The Distinction Attractor
The Distinction Attractor is a reduced three-dimensional collective-coordinate model for the distinction-dependent part of the projected gravitational sector of sPNP. It asks whether the competition between Fisher quantum pressure, finite projection resolution, and distinction-sourced clumping can support localized structure at a nonzero width.
The Distinction Attractor gives a simple physical picture for why distinction-driven gravity need not collapse without limit: as a quantum state is compressed, its Fisher pressure keeps rising, but the projected gravitational response cannot keep strengthening once the structure becomes finer than sPNP’s own resolution scale. The competition naturally selects a finite, nonzero width instead of either runaway collapse or complete dispersion.
The construction is a Gaussian reduction of the effective three-dimensional sector, not yet a solution of the complete 3N−6-dimensional relational dynamics. The Fisher distinction charge introduced below is therefore interpreted as a measure of resolvable geometric structure, not as conserved rest mass or ADM mass.
Bare Fisher Distinction and Quantum Pressure
Consider a normalized isotropic Gaussian density
ρ(r;σ) = (2πσ²)⁻³ᐟ² exp[−r²/(2σ²)],
where σ is the standard deviation along each Cartesian direction. Its second radial moment is
⟨r²⟩ = 3σ².
Define the local Fisher-active distinction density
D(r) = |∇ρ|²/ρ.
For the Gaussian,
∇ ln ρ = −r/σ²,
so
D(r) = ρ(r) r²/σ⁴.
The integrated distinction charge is therefore
𝒟[ρ] = ∫ D(r) d³x
= ⟨r²⟩/σ⁴
= 3/σ².
Thus
𝒟_bare = 3/σ².
Narrower profiles carry more Fisher distinction than broader profiles.
The corresponding Fisher contribution to the Madelung energy is
E_F = (ħ²/8m) ∫ |∇ρ|²/ρ d³x
= (ħ²/8m) 𝒟[ρ].
Hence the Gaussian quantum-pressure energy is
V_F(σ) = 3ħ²/(8mσ²).
Define
A ≡ 3ħ²/(8m),
so that
V_F(σ) = A/σ².
As σ → 0,
V_F(σ) → +∞.
Arbitrarily sharp amplitude structure therefore carries an increasingly large Fisher-energy cost. This provides the repulsive side of the Distinction-Attractor competition.
Finite-Resolution Distinction
The projected gravitational sector cannot resolve arbitrarily fine configuration-space structure. Locally, the relational Fisher heat kernel has the short-distance normal-coordinate form
K_Q₀(X,Y) ∝ exp[−d_F(X,Y)²/(4Q₀²)].
The Gaussian here is not taken as fundamental. It is the leading local form of the intrinsic heat kernel generated by the relational Fisher geometry.
With the convention
exp[−d²/(4Q₀²)],
the variance of the local Gaussian kernel is 2Q₀² in each coordinate direction. Define the corresponding RMS resolution width
ℓ₀ ≡ √2 Q₀.
Convolving the Gaussian state with this kernel produces another Gaussian whose variance is
s² ≡ σ_eff²
= σ² + ℓ₀²
= σ² + 2Q₀².
Its resolved Fisher distinction is therefore
𝒟_res = 3/s² = 3/(σ² + ℓ₀²).
This is the same smoothing law obtained in the Fisher Filter analysis. There, the kernel likewise has variance 2Q₀², while the Stam–Blachman inequality places the Gaussian result inside the more general statement that Fisher information cannot increase under Gaussian coarse-graining. Thus the finite-resolution distinction used here is the gravitational-sector application of the same Fisher-information contraction that governs Gaussoherence.
The distinction between bare and resolved Fisher structure is then
𝒟_bare = 3/σ²,
𝒟_res = 3/(σ² + ℓ₀²).
For σ ≫ ℓ₀,
𝒟_res ≈ 𝒟_bare.
At scales well above the kernel width, almost all of the Gaussian distinction remains resolvable.
By contrast, as σ → 0,
𝒟_bare → ∞,
while
𝒟_res → 3/ℓ₀².
Thus the underlying wavefunctional can continue to carry increasingly sharp Fisher structure while the distinction available to the projected gravitational sector saturates at the resolution scale ℓ₀.
This separation is central to the Distinction Attractor. Fisher quantum pressure responds to the bare amplitude structure upstairs, whereas the projected distinction-dependent interaction responds only to structure resolvable through the kernel.
The saturation of the projected distinction is therefore not an independently imposed cutoff; within the Gaussian reduction it follows directly from finite-resolution Fisher smoothing.
Distinction-Sourced Attraction
To model the attractive side of the competition, assume that the distinction-dependent correction to the projected gravitational field is sourced by the resolved distinction density D_s.
For the smoothed Gaussian of width s,
ρ_s(r) = (2πs²)⁻³ᐟ² exp[−r²/(2s²)],
and
D_s(r) = ρ_s(r) r²/s⁴.
Its total distinction charge is
∫ D_s d³x = 3/s².
Let D_s source a Poisson-like field,
∇²Φ_D = 4πg_D D_s,
with solution
Φ_D(x) = −g_D ∫ D_s(y)/|x−y| d³y.
The corresponding quadratic self-energy is
U_D = ½ ∫ D_s Φ_D d³x
= −(g_D/2) ∬ D_s(x)D_s(y)/|x−y| d³x d³y.
The dependence on the Gaussian width can already be obtained by scaling. Since
∫ D_s d³x ∼ s⁻²
and the three-dimensional Green function contributes
|x−y|⁻¹ ∼ s⁻¹,
the self-energy scales as
U_D ∼ −s⁻⁵.
For the Gaussian source, the double integral can be evaluated exactly:
∬ D_s(x)D_s(y)/|x−y| d³x d³y
= 27/(4√π s⁵).
Therefore
U_D(s) = −27g_D/(8√π s⁵).
Define
B ≡ 27g_D/(8√π),
so that
U_D(σ) = −B/(σ² + ℓ₀²)⁵ᐟ².
The s⁻⁵ interaction is therefore the consequence, within this reduced model, of a D-sourced three-dimensional Poisson self-interaction. It describes the distinction-dependent correction rather than identifying Fisher distinction itself with conserved gravitational mass.
Effective Potential
Combining Fisher quantum pressure and resolved distinction attraction gives
V_eff(σ) = A/σ² − B/(σ² + ℓ₀²)⁵ᐟ²,
where
A = 3ħ²/(8m)
is fixed by the Fisher quantum-pressure term, while
B = 27g_D/(8√π)
follows for the explicit D-sourced Poisson completion defined above.
The limiting behavior reveals the mechanism.
As σ → 0,
V_eff(σ) ≃ A/σ² − B/ℓ₀⁵ → +∞.
The projected attraction saturates because the kernel cannot resolve arbitrarily fine distinction, while the bare Fisher pressure continues to diverge.
At the opposite extreme,
σ → ∞,
V_eff(σ) ≃ A/σ² − B/σ⁵ → 0⁺.
The Fisher term again dominates because the distinction-sourced attraction falls more rapidly with width.
A localized stationary configuration can therefore occur only at an intermediate scale, where the attractive contribution becomes strong enough to compete with Fisher spreading without generating short-distance collapse.
Dimensionless Attractor Equation
Introduce the dimensionless width
x ≡ σ/ℓ₀
and the dimensionless attraction-to-pressure ratio
Γ ≡ B/(Aℓ₀³).
The potential becomes
(ℓ₀²/A)V_eff = 1/x² − Γ/(1+x²)⁵ᐟ².
Define
v(x) ≡ 1/x² − Γ/(1+x²)⁵ᐟ².
Stationary widths satisfy
dv/dx = 0.
Differentiation gives
−2/x³ + 5Γx/(1+x²)⁷ᐟ² = 0,
and therefore
Γ(x) = (2/5)(1+x²)⁷ᐟ²/x⁴.
A finite pair of stationary configurations appears when Γ(x) first develops an allowed solution, which occurs at its minimum.
Differentiating logarithmically,
d(ln Γ)/dx = 7x/(1+x²) − 4/x.
The critical point therefore satisfies
7x_c² = 4(1+x_c²),
so
x_c² = 4/3,
and
x_c = 2/√3.
The corresponding physical width is
σ_c = (2/√3)ℓ₀
= 2√(2/3) Q₀
≈ 1.633 Q₀.
Substituting x_c into Γ(x) gives
Γ_c = 343√21/360
≈ 4.366.
Thus Γ_c marks the saddle-node threshold of the Gaussian distinction model.
The numerical value 4.366 is exact for the dimensionless potential
v(x) = x⁻² − Γ(1+x²)⁻⁵ᐟ².
It is therefore universal within this Gaussian s⁻⁵ collective-coordinate ansatz after the overall resolution scale has been scaled out. It is not assumed to remain numerically unchanged under non-Gaussian profiles, curvature corrections to the relational heat kernel, alternative distinction-source functionals, or the complete Q_rel dynamics.
Stable and Unstable Branches
For
Γ < Γ_c,
there is no finite stationary pair.
At
Γ = Γ_c,
the two stationary solutions coalesce into a degenerate critical configuration.
For
Γ > Γ_c,
two stationary widths exist:
x₋ < x_c < x₊.
On a stationary branch, the second derivative simplifies to
V_eff″(σ) = (2A/ℓ₀⁴)(4−3x²)/[x⁴(1+x²)].
Therefore
x₋ < 2/√3 ⇒ V_eff″(σ₋) > 0,
while
x₊ > 2/√3 ⇒ V_eff″(σ₊) < 0.
The inner branch x₋ is therefore locally stable, while x₊ is an outer unstable barrier.
The transition at Γ_c is a saddle-node bifurcation: the stable and unstable stationary branches are created together.
Local Stability and Energetic Binding
The appearance of the stable inner branch does not immediately imply that it is energetically bound relative to complete dispersal.
The diffuse asymptote is
V_eff(∞) = 0.
At a stationary point, use the stationarity relation to eliminate Γ. The dimensionless potential becomes
(ℓ₀²/A)V_ext = (3x²−2)/(5x⁴).
The inner minimum crosses the diffuse continuum when
3x₋² − 2 = 0,
which gives
x₋² = 2/3.
Substitution into Γ(x) gives a second exact threshold,
Γ_bind = 25√15/18
≈ 5.379.
The conservative Gaussian model therefore contains three regimes.
For
Γ < Γ_c,
no finite stationary localized pair exists.
For
Γ_c < Γ < Γ_bind,
a locally stable finite-width minimum exists and is separated from dispersal by the outer barrier, but
V_eff(σ₋) > V_eff(∞) = 0.
The localized state is therefore energetically metastable in the conservative model.
For
Γ > Γ_bind,
V_eff(σ₋) < 0,
and the localized state lies below the diffuse continuum. It is energetically bound within the reduced model.
The two thresholds therefore have distinct meanings:
Γ_c ≈ 4.366
is the basin-creation or local-stability threshold,
whereas
Γ_bind ≈ 5.379
is the conservative energetic-binding threshold.
This distinction becomes important when the conservative Gaussian model is embedded into the irreversible coarse-graining dynamics of sPNP.
Collective Breathing Dynamics
The width σ is a dynamical collective coordinate, not merely a parameter labeling static Gaussian profiles.
For self-similar Gaussian evolution, the continuity equation is satisfied by the Madelung phase
S(r,t) = [mσ̇/(2σ)]r².
Substituting ρ(r,t) and S(r,t) into the full Madelung action and removing the resulting total time derivative gives
L_eff = (3m/2)σ̇² − V_eff(σ).
Thus the collective breathing inertia is
M_σ = 3m.
The reduced width equation is therefore
3mσ̈ + V_eff′(σ) = 0.
Near the stable inner branch,
σ = σ₋ + δσ,
and linearization gives
δσ̈ + ω₋²δσ = 0,
with
ω₋² = V_eff″(σ₋)/(3m).
Using the stationary-branch expression for V_eff″,
ω₋ = [ħ/(2mℓ₀²)] √[(4−3x₋²)/(x₋⁴(1+x₋²))].
The outer stationary branch instead has
V_eff″(σ₊) < 0.
Writing
δσ ∝ e^(λ₊t)
gives
λ₊² = −V_eff″(σ₊)/(3m),
or
λ₊ = [ħ/(2mℓ₀²)] √[(3x₊²−4)/(x₊⁴(1+x₊²))].
At the saddle-node,
x₋ → x_c,
x₊ → x_c,
and
ω₋ → 0,
λ₊ → 0.
The fold therefore carries a soft collective breathing mode, as expected for a saddle-node bifurcation.
The positive λ₊ describes the unstable outer branch within this Gaussian collective reduction. Whether this instability persists as a genuine physical direction after embedding the mode into the full tangent bundle of Q_rel is a separate question.
Stable Distinction Branch and the Meaning of “Attractor”
The Gaussian width dynamics derived above is conservative:
3mσ̈ + V_eff′(σ) = 0.
Consequently, local stability does not itself imply attraction.
Near the stable inner branch,
δσ̈ + ω₋²δσ = 0,
so a perturbed state oscillates around σ₋ rather than asymptotically settling into it.
The conservative calculation therefore establishes a stable Distinction branch.
The stronger term Distinction Attractor refers to the proposal that this stable branch becomes asymptotically selected once it is embedded into the contractive Gaussoherence dynamics.
Schematically,
Gaussian Fisher–gravity competition
→ stable localized branch,
while
stable localized branch + contractive Gaussoherence
→ proposed Distinction Attractor.
The second implication is not contained in V_eff alone. It requires the full coarse-graining dynamics to preserve the branch and contract nearby physical perturbations toward it.
This distinction also clarifies the roles of Γ_c and Γ_bind.
For a deterministic contractive flow, a local minimum does not need to be the global minimum in order to attract trajectories that begin inside its basin. Accordingly,
Γ_c
is the natural threshold for the existence of a local basin that could become dynamically attracting under Gaussoherence.
By contrast,
Γ_bind
answers the conservative energetic question of whether that localized state lies below the diffuse continuum.
Thus a configuration in the interval
Γ_c < Γ < Γ_bind
could be a genuine dynamical attractor under an idealized noiseless contractive flow while remaining metastable against additional processes not represented by that flow, such as quantum tunneling through the outer barrier, sufficiently large perturbations, or stochastic fluctuation channels.
The persistence of a Distinction Attractor therefore involves two logically separate questions:
-
Does Gaussoherence dynamically capture nearby states into the local basin?
-
Once captured, what escape channels remain in the complete theory?
The first is controlled primarily by the existence and stability of the local branch; the second requires physics beyond the conservative Gaussian reduction.
So, configuration density defines Fisher distinction; finite-resolution distinction competes with quantum spreading; that competition can produce a stable nonzero-width branch; and the projection of that surviving distinction structure supplies the distinction-dependent contribution to gravitational curvature. When embedded in contractive Gaussoherence, this stable branch becomes the candidate Distinction Attractor.
