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Cohesion Dynamics (CD)

P-DM1 — Dark Matter Halo Core Scaling

Abstract

This paper states a precise, falsifiable prediction for dark matter halo cores derived from Cohesion Dynamics.
The prediction concerns the scaling of halo core radius with total halo mass and the emergence of an approximately constant central surface density.

No empirical fitting, simulation, or data analysis is performed here.
This paper only states what must be observed if Cohesion Dynamics is correct.

1. Scope

This paper:

  • Introduces one prediction (P-DM1)
  • Fixes its mathematical form
  • Specifies falsification criteria

This paper does not:

  • Analyse observational data
  • Compare against other theories
  • Introduce fitting procedures
  • Discuss datasets or surveys

Those belong to application papers.

2. Physical Basis (Summary)

From G1–G4, gravity in Cohesion Dynamics arises from availability gradients in CIU constraint saturation.

Key consequences already established:

  • CIU constraint saturation plays the substrate role of mass–energy
  • Closure saturation limits central compaction
  • Extreme constraint saturation induces a self-limiting core, not a cusp
  • Geometry and dynamics respond universally to availability gradients

These mechanisms force a specific form for equilibrium dark matter halos.

2.1 Boundary-Limited Saturation and Surface Density Invariance

Core insight from G4: Reconciliation is a boundary process, not a volume process.

In Cohesion Dynamics:

  • CIUs reconcile with neighbours across interfaces
  • Mismatch flows across reconciliation surfaces
  • Closure delay accumulates along reconciliation chains

Nothing in G4 privileges volume. Therefore, the saturation condition must be expressible as:

constraint load per reconciliation surface≈constant\text{constraint load per reconciliation surface} \approx \text{constant}

Why surface density, not volume density:

Closure and reconciliation are boundary-driven processes. The critical condition for saturation is reached when:

Additional constraint reuse does not reduce marginal mismatch resolution cost.

This condition is interface-limited, not volume-limited, because:

  • Reconciliation happens across CIU boundaries
  • Closure cost accumulates along paths through the CIU network
  • No background manifold privileges volume integration

Therefore: The saturation condition forces a constraint:

Σc≡M(<rc)rc2≈Σ0=constant\Sigma_c \equiv \frac{M(<r_c)}{r_c^2} \approx \Sigma_0 = \text{constant}

where Σ0\Sigma_0 is the critical surface density at which boundary-limited saturation occurs.

This is not an assumption—it is forced by:

  • Relational ontology (only CIUs and reconciliation chains exist)
  • Boundary-driven reconciliation (closure is an interface process)
  • Absence of a background manifold (no volume privilege)

Deriving the √M law:

From the surface density invariant:

M(<rc)∼Σ0 rc2M(<r_c) \sim \Sigma_0 \, r_c^2

Solving for rcr_c:

rc∼MΣ0r_c \sim \sqrt{\frac{M}{\Sigma_0}}

The square-root scaling follows from:

  • Dimensional analysis (M∝r2M \propto r^2 for fixed surface density)
  • Boundary-based saturation (interface-limited, not volume-limited)
  • No volume privilege in relational ontology

This derivation involves:

  • ✅ No acceleration thresholds
  • ✅ No tolerance WW tuning
  • ✅ No empirical fitting
  • ✅ No modified inertia
  • ✅ No dark particles

Only: Closure saturation must be boundary-limited in a relational ontology where reconciliation is an interface process.

3. The Prediction

3.1 Core Radius Scaling

For an isolated dark matter halo of total mass ( M_{\text{halo}} ), the equilibrium core radius ( r_{\text{core}} ) satisfies:

rcore=r0Mhalo109M⊙r_{\text{core}} = r_0 \sqrt{\frac{M_{\text{halo}}}{10^9 M_\odot}}

where:

  • ( r_0 \approx 0.7 ,\text{kpc} )
  • The exponent ( \tfrac{1}{2} ) is fixed (not a fit parameter)

No free parameters may be tuned per galaxy.

3.2 Central Surface Density

The prediction further implies an approximately constant central surface density:

Σ0≡M(<rcore)rcore2≈140 M⊙ pc−2\Sigma_0 \equiv \frac{M(<r_{\text{core}})}{r_{\text{core}}^2} \approx 140 \, M_\odot \, \text{pc}^{-2}

This value:

  • Is independent of halo mass
  • Emerges from boundary-limited closure saturation (§2.1)
  • Is not imposed phenomenologically

Note: The √M law (§3.1) is algebraically equivalent to this surface density constraint. The physical content is the surface density invariance—the square-root scaling is a consequence, not the fundamental claim.

Boundary-limited definition: The absence of π\pi reflects that reconciliation is an interface process, not a volume process. The numerical value adjusts accordingly (≈140\approx 140 vs ≈100/π≈32\approx 100/\pi \approx 32 for the traditional volume-based definition).

4. Universality Claims

The prediction asserts:

  • The same scaling applies across:
    • Dwarf galaxies
    • Low surface brightness galaxies
    • Spiral galaxies
  • No dependence on:
    • Baryonic feedback tuning
    • Environment-specific parameters
    • Galaxy morphology

Any statistically significant deviation from the fixed scaling falsifies the prediction.

5. Falsification Criteria

P-DM1 is falsified if any of the following are robustly established:

  1. Core radius scales with halo mass using an exponent significantly different from ( \tfrac{1}{2} )
  2. Core radii show large intrinsic scatter incompatible with a single scaling law
  3. Central surface density varies systematically with halo mass
  4. Different parameter choices are required for different galaxy classes

6. Relationship to Empirical Work

This prediction is intended to be tested against observational datasets such as:

  • SPARC
  • Rotation curve halo decompositions
  • Independent halo core catalogues

All empirical testing must reference this prediction exactly, without retrofitting.

7. Status

Epistemic Role: Prediction
Normativity: Strong (falsifiable claim)
Adjustability: None — fixed once stated

P-DM1 stands or falls as stated.