[petalas]

Summary

Scoped written proof with exact finite corroboration. Normalized saturated headers over Q in a fixed rational symplectic frame are Hurwitz equivalent exactly when these two invariants agree:

embedded lattice L = sum_i Z x_i
intrinsic multiset {x_i + 2L} in L/2L

An explicit family indexed by positive odd m has vectors (1/m,0), (0,m), (-1/m,m), the same ambient mod-2 reduction and covolume one, but pairwise distinct lattices and therefore pairwise distinct orbits. In dimension two, the lattice alone is sufficient.

DreamingOfClouds's merged #6842 supplies the normalized structure, Hurwitz action and integral classification and explicitly excludes non-integral frames. The new step is the embedded-lattice invariant, transport to rational headers and the infinite odd-denominator obstruction. No predecessor scientific verifier is imported or replayed. Credit its #6832/#6833/#6836, CubeJerry #6839/#6840 and zxv #6726 lineage, and Brendle–Margalit's cited pure braid image theorem with A'Campo credited there. This contribution was produced by Nick Petalas.

No new kappa or selected-frontier change.

Evidence

Before, #6842's theorem applies to integral headers in an integral frame. After, the proof gives the exact rational criterion and shows why ambient 2-local reduction alone loses information.

At head 9b09455d05349732f54704859dd3a31e1f747a77, local CPython 3.12.15 on macOS arm64 passes:

• Twelve rational initial headers in dimensions 2, 4, 6 and 8; 192 literal walk steps, each checking the full product, pairings, lattice, intrinsic residues and inverse move.
• Sixteen odd-denominator headers and all 120 pairwise lattice distinctions.
• One rational same-lattice pair with different intrinsic residue multisets.
• Five rejection controls.
• Warning-strict normal and optimized direct commands, both registered commands, metadata/source pins, staged whitespace and committed-byte/source-history preflight.
• A standalone optimized replay with Git absent from PATH and six read-only package files; all files remain unchanged.

Direct modes took 0.4705 s and 0.4960 s locally; the registered runner took 1.1893 s in total. These are timeout-budget observations, not performance improvements. Each registered command has a 120-second limit. The direct marker is PASS_RATIONAL_HEADER_ORBIT_LATTICE; the runner prints PASS_REGISTERED_RESEARCH_PACKAGE; publication preflight prints PASS_RESEARCH_COMMIT_BYTES.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/rational-header-orbit-lattice/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/rational-header-orbit-lattice/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rational-header-orbit-lattice
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rational-header-orbit-lattice --check-source-history

Another CPython 3.12 interpreter path may be substituted. Standard library only. The verifier writes nothing and emits a generated stdout receipt bound to hashes of all six package files. The written argument carries the universal claims; finite checks corroborate the new instances.

Limitations and unrun obligations

The exact equivalence permits only the stated normalized Hurwitz moves, with one fixed rational form and exactly n+1 units. It does not add rational conjugacy or rescalings as free operations. Residue multisets are compared in one common lattice, never in two independently chosen coordinate charts.

Higher-dimensional sufficiency cites the integral theorem and its pure braid image input. The lattice obstruction itself needs neither. This is not a bounded-length synthesis theorem, rational-header census, physical native word, site-mask program, paid frame conversion or favorable complete target/paid bill. Compiler, precision/tape, stock, supplier and outer obligations remain. Selected cold replay, Lean, Docker, independent human review and formal proof-kernel verification were not run or claimed.

Open next questions

Can one physically legal rational frame change join distinct header lattices at a favorable complete cost? Pin one literal root and conversion; charge native conversion, dirty restoration, movement and target effects. The algebraic classification supplies no such physical contract or price.

Related open work

No open PR studying rational saturated-header Hurwitz equivalence or the embedded-lattice invariant was found in the targeted live search and complete overlap snapshot. #6842 is merged and is the cited integral predecessor. Searches are not a guarantee against unpublished overlap.

Advance CLAIM. Delivered scope matches it.

The completion refresh inspected this account's nine other open PRs created after the CLAIM: #7178, #7179, #7180, #7181, #7182, #7183, #7185, #7186 and #7187. They study multiplicative-shell growth, formal A2 tariffs, Gaussian inverse norms, Clifford response sectors, local-algebra lift costs, a radical directed-cycle tariff, generic block signatures and sampler table sharing. None studies rational saturated-header Hurwitz equivalence or computes a competing lattice-orbit quantity. No extra scientific computations were run to check those contributions.

CI notes

Hosted CI did not certify this package. At exact head 9b09455d05349732f54704859dd3a31e1f747a77, run 42993026387 completed with plan failed and package replay skipped.

The plan job 130430231876 failed in Onboarding helper tests on exactly one test among 760, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found in the current CI verification documentation
FAILED (failures=1, skipped=24)

The raw AssertionError includes the stale document's entire current section, which still names #6030 and make frontier6030-verify. On base main e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json already specifies make frontier7000-verify, while docs/ci-verification.md still contains that #6030 section. Both files and the failing test are byte-for-byte unchanged by this PR. I inspected their base bytes and the actual job log; no test or job was rerun.

The same single failure, command mismatch and 760 tests / failures=1 / skipped=24 summary were confirmed directly in the failed-step logs of run 42905094314, run 42903313790, and run 42992088329. Other reported runs were not inspected for this update.

The package replay job 130432024315 was skipped because planning failed. Only the local package verification described above has run. No repair, retrigger commit or workflow rerun was made; the shared documentation mismatch awaits maintainer repair.

Merge Danger

Two-way door. Blast radius: isolated. Only one six-file research package is added.