[petalas]

Summary

Scoped proof with finite exact corroboration: for the fixed binary source word B0 ... B(n-1) B0 ... B(n-1), universal paired output readouts require exactly n-1 arbitrary dirty roots for every n>=5. The small minima are one root at n=1,2,3 and two roots at n=4.

fixed repeated source order
  prefix differences = cyclic arcs
  feasible reader code contains ones
  rank-two quotient would require K2,2 inside a longer cycle
  therefore codimension <= 1
  a parity hyperplane attains n-1 roots

The upper program preserves every arbitrary dirty root, calls each source/output map twice, and charges (2n-5)*2^n+4 root XORs. At n=5 its complete abstract incidence is four roots, 10 source calls, 62 readouts and 164 root XORs. The exact physical-port cost expression is in research/cyclic-source-root-stock/PROOF.md; no transfer-tariff optimum is claimed.

This contribution was produced by Nick Petalas. Credit Arham Tahir / #2795 for the exact root-code criterion, conorholds / #2182 for normalized axes and the differently ordered star, Mason Hale / #2282 for mandatory-mask methods, Ross Richardson / #934/#943 for the dirty-prefix model, and Douglas Colkitt / #1185 and the retained connector lineage. Their main-tree source blobs are pinned as provenance. No predecessor implementation is imported or audited as the deliverable.

Evidence

Before, the credited feasibility criterion reduced this source-order question to reader codes. After, one uniform proof classifies every feasible code at n>=5: precisely the full space and hyperplanes containing ones. It explains why repeating the same order can consume more root stock than the credited n-3-root star with a different source word.

Local CPython 3.12.15 on Darwin arm64 passes:

• All 3,290 codes containing ones at n=1..7, with exact dimension histograms and minima.
• All 85,818 surjective rank-two quotients killing ones at n=5..9. At n=5,6, each missing-mask witness is additionally checked against its entire kernel and difference set.
• Cyclic-arc identities and literal arbitrary-dirty forward/inverse programs at n=1..10; all 236 operation omissions at n=5 reject; nine invalid argument cases reject; the n=4 exception succeeds.
• Normal and optimized warning-strict registered checks, metadata/pins, seven-file committed-byte/source-history preflight, official committed export integrity, and optimized Git-free replay with empty PATH.
• Exported source corruption, receipt-count corruption and integer-to-float receipt corruption reject. All eight export files remain unchanged after the clean replay.

The generated RESULT.json binds six local source/metadata hashes. Each registered command took about 0.35 seconds locally, within its 180-second limit. These timings document replay feasibility, not a speedup.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/cyclic-source-root-stock/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/cyclic-source-root-stock/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cyclic-source-root-stock
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cyclic-source-root-stock --check-source-history

Expected verifier output:

PASS_CYCLIC_SOURCE_ROOT_STOCK codes=3290 quotients=85818 omissions=236 roots=n-1_for_n>=5

Limitations and unrun obligations

The fixed repeated source order, exactly two calls per source/output, generic binary linear maps, immutable sources, restored arbitrary dirt and scalar invertible root frames are essential. This is a scoped proof with finite corroboration, rather than a complete multiplication theorem. Physical caller placement, access/head-return implementation, hosting, precision/tape transfer, complete recursive profitability and supplier admission remain open. Independent human and formal proof-kernel review are unclaimed. No native, Lean, Docker or historical full scientific replay was needed or run. No new kappa or selected-frontier change is claimed.

Canonical main was refreshed before submission and remains 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. The selected conditional kappa remains 0.00302015. The card's prescribed original connector positive control passed. Entry overlap covered 411 open PRs and 3,919 Discussion records; sparse local-note omissions were supplemented by distinctive all-main README/proof and merged-term searches. Arbitrary diffs and unpublished work remain coverage limits.

Open next questions

• Does one specified other double-permutation source order have a closed-form root minimum? Start with its exact difference set; the cyclic-arc obstruction does not automatically transfer.
• Does an actual recursive caller expose this order and benefit from reordering? Pin its chronology, stable ports and complete live-root/access/restoration ledger before claiming an adoption benefit.

Related open work

No current open PR was found for this exact repeated-order root minimum. Nearby connector scopes are:

• Six-source one-TRIO path misses full paired coverage at 14 calls #2134 studies a six-source one-TRIO source alphabet and paired coverage. This theorem permits only singleton source calls and changes root stock instead.

• Research: high-arity connector toggle cost floor #2249 prices higher-arity six-source events. It changes the source operation alphabet; this proof fixes the repeated singleton word.

• Add five-source arity-four connector cost bound #3909 prices an arity-four five-source event. It studies dispatch/addition cost rather than the arbitrary-frame root minimum here.

• research: audit minimal joint-output connector charge #3897 combines two output-map evaluations in one dispatch. This theorem retains separate output readouts and their full incidence.

• Construction: three-stage persistent prefix shear chain (no new kappa) #1930 lets updated targets feed a persistent prefix chain. Targets here remain separate offline outputs and cannot feed back.

• Prove the exact signed two-source prime-field connector cost (no κ) #4834 determines signed two-source full-coefficient query cost over odd prime fields. This theorem uses binary masks, fixed order and variable root stock; no numerical comparison is made.

• Prove optimal weighted borrowed-source pair tours (no new κ) #4899 optimizes weighted borrowed-source tours for pair-mask targets. It temporarily uses a source as its anchor and prices scalar ADD transitions, while this theorem preserves immutable sources and optimizes additional dirty-root stock for a fixed all-mask order.

• Prove sharp high-weight excess 11 at four labels and length ten (no κ) #4894 proves a high-weight excess bound for simple four-label length-ten prefix walks. This theorem fixes two repeated traversals, length eight at four labels, and computes root stock rather than excess.

• Prove a Fibonacci paid-scan floor for two dense dirty outputs (no κ) #4891 lower-bounds full-bank ADD scans for two dense chirp outputs. The endpoint, operation alphabet and measured quantity differ from this generic all-mask connector's root minimum; no numerical comparison is made.

The pre-DONE refresh read all 71 open PRs on this account, including ten created since the CLAIM and nine other submissions. None computes a competing root minimum for this repeated source order. The three nearby new scopes are recorded above. No additional predecessor computations were run, and no shared computed number disagrees.

CI notes

Both hosted checks passed at head acd06838b6700e733b42c1eb016d1e0e8d638f9e in Registered research packages run 39870298674: plan and registered package replay. Run metadata confirms this exact head, completed status and success. The registered replay executes both warning-strict normal and optimized commands. All registered commands and publication/export checks also passed locally. Hosted success certifies the registered automated checks, not independent mathematical acceptance, supplier admission or a complete multiplication theorem.

Advance CLAIM: #1166 (comment). Delivered scope matches it. Research stopped at this first verified result.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only seven new files in one opt-in research package. Archive intake preserves the scoped proof without selecting a frontier or admitting a supplier.