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Marked Prime Word Readout and Deletion

Abstract

Marked Prime Word Readout and Deletion.

A marked word consists of a prime word with product n and a set of k minus one internal positions. Its readout lists one, the prefix products at the sorted positions, and n. These prefix products form a strict divisor chain. The marked word retains the complete word; the strict chain records only the selected products. Each marked copy carries its word mass divided by the binomial number of mark selections, and the readout mass sums these copy masses over the corresponding fibre. The deletion kernel assigns one over k when one of the k internal vertices is omitted, and zero otherwise.

Theorem 1.1 (Exact deletion of marked prime word readouts).

Lean statement: D5/S3/Factorization/Combinatorics/MarkedPrimeWordReadoutDeletion.actual_readout_deletion

Proof. Machine-checked in Lean as D5/S3/Factorization/Combinatorics/MarkedPrimeWordReadoutDeletion.actual_readout_deletion (✓ std3). ∎

Source. Repository-derived.

Commentary.

For n greater than one and k between one and the total prime multiplicity of n minus one, every row of the internal-vertex deletion kernel sums to one. For any real mass on prime words, the readout mass at level k is obtained by applying this kernel to the readout mass at level k plus one. Each smaller mark selection has exactly the total prime multiplicity minus k extensions, and erasing a selected position removes precisely the corresponding prefix-product vertex. The binomial normalizations therefore give the same mass on each coarse readout. Applying the same identity to word masses multiplied by their Boltzmann factors gives the deletion identity for partition masses, with the kernel factor retained.

References