[petalas]

Summary

Scoped written obstruction with exact finite corroboration. Independent fixed zero-preserving scalar-role bijections conjugating a connected graph of unit ADDs into fixed scalar ADDs have exactly this form:

f_v = s_v H,      H additive,      b_(source,target) = s_target/s_source
one unchanged i-phase compatibility => H is Q(i)-linear
pointwise Gaussian-native word: Phi A Phi^-1 = D A D^-1

Every such nonlinear recoding has a scalar-gauge counterpart with the same native operators, ADD coefficients, chronology and arbitrary-input endpoint. Its native paid profile is identical. No continuity, measurability or differentiability is assumed. This closes a precise nonlinear recoding escape route; it supplies no new kappa.

The proof and checker were produced by Nick Petalas. Cinna Davis's merged #4983 supplies the shared-potential predecessor, cited without replaying its source audit. Cauchy's functional equation, scalar gauge consistency and linear commutation are classical; the contribution is their complete independent-role classification and Gaussian-native application. Only six files in research/nonlinear-role-recoding/ are added. Advance CLAIM.

Evidence

Before, the shared-map criterion leaves independent role maps and changed edge coefficients unspecified. After, spanning-tree weights and cycle consistency classify them completely, and one phase compatibility propagates the coefficient-field constraint across the graph.

Local CPython 3.12.15, macOS arm64, standard library only:

• All 5,632 map/label assignments on 54 connected directed three-role graphs over F3, and 1,105,920 assignments on a path and triangle over F5. Literal gate equations agree with the classification; 432 and 128 assignments respectively are compatible.
• All 40,320 zero-preserving F9 permutations: 48 additive, 32 phase-compatible, eight both. Explicit witnesses delimit the two conditions.
• 48 complete seven-event fixtures over Q(i,sqrt(2)), including every rational basis column and mixed dirty inputs. Non-scalar Q(i)-linear recoding, direct scalar gauges and recoded instruction execution agree on 288 scalar outputs, comprising 1,152 rational components. An independent monomial product checks 81 extension-arithmetic pairs. Nine graph/domain/hypothesis controls pass.
• Normal and optimized warning-strict direct checks pass in 0.457 and 0.476 seconds. Both registered commands pass; their 60-second limits leave ample replay margin. These are replay-budget measurements, not a speedup claim.
• Metadata/source pins, staged whitespace, committed bytes/modes and source-history preflight pass. The official committed archive passes input coverage. Read-only, Git-free normal and optimized exported replays preserve all seven exported files and their modes. Generated stdout receipts bind all package hashes.

Reproduce from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/nonlinear-role-recoding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/nonlinear-role-recoding/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/nonlinear-role-recoding
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/nonlinear-role-recoding --check-source-history --check-modes

Another CPython 3.12 interpreter may replace the absolute path. Expected markers are PASS_NONLINEAR_ROLE_RECODING, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The universal result rests on PROOF.md, not on finite-field testing.

Limitations and unrun obligations

The maps are fixed per role, pointwise, scalar and zero-preserving. ADD compatibility holds on all independent inputs; the graph is weakly connected and one role preserves multiplication by i. General nonlinear circuits, occurrence-dependent recodings, non-pointwise address maps, approximate/rounded identities and restricted reachable states are outside the claim.

This is equality of attainable native profiles within the declared recoding class. Physical encodings, boundary adapters, ordinary scalar costs, precision, storage, movement, a complete signed invoice and supplier/compiler/hosting admission remain separate. No selected-word scientific replay, Lean, Docker, complete multiplication theorem or selected-frontier update was performed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can occurrence-dependent recodings lower a complete helper invoice? Specify one changed pair of consecutive recodings and all replacement gates and boundary adapters, then price restoration, storage and the full signed bill. No such physical construction or budget is supplied here.

Related open work

• Exclude finite-menu adaptivity for exact helper endpoints (no new κ) #6015 studies data-dependent selection from a finite menu of fixed linear endpoint maps. This contribution instead classifies one fixed family of arbitrary role bijections through exact gate conjugacies; neither supplies the other's premises.

• Prove zero-preserving polynomial helpers cannot reduce linear call minima (no new κ) #6017 studies zero-preserving polynomial arithmetic words by degree-one extraction. This contribution permits nondifferentiable role bijections but requires fixed coordinatewise conjugacies and exact ADD compatibility. The mathematical objects and price comparisons differ.

• Prove a sharp ten-unit floor for the nonlinear six-frame seed #5944 bounds nonlinear finite-alphabet circuits in a fixed six-frame alphabet. This contribution concerns arbitrary complex scalar bijections under a different connected ADD/native contract and computes no competing finite-alphabet minimum.

• Prove sharp two-role dyadic helper-basis conditioning (no new κ) #6016 optimizes the Euclidean conditioning of a two-role dyadic linear basis completion. This contribution classifies fixed nonlinear role recodings and computes no condition number.

• Prove exact chronological helper spill and repair rank (no new κ) #6019 determines chronological leakage and repair rank for fixed linear helper replacements. This contribution has no replacement cuts or competing repair-rank value.

• Certify six-bit stochastic precision for the selected P104 prefix (no κ) #6026 gives stochastic precision bounds for a selected scalar prefix. This contribution requires exact gate conjugacies and supplies no rounded-error or precision-bit bound.

• Decide paired lattices with rank-one projector anchors (no new κ) #6027 decides common paired lattices for projector-anchored rational Borel words. This contribution uses scalar pointwise bijections and computes no invariant-lattice decision or rank floor.

The completion refresh inspected this account's 12 other open PRs created since the advance CLAIM: #6016, #6017, #6018, #6019, #6020, #6021, #6022, #6023, #6024, #6026, #6027 and #6028. Adjacent work is distinguished above. The others concern CRT ROM reuse, continuation headers, transpose replay, S3 convolution, H6 repair, physical convolution and XOR tape travel. None computes a competing independent-role recoding classification or native-profile minimum. No extra mathematical computation was run to audit these contributions or compare their numbers.

CI notes

Both hosted checks passed at exact head e15e504b2ce847923792cb6178f6f49903df472c: plan and replay (research/nonlinear-role-recoding), in run 40592582029. The completed check list was read once, and run metadata confirms completed success at the same head as the PR. Hosted replay certifies both registered normal and optimized finite checks. All listed local scientific, metadata, committed-byte/source-history/mode and portable-export checks also passed. The universal written theorem and missing physical obligations retain their stated scope; independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast Radius: package.

Only an isolated opt-in research package is added. Shared code, frozen predecessors and the selected result are unchanged.