertices, 28 edges, 12 boundary edges, and Euler characteristic 1. There are no interior cuts, holes, disconnected pieces or hidden panels. At the final state some *distinct material vertices* occupy the same spatial positions; they are not merged.

Exact motion and independent derivation

Let progress p∈[0,1], θ=pπ/2, c=cosθ and s=sinθ. The base stays in z=0. Each wall is a rigid rotation about its base edge. At the northeast base corner, define:

    A = (a, a, 0)
    B = (a+h c, a, h s)
    C = (a, a+h c, h s)
    D = (a+h d, a+h d, h e)
    d = (c − √2 s²) / (1+s²)
    e = s(2 + √2 c) / (1+s²)

The two gusset triangles are ABD and ADC. Reflect these coordinates across the x and y axes for the other corners. The chosen branch first raises the corner and then tucks it inward. It is not coordinate interpolation between flat and folded poses.

The two governing constraints are:

    2d²+e²=2
    cd+se=1

They are derived by requiring AD=√2h and BD=CD=h. Together with AB=AC=h they fix every distance in each triangular face. Solving these two equations yields the displayed branch. Wall edges, diagonals and areas are unchanged by rigid rotation; the base is fixed. The common denominator is ≥1, so the formulas are continuous on the complete interval. Signed dihedrals are computed from oriented normals and shared edge directions; an independent test oracle checks base-wall angle θ and diagonal gusset angle −2θ. Mountain/valley assignments are taken from a reference interior pose, so they do not flicker near p=0. Angles at fully coincident gussets retain −180°.

At p=0 the faces tile precisely the uncut square. At p=1 all four walls stand at 90°, B=C=(a,a,h), and D=(a−h/√2,a−h/√2,h). The paired gusset triangles intentionally coincide with opposite material normals. Their consistent layer order is inherited from the noncrossing approach; no finite-thickness offsets are introduced. The corner reaches height √2h at θ=π/4, then descends to h at the endpoint, and the camera bounds account for it.

Analytic no-crossing argument

For 0<θ<π/2:

1. At one corner det(B−A,C−A,D−A)=√2h³cs>0. Thus the two gusset planes are distinct; their triangles intersect only on common edge AD.
2. For the east-wall plane E(P)=s(Px−a)−cPz, E(A)=E(B)=0, E(C)=−hcs<0 and E(D)=−hs(c+√2)/(1+s²)<0. Convexity puts the gusset interiors strictly on one side of that wall plane, with only their shared hinge permitted to touch it. The north-wall case is symmetric.
3. Every gusset point other than A has positive z, separating it from the base.
4. d≥−1/√2 follows from d+1/√2=c(1+c/√2)/(1+s²)≥0. Every northeast gusset point therefore has x,y≥a−h/√2>0. Reflections put the other gussets in separated quadrants and separate every gusset from its two distant walls.
5. Opposite walls occupy disjoint ranges. Adjacent walls share only their base corner before the endpoint.

At p=0 all cells have disjoint interiors in the plane. At p=1 neighboring wall edges meet and same-corner gussets make intentional double-layer contact. No transverse crossing is introduced. The claim is crossing-free ideal motion with intended endpoint layer contact, not completely disjoint geometry at every pose. This construction-specific analytic result does not make the runtime checker a general origami collision solver.

Rendering and export

The canvas uses an original orthographic software depth renderer. All 13 panels are rasterized at every pose. Occlusion is ordinary depth testing; it is not a face-visibility animation. Camera scale and center stay constant through the motion (except explicit zoom or viewport/parameter changes). Colors and reverse-side shading are visual only. The mesh has no thickness or render-induced vertex offsets.

FOLD exports the actual immutable material topology, flat coordinates, current-pose coordinates and signed hinge angles. The endpoint has contact metadata and a description; general layer-order arrays and thickness offsets are not supplied, so external importers may render coincident faces differently. SVG shows the same crease assignments viewed from +Z, a square boundary and a 20 mm scale check. Its 210×250 mm page fits A4 portrait with zero print margins; the drawing itself has ≥10 mm horizontal paper margin. Browser/native print settings and physical output still need verification on the user's printer.

Limits and validation

The supported surface is a geometric open tray, not a self-locking or load-tested object. No claims of practical hand-foldability, beginner suitability, finite-thickness clearance, stability, paper strength, printing accuracy or physical performance follow from these tests. Physical test folding remains TASK-151456.

Deterministic tests cover the complete parameter bounds, dense and seeded-random motion samples, all face pair distances, triangle/quad areas, planarity, fixed topology, hinge directions, an independent trigonometric oracle, endpoint identities, continuity, construction-specific separation inequalities, malformed inputs, matching FOLD/SVG data, renderer geometry, slider endpoints, resets, repeated/interrupting playback, view switches and small canvas widths. Dense samples are regression checks, not the basis of the continuous proof above. Exact release verification and browser QA are recorded separately.