9.7 KiB
Goxel Audit Bridge
Status: HOLD / workbench projection
Authority: bridge spec; not canonical proof
Related: docs/gcl/AutopoieticNScalarField.md, docs/gcl/GCLCompleteSurface.md, docs/gcl/MassNumberGCLSubset.md
Purpose
The Goxel Audit Bridge explains how N-space geometric shapes can inhabit existing voxel, microvoxel, mesh, SDF, CAD, and rendering workflows while keeping their derived mathematics auditable under a declared field regime.
The goal is not to replace existing voxel work. The goal is to let richer geometric-volume elements dock into it.
N-space shape
-> Goxel geometric-volume element
-> voxel-like editable structure
-> microvoxel / mesh / SDF / WebGPU projection
-> scalar-field audit
-> receipts or HOLD
Canonical definition
A Goxel is an N-space shape that inhabits a geometric volume.
More formally, a Goxel is a finite geometric-volume element: a bounded scalar sub-manifold with local geometric intent.
G = { v in R^n : Phi_G(v) <= iso }
where:
G = Goxel domain
v = coordinate in active n-space
R^n = finite active ambient regime
Phi_G = local Goxel potential function
iso = level/sublevel threshold
A Goxel is not merely a container of data. It is a topological domain inside n-space.
It is a packet of the field equation carrying local geometric intent.
Core idea
voxel: discrete sample cell / occupancy slot
hoxel: 4D unit block / spatiotemporal grid element
goxel: geometric-volume element / scalar sub-manifold
A voxel asks:
Where is occupancy sampled?
A Goxel asks:
What N-space shape inhabits this geometric volume?
A microvoxel asks:
Where must local detail be materialized?
Element evolution
| Element | Space representation | Dimensionality | Mathematical nature |
|---|---|---|---|
| Voxel | point sample / grid cell | 3D | discrete value |
| Hoxel | unit block | 4D | spatiotemporal grid |
| Goxel | geometric volume | n-dimensional | scalar sub-manifold |
Why this matters
Existing voxel systems are useful because they provide:
- edit locality
- chunking
- collision approximation
- storage layouts
- rendering pipelines
- user-understandable world editing
But pure voxels tend to flatten geometry into cells.
Goxels preserve the existing workflow while allowing the underlying object to be:
- an implicit surface
- a signed-distance fragment
- a spline patch
- a capsule or convex primitive
- a meshlet
- a field sample packet
- a higher-dimensional projected shape
- a repair patch derived from scalar-field residuals
- a bounded scalar sub-manifold
Informational DNA
Every Goxel carries a local potential function:
Phi_G : Omega_G -> R
This local potential is the Goxel's informational DNA.
It defines:
- boundary behavior
- curvature
- local topology
- field contribution
- fusion/repulsion behavior
- zero-crossing or sublevel-set structure
- audit obligations
N-space inhabitation rule
An N-space shape may inhabit an existing editable world only through a declared projection.
Shape_n in R^n
-> chart selection
-> Goxel encoding
-> projection to local world coordinates
-> voxel-like occupancy / mesh / SDF / microvoxel detail
No N-space shape may silently pretend to be native 3D geometry without declaring the projection that made it visible or editable.
Fusion rule
When multiple Goxels occupy compatible regions of n-space, they compose through a declared field-fusion operator.
Draft default:
Phi_Total = SmoothMax(Phi_G1, Phi_G2, ..., Phi_Gk)
SmoothMax is a projection/operator choice, not automatically a theorem. It must declare its regime, smoothing parameter, continuity class, and audit obligations.
Goal:
compatible Goxel junctions
-> smooth transition
-> low or zero local Anti-Music residual
Do not globally assume every SmoothMax fusion has zero residual. Zero residual is a gate result, not a default property.
Mass-number and metabolic cost
Because a Goxel inhabits geometric volume, it carries field inertia in the accounting sense.
m_A(G) = mass-number / metabolic cost proxy for the Goxel domain
Mass-number may depend on:
- occupied geometric volume
- scalar curvature / residual
- topological complexity
- internal zero-crossing complexity
- support size
- projection complexity
- GPU/compute budget
- route cost
Boundary:
m_A is finite accounting mass.
m_A is not SI physical mass.
m_A is not automatically distance.
Audit rule
Any math derived from a Goxelized N-space shape must be auditable in a declared field.
DerivedMath(Goxel)
-> declare source shape
-> declare projection
-> declare scalar field / chart
-> compute gates
-> attach receipts
If the audit cannot close, the result remains HOLD or routes to NaNMass / ProjectionArtifact / RegimeMismatch.
Proven field boundary
A “proven field” does not mean the rendered shape proves reality.
It means the field has declared enough structure to audit claims locally:
ProvenField_R iff
finite_regime(R)
and declared domain Omega
and declared scalar field Phi
and declared projection Pi
and declared regularity gate
and declared budget/mass accounting
and declared receipt requirements
Within such a field, a Goxel-derived claim can be checked against:
- regular level-set conditions
- collision / CB2 gates
- mass-number budget
- projection consistency
- source-seed provenance
- residual / anti-music cost
- closure state
CB2 collision audit
CB2 is a collision or contradiction detector. For Goxels, it checks whether combined domains create invalid topology.
Possible failure modes:
- non-manifold singularity
- invalid overlap
- contradictory scalar assignment
- Sidon collision
- regime mismatch
- projection collision
Fusion / repulsion response
If two Goxels attempt to occupy the same n-space in a structurally incompatible way, the gate triggers an incompatibility response.
Allowed responses:
Fuse:
morph volumes into one continuous scalar surface when compatibility is established
Repel:
adjust local potentials to maintain a gap when fusion would create invalid topology
Hold:
preserve the conflict as unresolved until receipts exist
Quarantine:
block the construction when it is unsafe, overbroad, or misleading
Corrected Goxel boundary condition
Do not write this as an unconditional theorem:
CB2(Goxel_A union Goxel_B) = 0 implies the volumes are topologically fused.
That is too strong.
Use the scoped gate form:
FusionAllowed_R(G_A, G_B) iff
CB2(G_A union G_B) = 0
and Compatible_R(G_A, G_B)
and Regular(boundary(G_A union G_B))
and m_A(G_A union G_B; R) <= Budget_R
and ReceiptsRequired(fusion) are present or explicitly marked missing
Then:
FusionAllowed_R(G_A, G_B)
-> may construct fused candidate G_AB
The result is a candidate fused Goxel, not universal proof of all topology.
Goxel object
type Goxel = {
goxel_id: string;
source_shape_id: string;
source_space: `R^${number}` | string;
target_chart: string;
projection: string;
domain_definition: string;
local_potential: string;
primitive:
| "plane_patch"
| "sdf_fragment"
| "convex_cell"
| "spline_patch"
| "capsule"
| "implicit_blob"
| "meshlet"
| "field_sample_packet"
| "nspace_projected_shape"
| "bounded_scalar_submanifold";
parameters: Record<string, unknown>;
boolean_role?: "fill" | "cut" | "blend" | "constraint" | "repair" | "fuse" | "repel";
support_region: string;
mass_number_cost?: number;
derived_math_refs: string[];
audit_status: "raw" | "held" | "audited" | "closed" | "quarantined";
receipts: string[];
};
Derived math object
type GoxelDerivedMath = {
math_id: string;
goxel_id: string;
expression: string;
variables: string[];
source_field: string;
projection: string;
assumptions: string[];
gates: string[];
claim_state: "U_scope" | "HOLD" | "V_scope" | "REVIEWED" | "CANONICAL_LEAN" | "QUARANTINE";
failure_mode?:
| "projection_artifact"
| "regime_mismatch"
| "missing_receipt"
| "regularity_failure"
| "collision_failure"
| "sidon_collision"
| "nan_mass";
};
Field audit pipeline
1. Receive or synthesize N-space shape.
2. Encode it as one or more Goxels.
3. Declare chart and projection into editable world space.
4. Materialize only the needed voxel/microvoxel/mesh/SDF view.
5. Derive local math from the Goxelized shape.
6. Audit derived math in declared scalar field.
7. Apply gates: finite regime, regularity, CB2, mass-number budget, projection consistency.
8. If gates pass: attach receipts and mark locally valid.
9. If gates fail: HOLD, repair, quarantine, or route to NaNMass / ProjectionArtifact.
Final thesis
By defining shapes as Goxels, the editor becomes a geometric fluid.
Volumetric sculpting:
editing injects volumes of intent into n-space
Continuous topology:
scalar-defined boundaries can remain smooth and resolution-independent
Audited organism:
the Goxel collection forms a field body, with OTOM acting as immune/audit system
Canonical warning
Goxelized shape != proof
projection != proof
voxel occupancy != source geometry
microvoxel detail != ontology
rendered N-space shadow != N-space truth
CB2 = 0 alone != complete manifold viability
SmoothMax alone != guaranteed zero residual
A Goxel makes N-space shape usable inside existing work. It does not make the shape true.
Operating sentence
A Goxel is an N-space shape inhabiting a geometric volume, expressed as a bounded scalar sub-manifold and admitted into ordinary editing workflows only through declared projection, audit, and receipt gates.