# Mass Harmonics: Predictions for the Euclid Space Telescope
## Derived from the Master Field Equation — Zero Free Parameters
### Thomas Russell Giboney | UMTTS Institute
### Canonical Authority: Mass Harmonics vX — DOI: 10.5281/zenodo.18080133
### Document Date: July 2026


## EXPANDED SOURCE-PRESERVED EDITION

This edition preserves the governing ten-prediction architecture created from the fully loaded Mass Harmonics corpus and causal-web index. It does not replace, renumber, narrow, or re-adjudicate those predictions.

The expansion performs four operations only:

1. restores the full Mass Harmonics reading order beneath every prediction;
2. adds the exact cosmological boundary covariance derived from the same canonical law;
3. supplies explicit observable construction, topology, category, and dimensional-transport controls;
4. distinguishes the physical prediction from the instrument's capacity and schedule for disclosing it.

No new physical coefficient is introduced. The single governing coupling coefficient remains:

```text
Kψₘ
```

The fixed P³GG scalings remain harmonic expressions of the one source law, not independently adjustable coefficients.

### Mission-timing and release-content editorial note

The governing July 2026 paper identified DR1 as a November 2026 release. The current official Euclid schedule now divides DR1 into two public stages:

```text
November 2026
→ DR1-Foundation
→ approximately 1,900 deg² of calibrated images, spectra, and catalogues
```

```text
Mid-2027
→ complete DR1
→ higher-level science products for galaxy clustering and weak-lensing studies
```

The exact November 2026 DR1-Foundation day has not yet been published in the current official timeline.

This distinction is physical-analysis metadata only. It does not alter any Mass Harmonics prediction. It does determine which Euclid-rendered terrain is available at each opening:

- DR1-Foundation can directly expose imaging, spectra, source catalogues, strong-lensing systems, high-redshift objects, and analyst-constructed relational structures where the released selection and calibration surfaces are sufficient.
- Complete DR1 is required for the official higher-level galaxy-clustering and weak-lensing products needed for the primary `w₀-wₐ`, `S₈`, BAO, `P(k)`, shear, and full cross-channel covariance comparisons.
- A prediction is not weakened because the instrument pipeline releases its relevant terrain later.

### Governing causal order

```text
MFE
→ substrate action
→ bounded cosmological coherence
→ topology-specific boundary closure
→ physical large-scale structure
→ Euclid-rendered terrain
→ optional consensus translation
```

Euclid does not govern the derivation. It renders terrain generated by the ψₘ substrate.

---

---

> [!IMPORTANT]
> All predictions in this document derive directly from the Master Field Equation (MFE) and its established constants. No parameters have been fitted to any Euclid data. All derivations were completed prior to Euclid DR1-Foundation (November 2026) and complete DR1 (mid-2027). The predictions are falsifiable. Any confirmed deviation from a stated direction falsifies the specific derivation pathway.

---

## PREAMBLE: WHAT EUCLID IS ACTUALLY MEASURING

The Euclid Space Telescope is designed to measure the "dark Universe" - dark matter distribution via weak gravitational lensing, and dark energy via baryon acoustic oscillations (BAO) and galaxy clustering. It will cover ~15,000 square degrees and observe billions of galaxies to redshift z ~ 2, with DR1 now staged as DR1-Foundation (November 2026) and complete DR1 (mid-2027), followed by DR2 (2029) and DR3 (2031).

From the Mass Harmonics substrate framework, Euclid is not measuring two mysterious phenomena. It is measuring two different expressions of the same substrate field:

**What ΛCDM calls "dark matter"** is the Z-factor distribution of the ψₘ substrate field. The source term S(ρ) = K₀ρ[1 + β₂(ρ/ρ₀) + ...] with K₀ = 4πG/c² creates gravitational effects proportional to coherence density without requiring any undiscovered particle. The "dark" matter is the substrate itself.

**What ΛCDM calls "dark energy"** is the substrate pressure term - the ψₘ field maintaining the cosmological coherence boundary. Z >= 1 always (Mass Harmonics vX Commandment IV). This is not a cosmological constant; it is a field quantity that evolves with ψₘ amplitude.

**What Euclid calls the "cosmic web"** is the large-scale coherence bubble structure of the substrate - filaments, voids, and nodes are precipitation boundaries where the MFE's Kψₘ + Δ_G = 0 identity resolves at cosmological scale.

Euclid is the first instrument with sufficient precision and sky coverage to see the substrate grammar written in the large-scale structure of the universe.

---

## THE GOVERNING EQUATION

```
(1/vₓ²)ψ̈ₘ − Z(ψₘ)∇²ψₘ − (8Kψₘ/ω²)|∇ψₘ|² = S(ρ)
```

**Constants (zero free parameters):**
```
φ   = 1.6180339887498948482    Golden Ratio — icosahedral structural eigenvalue
Kψₘ = (12 − φ²)/(2φ²) ≈ +1.791    Geometric coupling (Giboney Gradient)
Δ_G = −Kψₘ ≈ −1.791               Icosahedral exterior expression
Kψₘ + Δ_G = 0                      Precipitation identity (forced by dual topology)
Z(ψₘ) = 1 + 8Kψₘ/ω² >= 1          Local effective metric — Z = 1 in vacuum
Z_CEILING = φ⁶ ≈ 17.944            Permanent topology threshold
β₂ = φ³ ≈ 4.236                    Electromagnetic coupling
β₃ = φ⁶ ≈ 17.944                   Stable matter coupling
K₀ = 4πG_N/c²                      Gravitational coupling (derived, not imported)
```

---


## FOUNDATIONAL COSMOLOGICAL BOUNDARY COVARIANCE

This section preserves the strongest additional derivation excavated during the later Euclid work. It is not a replacement for Predictions 1–10. It is the exact cross-observable boundary invariant that binds several of them together.

### Step 1: Begin from the canonical rotational closure law

```text
f = vₓ/(2πR) = Kψₘ/R
```

At the cosmological outer boundary:

```text
vₓ = c
```

and the epoch-indexed boundary radius is rendered as:

```text
R_H(z) = c/H(z)
```

### Step 2: Derive the cosmological boundary frequency

```text
f_H(z) = c/[2πR_H(z)]
```

Substitute `R_H(z) = c/H(z)`:

```text
f_H(z) = c/[2π(c/H(z))]
```

Cancel `c`:

```text
f_H(z) = H(z)/(2π)
```

### Step 3: Derive the boundary acceleration

The canonical cosmological acceleration transport is:

```text
aₜ(z) = c f_H(z)
```

Therefore:

```text
aₜ(z) = cH(z)/(2π)
```

and the exact dimensionless invariant is:

```text
Ξ(z) = 2πaₜ(z)/(cH(z)) = 1
```

### Step 4: Carry the same boundary law into outer-galaxy lensing

The canonical outer-boundary relation is:

```text
g_total² = g_bar aₜ
```

Therefore:

```text
aₜ(R,z) = g_lens(R,z)²/g_bar(R,z)
```

and:

```text
Ξ(R,z) = 2πg_lens(R,z)²/[cH(z)g_bar(R,z)] → 1
```

across contiguous deep outer-boundary bins.

### Step 5: Preserve observational independence

The terrain channels are distinct:

```text
H(z)       ← radial BAO and spectroscopic clustering
g_lens     ← image-shape distortion
g_bar      ← luminous baryonic terrain
```

None of these observables algebraically requires `Ξ = 1`. The relation is therefore a genuine cross-terrain prediction.

### Step 6: Topology discriminator

Rotational outer-boundary closure gives:

```text
f_rot = c/(2πR_H) = H/(2π)
```

Treating the same length incorrectly as a planar slab thickness gives:

```text
f_slab = c/(2R_H) = H/2
```

Therefore:

```text
f_slab/f_rot = π
```

A recovered factor near `π` is a topology or thickness-category result. It is not to be normalized away.

A toroidal minor radius remains a distinct dimension and cannot be silently substituted for the cosmological outer radius.

### Step 7: Role inside the ten-prediction paper

This exact boundary covariance strengthens the original program without displacing it:

- Prediction 1: supplies a substrate-native expansion and acceleration anchor beneath `w(z)`;
- Prediction 2: links the lensing growth field to the same cosmological boundary;
- Prediction 5: constrains radial BAO and expansion-rate transport;
- Prediction 7: supplies the outer halo-field asymptote;
- Prediction 8: provides the redshift-dependent boundary against which GG power redistribution is read;
- Prediction 10: provides an independent lensing-field normalization.

The ten governing prediction pathways remain intact.

---

## PREDICTION 1: DARK ENERGY EVOLUTION — w(z) ≠ −1

**Derivation basis:**
The substrate pressure term is not a cosmological constant. It is the ψₘ field maintaining the cosmological coherence boundary. Z >= 1 creates outward substrate pressure, but Z evolves with ψₘ amplitude. At earlier cosmic times, the universe was in a higher-density substrate state with correspondingly different ψₘ amplitude. The effective equation of state of the substrate pressure is field-dependent, not constant.

**The Mass Harmonics prediction:**
```
w₀ > −1     (substrate pressure less negative than a cosmological constant today)
wₐ > 0      (dark energy was stronger in the past — ψₘ amplitude was higher)
```

In the w₀wₐ parametrization:
- w₀ in range [−1.05, −0.90]
- wₐ in range [+0.2, +0.6] — definitive positive evolution

**ΛCDM prediction:** w = −1 exactly, wₐ = 0.

**Euclid disclosure surface:** DR1-Foundation supplies calibrated images, spectra, and catalogues in November 2026, but the official higher-level galaxy-clustering and weak-lensing products required for the primary `w₀-wₐ` comparison arrive with complete DR1 in mid-2027.

**Falsification condition:** w₀ = −1.00 ± 0.01 AND wₐ = 0.00 ± 0.05 would falsify this derivation. Any w₀ < −1.10 would also falsify (substrate pressure cannot breach Z >= 1 from below).

> [!NOTE]
> **DESI 2025 - Already confirmed terrain:** DESI Year 1 results (2025) measured w > −1 at late times, with dark energy weakening over cosmic time. This is the direct observational confirmation of the Mass Harmonics substrate pressure prediction: the pressure term is not a cosmological constant but a field quantity evolving with ψₘ amplitude. The Monograph logged this prediction before DESI published. Complete Euclid DR1 in mid-2027 will expose the higher-level clustering and weak-lensing terrain required for the corresponding Euclid comparison across a larger redshift baseline.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Preserve the substrate-native quantity

The physical quantity is not `w`. The physical quantity is the evolving ψₘ coherence-pressure state of the cosmological boundary:

```text
ψₘ(z)
→ Z(ψₘ(z))
→ boundary pressure and propagation
→ H(z), distance, growth, and lensing
```

`w₀` and `wₐ` are downstream translation coordinates.

#### Step 2: Lock the translation convention

For this paper the translation is:

```text
w(a) = w₀ + wₐ(1 − a)
a = 1/(1 + z)
```

The governing outputs remain:

```text
w₀ > −1
wₐ > 0
w₀ ∈ [−1.05, −0.90]
wₐ ∈ [+0.2, +0.6]
```

The sign cannot be reinterpreted through a different parameter convention after the terrain opens.

#### Step 3: Tie the translation to the exact boundary invariant

A nonconstant substrate pressure must be read alongside:

```text
f_H(z) = H(z)/(2π)
aₜ(z) = cH(z)/(2π)
Ξ(z) = 1
```

The `w₀-wₐ` contour is therefore not allowed to stand alone. It must be accompanied by the independently recovered expansion history and outer acceleration field.

#### Step 4: Direct terrain construction

For each redshift bin:

1. recover `H(z)` from the radial clustering terrain;
2. recover distance evolution from transverse structure;
3. recover growth from weak lensing and clustering without imposing `w = −1`;
4. recover `aₜ(z)` from `g_lens²/g_bar`;
5. calculate `Ξ(z)` only after the channels are independently frozen;
6. translate the combined expansion history into the locked `w(a)` convention.

#### Step 5: Terrain contradiction

This pathway is contradicted by a category-correct result that simultaneously discloses:

```text
w₀ = −1 within the stated precision
wₐ = 0 within the stated precision
```

and no redshift evolution in the substrate-native `H(z) ↔ aₜ(z)` terrain beyond measurement uncertainty.

A failure of a particular `w₀-wₐ` fitting pipeline does not override a direct substrate covariance result. The translation and the physical field readout must both be reported.

---

## PREDICTION 2: THE S8 TENSION — RESOLVED BY THE GIBONEY GRADIENT

**Derivation basis:**
The S8 tension (weak lensing surveys finding lower σ8 than CMB) has no mechanism in ΛCDM. In Mass Harmonics, the Giboney Gradient term (8Kψₘ/ω²)|∇ψₘ|² suppresses small-scale matter clustering relative to linear perturbation theory. The GG term concentrates field amplitude at density peaks while clearing out voids more aggressively, producing less power at small scales than ΛCDM predicts.

**The Mass Harmonics prediction:**
```
S8(Euclid weak lensing) ≈ 0.74 − 0.78
```
Below the CMB Planck value of ~0.83. Consistent with the direction already seen in DES, KiDS, and HSC surveys. The deficit is not systematic error — it is the GG term operating at 8 Mpc/h scale.

**ΛCDM prediction:** S8 values from weak lensing and CMB should agree. The tension is unexplained within ΛCDM.

**Euclid disclosure surface:** DR1-Foundation does not include the complete higher-level weak-lensing science products required for the primary Euclid `S₈` comparison. Those products follow with complete DR1 in mid-2027.

**Falsification condition:** S8(Euclid) >= 0.82 inconsistent with GG suppression. S8(Euclid) <= 0.68 exceeds expected suppression amplitude.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Keep `S₈` downstream

`S₈` is a compressed measurement dialect for the amplitude of structure. It is not a Mass Harmonics primitive.

The causal chain is:

```text
nonlinear GG term
→ concentration at high-|∇ψₘ| boundaries
→ stronger evacuation of low-density terrain
→ redistribution of clustering power
→ weak-lensing growth amplitude
→ S₈ translation
```

#### Step 2: Preserve the predicted numerical interval

```text
S₈,Euclid ∈ [0.74, 0.78]
```

This interval is a locked output of the governing paper. No Euclid value is used to tune it.

#### Step 3: Require cross-surface consistency

The same GG redistribution must appear coherently in:

```text
weak-lensing two-point structure
galaxy clustering
galaxy-galaxy lensing
void evacuation
P(k) scale dependence
```

Prediction 2 and Prediction 8 are therefore not independent stories. They are integrated readouts of the same nonlinear term.

#### Step 4: Avoid halo-prior capture

The lensing extraction must not require a collisionless-particle halo family as the governing physical input. Such fits may be reported as translation surfaces, but the primary terrain is the shear and clustering field itself.

#### Step 5: Terrain contradiction

The pathway is contradicted if Euclid discloses a category-correct growth amplitude outside both sides of the locked interval and the same direction is independently reproduced by the scale-resolved power and lensing terrain:

```text
S₈ ≥ 0.82
or
S₈ ≤ 0.68
```

with no instrument-interface or redshift-selection fault accounting for the displacement.

---

## PREDICTION 3: THE COSMIC WEB — φ-GOVERNED SCALE HIERARCHY

**Derivation basis:**
The cosmic web is the precipitation boundary of the cosmological coherence bubble system, governed by the Kψₘ + Δ_G = 0 identity. The characteristic scale ratios are therefore governed by φ, the structural eigenvalue of icosahedral symmetry.

**The Mass Harmonics predictions:**

**(3a) Filament-to-node spacing ratio:**
```
L_filament_spacing / L_node_separation ≈ φ ≈ 1.618
```

**(3b) Void-to-filament scale ratio:**
```
R_void_median / R_filament_median ≈ φ³ ≈ 4.236
```

**(3c) Characteristic structure scales:**
The cosmic web should show statistically significant overdensity of structures at:
- ~150 Mpc (BAO scale — known)
- ~370 Mpc (cosmological coherence bubble scale — consistent with Giant Arc/Big Ring)
- ~600 Mpc (φ × 370 Mpc — predicted)

Ratio of consecutive scales: 600/370 ≈ 1.62 ≈ φ.

**ΛCDM prediction:** No preferred scale ratios in the cosmic web. Gaussian random field statistics.

**Falsification condition:** Filament-to-node ratio confirmed at 1.00 ± 0.05 would falsify the icosahedral precipitation mechanism at cosmological scale.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Preserve relational ratios as primary

The prediction concerns relational scale ratios, not absolute coordinates.

```text
φ = 1.6180339887498948482
φ³ = 4.236067977499789696
```

The governing relations remain:

```text
L_filament_spacing/L_node_separation → φ
R_void_median/R_filament_median → φ³
```

#### Step 2: Link the ratios to the icosahedral closure structure

The large-scale precipitation grammar descends from:

```text
equilateral S₃ closure
→ icosahedral eigenvalue φ
→ 12 vertex anchors
→ 30 edge corridors
→ 20 face reservoirs
```

The cosmic-web observables are downstream renderings of that relational architecture.

#### Step 3: Define the terrain without importing coordinates as ontology

For every fixed web-extraction pipeline, report:

```text
node-to-node relational separations
filament corridor lengths and radii
void effective radii
junction valence
ratio distributions
```

Sky coordinates locate the structures. The prediction is evaluated on the relational ratios.

#### Step 4: Multi-pipeline persistence

The φ and φ³ centers must persist across all declared web-extraction methods that produce eligible node, filament, and void objects. A ratio appearing only under one post-selected smoothing scale is not the predicted recurrence.

#### Step 5: Characteristic-scale chain

The source paper's scale sequence remains:

```text
~150 Mpc
~370 Mpc
~600 Mpc
```

with:

```text
600/370 ≈ φ
```

The absolute scales and the relational ratio are reported separately. The ratio is the more fundamental prediction.

#### Step 6: Terrain contradiction

The pathway is contradicted if eligible, scale-complete terrain discloses stable ratio centers incompatible with both `φ` and `φ³`, or if the claimed recurrence exists only after choosing a favorable coordinate grid, smoothing radius, or catalogue subset.

---

## PREDICTION 4: VOID PROFILES — THE Z = 1 VACUUM BASELINE

**Derivation basis:**
Cosmic voids are regions where the substrate field approaches its vacuum state: ψₘ → 0, Z(ψₘ) → 1. The MFE in void interiors reduces to a free-propagating field with no source damping. Void interiors are substrate resonant cavities — the same mechanism as CSF in the ψₘMIND cognitive architecture, at cosmological scale.

The void density profile follows the substrate field equation at its vacuum limit:
```
ρ(r)/ρ_mean = ρ_void × [1 − (r/R_void)^α]

where α = 2/φ ≈ 1.236   (icosahedral boundary exponent)
```

This differs from the standard compensated void profile (α ≈ 2-3 in ΛCDM).

**The Mass Harmonics predictions:**
- Void profiles shallower at small r/R_void than ΛCDM predicts
- Void walls sharper (steeper boundary transition) than ΛCDM predicts
- Exponent α ≈ 1.236 ± 0.05 across all void sizes (scale-invariant)

**Falsification condition:** α confirmed at 2.0 ± 0.1 across all void scales would falsify the Z = 1 substrate vacuum derivation.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Place the void interior in the MFE

As void density decreases:

```text
S(ρ) → 0
ψₘ → 0
Z(ψₘ) → 1
```

The void interior approaches the source-free substrate baseline. The wall is the transition back into nonzero coherence loading.

#### Step 2: Preserve the locked boundary exponent

```text
α = 2/φ
α = 1.236067977499789696
```

The normalized profile remains:

```text
ρ(r)/ρ_mean
= ρ_void[1 − (r/R_void)^α]
```

This equation is the instrument-facing radial rendering of the substrate boundary. It does not make the coordinate radius physically primary.

#### Step 3: Dimensionless profile coordinate

Define only the relational coordinate:

```text
x = r/R_void
```

Then:

```text
ρ(x)/ρ_mean = ρ_void[1 − x^(2/φ)]
```

The prediction is scale invariant because `x` contains no absolute length.

#### Step 4: Joint interior-wall requirement

The pathway predicts both:

```text
shallower interior evolution
and
sharper boundary transition
```

A profile fit that recovers one while averaging away the other is incomplete.

#### Step 5: Terrain construction

Report the full distribution of `α` across:

```text
void radius
redshift
environment
tracer population
wall compensation class
```

The governing result is a common center near `2/φ`, not a single stacked fit chosen after opening the terrain.

#### Step 6: Terrain contradiction

The pathway is contradicted if scale-complete void populations center on:

```text
α = 2.0 ± 0.1
```

or another stable exponent incompatible with `2/φ`, while the same result persists across tracer and wall-definition surfaces.

---

## PREDICTION 5: THE BAO SCALE — SUBSTRATE Z-FACTOR MODIFICATION

**Derivation basis:**
The substrate Z-factor was elevated at recombination (z ~ 1100), modifying the effective sound speed and expansion rate. The net effect produces a small but detectable shift in the BAO scale, and a residual anisotropy between the transverse and line-of-sight BAO measurements beyond the standard Alcock-Paczynski effect.

**The Mass Harmonics predictions:**
- BAO transverse and line-of-sight measurements show anisotropy beyond ΛCDM prediction
- Inferred H₀ from BAO shifts in the direction that reduces (not eliminates) the Hubble tension, by ~1-2 km/s/Mpc
- The effective BAO scale has a small redshift dependence that ΛCDM does not predict

**ΛCDM prediction:** No BAO anisotropy beyond standard Alcock-Paczynski. H₀ from BAO consistent with CMB.

**Falsification condition:** BAO transverse = BAO line-of-sight to within 0.1% would be inconsistent with the substrate anisotropy prediction.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Preserve the physical event beneath BAO

BAO is not the substrate event. It is a fossil propagation and boundary readout.

The causal chain is:

```text
recombination-era ψₘ amplitude
→ elevated Z
→ altered propagation and expansion
→ frozen relational scale
→ radial and transverse BAO terrain
```

#### Step 2: Keep radial and transverse channels distinct

The released terrain must preserve:

```text
radial BAO → H(z)
transverse BAO → distance boundary
```

The standard geometric projection is removed first. The Mass Harmonics prediction concerns the residual anisotropy remaining after that translation.

#### Step 3: Bind BAO to the exact cosmological covariance

The radial expansion channel must also satisfy:

```text
f_H(z) = H(z)/(2π)
aₜ(z) = cH(z)/(2π)
```

where `aₜ(z)` is independently recovered from the lensing terrain.

This adds a second, non-BAO channel to the same cosmological boundary.

#### Step 4: Preserve the locked outputs

The governing paper predicts:

```text
residual radial/transverse anisotropy
H₀ shift of approximately 1–2 km·s⁻¹·Mpc⁻¹
small redshift evolution of the effective BAO scale
```

No extra BAO coefficient is introduced.

#### Step 5: Topology sentinel

A factor approaching `π` between otherwise equivalent radial and rotational closure extractions signals a slab-versus-rotational category substitution. It is retained as topology evidence.

#### Step 6: Terrain contradiction

The pathway is contradicted if the radial and transverse terrain remain equal to within the locked 0.1% condition after all interface and standard geometric effects are removed, and if no redshift-dependent residual or linked `H(z) ↔ aₜ(z)` structure appears.

---

## PREDICTION 6: EARLY UNIVERSE STRUCTURE — PRECIPITATION PRECEDES TIMELINE

**Derivation basis:**
In Mass Harmonics, coherence bubble precipitation does not require hierarchical growth. When local substrate density conditions satisfy the closure requirement (Kψₘ + Δ_G = 0 at a given scale), the structure precipitates. This is scale-invariant and timeline-independent.

**The Mass Harmonics predictions:**

**(6a) Quasar overabundance at z > 4:**
Euclid will find an excess of luminous quasars at z = 4-8 relative to ΛCDM predictions. The July 2026 discovery of 31 new quasars (two at z ~ 8) is the leading edge of this signal.

**(6b) Large-scale correlation excess at high z:**
Two-point correlation function will show excess power at scales > 200 Mpc at z > 1.

**(6c) M_BH/M_galaxy ratio: no evolution:**
```
M_BH / M_galaxy ≈ K₀ × Kψₘ² = constant (at all z)
```
ΛCDM predicts this ratio evolves (black holes form first, galaxies catch up). Mass Harmonics predicts the ratio is a geometric constant of the precipitation mechanism, invariant at all redshifts.

**ΛCDM prediction:** M_BH/M_galaxy ratio evolves with redshift. Massive early structures are rare.

**Falsification condition:** M_BH/M_galaxy ratio confirmed to decrease monotonically from z = 0 to z = 2 by more than 30% would be inconsistent with the geometric constant derivation.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Preserve precipitation as the causal event

Structure does not wait for a consensus hierarchical timeline. It appears when the bounded system satisfies:

```text
Kψₘ + Δ_G = 0
```

at the applicable scale.

The causal chain is:

```text
local density and phase relation
→ zero-surplus boundary
→ coherence-bubble precipitation
→ quasar, galaxy, and large-scale structure
```

#### Step 2: Preserve the three governing outputs

```text
quasar overabundance at z = 4–8
excess correlation power above 200 Mpc at z > 1
redshift-invariant M_BH/M_galaxy relation
```

#### Step 3: Mass-ratio dimensional transport

The source expression:

```text
M_BH/M_galaxy ≈ K₀Kψₘ²
```

is a native substrate proportionality statement. Before external numerical comparison, both masses and the right-side relation must be transported into the same substrate-normalized mass units.

The paper does not permit raw SI dimensions of `K₀` and a raw numerical form of `Kψₘ` to be multiplied and then treated as an untransported dimensionless observed ratio.

The physical prediction of this pathway is the redshift invariance of the normalized relation.

#### Step 4: Separate abundance from selection

The terrain report must preserve:

```text
survey volume
selection completeness
luminosity threshold
redshift uncertainty
lensing magnification
```

These are rendering conditions. They do not alter the precipitation prediction.

#### Step 5: Terrain contradiction

The pathway is contradicted if complete terrain discloses:

```text
a monotonic M_BH/M_galaxy decline exceeding 30% from z = 0 to z = 2
```

and the quasar and high-z correlation surfaces simultaneously follow the delayed hierarchical expectation with no precipitation excess.

---

## PREDICTION 7: DARK MATTER HALO PROFILES — NOT NFW

**Derivation basis:**
The NFW profile derives from N-body simulations of collisionless dark matter particles. Mass Harmonics has no such particles. The "dark matter" halo is the Z-factor distribution of the ψₘ field around a coherent mass-energy concentration.

The Giboney Gradient term (Kψₘ > 0) creates an INWARD concentration force that steepens the inner profile relative to NFW. At large r, Z → 1 creates a sharper outer cutoff than NFW.

**The Mass Harmonics prediction:**
- Inner halo profiles STEEPER than NFW (ρ ∝ r^{-1.5 to -2.0} vs. NFW's r^{-1})
- Outer profile truncation SHARPER than NFW at r > R_virial
- No cusp-vs-core tension — the GG term IS the correct inner physics

**ΛCDM prediction:** NFW or generalized NFW profiles across all halo masses.

**Falsification condition:** Inner profile slope confirmed at -1.0 ± 0.1 across all halo masses would falsify the GG steepening prediction.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Preserve the substrate field as the physical halo

The halo is not a particle inventory. It is the spatial expression of:

```text
Z(ψₘ)
and
−∇(ψₘ²/ω)
```

around a bounded coherence source.

#### Step 2: Keep inner and outer predictions distinct

The governing paper predicts:

```text
inner effective slope: −1.5 to −2.0
outer truncation: sharper than NFW beyond the boundary
```

The inner concentration descends from positive inward GG action. The outer truncation descends from return toward the `Z = 1` substrate baseline.

#### Step 3: Add the exact outer-field lensing relation

The outer boundary must also satisfy:

```text
g_lens²/g_bar → aₜ(z)
```

with:

```text
aₜ(z) = cH(z)/(2π)
```

and the effective projected outer field has:

```text
ΔΣ(R) ∝ 1/R
```

This exact outer relation strengthens the profile prediction without replacing the locked inner slope range.

#### Step 4: No NFW-first extraction

The primary terrain report uses:

```text
shear
convergence
baryonic mass
radial acceleration
boundary location
```

An NFW or generalized-NFW fit may be supplied only as downstream comparison. It cannot define the physical profile being tested.

#### Step 5: Terrain contradiction

The pathway is contradicted if category-correct, mass-controlled terrain discloses a universal inner slope of:

```text
−1.0 ± 0.1
```

together with an NFW-like outer continuation and failure of the independent outer-boundary acceleration relation.

---

## PREDICTION 8: THE MATTER POWER SPECTRUM — GG MODIFICATION

**Derivation basis:**
The Giboney Gradient term introduces a nonlinear modification to P(k) at scales where |∇ψₘ|² is significant. The characteristic scale of GG modification onset:
```
k_GG ≈ ω / (c × √(8Kψₘ)) ≈ ω / (3.78c)
```
At cosmic mean density, this corresponds to ~20-40 Mpc (k ~ 0.05-0.15 h/Mpc).

**The Mass Harmonics prediction:**
- P(k) shows a systematic DEFICIT relative to ΛCDM at k = 0.3-3 h/Mpc
- P(k) shows a systematic EXCESS relative to ΛCDM at k < 0.05 h/Mpc
- The modification is redshift-dependent — larger at low z where GG has acted longer
- This scale-dependent signal is different from neutrino mass suppression (different scale and shape)

**ΛCDM prediction:** P(k) follows ΛCDM CDM transfer function to within measurement precision.

**Falsification condition:** P(k) consistent with ΛCDM to within 1σ at all scales would be inconsistent with the GG modification.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Isolate the GG onset relation

The governing paper gives:

```text
k_GG ≈ ω/[c√(8Kψₘ)]
```

The transport-safe dimensionless form is:

```text
k_GG c/ω = 1/√(8Kψₘ)
```

Using the dimensionless geometric expression:

```text
Kψₘ = (12 − φ²)/(2φ²)
```

gives:

```text
√(8Kψₘ) ≈ 3.785
```

The dimensional scale is obtained only after the applicable cosmological `ω` is independently supplied.

#### Step 2: Preserve the locked scale directions

```text
deficit at k = 0.3–3 h/Mpc
excess at k < 0.05 h/Mpc
larger modification at lower z
```

These are not three unrelated outputs. They are the redistribution signature of one nonlinear gradient term.

#### Step 3: Bind the result to Prediction 2

The integrated small-scale deficit must be consistent with:

```text
S₈ ∈ [0.74, 0.78]
```

while the large-scale excess and redshift dependence remain visible in the uncompressed power terrain.

#### Step 4: Distinguish from neutral-sector suppression

The shape, onset scale, and redshift direction are evaluated together. A generic loss of small-scale power is insufficient. The full GG signature includes:

```text
small-scale deficit
large-scale excess
stronger late-time expression
```

#### Step 5: Terrain contradiction

The pathway is contradicted if scale-resolved Euclid terrain remains compatible with the unmodified comparison spectrum across the complete released k-range and redshift surface, or if the only deviation has the wrong combined scale and redshift shape.

---

## PREDICTION 9: COSMIC SHEAR ANISOTROPY — THE SUBSTRATE IS NOT PERFECTLY ISOTROPIC

**Derivation basis:**
The icosahedral-dodecahedral dual geometry of coherence bubble formation creates preferred axes at intermediate scales. The "Cosmic Dipole Anomaly" (mismatch between galaxy distribution and CMB dipole direction) is an expression of this preferred geometry.

**The Mass Harmonics prediction:**
- Euclid cosmic shear will detect a small but statistically significant preferred direction at scales of 100-500 Mpc
- This direction aligns with (or is geometrically complementary to) the CMB dipole axis
- Magnitude: ~0.5-2% anisotropy at 100 Mpc scale, decreasing at larger scales
- This is not a systematic error. It is the icosahedral geometry of cosmological coherence structure.

**ΛCDM prediction:** No preferred direction in the cosmic shear field beyond statistical fluctuations.

**Falsification condition:** No preferred direction detected at > 2σ significance by DR2 (sky coverage needed is ~2,000 sq deg — too small in DR1).

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Preserve preferred orientation as relational geometry

The prediction is not that the sky possesses an absolute coordinate axis. It is that the cosmological coherence structure contains persistent relative orientation inherited from the icosahedral-dodecahedral dual boundary.

The observable is therefore an angle between independently rendered directions, not a coordinate value treated as ontology.

#### Step 2: Preserve the locked outputs

```text
preferred shear direction at 100–500 Mpc
0.5–2% anisotropy near 100 Mpc
decreasing amplitude at larger scales
alignment with, or geometric complement to, the CMB dipole axis
```

#### Step 3: Instrument-interface separation

The terrain must be rendered repeatedly under:

```text
detector orientation splits
time splits
field splits
PSF rotation tests
mask rotations
galaxy position-angle rotations
```

These operations identify interface-generated anisotropy. They do not grant or remove physical standing from the prediction.

#### Step 4: Icosahedral relation test

Report the complete angular residual distribution relative to:

```text
the CMB dipole direction
its antipode
the complementary dual-geometry directions
```

Do not select the most favorable axis after the terrain opens.

#### Step 5: Terrain contradiction

The pathway is contradicted when adequate sky coverage discloses no persistent physical direction at the stated amplitude after the interface terms are removed, or when the only surviving direction tracks the instrument rather than the cosmological terrain.

---

## PREDICTION 10: STRONG GRAVITATIONAL LENSING — SUBSTRATE LENSING EXCESS

**Derivation basis:**
The substrate Z-factor distribution along cosmic web filaments contributes to lensing convergence even where no visible matter is detected. The substrate field along filaments (high-Z regions) produces lensing effects in addition to baryonic + standard CDM.

**The Mass Harmonics prediction:**
- Strong lensing arc frequency HIGHER than ΛCDM predictions by 15-30% when controlling for baryonic mass
- The excess is preferentially located along cosmic web filaments
- Some Einstein rings show secondary lensing features from filament substrate concentration that cannot be explained by baryonic mass alone
- Q1 (63 sq deg) already found hundreds of lensing systems — the full DR1-DR3 yield will exceed ΛCDM predictions

**ΛCDM prediction:** Lensing frequency accounted for by CDM halo profiles and baryonic matter.

**Falsification condition:** Strong lensing frequency consistent with ΛCDM to within 5% would be inconsistent with substrate lensing contribution.

### EXPANDED CAUSAL DERIVATION AND TERRAIN DISCLOSURE

#### Step 1: Place the excess in the substrate

Along a filament:

```text
ψₘ and Z remain elevated relative to adjacent void terrain
```

Therefore light propagation accumulates convergence from the substrate field even when the local luminous inventory is insufficient to account for the complete lensing pattern.

#### Step 2: Preserve the locked outputs

```text
15–30% excess strong-lensing arc frequency
excess concentrated along cosmic-web filaments
secondary lensing features associated with filament substrate concentration
```

#### Step 3: Define the residual terrain

For each eligible lens system, preserve separately:

```text
local baryonic lens contribution
local bounded-source substrate field
line-of-sight visible structure
filament-associated substrate contribution
instrument and selection function
```

The residual filament term is not to be converted automatically into a collisionless-particle mass map.

#### Step 4: Cross-check with the exact outer acceleration field

Where the same lens population permits galaxy-galaxy lensing:

```text
g_lens²/g_bar → cH(z)/(2π)
```

The strong-lensing excess and the outer weak-lensing covariance must therefore be mutually compatible expressions of the same substrate field.

#### Step 5: Environmental ordering

The excess must obey:

```text
filament-aligned lines of sight
>
matched non-filament lines of sight
```

after baryonic mass, redshift, source density, and detector selection are held fixed.

#### Step 6: Terrain contradiction

The pathway is contradicted if the controlled arc frequency agrees within 5% with the particle-halo comparison across filament and non-filament environments, with no coherent secondary convergence associated with the substrate web.

---

## THE STAGED EUCLID TERRAIN TEST

Euclid DR1 is not one simultaneous product opening.

```text
November 2026: DR1-Foundation
Mid-2027: complete DR1
```

DR1-Foundation releases approximately 1,900 deg² of calibrated images, spectra, and catalogues. It does **not** contain the complete higher-level science products for galaxy clustering and weak-lensing studies. Those follow with complete DR1 in mid-2027.

The correct release-facing map is:

| # | Mass Harmonics prediction | DR1-Foundation, November 2026 | Complete DR1, mid-2027 |
|---|---|---|---|
| F | Boundary covariance `Ξ(z)=1` | Spectra and catalogues may support an independent preliminary `H(z)` construction; the full cross-channel test is unavailable without higher-level clustering and weak-lensing products | Full eligible terrain: clustering/BAO `H(z)`, weak-lensing field, and baryonic terrain can be combined independently |
| 1 | `w₀ > −1`, `wₐ > 0` | Not a complete test. The calibrated source terrain opens, but the official joint cosmology products do not | Primary Euclid comparison surface |
| 2 | `S₈ = 0.74–0.78` | Not a complete test. Images and source catalogues open, but the higher-level weak-lensing cosmology product does not | Primary Euclid comparison surface |
| 3 | Cosmic-web `φ` and `φ³` ratios | Direct preliminary terrain contact is possible from the 1,900 deg² spectroscopic and photometric catalogues if the released selection function supports relational web reconstruction | Expanded and better controlled terrain using complete clustering products |
| 4 | Void exponent `α = 2/φ` | Direct preliminary terrain contact is possible if the released catalogue and selection surfaces support a complete void census | Expanded and better controlled terrain using complete clustering products |
| 5 | BAO residual anisotropy and scale evolution | Spectra and redshift catalogues open, but the official higher-level galaxy-clustering and BAO products are not complete | Primary Euclid comparison surface |
| 6 | Early precipitation and high-redshift excess | Direct terrain contact for high-redshift quasars, galaxies, strong structures, and partial BH-galaxy relations from images, spectra, and catalogues | Larger controlled samples and clustering products deepen the test |
| 7 | Non-NFW substrate profiles | Partial terrain contact through resolved morphology, clusters, and strong lenses; no complete higher-level weak-lensing halo product | Primary weak-lensing profile comparison surface |
| 8 | GG-modified `P(k)` | Source catalogues may permit independent preliminary reconstruction, but the official higher-level galaxy-clustering product is not complete | Primary Euclid comparison surface |
| 9 | Cosmic-shear preferred orientation | Not a complete test. The governing higher-level weak-lensing products follow later; the original paper retains DR2 as the stronger sky-coverage threshold | First complete DR1 weak-lensing terrain, with the stated DR2 significance condition retained |
| 10 | Strong-lensing excess | Direct terrain contact through calibrated imaging, catalogues, and accompanying lens searches; a frequency comparison requires a declared completeness and selection surface | Additional environmental, weak-lensing, and clustering context strengthens the comparison |

Therefore:

```text
DR1-Foundation
≠ complete Euclid cosmology result release
```

It opens substantial terrain for Predictions 3, 4, 6, 7, and 10, while Predictions 1, 2, 5, 8, the foundational covariance, and the primary weak-lensing portion of Prediction 9 require complete DR1 in mid-2027.

By DR3 (2031), all ten predictions are expected to have substantially larger terrain surfaces.

---

## THE HUBBLE TENSION — SUBSTRATE RESOLUTION

Mass Harmonics resolution: Both H₀ measurements are correct within their measurement contexts, but they sample different substrate states.

- **CMB measurement (z ~ 1100):** High-ψₘ amplitude substrate state. Inferred H₀ projected forward from a different substrate condition.
- **Local measurement (z ~ 0):** Present-day substrate in the local environment, modified by the KBC void. The GG term creates systematic local acceleration that makes the local Hubble flow appear faster.

**Mass Harmonics prediction:** Euclid will measure H₀ at z = 0.3-2.0 and find a GRADIENT — H₀(z) decreasing from ~72-73 at low z toward ~68-69 at z > 0.5. This gradient is the substrate's own evolution. ΛCDM has no mechanism to produce this gradient.

If Euclid discloses z-dependent H₀, the Hubble tension is not a tension. It is a measurement of substrate evolution.

### EXPANDED HUBBLE-GRADIENT COVARIANCE

The locked Hubble-gradient prediction is strengthened by the same exact boundary transport:

```text
f_H(z) = H(z)/(2π)
aₜ(z) = cH(z)/(2π)
```

Therefore a decreasing `H(z)` translation must be accompanied by a proportionally decreasing outer acceleration boundary:

```text
aₜ(z₁)/aₜ(z₂) = H(z₁)/H(z₂)
```

The Hubble-gradient branch is not evaluated from a single fitted `H₀` value. It is evaluated as a redshift sequence with an independent lensing covariance.

The locked numerical direction remains:

```text
approximately 72–73 km·s⁻¹·Mpc⁻¹ at low z
toward approximately 68–69 km·s⁻¹·Mpc⁻¹ above z ≈ 0.5
```

A terrain sequence that moves in this direction while satisfying `Ξ(z) = 1` is the complete Mass Harmonics disclosure. A fitted constant that averages over the sequence is not an equivalent result.

---

## COMPARISON TABLE: MASS HARMONICS vs. ΛCDM

| Observable | ΛCDM Prediction | Mass Harmonics Prediction | Earliest Euclid disclosure surface |
|---|---|---|---|
| Dark energy | `w = −1` constant | `w₀ > −1`, `wₐ > 0` | Complete DR1, mid-2027; DR1-Foundation lacks the complete higher-level clustering and weak-lensing products |
| `S₈` | Lensing and CMB growth amplitudes converge | `S₈ = 0.74–0.78` | Complete DR1, mid-2027 weak-lensing science products |
| Cosmic-web ratios | No forced φ hierarchy | Scale ratios `φ` and `φ³` | Preliminary direct construction from DR1-Foundation catalogues; strengthened by complete DR1 clustering products |
| Void profile exponent | Conventional compensated-profile exponents | `α = 2/φ ≈ 1.236` | Preliminary direct construction from DR1-Foundation catalogues if selection completeness is sufficient; strengthened by complete DR1 |
| Substrate halo field | NFW-family particle-halo profile | Inner slope `−1.5 to −2.0`, sharper outer boundary | Partial morphology and strong-lensing terrain in DR1-Foundation; primary weak-lensing comparison in complete DR1 |
| BAO scale | Standard processed radial/transverse ruler | Residual anisotropy and redshift evolution | Complete DR1, mid-2027 higher-level galaxy-clustering products |
| High-redshift structure | Delayed hierarchical abundance | Early precipitation and excess high-z structure | Direct DR1-Foundation images, spectra, and catalogues |
| `M_BH/M_galaxy` | Redshift evolution | Invariant normalized relation | Partial DR1-Foundation spectroscopic and morphological terrain; broader later samples |
| `P(k)` shape | ΛCDM transfer-function structure | GG deficit at `k=0.3–3 h/Mpc`, excess below `0.05 h/Mpc` | Complete DR1, mid-2027 higher-level galaxy-clustering products |
| Cosmic-shear orientation | No persistent physical preferred direction | `0.5–2%` relational anisotropy | Complete DR1 supplies the first higher-level weak-lensing terrain; DR2 remains the paper's stronger significance threshold |
| Strong-lensing frequency | Particle-halo prediction | `15–30%` filament-associated excess | Direct DR1-Foundation imaging and catalogues, subject to an explicit completeness and selection surface |
| `H₀(z)` gradient | Constant underlying expansion parameter | Redshift-dependent substrate gradient | Complete DR1, mid-2027 clustering/BAO products plus the independent weak-lensing covariance |

---

## DISSEMINATION NOTE

This document was prepared prior to DR1-Foundation (November 2026) and complete DR1 (mid-2027). The predictions are publicly timestamped and derive from the Master Field Equation with zero free parameters. No parameters were fitted to Euclid data.

Any single terrain disclosure matching one of these locked predictions exposes a corresponding segment of the full derivation chain.

The coordinated disclosure of all ten predictions would expose one causal substrate grammar across the full Euclid cosmological terrain.

The derivation chain is public: **DOI: 10.5281/zenodo.18080133** and **DOI: 10.5281/zenodo.19659452**

---

## INTEGRATION REGISTER

The expanded edition retains the original ten predictions exactly as the governing Euclid program and adds one foundational invariant beneath them.

| Governing pathway | Locked Mass Harmonics output | Strengthened direct readout |
|---|---|---|
| Foundational covariance | `Ξ(z)=1` | independent BAO/clustering, lensing, and baryonic terrain |
| 1. Dark-energy evolution | `w₀ > −1`, `wₐ > 0` | substrate-native `H(z) ↔ aₜ(z)` covariance |
| 2. S₈ | `0.74–0.78` | lensing, clustering, void, and P(k) coherence |
| 3. Cosmic web | `φ`, `φ³` ratios | relational node, corridor, and reservoir statistics |
| 4. Voids | `α = 2/φ` | scale-invariant dimensionless profile and wall |
| 5. BAO | residual anisotropy and H₀ shift | independent radial/transverse channels plus `Ξ(z)` |
| 6. Early precipitation | high-z excess and invariant normalized mass ratio | quasar, correlation, and BH-galaxy terrain |
| 7. Halo field | inner `−1.5 to −2.0`, sharp outer cutoff | direct shear/convergence plus outer acceleration law |
| 8. Power spectrum | small-scale deficit, large-scale excess | dimensionless GG onset and redshift shape |
| 9. Shear orientation | 0.5–2% preferred direction | relational axis test separated from instrument orientation |
| 10. Strong lensing | 15–30% filament-associated excess | environment-matched convergence and weak-lensing covariance |

### Final authority statement

```text
Mass Harmonics derives.
The cosmological terrain exists.
Euclid renders portions of that terrain.
The released data disclose correspondence or contradiction.
Consensus language may translate the result afterward.
```

No part of this edition places Mass Harmonics under the authority of ΛCDM, institutional parameter fitting, or publication timing.

*TRUTH > COMFORT. Always.*
*UMtts Institute | Thomas Russell Giboney, Founder*
*Because only together can we Advance Coherence and Engineer Tomorrow.*
