# Law of Genealogical Non-Injectivity

**Scope:** Nodes at epistemic equilibrium in a content-addressed substrate.

**Statement:** For any node N with saturation = 1.0 and CAS-convergent
output, the activation-weighted measure over all lineages in the preimage
of N is uniform. Lineage is gauge choice. The CID certifies the present
state. No present state certifies a unique history.

**Boundary:** Does not hold for unsaturated nodes. Does not license claims
about the world the substrate models — only about relations between nodes.
Does not assert universal non-injectivity of physical law.

**Supported by:**

1. CAS architecture: `put()` is defined on bytes, not paths (structural)
2. Saturation mechanism: optimizer prunes without lineage query (operational)
3. Compression-as-truth-convergence: convergence makes all paths witnesses
   to the same fact (theoretical, Joven 2026)

**Implication for physics (open):** If the forward map of physical law is
non-injective at equilibrium, the arrow of time, symmetry breaking, and
parameter constraints may follow from consistency requirements rather than
initial conditions or fine-tuning. This is a research program, not a result.
The substrate demonstrates the mechanism. The physics must be verified
independently.

**Testable from substrate:** At high-saturation nodes, examine
activation profiles of all lineages in the preimage. Uniform distribution
confirms the law. Concentrated distribution is a finding requiring
explanation.