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Initial long-distance scan. Choose the equation and four structural settings that define the mathematical planetary body, then travel into that same solved world.
The actual mathematical organs behind the live world, followed by a non-destructive parameter-space instrument for finding candidate regime boundaries.
A regime is a connected region of parameter space whose solutions share a recognisable qualitative structure. This first instrument samples a reduced shadow sphere and marks candidate boundaries where the empirical signature changes sharply. It never changes the live planet.
Cell colour is the provisional regime class. Optional topology miniatures show a mathematically honest shadow witness for each sampled world, either as a planet-style 50 × 50 patch or the original six-face quilt. Bright internal borders are candidate regime boundaries. The cross marks the current live parameter pair when it lies inside the scan.
Record the present equation-world as a provenance-rich evidence object, compare controlled specimens, let a transparent heuristic propose a bounded interpretation, then decide whether it belongs in the Claims Atlas or should become a new hypothesis.
Freeze the current numerical state and provenance. The latest Probe Analysis summary is attached when available; the heavy raw cube remains a separate export.
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The integer seed fixes a set of spherical kernels and a base phase. The user phase translates the equation without changing the seed lineage.
p = 2π · frac(seed / 100000) + phaseEach family defines a scalar F(n; θ) over a unit direction n=(x,y,z). Natural Functions, Pythagorean, waves, geometry, fractals, chaos and probability families share the same spherical projection and witnesses.
F : S² × Θ → ℝThe raw equation is bounded, passed through a sigmoid and mixed with a small deterministic memory field. This produces a stable height in [0,1].
h(n) = clamp[ 0.9 σ(3.1(F(n)-0.16)) + 0.1 M(n), 0, 1 ]The sea cutoff is a threshold, not a separate terrain generator. Moving it can change topology suddenly because connected land and water components may split or merge.
land(n) ⇔ h(n) ≥ cₛₑₐThe epsilon/error-link organ adds a structured deterministic perturbation rather than uncorrelated visual noise. Planted epsilon kernels are radial functions on the sphere, so they alter the equation field before rendering and can create local bifurcation experiments.
F̃(n) = F(n) + Σⱼ 0.16 wⱼ exp[-(1-n·vⱼ)(10+22τⱼ)] + 0.08 εE(n)This catalogue is generated from the implemented equation families. It records the exact finite forms used to create Gloxel's raw scalar field before seed kernels, epsilon and the common height projection are applied.
Neighbour differences estimate local slope. A grid-normalisation factor keeps the interpretation reasonably comparable across resolutions.
|∇h| ≈ √[((hᵣ-hₗ)/2)² + ((h_d-hᵤ)/2)²] · s_gThe curvature witness measures local departure from the neighbour mean and contributes to relief, fault evidence and transport interpretation.
κ ≈ |hₗ+hᵣ+hᵤ+h_d-4h| · s_g²Cells close to sea cutoff are structurally sensitive: a small parameter change may change land/water membership and therefore connectivity.
Bδ = { n : |h(n)-cₛₑₐ| < δ }Component counts, extrema, roughness, antipodal symmetry and cutoff-band population become a finite signature for comparing solution families.
Rₜ = (f_land, N_land, N_water, σ_h, ⟨|∇h|⟩, f_cutoff, S_antipodal, N_extrema)The hydrology ledger is a six-reservoir closed system except when an explicit external event—such as Natural Global Rain—is recorded. Local spikes can therefore be distribution phenomena even when the global total is exceptionally well conserved.
H = W_surface + W_soil + W_vapour + W_cloud + W_falling + W_iceSurface water moves down differences in free-surface level, limited by gravity, equalisation, damping, source availability and cross-face geometry.
ηᵢ = hᵢ + wᵢ qᵢ→ⱼ ∝ max(0,ηᵢ-ηⱼ) · gravity · equalise · dampingCube-sphere cells have unequal spherical areas. Transfers are computed as amounts and committed back as depths using source and destination area weights.
A = q · a_source w_source -= A/a_source w_dest += A/a_destLand capacity depends on landness, lowland position and shore proximity. Overflow returns honestly to surface water.
0 ≤ W_soil(n) ≤ C_soil(h,cₛₑₐ,mode)The stage witness distinguishes intentional sources from unexplained drift. The accepted project criterion is bounded chaotic conservation, not artificial exact zero.
|H_current - H_baseline - H_external| < e/π² ≈ 0.2754195323A completed weather cycle is an ordered composition of operators. Each stage changes distribution while the amount-space repairs preserve the closed reservoirs.
T_weather = T_current ∘ T_water ∘ T_freeze ∘ T_fall ∘ T_rain ∘ T_bridge ∘ T_coriolis ∘ T_pressure ∘ T_lattice ∘ T_mix ∘ T_evapAtmospheric capacity, pressure, temperature, lift and condensation potential are functions of terrain, water, soil wetness, epsilon, solar exposure and local differential structure.
Vapour and cloud gradients are mixed conservatively over one or more explicit passes. Lattice transport follows the configured pressure/hybrid route.
Pressure gradients, wind, coast response and lift produce directional redistribution, including cross-face movement.
Moisture is allowed to cross the land/sea boundary through an area-corrected bridge, preventing the ocean atmosphere from becoming mathematically isolated.
Condensation depends on supersaturation, terrain lift, land wetness and temperature. Rain release occurs only when the cloud reservoir exceeds a local capacity.
Cloud-to-fall and fall-to-surface are separate reservoirs, preserving the time between cloud release and ground arrival.
Heating uses the dot product between surface normal and sun direction, broadened across the daylight hemisphere and scaled by inverse-square orbital distance.
I(n)=K_b(max(0,n·s)) · solarHeat / orbitDistance²Dry land responds fastest; wet land and open water respond slowly. The relaxation coefficient is derived from material response and the thermal-memory control.
Tₙ₊₁ = Tₙ + α(material,memory)·(T_target-Tₙ)Cold bias, elevation and epsilon evidence move water and soil moisture into ice; warming and atmospheric conditions return it to surface water.
Physical rotation turns tangent wind and optionally transports moisture. It is independent of visual globe spin.
v' = R_axis(Ω, latitude, coupling) vA persistent tangent vector field exists over all wet cells. Wind stress, memory, drag, slope/pressure equalisation and rotation evolve the field; heat is transported conservatively.
Runoff may remove terrain from strong flows and deposit it downstream. It changes topography, not the hydrology ledger.
Each face is a square parameter chart. Edge steps are projected through a canonical 3-D direction and resolved on the adjacent face, including rotation and reflection.
Reference-grid spherical weights correct unequal cell area for totals and conservative transport.
Changing grid samples the old world through canonical directions, blends seam candidates and then restores each closed reservoir’s integrated amount.
Live fields are compact Float32 arrays. Critical amount-space water work uses reusable Float64 buffers before one paired Float32 commit per cell.
Texture and presentation meshes reveal a sampled solved world. They do not create new simulation cells or feed values back into the equations.
The shared probe is the same geometry, texture and depth-buffered renderer under a local tangent camera. Its direction is parallel-transported over the sphere.
No sampled weather cycle yet.
No local analysis pack yet.