CHAMBER GRAV-I v2.0 — Structural Phase Diagnostics

GUIDE — WHAT THE CHAMBER MEASURES · DATASETS · CLASSIFICATION ▶

WHAT THE CHAMBER MEASURES

NOT ASKING

“Does the gravity field have structure?”

ACTUALLY ASKING

“Does the field have a dominant global axis direction?”

The chamber computes J(θ,φ;L) — the fraction of total power concentrated in the zonal mode under rotation to axis (θ,φ) at truncation depth L. At each L it finds the best axis (J₁) and the second-best distinct axis (J₂). Classification uses the spectral dominance gap:

SPECTRAL DOMINANCE GAP

gap = J₁ − J₂
J₁ = best-axis score J₂ = runner-up score (distinct axis)
Large gap → one axis strongly dominates Small gap → axes compete; no structural preference

Why not J₁ alone? Every scalar field on S² has a maximum — so a high J₁ is always achievable. The runner-up J₂ provides the baseline. Only the gap between them reveals whether one axis genuinely dominates or whether multiple competing directions share the power equally.

WHY PLANETARY GRAVITY FIELDS GIVE WEAK AXIS DOMINANCE

Planetary gravity fields are not axisymmetric objects. They are composed of many competing large-scale structures that produce comparable contributions at different orientations. The spectral landscape looks like J₁ ≈ J₂ ≈ J₃, which keeps the gap small.

BODY

COMPETING STRUCTURES

EXPECTED CLASS

EARTH

Equatorial bulge (rotation) · continental mass asymmetry · ocean basins · mantle convection patterns · large-scale density anomalies

CLASS II / III — gap ≈ 0.001–0.05
Competing mechanisms distribute power across many directions

MARS

Tharsis volcanic plateau · Hellas impact basin · crustal dichotomy (north/south) · Valles Marineris · polar ice mass

CLASS II / III — gap ≈ 0.001–0.05
Tharsis is large but competing structures reduce net dominance

MOON

Near-side mascon basins (Mare Imbrium, Serenitatis) · far-side crustal thickness · tidal-locking asymmetry · polar gravity anomalies

CLASS II — gap ≈ 0.005–0.02
Near-side mascons create preference without full dominance

SYNTHETIC

Random spherical harmonic coefficients — no physical mechanism; no structural coherence by construction

CLASS 0 — gap ≈ 10⁻&sup4;
Noise floor baseline

STRUCTURED vs. AXISYMMETRIC

STRUCTURED

mass distribution not random

The field shows coherent spatial organisation at large scales — gravity is not white noise.

All four planetary datasets are structured. Even the synthetic field could be structured if given correlated coefficients.

AXISYMMETRIC

dominated by one direction

The field has a single preferred axis around which it is approximately symmetric — power concentrates in one zonal orientation.

No planetary dataset in this chamber is expected to be axisymmetric. Earth comes closest due to rotational flattening, but continental mass distribution, ocean basins, and mantle structure all compete with the rotational term.

KEY POINT

GRAV-I tests axisymmetry, not structure.
Planetary gravity fields are structured but not strongly axisymmetric. CLASS III does not mean "no structure" — it means "no single dominant axis." Distributed mechanisms compete.

WHAT GRAV-I DETECTS BEYOND THE GAP

AXIS TRAJECTORY STABILITY

The axis path θ(L), φ(L) stabilizes as L increases. Deeper harmonics reinforce the same directional tendency. Random fields do not do this — their axis wanders chaotically with no convergence.

STABILITY DEPTH

Early stabilization (low L) means the directional preference is encoded in low-degree harmonics — large-scale structural features. Late stabilization indicates the preference only emerges from fine-scale detail.

TRANSITION STRUCTURE

Real planetary fields show limited axis jumps and structured degeneracy windows — regime transitions occur at specific L values, not randomly. Synthetic fields show chaotic, unpatterned axis motion.

NOISE FLOOR & FALSIFICATION CONTEXT

The synthetic random dataset is the null-hypothesis control. It establishes the noise floor:

Synthetic (random)

gap ≈ 10⁻&sup4; → CLASS 0 (noise)

Real planetary data

gap ≈ 10⁻³ – 10⁻² → CLASS II / III (above noise floor)

The chamber confirms that real fields sit measurably above the noise baseline — not because they are axisymmetric, but because physical mechanisms impose persistent directional coherence. That coherence is what the UNNS substrate framework predicts.

UNNS SUBSTRATE IMPLICATION

Real physical mechanisms are multi-constraint structures, not single-axis attractors. Planetary gravity is a competition of structural operators — rotational, tectonic, convective — each imposing partial directional preference without any single one dominating.

So weak axis dominance is physically correct. The important result is not "Earth has a strong axis." It is:

Planetary gravity fields show persistent directional coherence without strong axis dominance.

That is a structural signature of distributed physical mechanisms — exactly what the UNNS substrate framework predicts for fields generated by multi-operator competition rather than single-mode attractors.

CONNECTED MANUSCRIPT

📄

Directional Rigidity of Planetary Gravity Fields Under Harmonic Extension

UNNS Research Programme — Empirical certification of axis emergence under truncation operators across planetary bodies

⬇ PDF

INJECTION DATASETS

EARTH

EIGEN-6C4
L720 Subset

Lmax: 720 | model: EIGEN-6C4
norm: fully_normalized
tide: tide-free

Expected: CLASS II / III — distributed structure; axis stabilises at low-to-mid L but gap remains modest due to competing mechanisms

⬇ JSON

MARS

JGM85F01
L85 Subset

Lmax: 85 | model: JGM85F01
norm: fully_normalized
asymmetric crustal field

Expected: CLASS II / III — later stabilisation; more transitions than Earth; Tharsis visible but not singularly dominant

⬇ JSON

MOON

GRGM1200
L300 Subset

Lmax: 300 | model: GRGM1200
norm: fully_normalized
tidally locked satellite

Expected: CLASS II — early stable axis; low transition count; gap ≈ 0.005–0.02

⬇ JSON

SYNTHETIC

Random SH
L300 seed 1337

Lmax: 300 | random coefficients
seed: 1337
null-hypothesis control

Expected: CLASS 0 — no structural axis; gap ≈ 10⁻⁴; spectral noise floor

⬇ JSON

AXIS DOMINANCE CLASSIFICATION

Classification is based on the spectral dominance gap gap = J₁ − J₂ at L_max. Every scalar field on S² has a maximum, so axis existence is trivially true for all inputs. What matters is whether the best axis dominates spectrally over the runner-up.

CLASS I gap ≥ 0.05

Strong global axis dominance — the field contains a clearly preferred orientation. One direction spectrally outcompetes all others.

CLASS II gap ≥ 0.01

Moderate axis anisotropy — a directional bias exists but competes with other orientations. The field is structured and non-random.

CLASS III gap ≥ 0.001

Weak axis anisotropy — the field is structured but dominated by multiple competing directions. Does not mean weak structure. Typical for distributed physical mechanisms.

CLASS 0 gap < 0.001

No detectable directional anisotropy — consistent with a random field. Degenerate windows (gap ≤ ε_gap) also force CLASS 0.

How to use: Download a dataset above, load it via DATA INPUT in Layer III, configure L_max and grid resolution, then click RUN SWEEP. The Layer I structural state panel will classify the result immediately on completion.

LAYER I — STRUCTURAL STATE

AXIS DOMINANCE

—

Load data and run sweep to diagnose

STABILITY DEPTH

—

first L, 10-step window

TRANSITIONS

—

axis jump events

DEGEN WINDOWS

—

gap ≤ ε contiguous

J₁ FINAL

—

score at L_max

LAYER II — PHASE EVOLUTION

AXIS TRAJECTORY — S² n̂(L) path · drag to rotate

L SCRUBBER L=—

AXIS VECTOR at L=—

CARTESIAN

n̂ₓ—

n̂ᵧ—

n̂₂—

SPHERICAL

θ—

φ—

J₁—

GAP

—

DEGEN

—

JUMP

—

Δaxis°

—

AXIS PATH θ(L), φ(L) cyan=θ · purple=φ · red=jump events

Colatitude θ and azimuth φ of best axis vs truncation depth L. Shaded columns = jump events.

DEGENERACY PHASE MAP green=stable · amber=degenerate · red=jump

Structural regime across L. Boundary events mark phase transitions in axis geometry.

LAYER III — ENGINE DIAGNOSTICS

DATA INPUT EIGEN-6C4 JSON SUBSET

Awaiting grav_i_eigen6c4_L720_subset.json …

model—

GM—

R—

norm—

tide_system—

Lmax_included—

n_coeff loaded—

CONFIGURATION

L_min L_max grid_deg

eps_gap jump_thresh °

Load data to enable sweep.

GRID

—

OPS EST.

—

ETA

—

RAW METRICS at scrubber L position

J₁—

J₂—

gap = J₁−J₂—

degenerate—

δ_axis (rad)—

transition_jump—

degen_enter—

degen_resolve—

MATHEMATICAL PROTOCOL

Rotation: Passive ZYZ convention R = R_z(φ) R_y(θ) R_z(0); axis (θ,φ) parameterizes n̂ ∈ S². Score: J(θ,φ;L) = Σ_{ℓ=1}^{L} |a′_{ℓ,0}(θ,φ)|² / E_total(L). Rotated zonal coefficient (real field): a′_{ℓ,0}(θ,φ) = d^ℓ_{0,0}(θ)·C_{ℓ,0} + Σ_{m=1}^ℓ √2·d^ℓ_{0,m}(θ)·[C_{ℓ,m}·cos(mφ) − S_{ℓ,m}·sin(mφ)]. d^ℓ_{0,m}: (−1)^m P̅^m_ℓ(cosθ) / √(2ℓ+1). Requires both C and S coefficients and φ mixing. Grid: θ ∈ [0,π], φ ∈ [0,2π), same step = grid_deg. Axis undirected: δ = min(angle, π−angle). ALF recurrence: 4π-normalized P̅^m_ℓ(cosθ) via Colombo recurrence; precomputed for all θ grid values.