Amenable groups with nearly exponential sofic profile, and quantum channels that need nearly linear memory

How far from finite can an amenable group be? Every finite piece of an amenable group can be imitated by permutations of a finite set, and Cornulier's sofic profile counts how many points such an imitation needs at accuracy 1/r. We build a finitely presented elementary amenable group with a finite piece whose profile is exp(r^(1+o(1))), close to the most our counting method can ever detect. The construction is a lamplighter with its lamps on configurations rather than on points. Houghton's group moves the points of three rays, finitely supported affine maps act on their binary configurations, and a copy of S_3 sits on every configuration. Then N points carry 2^N lamps, yet any two lamps can be brought together by moves that each involve at most three points. The group embeds in Brin's group 3V, so 3V too has a finite piece of nearly exponential profile. The group also gives an explicit quantum channel on C^873 that is cheap to use once and expensive to use many times. A device that applies it n times, releasing each output before the next input arrives, needs about n/B qubits of memory when it may spend B bits of purity, and exact devices achieve this up to polylogarithmic factors. With exchange at the optimal rate, the least memory is about sqrt(n), attained with purity of the same order. The channel is factorizable and lies in the closure of channels with finite maximally mixed baths, yet any such bath that imitates it to accuracy u needs dimension exp(u^(-1+o(1))), although its minimal Stinespring dilation implements it exactly with a pure environment of dimension 130. The engine is a one-round theorem: after a single use, the memory of any such device carries an approximate representation of the group. Version 1.0.1 states the exchange condition of the rank-rate result in the introduction, adds a table of the results used from the two companion papers with their theorem numbers, and adds a single command that runs every check. The mathematical results are unchanged. Version 1.0.2 corrects attributions: the bounded-or-linear dichotomy is credited to Cornulier (sharpened in the companion paper), the embedding of Houghton groups into Thompson's group V to Roever, and finite presentability of the lamplighter over Thompson's group F to Cornulier's criterion; the dependency table now lists every companion result used, including the weighted rounding and extraction lemmas; the causal-memory model is credited to Kretschmann-Werner and Rybar-Ziman; known clauses are moved out of numbered statements. The mathematical results and the numbering are unchanged. Version 1.1.0 rewrites the paper for readers. The abstract and introduction are new; each half now runs as one line from definitions to its main theorem, with side results and long proofs in appendices; four figures and two worked examples are added. New content: a theorem on finite tracial approximations of the channel (Theorem C, Corollary 14.1); the exact Choi rank, 130, computed by two independent engines (verification/choi_rank_check.py); and the proofs of the counting criterion and of sequential extraction, reproduced from the companion papers so that the main lower bounds can be checked from this paper. The paper now notes that Theorem A contradicts the conclusion drawn from Cornulier's Theorem 3.15, that the Connes embedding problem has a negative answer, and credits Musat and Rordam, Rybar and Ziman, Gutoski and Watrous, Chiribella, D'Ariano and Perinotti, and Anantharaman-Delaroche. A question left open in version 1.0.2 is answered from the companion paper. The theorem numbering has changed; the statements of the results of version 1.0.2 are unchanged, except that the Choi rank is now stated exactly. Version 1.1.1 makes the corrections found in final readings before arXiv submission. The reading guide now says that the profile bounds do not use Sections 7 and 8; the companion paper's results on far commutators are cited as its Theorems 3.2 and 3.3; the definition of a causal service states that each round has its own unitary; the set of gadget coefficients is stated exactly; the factorizability of the gadget channel is noted where the channel is defined; the dependency table lists two further results of the companion paper; the Choi-rank count for the grid host is explained; and several notation clashes are removed. The mathematical results and the numbering are unchanged. Version 1.1.2 places the paper in the literature of both fields. The introduction now relates the results to the sofic groups of Gromov and Weiss and the non-sofic groups found in 2026 (OpenAI, Fournier-Facio, Kun and Thom), to the lower bounds of Slofstra and of Slofstra and Vidick, and to Arzhantseva and Cherix; and, on the quantum side, to the quantum reverse Shannon theorem, storage and retrieval of unitary channels, memory costs of sequential measurements and of quantum models of processes, collision models, process tensors and the resource theory of purity. The reading guide now refers to the parts of Theorem A as stated in the introduction. The mathematical results and the numbering are unchanged. This preprint archive includes the PDF, TeX source and finite verification scripts. The scripts are regressions for the stated certificates; the general statements are proved in the manuscript. The grid variant described in a remark is only sketched.