# Beyond polynomial germs: Cairn integration **User-supplied research draft, 20 September 2026. Proposed proofs; independent review, publication priority and Lean verification remain outstanding.** This is an edited integration of *Beyond polynomial germs: orbit-separated mapping tori, recurrence profiles, and simple groups*, supplied in the conversation. The unavailable sandbox download is not claimed to have been copied byte for byte. The complete mathematical integration is split into [mapping tori and the compact-core argument](mapping-tori-and-compact-core.md) and [recurrence, matrix profiles and their consequences](recurrence-and-matrix-proofs.md). These proof artifacts preserve the proposed constructions and their proof gates; the present file is their index, not the original 8,700-word note. ## Proposed statements 1. **Observed automorphism mapping tori.** For k≥2 or k=∞, let D have type F_k, let φ be an automorphism, and let ρ:D→V be a homomorphism satisfying both ∩_{j≥0}ker(ρφ^j)=1 and ∩_{j≥0}ker(ρφ^{-j})=1. Then T(D,φ)=⟨D,t | t⁻¹gt=φ(g)⟩ embeds in a simple F_k group generated by two finite-order elements. This includes every automorphism of every F_k subgroup of V, but does not require ρ injective. The same host contains the specified twisted restricted wreath product V^(Z)⋊T(D,φ). 2. **Recurrence-twisted nilpotent lamps.** Finitely many bilateral integer profiles with independently eventually recurrent positive and negative tails, together with the constant profile and their translates, act on UT_m(Z[1/q]) lamps by grading dilations. The resulting finitely generated semidirect product has a proposed two-generated simple F∞ host. 3. **Noncommuting matrix profiles.** For vector lamps Z[1/q]^r, allow profiles q^{a(n)}U(n), U(n)∈UT_r(Z), with recurrent scalar and matrix-entry tails, and include constant UT_r(Z) and qI. Weighted finite coefficient lattices control the noncommuting products and supply the proposed simple F∞ host. 4. **Exact scalar-germ criterion.** For finitely many eventually integer one-sided profiles, let M contain their germs, the constant germ and all positive/negative shifts. For J=M⋊Z, eventual recurrence of every profile, finite rational rank of M, finite presentation of J, and type F∞ of J are proposed equivalent. This is not a criterion for BH of arbitrary groups containing those germs. The concrete subgroup G_{q,a}=⟨A,X,T⟩, q,a≥2, has unit lamp A at zero, shift X, and T scaling position n by q^{a^{|n|}}. The draft proves algebraically that it is torsion-free, has derived length exactly three, is nonlinear over every field, and has infinite-rank global profile module. The recurrence envelope conclusion depends on the new geometric gates. The integration separates those two claims instead of using formula checks to certify the host. ## Why this differs from the polynomial route The shared new gate proves finiteness of every required SingFix_Γ(M,P), M⊆P finite, by an actual finite-type core and supported contracting HNN letters. Their depth characters prove that the abstract HNN normal forms inject into homeomorphisms. This supplies **BHM Theorem 2.1**; it does not require the successive normal germ filtration of the earlier note. The original base isotropy must be contained in each enlarged profile group. Finite-forest restriction groups supply the clopen stabilizers required by BZ. For mapping tori, eventually-zero and eventually-one binary points are kept as distinct V-orbits, carrying φ- and φ⁻¹-profiles respectively. Combining these noncommuting families at interchangeable endpoints would introduce an uncontrolled profile group. The binary shell shift is the exact prefix map 0w→00w, 10w→01w, 11w→1w; the full profiles satisfy X⁻¹ĝX=φ(g)̂ on every shell, including the bridge. For recurrences, fix one cutoff and take the **forward** integer span of all tail shifts. Its first d values inject it into Z^d. This finite lattice must not be replaced by all eventually integral germs in its rational span. The matrix version uses coefficient groups C_j spanned by bounded products, with C_i C_j⊆C_{i+j}; the upper-triangular weights bound j. An unrestricted coefficient ring could have infinite additive rank. ## Prior work, limits and verification The archive antecedents are `dynamically-v-separated-groups-satisfy-boone-higman`, `dynamically-v-separated-host-singfix-proof`, `zoom-tower-germ-groups-give-f-infinity-germ-extensions`, `block-power-germ-schedules-with-fp-germ-group-are-exponential`, and `annular-free-factor-germ-extensions-fail-bhm-singfix`. Their existing arguments and status labels are not new discoveries or certificates for the strengthened statements. In particular, a one/two-point F₂ stabilizer proof is not already the all-finite-configurations F_k theorem. The draft does not prove arbitrary linear mapping-torus embeddings, injective-endomorphism HNN closure, all solvable or residually finite groups, full BH, failure of F_{k+1} for a host, or quasi-retractions. An automorphism is needed for negative-time profiles. A finite-image observation cannot separate an infinite finitely generated group in this manner. Factorial and 2^{n²} profiles fall outside the scalar finite-presentation criterion, without implying that their lamp groups fail BH. Classical inputs are [BHM, Theorem 2.1](https://arxiv.org/html/2407.03149v1), [BZ, Theorems D and 3.4](https://arxiv.org/html/2001.04579), and [SZ, Corollary 4.28](https://arxiv.org/html/1709.06524v2), with the finite-forest adaptation explicitly proposed in the proof artifacts. The noninjective example's separation from subgroups of V uses [Bleak–Salazar-Díaz](https://arxiv.org/abs/0911.0979). The supplied literature audit distinguishes already-known free-by-cyclic and Aut(F_n) cases in the [BH survey](https://arxiv.org/html/2306.16356), and the stronger [complex-linear quasi-retraction problem](https://arxiv.org/abs/2510.01952). No fresh priority determination is represented here. The supplied [script](check_beyond_polynomial_germs.py) and [reported results](supplied-check-results.json) are retained separately from the fresh MSI replay. The reported total is 5,678 exact finite assertions; none checks infinite-group finiteness, simplicity, nonlinearity or embeddings. The script computes hashes of itself and this **edited index** when replayed. In particular its fresh note hash is not the supplied original manuscript hash. Any mismatch must be recorded, not relabeled as reproduction of unavailable bytes.