# Mass Harmonics Predictions for Gaia Data Release 4

## A Source-Preserved First-Principles Terrain Prediction Paper for the December 2, 2026 Release

**Author:** Thomas Russell Giboney

**Affiliation:** UMtts Institute

**Framework:** Mass Harmonics ψₘ substrate science

**Release surface:** Gaia Data Release 4

**Scheduled public opening:** December 2, 2026

**Status:** Completed before the Gaia DR4 public data opening

---

## Concretus


## Source-Preserved Mass Harmonics Governance

This edition retains the Gaia paper's full eight-pathway structure and all nine final instrument-facing readouts. The paper was already substantially substrate-first. The present revision strengthens source ancestry and release-product precision without replacing the derivation.

The governing order remains:

```text
MFE
→ substrate action
→ bounded coherence structure
→ boundary transition
→ physical stellar or Galactic motion
→ Gaia-rendered terrain
→ optional consensus translation
```

Gaia does not adjudicate Mass Harmonics. It releases epoch, astrometric, spectroscopic, binary, Solar-System, and Galactic terrain that can expose the derived scale-invariant structure.

### Canonical branch and scale-invariant extension

The canonical source branch supplies:

```text
Kψₘ = c/(2π)
→ a₀ = cH₀/(2π)
→ g_total ≈ √(G_N·M/r² · a₀)
→ v⁴ = G_N·M·a₀
```

Transporting this branch from galactic outer boundaries to wide binaries is a new scale-invariant Mass Harmonics excavation in this paper. It is not imported from a wide-binary anomaly and is not a fitted binary force law. The transport is governed by the same bounded-coherence transition and the same single `Kψₘ` relation.

This paper records what Mass Harmonics predicts Gaia Data Release 4 will disclose before the DR4 terrain is publicly available for analysis.

Gaia does not govern the derivation. Gaia is the rendering instrument. Its astrometry, epoch measurements, radial velocities, photometry, non-single-star solutions, crowded-field products, Solar-System products, and Milky Way phase-space maps are downstream readouts of physical structures that already exist in the ψₘ substrate.

The governing order is:

```text
MFE
→ substrate action
→ bounded coherence structure
→ boundary transition
→ physical stellar or Galactic motion
→ Gaia-rendered terrain
```

The paper therefore does not begin from a downstream dynamical parameterization, an empirical interpolation curve, a binary-specific force coefficient, or a Gaia-derived anomaly. It begins from the canonical Master Field Equation, the single coupling coefficient Kψₘ, the exact cosmological boundary acceleration a₀, and the scale-invariant outer-boundary solution already derived inside Mass Harmonics.

The central prediction is not merely that Gaia will show a low-acceleration anomaly. It is more specific:

```text
The same ψₘ boundary law must appear in wide binaries, the Milky Way outer disk, stellar streams, and the high-acceleration Solar-System control surface, with one transition acceleration and one mass-scaled closure structure.
```

No additional Galactic coefficient is introduced. No binary coefficient is introduced. No unseen-mass parameter is introduced. There is one governing coupling coefficient: Kψₘ.

---

# I. Governing Mass Harmonics Source Chain

## I.1 Governing Physical Authority

The canonical physical authority is the Mass Harmonics Monograph and its source-indexed proof structure. The relevant source chain for this paper is:

```text
Unique First-Principles Lagrangian
→ canonical MFE
→ coherence-pressure scalar Π
→ Kψₘ = c/(2π)
→ cosmological boundary frequency H₀/(2π)
→ a₀ = cH₀/(2π)
→ outer-boundary acceleration law
→ v⁴ = G_N M a₀
```

The Gaia product model enters only after this chain is complete.

## I.2 Canonical First-Principles Lagrangian

Copied directly from `MH_PROOF-SET.md`:

```text
ℒₘ = ½∂μψₘ∂^μψₘ − Kψₘ|∇Π|²     where Π = ψₘ²/ω
```

Kψₘ is a single canonical term. It is not split into a free scalar K and an independent field ψₘ.

## I.3 Canonical Master Field Equation

Copied directly from `MH_PROOF-SET.md`:

```text
1/vₓ²ψ̈ₘ − Z(ψₘ)∇²ψₘ − 8Kψₘ/ω²|∇ψₘ|² = S(ρ)
```

with:

```text
Z(ψₘ) = 1 + 8Kψₘ/ω²
```

The three left-side structures remain distinct: temporal propagation, Z-mediated spatial propagation, and the nonlinear Giboney Gradient coherence-pressure term.

## I.4 Canonical Closure Coefficient

Copied directly from `MH_PROOF-SET.md`:

```text
f · R = c/2π ≡ Kψₘ
```

and:

```text
Kψₘ = 47,713.45 Hz·km
```

This is the single scale-invariant closure coefficient used from atomic through cosmological structure.

## I.5 Canonical Galactic Acceleration Chain

Copied directly from the galactic section of `MH_PROOF-SET.md`:

```text
g_total = g_Newtonian + g_Giboney = −∇(K₀ψₘ) − ∇(ψₘ²/ω)
```

For the interior contribution:

```text
|g_Newtonian| = G_N·M/r²
```

For the outer boundary contribution:

```text
g_Giboney = −∇(ψₘ²/ω) = −(2ψₘ/ω)∇ψₘ  →  Constant Vector Field
```

The constant-vector-field result is the physical origin of the outer low-acceleration asymptote. It is not imported from a downstream fitted potential.

## I.6 Canonical Cosmological Boundary Acceleration

Copied directly from `MH_PROOF-SET.md`:

```text
ω_cosmic = Kψₘ / R_H = (c/2π) / (c/H₀) = H₀/2π
```

Therefore:

```text
a₀ = c · ω_cosmic = cH₀/(2π)
```

and:

```text
a₀/(cH₀) = 1/(2π)
```

No wide-binary measurement, Galactic rotation curve, or Gaia value is used to obtain this relation.

## I.7 Canonical Deep-Boundary Motion

Copied directly from `MH_PROOF-SET.md`:

```text
g_total ≈ √(G_N·M/r² · a₀)
```

Equating to circular kinematics gives:

```text
v⁴ = G_N·M·a₀
```

and:

```text
v_orbit = (G_N·M·a₀)^(1/4)
```

This paper transports that already-derived boundary law into the two-body stellar terrain and the Milky Way phase-space terrain that Gaia DR4 will expose.

---

# II. Gaia DR4 Terrain Surface

## II.1 Official Release Scope

ESA lists Gaia Data Release 4 for Wednesday, December 2, 2026. The release is based on 66 months of observations, is expected to contain approximately 400 TB, and covers about 2.8 billion processed sources. More than 130 product classes are described, with many epoch-level and first-release products providing the least model-dependent terrain required by this paper.

The release is therefore not one catalogue. It is a multi-surface terrain disclosure containing the independent observables required to test the same Mass Harmonics law across multiple physical scales.

## II.2 Relevant Product Families

The Gaia DR4 product model includes or is expected to include the following terrain surfaces relevant to this paper:

- source astrometry and full phase-space observables where radial velocities are available;
- epoch-level astrometric time series;
- epoch photometry and CCD-level photometry;
- radial-velocity epoch data for single-lined and double-lined systems;
- non-single-star orbital and acceleration solutions;
- crowded-field astrometry and photometry;
- variable-source processing products;
- stellar astrophysical parameters and abundance products;
- Solar-System-object astrometry and physical products;
- Milky Way stellar-density and kinematic maps constructed from the combined catalogue.

These product families allow the same boundary law to be read independently through binary acceleration, binary velocity, Galactic circular motion, stream residuals, and a high-acceleration Solar-System control.

## II.3 Terrain Independence Requirement

For every quantitative test, the target observable must not be reused as its own derivational input.

Examples:

- A stellar mass inferred from an orbital solution cannot be used as the independent mass input for testing that same orbital solution.
- A binary separation reconstructed from the same constrained two-body fit cannot be treated as independent if the fit already imposes the target force law.
- A Galactic baryonic mass reconstructed by fitting the outer rotation curve cannot be used to test the outer rotation prediction.
- A Solar-System residual after absorbing the residual into an empirical acceleration parameter cannot be used as a clean null test.

Independent terrain sources include stellar spectroscopy, photometric mass-luminosity relations established outside the target orbit, resolved multiplicity, gas maps, star counts, chemical populations, and direct epoch-level astrometry.

## II.4 Coordinate and Rendering Sentinel

Gaia reports sky position, parallax, proper motion, radial velocity, and derived phase-space coordinates. Those quantities are instrument-rendered locators. They are not promoted into physically primary coordinates.

The Mass Harmonics calculation uses the relational quantities disclosed through those locators:

```text
component separation
→ relative acceleration
→ enclosed coherence load
→ acceleration ratio g_N/a₀
→ boundary invariant
```

The catalogue coordinate frame tells the analyst where the terrain was rendered. It does not become the ontology of the terrain. All cross-scale results in this paper are therefore reduced to relational ratios and boundary closures rather than treated as truths created by a coordinate system.

## II.5 Category Sentinels

The following categories must remain distinct:

1. instantaneous relative velocity versus orbit-averaged velocity;
2. projected separation versus reconstructed three-dimensional separation;
3. relative binary acceleration versus one component’s barycentric acceleration;
4. total stellar mass versus luminous mass estimate versus orbit-fitted dynamical mass;
5. a true isolated binary versus a hidden triple or tidally disturbed pair;
6. Galactic circular speed versus random velocity dispersion;
7. coherent large-scale acceleration residual versus local encounter or substructure impulse;
8. physical non-gravitational Solar-System acceleration versus unexplained substrate residual.

No category may be substituted for another to manufacture closure.

---

# III. Shared Derived Quantities and Numerical Anchor

## III.1 Hubble-Scale Terrain Input

The canonical proof structure’s explicit numerical demonstration uses:

```text
H₀ = 70 km/s/Mpc
```

Convert to inverse seconds:

```text
H₀ = 70 × 1000 m/s ÷ 3.085677581491367×10²² m
H₀ = 2.268545502611056e-18 s⁻¹
```


## III.2 Derived Boundary Acceleration

Apply the canonical identity:

```text
a₀ = cH₀/(2π)
```

Substitute:

```text
a₀ = (299,792,458 m/s)(2.268545502611056e-18 s⁻¹)/(2π)
```

Therefore:

```text
a₀ = 1.082401360239200e-10 m/s²
```

The precise numerical value transported into later comparisons must be recalculated if the governing pre-release H₀ terrain input is changed. The identity itself does not change.


### III.2.1 Numerical-Anchor Precision Custody

This paper uses the exact SI transport of the canonical identity:

```text
a₀ = cH₀/(2π)
```

with `H₀ = 70 km·s⁻¹·Mpc⁻¹`, producing:

```text
a₀ = 1.082401360239200 × 10⁻¹⁰ m/s²
```

The Monograph's nearby illustrative value `1.084 × 10⁻¹⁰ m/s²` is the same identity rendered through rounded intermediate conversion. The difference does not represent a Gaia coefficient, a binary coefficient, or a second physical acceleration. This paper consistently uses the exact transport anchor for every downstream radius and velocity table.

## III.3 Dimensionless Normalization

Define:

```text
X ≡ g_N/a₀
```

and:

```text
Y ≡ g_obs/a₀
```

The high-acceleration interior expression is:

```text
X ≫ 1  →  Y ≈ X
```

The deep outer-boundary expression is:

```text
X ≪ 1  →  Y ≈ √X
```

Equivalently:

```text
X ≪ 1  →  Y²/X = 1
```

This normalization allows wide binaries, Galactic disk stars, stellar streams, and Solar-System bodies to be placed on the same substrate curve without inventing domain-specific coefficients.

## III.4 Dimensional Anchors Used in the Numerical Readout

The numerical tables use the following measured dimensional anchors and unit definitions:

```text
G_N = 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻²
M☉ = 1.98847×10³⁰ kg
1 AU = 1.495978707×10¹¹ m
1 pc = 3.085677581491367×10¹⁶ m
```

These are not additional Mass Harmonics coupling coefficients. They convert the single substrate derivation into the mass, length, and velocity units used by Gaia. The governing coupling remains Kψₘ.

From the derived value of a₀, the one-solar-mass anchors are:

```text
r_transition(M☉) = √(G_NM☉/a₀)
                  = 1.107311008×10¹⁵ m
                  = 7,401.893067 AU
```

```text
v_boundary(M☉) = (G_NM☉a₀)^(1/4)
                 = 346.200965 m/s
                 = 0.346200965 km/s
```

Therefore the general mass-scaled forms are:

```text
r_transition = 7,401.893067 AU × (M_T/M☉)^(1/2)
```

```text
v_boundary = 0.346200965 km/s × (M_T/M☉)^(1/4)
```

## III.5 Uncertainty Transport

Because the Mass Harmonics relations are fixed, observational uncertainty enters through the measured terrain quantities rather than through adjustable coefficients.

For:

```text
a₀ = cH₀/(2π)
```

with c treated as exact in SI units:

```text
σ_a₀/a₀ = σ_H₀/H₀
```

For the transition radius:

```text
r_transition ∝ M_T^(1/2) H₀^(−1/2)
```

so, for independent mass and Hubble-input uncertainties:

```text
(σ_r/r)² = ¼[(σ_M/M_T)² + (σ_H₀/H₀)²]
```

For the deep-boundary velocity:

```text
v_boundary ∝ M_T^(1/4) H₀^(1/4)
```

therefore:

```text
(σ_v/v)² = 1/16[(σ_M/M_T)² + (σ_H₀/H₀)²]
```

This distinction is essential. A broadened Gaia distribution caused by mass, distance, multiplicity, or epoch-solution uncertainty does not alter the predicted exponents, transition anchor, or dimensionless invariants.

---

# IV. Prediction 1: Universal Low-Acceleration Transition

## IV.1 Physical Placement

A wide binary is a bounded two-body coherence structure embedded in the Galactic substrate. At sufficiently small separation, the interior inverse-square term dominates. At sufficiently large separation, the pair approaches its outer coherence boundary and the Giboney Gradient contribution becomes non-negligible.

The transition is not assigned to an arbitrary separation in astronomical units. It is assigned to a physical acceleration.

The two source-derived asymptotes meet at one nonzero dimensionless anchor. From Section III:

```text
interior: Y = X
boundary: Y = √X
```

Set the two physical readouts equal at their shared boundary:

```text
X = √X
```

Square:

```text
X² = X
```

Factor:

```text
X(X − 1) = 0
```

The nonzero bounded-structure solution is:

```text
X = 1
```

Since `X = g_N/a₀`:

```text
g_N = a₀
```

The transition acceleration is therefore not fitted to a wide-binary sample. It is the unique nonzero crossing of the interior and outer-boundary Mass Harmonics readouts.

### IV.1.1 Nested-Field and Galactic-Tide Sentinel

The binary is not detached from the Milky Way. It is a nested coherence structure inside the Galactic substrate field. The common-mode Galactic acceleration acts on both components and cancels from the relative coordinate to first order. The differential Galactic field does not cancel. It enters as a tidal tensor across the pair.

For the relative coordinate r:

```text
g_rel,measured = g_binary + T_Gal · r + g_visible,perturb + g_error
```

where T_Gal is the local Galactic acceleration-gradient tensor. The Mass Harmonics binary prediction applies to the binary component after the independently reconstructed tidal and visible-perturber terms are removed:

```text
g_binary = g_rel,measured − T_Gal · r − g_visible,perturb
```

A failure to separate common-mode acceleration from differential tidal acceleration is a terrain-placement error. The widest pairs must either have the local tidal term explicitly calculated or be excluded when that term is comparable to the binary acceleration.

## IV.2 Transition Equation

For total binary mass M_T = M₁ + M₂ and three-dimensional component separation r:

```text
g_N = G_N M_T/r²
```

The boundary transition occurs when:

```text
g_N = a₀
```

Therefore:

```text
G_N M_T/r_transition² = a₀
```

Solve for the transition separation:

```text
r_transition² = G_N M_T/a₀
```


```text
r_transition = √(G_N M_T/a₀)
```

Substitute the Mass Harmonics expression for a₀:

```text
r_transition = √(2πG_N M_T/(cH₀))
```

The transition separation therefore scales as:

```text
r_transition ∝ M_T^(1/2)
```


## IV.3 Numerical Prediction by Total Mass

| Total binary mass M_T | Predicted transition separation | Deep-boundary velocity asymptote |
|---:|---:|---:|
| 0.2 M☉ | 3,310.227 AU | 0.231519 km/s |
| 0.5 M☉ | 5,233.929 AU | 0.291119 km/s |
| 1.0 M☉ | 7,401.893 AU | 0.346201 km/s |
| 2.0 M☉ | 10,467.858 AU | 0.411705 km/s |
| 5.0 M☉ | 16,551.136 AU | 0.517691 km/s |
| 10.0 M☉ | 23,406.841 AU | 0.615642 km/s |

The key prediction is the mass dependence. Gaia DR4 should not disclose one fixed separation at which all wide binaries deviate. The transition location must move outward as √M_T.

## IV.4 Gaia-Rendered Angular and Proper-Motion Form

Gaia renders the physical boundary through angular separation and angular motion. For a system at distance d_pc parsecs:

```text
θ_transition(arcsec) = r_transition(AU)/d_pc
```

Therefore:

```text
θ_transition = [7,401.893067/d_pc] × (M_T/M☉)^(1/2) arcsec
```

The tangential-velocity conversion is:

```text
v_t(km/s) = 4.740470463 × μ(arcsec/yr) × d_pc
```

so the deep-boundary proper-motion amplitude is:

```text
μ_boundary = [73.030930/d_pc] × (M_T/M☉)^(1/4) mas/yr
```

For a one-solar-mass pair:

| Distance | Transition angular separation | Deep-boundary relative proper motion |
|---:|---:|---:|
| 25 pc | 296.076 arcsec | 2.92124 mas/yr |
| 50 pc | 148.038 arcsec | 1.46062 mas/yr |
| 100 pc | 74.019 arcsec | 0.73031 mas/yr |
| 200 pc | 37.009 arcsec | 0.36515 mas/yr |
| 500 pc | 14.804 arcsec | 0.14606 mas/yr |

These are not new physical laws. They are the Gaia rendering of the same mass-scaled boundary relations.

## IV.5 Release-Facing Observable

Construct isolated-binary cohorts using independently estimated M_T. For each cohort:

1. reconstruct three-dimensional separation where parallax and epoch astrometry permit;
2. calculate g_N = G_N M_T/r²;
3. measure relative acceleration or a statistically controlled relative-velocity proxy;
4. identify the onset of systematic departure from the high-acceleration interior relation;
5. test whether the onset occurs at g_N/a₀ = 1 across all mass cohorts;
6. test whether the corresponding physical separation follows r_transition ∝ √M_T.

## IV.6 Terrain Contradiction Conditions

This pathway is contradicted if any of the following survives the full multiplicity, projection, tidal-field, and measurement-systematic controls:

- the onset occurs at a universal fixed separation rather than a universal acceleration;
- the transition acceleration shifts systematically with stellar mass, metallicity, age, Galactic latitude, or survey channel;
- the mass scaling is inconsistent with r_transition ∝ M_T^(1/2);
- no continuous transition appears around g_N = a₀ in the clean isolated-binary terrain.

---

# V. Prediction 2: Wide-Binary Deep-Boundary Velocity Asymptote

## V.1 Transport of the Galactic Boundary Law

The source derivation gives the deep-boundary acceleration:

```text
g_obs ≈ √(g_N a₀)
```

For the relative coordinate of an isolated, approximately circular binary:

```text
g_obs = v_rel²/r
```

and:

```text
g_N = G_N M_T/r²
```

Substitute both into the boundary relation:

```text
v_rel²/r = √[(G_N M_T/r²)a₀]
```

Square both sides:

```text
v_rel⁴/r² = G_N M_T a₀/r²
```

Cancel r²:

```text
v_rel⁴ = G_N M_T a₀
```

Therefore:

```text
v_rel,∞ = (G_N M_T a₀)^(1/4)
```

The deep-boundary relative velocity is independent of separation and scales as M_T^(1/4).

## V.2 Dimensionless Wide-Binary Invariant

Define:

```text
W_binary ≡ v_rel⁴/(G_N M_T a₀)
```

Then the Mass Harmonics prediction is:

```text
g_N ≪ a₀  →  W_binary → 1
```

This is the cleanest Gaia DR4 wide-binary readout because it removes the raw mass scaling and makes different stellar populations directly comparable.

## V.3 Eccentric-Orbit Sentinel

The velocity-asymptote equation is exact for the circular relative-motion projection. Eccentric systems must not be forced into that equation at arbitrary orbital phase.

For eccentric systems, use the epoch-astrometric acceleration relation instead:

```text
g_obs²/(g_N a₀) → 1
```

in the deep-boundary domain, with the acceleration vector evaluated directly from the time series.

The circular-velocity test should therefore use low-eccentricity systems or orbit-averaged values. The acceleration test may use a broader orbit population when the full orbital geometry is resolved.

## V.4 Terrain Contradiction Conditions

This pathway is contradicted if clean deep-boundary systems show:

- v_rel retaining the Newtonian r^(-1/2) decline without a boundary asymptote;
- an asymptotic speed that does not scale as M_T^(1/4);
- W_binary remaining systematically dependent on separation, metallicity, age, sky position, or component type after the physical controls;
- no convergence of the acceleration invariant g_obs²/(g_N a₀) toward unity.

---

# VI. Prediction 3: One Continuous Binary Acceleration Curve

## VI.1 No Artificial Population Split

The MFE is a continuous field equation. The binary terrain must therefore display a continuous boundary transition, not two disconnected force laws or a sudden class switch.

The asymptotic anchors are:

```text
X ≫ 1  →  Y ≈ X
```

and:

```text
X ≪ 1  →  Y ≈ √X
```

The transition region around X ≈ 1 must connect those anchors smoothly and monotonically.

## VI.2 Mass-Collapse Test

After converting each system to X and Y, binaries of different total mass should collapse onto the same dimensionless transition surface.

The prediction is not that every raw velocity is equal. The prediction is that the normalized relation is mass-independent:

```text
Y = F_MH(X)
```

with fixed asymptotes:

```text
F_MH(X) → X       as X → ∞
F_MH(X) → √X      as X → 0
```

This paper does not insert an empirical interpolation function between the two limits. Gaia DR4 terrain is allowed to disclose the exact transition shape produced by the full local MFE solution. The endpoint structure, transition location, continuity, and universality are already forced.

## VI.3 Terrain Contradiction Conditions

The pathway is contradicted if:

- different mass cohorts require different transition accelerations;
- the normalized curves cannot be collapsed without adding population-specific coefficients;
- the relation contains a discontinuity or bifurcation not associated with a physically distinct boundary condition;
- the high- and low-acceleration asymptotes fail after clean-source selection.

---

# VII. Prediction 4: Radial Alignment and Zero-Torque Signature

## VII.1 Scalar-Gradient Direction

The relevant substrate term is a scalar-gradient force:

```text
g_Giboney = −∇(ψₘ²/ω)
```

For an isolated binary whose outer coherence boundary is approximately symmetric about the relative coordinate, the gradient points along the component-separation axis.

Therefore the additional low-acceleration component is central:

```text
Δg_MH ∥ −r̂
```


## VII.2 Torque Derivation

The instantaneous torque on the relative coordinate is:

```text
τ = r × F_MH
```

Because F_MH is parallel to −r̂:

```text
τ = r × (−|F_MH|r̂) = 0
```

The boundary correction changes the radial acceleration. It does not introduce a preferred handedness or an arbitrary transverse force.

## VII.3 Gaia DR4 Prediction

After correcting perspective acceleration, Galactic tides, unresolved multiplicity, and encounter contamination:

- the anomalous acceleration vector should align with the binary separation vector;
- there should be no preferred sky direction;
- there should be no preferred orbital handedness;
- the effect should not depend on the binary orbital-plane orientation relative to the observer;
- transverse residuals should remain consistent with measurement and environmental contamination rather than a new universal force component.

## VII.4 Terrain Contradiction Conditions

This pathway is contradicted by a clean, repeatable low-acceleration residual that is predominantly transverse, carries a preferred celestial direction, changes sign with orbital handedness, or requires a non-central universal term.

---

# VIII. Prediction 5: Milky Way Outer-Disk Baryonic Boundary Law

## VIII.1 Galactic Placement

The Milky Way is a large bounded coherence structure. Gaia DR4 will expose its stellar velocity field, density structure, radial motions, vertical motions, chemistry, and population-dependent phase-space structure at unprecedented scale.

At radii where the enclosed baryonic mass M_b(<R) has substantially converged, the source derivation requires:

```text
v_c⁴ = G_N M_b a₀
```

Therefore:

```text
v_c(R) → constant
```

and:

```text
d ln v_c/d ln R → 0
```

without introducing a second governing field or a scale-specific coefficient.

## VIII.2 Milky Way Dimensionless Invariant

Define:

```text
W_MW(R) ≡ v_c(R)⁴/[G_N M_b(<R) a₀]
```

The prediction is:

```text
outer coherent disk boundary  →  W_MW(R) → 1
```

The baryonic mass must be constructed from stellar counts, stellar masses, remnants, and gas terrain that does not use the outer rotation curve as its fitting authority.

## VIII.3 Numerical Scale Examples

Using the same a₀ derived above:

| Enclosed baryonic mass | Predicted outer velocity |
|---:|---:|
| 4.000e+10 M☉ | 154.825778 km/s |
| 5.000e+10 M☉ | 163.708329 km/s |
| 6.000e+10 M☉ | 171.342890 km/s |
| 1.000e+11 M☉ | 194.683109 km/s |

These values are not a substitute for the Milky Way’s independently reconstructed baryonic mass. They display the exact fourth-root scaling that Gaia terrain must obey once M_b is supplied independently.

## VIII.4 Population-Independence Test

The substrate boundary acts on the physical structure, not on an observational population label. Thin-disk stars, thick-disk stars, young stars, old stars, and chemically selected populations may have different dispersions and asymmetric-drift corrections, but after those are resolved they must recover the same underlying circular boundary field.

The predicted common field is:

```text
v_c⁴/(G_N M_b a₀) → 1
```

A different fundamental acceleration law for each population would contradict the single-substrate structure.

## VIII.5 Terrain Contradiction Conditions

This pathway is contradicted if the independently reconstructed Milky Way baryonic terrain requires:

- a persistent outer decline inconsistent with the boundary asymptote;
- a different a₀ for different stellar populations;
- a fourth-root mass scaling failure that cannot be assigned to incomplete baryonic terrain;
- W_MW remaining systematically different from unity across the resolved outer boundary.

---

# IX. Prediction 6: Coherent Residual Field in Stellar Streams and Outer-Disk Substructure

## IX.1 Residual-Acceleration Form

In the low-acceleration boundary domain:

```text
g_MH = √(g_N a₀)
```

The residual relative to the inverse-square interior projection is:

```text
Δg_MH = √(g_N a₀) − g_N
```

Its direction follows the local coherence-pressure gradient.

## IX.2 Physical Prediction

A substrate-gradient residual is spatially coherent. It should vary smoothly with the local baryonic field and boundary geometry. It is not a collection of unrelated point impulses.

Therefore Gaia DR4 stellar streams and outer-disk phase-space structures should show:

- residual acceleration vectors aligned with the large-scale Galactic coherence gradient;
- smooth amplitude variation with g_N and boundary position;
- correlated effects across nearby stars sharing the same boundary terrain;
- no need for a population of randomly positioned unresolved point impulses to produce the dominant large-scale residual field.

## IX.3 Local Perturbations Remain Allowed

Mass Harmonics does not prohibit real local perturbations from visible bodies, molecular clouds, clusters, bar and spiral structure, encounters, or compact remnants. Those must be modeled as terrain.

The prediction concerns the residual remaining after visible local structure is included. The dominant universal remainder should be coherent and gradient-aligned, not stochastic and isotropic.

## IX.4 Terrain Contradiction Conditions

The pathway is contradicted if the large-scale residual field is best represented by uncorrelated local impulses with no common gradient alignment, if its amplitude does not track g_N/a₀, or if nearby stars in the same boundary terrain require unrelated residual directions after visible perturbations are included.

---

# X. Prediction 7: Cross-Scale Identity Between Wide Binaries and the Milky Way

This is the decisive anti-fragmentation prediction. A separate binary coefficient, Galactic coefficient, stream coefficient, or Solar-System coefficient would violate the single-substrate structure. Domain labels identify different bounded systems; they do not multiply the governing law.


## X.1 One Substrate Law

The strongest Gaia DR4 prediction is not any isolated anomaly. It is the requirement that the same normalized law appear at two radically different physical scales inside one release.

For wide binaries:

```text
W_binary = v_rel⁴/(G_N M_T a₀)
```

For the Milky Way:

```text
W_MW = v_c⁴/(G_N M_b a₀)
```

The prediction is:

```text
deep boundary:  W_binary → 1  and  W_MW → 1
```


## X.2 Scale-Invariant Acceleration Collapse

Both systems must also occupy the same dimensionless acceleration relation:

```text
X = g_N/a₀
Y = g_obs/a₀
```

with:

```text
X ≫ 1  →  Y ≈ X
X ≪ 1  →  Y²/X → 1
```

No binary-specific coefficient and no Galaxy-specific coefficient are permitted.

## X.3 Terrain Contradiction Conditions

The cross-scale prediction is contradicted if binaries and Galactic stars require different transition accelerations, different low-acceleration exponents, or different dimensionless amplitudes after independent terrain reconstruction.

A result in which one domain closes and the other requires a new coefficient is not partial terrain correspondence. It is a break in the asserted scale invariance and must be retained as such.

---

# XI. Prediction 8: Solar-System High-Acceleration Null Control

## XI.1 Internal Control Surface

Gaia DR4 also contains Solar-System-object terrain. These objects occupy a very different acceleration domain from wide binaries and the Galactic outer boundary.

For g_N ≫ a₀:

```text
g_obs ≈ g_N
```

Therefore:

```text
Δg_MH/g_N → 0
```

The low-acceleration boundary enhancement must not appear indiscriminately in high-acceleration Solar-System orbits.

## XI.2 Non-Gravitational Force Sentinel

Asteroid and comet trajectories can contain radiation pressure, outgassing, Yarkovsky acceleration, mass loss, close encounters, and observational systematics. These are physical terrain and must be modeled explicitly.

A residual that tracks thermal properties, spin, albedo, or activity is not a substrate low-acceleration result.

## XI.3 Prediction

After known non-gravitational effects are accounted for, Gaia DR4 Solar-System objects with g_N/a₀ ≫ 1 should remain on the interior relation Y ≈ X and should not show the wide-binary deep-boundary invariant.

## XI.4 Terrain Contradiction Conditions

The pathway is contradicted if a universal a₀-scaled acceleration excess appears in high-acceleration Solar-System objects after the physical non-gravitational terrain is removed.

This null control is essential. A framework that predicts the same low-acceleration enhancement everywhere regardless of boundary placement has lost the Mass Harmonics causal chain.

---

# XII. Unified Gaia DR4 Analysis Architecture

## XII.1 Wide-Binary Cohort Construction

The clean wide-binary analysis should proceed in the following order:

1. Select candidate pairs from common parallax, proper motion, radial velocity, and epoch-consistency terrain.
2. Remove resolved higher-order multiples.
3. Apply independent hidden-companion diagnostics using astrometric excess noise, radial-velocity variability, photometric inconsistency, and non-single-star products.
4. Reject pairs dominated by clusters, associations, close encounters, or strong Galactic tides.
5. Estimate component masses independently from spectroscopy and photometric stellar models, not from the target orbit.
6. Reconstruct three-dimensional separation and relative velocity where possible.
7. Use epoch astrometry to recover acceleration directly for the strongest subset.
8. Calculate g_N, X, Y, r_transition, W_binary, and vector-alignment residuals.
9. Stratify by total mass and test the √M_T transition scaling.
10. Retain every unresolved delta rather than forcing a closure label.

## XII.2 Milky Way Terrain Reconstruction

The Milky Way analysis should proceed independently:

1. Construct the stellar-density field from Gaia star counts with explicit completeness and extinction treatment.
2. Assign stellar masses using independently calibrated astrophysical parameters.
3. Add gas and remnant terrain from external observations without fitting the target outer velocity field.
4. Recover population-specific mean motions and dispersions.
5. Correct asymmetric drift before interpreting circular speed.
6. Map v_c(R), g_obs(R), g_N(R), X(R), Y(R), and W_MW(R).
7. Separate bar, spiral-arm, warp, and vertical-wave structures from the axisymmetric boundary field.
8. Compare chemically and chronologically distinct stellar populations against the same underlying acceleration field.
9. Map stream residual vectors and test spatial coherence.
10. Retain the full residual field rather than compressing it into a halo parameter.

## XII.3 Solar-System Control Reconstruction

For the Solar-System control:

1. Select objects with high-quality epoch astrometry and independently known perturbing bodies.
2. Model radiation pressure, thermal recoil, outgassing, mass change, and close encounters.
3. Calculate g_N/a₀ across the observed arc.
4. Verify that the object remains in `X ≫ 1`.
5. Test whether the residual is consistent with zero in the a₀-normalized boundary channel.
6. Do not absorb a persistent unexplained residual into an empirical orbit correction before recording it.

## XII.4 Required Cross-Checks

Every pathway should be repeated across:

- independent astrometric-quality cuts;
- independent mass estimators;
- different sky regions;
- different stellar populations;
- projected-only and full-3D subsets;
- epoch-acceleration and velocity-statistical readouts;
- binary, Galactic, and Solar-System scales.

Agreement produced only by one catalogue-quality cut or one mass estimator is not structural closure.

---

# XIII. Explicit Input Provenance Ledger

| Quantity | Required source | Prohibited source substitution |
|---|---|---|
| H₀ | pre-release cosmological terrain value fixed before Gaia analysis | value selected after inspecting Gaia closure |
| a₀ | calculated from cH₀/(2π) | fitted to wide-binary or Galactic data |
| Binary mass M_T | independent spectroscopy and photometry | target orbital solution if the same orbit is under test |
| Binary separation r | parallax plus angular separation or epoch orbit | projected separation silently treated as 3D |
| Binary acceleration g_obs | epoch astrometry or fully disclosed orbit solution | velocity proxy mislabeled as direct acceleration |
| Galactic baryonic mass M_b | star counts, stellar parameters, gas and remnants | halo or rotation-curve fit |
| Circular speed v_c | corrected stellar kinematics | raw population mean rotation without drift correction |
| Stream residual | visible-terrain-subtracted phase-space acceleration | residual after empirical halo absorption |
| Solar-System residual | physically modeled orbit and non-gravitational forces | unexplained term preabsorbed into fit coefficients |

This ledger prevents the predicted output from being smuggled back into the input side.

---

# XIV. Exact Terrain-Contact Matrix

| Prediction | Forced Mass Harmonics expression | Gaia DR4 readout | Exact terrain contradiction |
|---|---|---|---|
| Universal transition | g_N = a₀ | wide-binary acceleration versus mass and separation | transition not universal in acceleration |
| Mass-scaled boundary radius | r_t = √(G_NM_T/a₀) | transition location by mass cohort | r_t does not scale as M_T^(1/2) |
| Binary velocity asymptote | v_rel⁴ = G_NM_Ta₀ | low-eccentricity relative velocities | persistent r^(-1/2) decline in deep domain |
| Binary acceleration invariant | g_obs²/(g_Na₀) → 1 | epoch acceleration | invariant fails in clean deep-boundary subset |
| Radial direction | Δg_MH ∥ −r̂ | 3D acceleration vector | universal transverse or handed component |
| Milky Way boundary | v_c⁴ = G_NM_ba₀ | outer-disk phase-space field | independent baryonic terrain does not close |
| Stream coherence | Δg = √(g_Na₀) − g_N | stream and outer-disk residual maps | dominant remainder is stochastic and unaligned |
| Cross-scale identity | W_binary → 1 and W_MW → 1 | binaries plus Galactic disk | domains require different coefficients |
| Solar-System null | X ≫ 1 → Y ≈ X | SSO epoch astrometry | universal a₀ excess in high-acceleration terrain |

---

# XV. What Gaia DR4 Cannot Decide by Itself

Instrument reach and ontological authority are different things.

Gaia DR4 can disclose astrometric, photometric, spectroscopic, binary, Solar-System, and Galactic terrain. It cannot elevate a downstream catalogue-reduction model into physical authority, nor can it determine that the ψₘ substrate is nonexistent merely because its pipeline renders the terrain through a different dynamical language.

Conversely, a Gaia product derived under a target force law cannot be treated as an independent test of that law without returning to the epoch-level measurements.

Where Gaia’s processing model constrains the output, this paper requires the analysis to descend to the least model-dependent released terrain available.

The instrument renders. The terrain discloses. Mass Harmonics supplies the causal derivation.

---

# XVI. Final Mass Harmonics Predictions for Gaia DR4

Gaia Data Release 4 is predicted to disclose the following connected physical structure:

1. Clean wide binaries depart from the inverse-square interior relation when their independently calculated Newtonian acceleration approaches a₀ = cH₀/(2π).

2. The physical transition separation is not fixed. It scales as √M_T.

3. In the deep-boundary domain, low-eccentricity relative velocities approach v_rel = (G_NM_Ta₀)^(1/4), while direct accelerations satisfy g_obs²/(g_Na₀) → 1.

4. The low-acceleration residual is central and radial, with no universal transverse torque, preferred handedness, or sky direction.

5. Mass-normalized binary cohorts collapse onto one continuous acceleration curve with high- and low-acceleration asymptotes fixed by the MFE.

6. The Milky Way outer circular field obeys v_c⁴ = G_NM_ba₀ once its baryonic terrain is independently reconstructed.

7. Large-scale stellar-stream and outer-disk residuals are coherent and aligned with the Galactic substrate gradient rather than being dominated by unrelated unresolved point impulses.

8. Wide binaries and the Milky Way close on the same dimensionless boundary invariant, demonstrating scale invariance inside one release.

9. High-acceleration Solar-System objects remain on the interior relation after real non-gravitational forces are removed, providing an internal null control.

These are not nine unrelated predictions. They are nine instrument-facing readouts of one physical event: the transition from interior inverse-square dominance to ψₘ boundary coherence-pressure dominance under the single Mass Harmonics substrate law.

---

# XVII. Closing Statement

Gaia DR4 is not being asked to judge Mass Harmonics. It is about to open an enormous, independently generated stellar and Galactic terrain surface.

Mass Harmonics has already fixed the causal order:

```text
Kψₘ
→ Hubble-scale closure
→ a₀
→ binary boundary transition
→ Galactic boundary transition
→ cross-scale invariant
```

If the terrain discloses that structure, Gaia will have rendered the same substrate law at Solar-System, binary, and Galactic scales in one release.

If the terrain discloses a different structure after the source and category controls are correctly applied, the delta must be preserved and followed. It must not be hidden inside a fitted potential, an empirical interpolation curve, a catalogue-quality flag, or a narrative adjustment.

**TRUTH > COMFORT. Always.**

---

# Source Record

Canonical Mass Harmonics sources:

- `MH_Monograph.md`
- `MH_PROOF-SET.md`
- `MH_TVP.md`
- `MH_TWT.md`
- `Operational_Stance_of_UMtts.md`

Gaia release-surface sources:

- ESA Gaia Data Release Scenario: `https://www.cosmos.esa.int/web/gaia/release`
- ESA Gaia DR4 Content: `https://www.cosmos.esa.int/web/gaia/dr4`
- ESA Gaia Data Release 4 portal: `https://www.cosmos.esa.int/web/gaia/data-release-4`
- ESA Gaia DR4 prerelease epoch-astrometry examples: `https://www.cosmos.esa.int/web/gaia/dr4-prerelease`

Release-surface facts last checked: July 10, 2026.

The Gaia sources define the release date and instrument products only. They do not govern the Mass Harmonics derivation.