5.9 KiB
Sidon Matrix over Goxel Domains
Status: HOLD / workbench projection
Authority: modeling spec; not canonical proof
Related: docs/gcl/GoxelAuditBridge.md, docs/gcl/AutopoieticNScalarField.md, docs/gcl/MassNumberGCLSubset.md
Purpose
This document defines how Goxels make the Sidon matrix cleaner.
A Goxel is an N-space shape inhabiting a geometric volume:
G = { v in R^n : Phi_G(v) <= iso }
Because each Goxel has a bounded scalar-domain definition, collisions can be modeled as structured field-domain interactions instead of ambiguous voxel-cell overlaps.
Core insight
Voxel collision asks:
Do two discrete cells overlap?
Goxel collision asks:
Do two bounded scalar sub-manifolds create an inadmissible intersection, resonance, degeneracy, or non-manifold junction under regime R?
The Sidon matrix becomes the audit surface for those pairwise and higher-order interactions.
Sidon matrix definition
Let the active Goxel set be:
G_set = { G_1, G_2, ..., G_k }
Define the Sidon matrix under regime R as:
S_R[i,j] = SidonAudit_R(G_i, G_j)
where each matrix entry is not merely boolean. It is a typed audit record.
type SidonMatrixEntry = {
left_goxel: string;
right_goxel: string;
regime: string;
intersection_class:
| "disjoint"
| "tangent"
| "transverse_intersection"
| "nested"
| "fusable"
| "repulsive"
| "singular"
| "unknown";
cb2_status: "pass" | "fail" | "unknown";
compatibility: "compatible" | "incompatible" | "needs_closure";
residual_score?: number;
mass_budget_delta?: number;
recommended_response:
| "no_action"
| "fuse"
| "repel"
| "hold"
| "quarantine"
| "request_receipts";
receipts: string[];
};
Why Goxels clean up the matrix
With voxels, the matrix tends to collapse into occupancy logic:
occupied / empty
same cell / different cell
collision / no collision
With Goxels, the matrix can represent geometric relation classes:
G_i disjoint from G_j
G_i tangent to G_j
G_i nested inside G_j
G_i smoothly fuses with G_j
G_i intersects transversely with G_j
G_i creates a singular junction with G_j
G_i must repel from G_j under regime R
That makes Sidon analysis about shape-domain compatibility, not just cell occupancy.
Scalar-field form
Each Goxel carries a local potential:
Phi_i = Phi_Gi
Phi_j = Phi_Gj
A pairwise relation can be evaluated through:
I_ij = { v in Omega_R : Phi_i(v) <= iso_i and Phi_j(v) <= iso_j }
The pair is easy if:
I_ij = empty
The pair needs audit if:
I_ij != empty
or if the boundaries approach below a regime threshold:
dist_R(boundary(G_i), boundary(G_j)) <= epsilon_contact
CB2 relationship
CB2 is the contradiction/collision detector. In this model, CB2 becomes one field of the Sidon matrix, not the whole matrix.
CB2(G_i union G_j) = 0
means only:
no detected collision contradiction under the current CB2 predicate
It does not, by itself, prove valid fusion.
Fusion row condition
A Sidon matrix entry may recommend fusion only if:
FusionCandidate_R(G_i, G_j) iff
S_R[i,j].cb2_status = pass
and S_R[i,j].compatibility = compatible
and Regular(boundary(G_i union G_j))
and m_A(G_i union G_j; R) <= Budget_R
Then the system may construct:
G_ij = Fuse_R(G_i, G_j)
The fused result remains a candidate until receipts are attached.
Repulsion row condition
A Sidon matrix entry may recommend repulsion when:
RepulsionCandidate_R(G_i, G_j) iff
S_R[i,j].cb2_status = fail
or S_R[i,j].compatibility = incompatible
or S_R[i,j].intersection_class in {singular, transverse_intersection}
Then the system may adjust local potentials:
Phi_i' = Phi_i + Delta_repulse_i
Phi_j' = Phi_j + Delta_repulse_j
subject to budget and regularity gates.
Higher-order Sidon tensor
The pairwise Sidon matrix is the first pass.
Some failures only appear when three or more Goxels interact.
Define optional higher-order audit tensors:
S_R[i,j,l] = SidonAudit3_R(G_i, G_j, G_l)
S_R[i1,...,im] = SidonAuditM_R(G_i1, ..., G_im)
Use these only when pairwise passes but global field fusion still produces residual, singularity, or budget failure.
Sidon matrix as graph
The Sidon matrix induces a typed graph:
nodes = Goxels
edges = SidonMatrixEntry relations
Edge labels:
disjoint
fusable
repulsive
singular
needs_closure
quarantined
This graph is projection-only. It helps route repair and audit work; it is not proof.
Mass-number integration
Each row/column can carry aggregate metabolic load:
row_cost(G_i) = sum_j cost(S_R[i,j])
A high row cost means the Goxel is interaction-heavy. It may indicate:
- high geometric centrality
- unresolved collision pressure
- excessive fusion/repulsion obligations
- mass-number budget stress
- need for decomposition into smaller Goxels
- need for quotient/closure repair
High row cost is an audit priority, not evidence of truth.
Cleaner modeling sentence
The Sidon matrix is no longer a table of voxel collisions. It is a typed compatibility matrix over N-space geometric-volume elements.
Failure modes captured
The Goxel-based Sidon matrix can represent:
cell overlap
boundary tangency
non-manifold singularity
projection collision
regime mismatch
unclosed mass interaction
SmoothMax residual
false fusion
false separation
higher-order junction failure
Validator requirements
Every Sidon matrix entry must declare:
left_goxel
right_goxel
regime
intersection_class
cb2_status
compatibility
recommended_response
receipt status
If any of those are missing, the row is HOLD.
Operating sentence
Goxels clean the Sidon matrix by turning collision detection into typed compatibility auditing over bounded scalar sub-manifolds.