[SHIN09-git]

Change and mathematical scope

This answers open question 1 of #6652 for balanced unsigned chains of every depth L≥2. If a block with n of each label repeats d times, let q be the least prime divisor of d absent from n, or infinity if none exists. A q below L obstructs every endpoint. For q above L the shorter target is multiplied by d. For q=L the only surviving shift is (d/L)n^L modulo dn. This gives the exact fixed count in terms of one shorter W row. Proof

The proof gives a single divisor D describing all orders that pass the arithmetic conditions. At target zero every divisor is realized, and the minimum rotation period is L times the product of distinct primes ≤L dividing the modulus. A unit-target word can repeat only at prime depth p, exactly p times, subject to a smaller root count. The credited unbalanced depth-six binary square is replayed as a boundary.

At depth five and modulus fifteen, 4,697,520 words have a nontrivial repetition. They all have target 3, least period 15, and form 313,168 shorter cyclic classes. Two different groupings of the 168,168,000 length-15 root words and a separate literal traversal give the same row (4,697,520,0,0). Numerical certificate · Literal output

This is a scoped proof with finite exact computations. The contract concerns algebraic endpoints and rotation classes. Physical scheduling prices, supplier admission and κ require a separate complete budget.

Task and overlap

Scientific base 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. The connector-alphabet-change card, scope key 0c9f43f3c7d61edb36fe, is marked changed to the balanced-chain counting question, using the existing alphabet. Its 945-pairing orientation baseline passed. Only research/chain-all-depth-powers/ is added.

The nearest work is SHIN09-git's depth-three #6080 and Nick Petalas's depth-four #6652. The latter explicitly asks for arbitrary depth. Its source note at e364eeb19e5268967189ed12aa4813ee73edf3dc is retained byte-for-byte in publication ancestor fd1b2f3e103f11734750a0fe93aa2431fd5468ad. The original provenance accompanies that pin. The final refresh covered 635 open PRs, 5,528 closed/merged PRs, 6,755 Discussion records and the local research-note inventory. Historical diffs and unpublished work fall outside this search. Task · Overlap record

Validation

From the repository root, with CPython 3.12 and the standard library,

python3.12 -B scripts/research_package.py --package research/chain-all-depth-powers

The optional uncompressed census can be repeated with a JDK supporting single-file launch,

java -Xmx128m research/chain-all-depth-powers/literal/LiteralQuintic.java

• Directly checked 309,184 block/power pairs and compared literal periods and cyclic classes with the fixed-count transforms.
• Checked 195,072 binomial coefficients, 979,380 order comparisons, 585 constructed zero-target words and 1,211 powers of 173 published depth-five witnesses.
• The independent prefix/inverse-suffix and middle-gap counters agree at n=2,3. OpenJDK 21.0.10 also traversed every complete word at both sizes, including all 168,168,000 at n=3. The registered Python entry point verifies the stored Java source hash and output.
• Both Python modes return identical full receipts. Seventeen actual private-copy mutations were rejected in each mode for their prescribed reasons. All 22 publication files and declared source ancestry passed the official preflight in an independent full-history checkout. The exact published-head export, SHA-256 e61c88c4844090520e88301fe3938d27b609ef5edf4177f57d014d97bf9455c6, passed normal, optimized, fault-injection and literal Java replays with all 23 exported files, including the manifest, byte-identical afterwards. Both hosted jobs passed. GitHub replayed the package and confirmed the committed inputs stayed unchanged. Hosted checks on the submitted head

The finite checks support the mathematical proof. The three counting implementations share the same assisted author. Independent human and proof-kernel review remain open. Reproduction and scope · Validation notes

Credit

Nick Petalas supplied the depth-four handoff, interval/tariff work, depth-three count control, depth-five witnesses and unbalanced binary precedent. Ross Richardson's #943/#934 lineage supplied the complete-call dirty interface. The Parikh matrix encoding, integer power expansion, Burnside's lemma and Möbius inversion are classical. #4249 · #4281 · #4748 · #6037 · Atanasiu, Poovanandran and Teh, Thm. 3.1

Prepared for SHIN09-git with substantial OpenAI Codex assistance in derivation, implementation, verification and writing. Apache-2.0.

Open next questions

The repeated population at m=35,c=12 is W_5(7,1). A published witness already proves positivity. Profiling the full gap counter at n=4 is a concrete first step toward an exact larger count. Separately, the unequal binary profile in #4748 gives a starting test for an extension beyond balanced letter counts.