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Geometry and computability theorem

FPRD-T117

Intrinsic geometry and representation-sensitive computability

Exact statement

At c=2 the intrinsic metric is complete, separable, zero-dimensional, homogeneous, isosceles-free, and has full distance spectrum [0,infinity); the product is computable from digit names but no metric-group product is total Type-2 computable from ordinary Cauchy names or o-minimal definable.

StatusProved and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Computability depends on the name supplied for a real. Digit names expose the transported XOR coordinates; ordinary Cauchy names expose the Euclidean presentation, where continuity is unavoidable for total Type-2 computation.

Proof or evidence

The readout is an isometry from Baire-space XOR distance to the intrinsic metric. Ordinary-name computability would force forbidden ordinary continuity, and o-minimal definability would force tame discontinuity behavior incompatible with the dense seam.

Intrinsic and ordinary representations →

Verification notes

The topological properties, distance spectrum, digit-name algorithm, ordinary-name obstruction, and definability boundary were reconciled separately.

Limitations

  • The positive computability claim is stated for c=2 and extends only to effectively presented computable parameters with a modulus.
  • The combined nested-XOR construction has no documented novelty determination.

Open work

Formalize the representation interfaces and seek review of the Type-2 and o-minimal no-go arguments.

Notes

The abstract Boolean-norm classification and the c=1 stream encoding are prior art. Novelty of the full nested-XOR full-spectrum combination is unresolved. Computability is asserted for c=2, or more generally only for an effectively presented computable parameter with an effective tail modulus, not for every set-theoretic real c >= 1.