Hughes iteration-depth no-gap conjecture

For a finite alphabet and arbitrary languages A and B, fix a positive insertion degree k throughout each iteration history. A word has iteration depth m when m is the least stage at which the word occurs for some such degree. Hughes's Section 10 conjecture states that if depth r occurs, every depth from zero through r occurs. The formalization also proves this statement over arbitrary alphabets, with no finiteness or countability assumption.

This is the arbitrary-language iteration-depth conjecture on page 8. It is not the finite-language Theorem 8 and not the insertion-degree conjecture in Section 11.

The frozen word-power recurrence records repeated concatenation of finite words. Literal insertion is the next operation considered here: it alternates source and base factors and then repeats that operation at a fixed degree. The published question asks whether the first-occurrence stages of this richer hierarchy still form an initial interval.

The paper proves the no-gap property for finite languages by maximum word length, then asks on page 11 whether the finite case implies the arbitrary case. That maximum-length argument is unavailable for arbitrary languages.

The bounded prior-art inspection found no exact resolution in the searched repository, pinned Mathlib, or inspected external sources. The 2025 Ibarra–McQuillan accepted manuscript was inspected through its definitions, 15 propositions, seven corollaries and conclusion; its binary-operation decision, closure and transducer results do not state the repeated fixed-degree minimum-depth spectrum theorem. The relevant portions of the 2007 and 2023 UCF notes and arXiv:2603.26162v2 likewise concern different decision or shuffle questions. Source URLs, inspected ranges and hashes are recorded in the library note. This is a bounded search result, not a worldwide absence claim.

Pad any degree-j insertion factorization to a larger degree K by appending empty (x,y) factor pairs. This preserves the concatenated base word, source word, and output. Induction lifts every stage of the fixed-degree iteration from j to K.

Suppose a word first occurs at stage r+1 through degree k, and extract the base predecessor v from its final factorization. If v occurred at a stage s<r through degree j, lift that predecessor occurrence to K=max(k,j), pad the final degree-k insertion to degree K, and obtain the original word at stage s+1<r+1, contradicting minimality. Thus v first occurs at r. Induction on r supplies every smaller attained depth.

An alphabet, arbitrary languages A,B, and a word of minimum depth r>0 for which no word has minimum depth r-1 would refute the theorem. A failure of empty-pair padding to preserve any of the three literal source conditions would invalidate the proof route. A formalization using a varying degree within one history, nonempty-factor assumptions, finite languages, or the Section 11 degree spectrum would not settle this problem.

The primary PDF has SHA-256 dd840a73b56821f5342811bcae34904356c7aa2553e624e7fea5183843dfa1cf. The literal insertion definition is on page 2, minimum depth is on page 7, the arbitrary-language conjecture is on page 8, and the related open-question discussion is on page 11.

D5/S1/Words/IterationDepthNoGap.result proves the conjecture for every alphabet and arbitrary languages A,B: if a natural depth r is attained, then every natural q ≤ r is attained. The literal insertion factorization consists of k source factors and k+1 base factors, each possibly empty, with concatenations in A and in the preceding stage respectively; the output is the interleaving x₁y₁…xₖyₖxₖ₊₁. Stage zero is B. The existential positive degree can vary between witnesses but is fixed throughout each witness's iteration history. Minimum depth means absence at every earlier stage for every positive degree. The proof requires no finiteness or countability of the alphabet, finiteness of the languages, epsilon membership in A, or monotonicity in depth. Empty alphabets, empty languages and epsilon-only languages are included; it asserts no attained depth when the spectrum is empty.

The result is stated at closedSourceZero; its spectrum is extensionally equal to the published zero-based spectrum.

The companion D5/S3/ConceptDynamics/InformationEscape/IterationDepthNoGapRegistration registers the theorem's source depth origin through a finite two-origin arena, with an inline realization bridge and a shifted-origin counterfactual for variation and sensitivity. The arena's finiteness does not impose finiteness on the alphabet or either language. Only the S1 result Describe node carries the typed Proved claim for this problem; the registration is not a second resolution.

Hughes retains credit for the conjecture, and the released project lead is credited for selecting this already-screened target. The bounded prior-art findings and access limits below do not establish worldwide priority.

theorem. The result supplies the universal proof for arbitrary alphabets and arbitrary languages, including empty and epsilon-only cases.

The final publisher version of Ibarra–McQuillan (2025) was not compared with the accepted manuscript. The Ito–Sugiura full chapter, Ito's older book, the 1981 Hughes–Selkow original and the 2005 CS-TR05 original remain outside the verified literature scope; later public notes do not replace them. Semantic Scholar was rate-limited and general search engines were unusable. No claim is made about sources beyond the bounded search.