A sharp nested-readout family of metric-group products

Exact statement

For every c>=1, the nested readout generated by h_c(t)=t/(c+t) is a continuous bijection from digit streams N^N onto the nonnegative reals, is subadditive under coordinatewise nim-XOR, and therefore transports XOR to an explicit metric-group product; for 0<c<1 the readout is discontinuous and misses (0,1-c].

StatusProved and internally audited

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The construction exposes a presentation pipeline: digit stream, coordinatewise XOR, nested real readout, and transported product.

Hypotheses and scope

The parameter c is positive; the positive theorem assumes c>=1.

Proof or evidence

Finite nests converge uniformly with tail 1/(N+1) at c=1 and (c-1)/(c^(N+1)-1) for c>1. Greedy decoding gives bijectivity, and digitwise XOR subadditivity transports the Boolean group law and metric bound.

Nested construction →

Verification notes

The endpoint c=1, the below-threshold gap, finite-nest identities, XOR subadditivity, and integer-parameter round trips were rechecked. Executable probes are regression evidence, not a replacement for the written proof.

Limitations

The threshold is sharp for this nested family, not for every possible metric-group construction.
No novelty or external review is claimed.

Open work

Seek specialist review of the infinite nested-readout argument and its sharp endpoint treatment.

Notes

Three independent first-pass agents and two deep construction audits reproduced the proof. The exact finite checker is regression evidence, not a proof of the infinite statement. The endpoint c=1 uses the harmonic tail modulus 1/(N+1), not strict Lipschitz contraction. The threshold is for this nested h_c family and is not a universal barrier.