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Uniqueness theorem

FPRD-T120

XOR is forced on the nonnegative integers

Exact statement

Bitwise XOR is the unique Boolean group law circle on the nonnegative integers with identity zero satisfying p circle q <= p+q for every p and q; equivalently, it is the unique metric-group product on the nonnegative integers.

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

Context

The integer carrier is a discrete calibration: it shows that the metric domination inequality alone recovers the familiar binary coordinate law.

Hypotheses and scope

  • The carrier is exactly the nonnegative integers.
  • The operation is a Boolean group law with identity zero and satisfies p circle q <= p+q everywhere.

Proof or evidence

Induction on H_k={0,...,2^k-1} forces translation by 2^k to send i to 2^k+i. Associativity and exponent two then extend the XOR table from H_k to H_(k+1).

Integer rigidity →

Verification notes

The block induction, unused-value argument, Boolean-law closure, and finite tables were independently rechecked. Elliott's nearby lattice-valued result has different hypotheses.

Limitations

  • The theorem concerns the discrete carrier N_0, not uniqueness of metric-group products on the real half-line.
  • The bounded literature search does not support a novelty claim.

Open work

Seek specialist review and complete the literature comparison with Elliott's Boolean-algebra uniqueness theorem.

Notes

The proof inducts over the binary initial blocks [0,2^k). The numerical domination forces translation by 2^k to map i to 2^k+i, after which the Boolean group laws force the full XOR table. Elliott's 1953 symmetric-difference uniqueness theorem is close prior art under Boolean-lattice and Boolean-operation hypotheses; the purely numerical characterization here was not found in the bounded search, which is not a novelty claim.