{"id":"1560f413-a2b4-4c16-b570-746eb8fde968","arxiv_id":"2606.06241","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"In a perfectoid ring R, I-torsion is I_perfd-almost zero for any ideal I, giving an excision-type decomposition of R along its I-torsion part.","lead":"The authors prove that for a perfectoid ring R and any ideal I, the I-torsion in R is almost zero after perfectoidizing I, which yields an excision-style decomposition of R. This is a structural tool for how perfectoid rings break along general ideals in p-adic and arithmetic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the abstract-only limitation already noted by the reader.","rationale":"The reader’s weakest_assumption already isolates the single load-bearing point that can be seen from the abstract: applicability of the excision square to general ideals. With no full text available, no sharper technical objection (e.g., a missing flatness hypothesis, an incorrect almost-zero statement, or a gap in the use of André’s lemma) can be formulated. The correct posture is therefore to leave the verdict UNVERDICTED and the confidence LOW, exactly as the reader did. Manufacturing an additional concern would violate the good-faith and non-manufacturing rules. The concrete test simply operationalizes the check that would resolve the existing uncertainty once the paper body is in hand.","tokens_in":1895,"tokens_out":430,"duration_ms":4855,"concrete_test":"Obtain the full text (or at least the statement of the main theorem and the paragraph invoking the excision square). Verify whether the square is applied to an arbitrary ideal I or only after reducing to a finitely generated / perfectoid ideal via André’s lemma; if the latter reduction is missing or incomplete, the general-I claim does not go through.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract-only status means the precise hypotheses under which the excision square for perfectoidization (via p-complete arc descent) applies to a completely general ideal I cannot be checked. That is exactly the soft spot the reader already flagged: if the square requires finite generation, perfectoidness of I, or other restrictions not stated in the abstract, the tameness claim (I-torsion is I_perfd-almost zero) and the resulting decomposition fail to hold in the stated generality. No further internal inconsistency or hidden circularity is visible from the abstract; the tools named (André’s lemma, p-complete arc descent) are standard and the outline is coherent. The concern is therefore purely one of unverifiable scope, not of a concrete mathematical flaw that can be isolated without the body of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims a tameness theorem for torsion in perfectoid rings: if R is perfectoid and I any ideal, the I-torsion submodule of R is I_perfd-almost zero. The argument is said to combine André’s lemma with the excision square for perfectoidization arising from p-complete arc descent, and the almost-vanishing is then used to produce an excision-type decomposition of R along its I-torsion part. Additional structural remarks on (semi)perfectoid rings and perfectoid ideals are announced.","tokens_in":2029,"tokens_out":570,"duration_ms":14459,"significance":"A decomposition of an arbitrary perfectoid ring along the torsion with respect to a completely general ideal would be a useful structural tool, extending known almost-mathematics results that typically impose finite-generation or other restrictions on I. The named ingredients (André’s lemma, p-complete arc descent) are standard and appropriate; if the claimed generality holds, the paper supplies a clean, parameter-free tameness statement that subsequent work on perfectoidization and prismatic cohomology could cite.","major_comments":[{"comment":"The abstract asserts the tameness statement for an arbitrary ideal I. The only indicated tools are André’s lemma and the excision square coming from p-complete arc descent. Without the body of the paper it is impossible to verify whether that square is established (or even stated) for non-finitely-generated ideals; if the square requires finite generation, perfectoidness of I, or other unstated hypotheses, the main claim fails in the advertised generality. This is a load-bearing scope issue.","section":"Abstract"},{"comment":"The abstract gives no indication of the precise almost-zero estimates or of the intermediate lemmas that convert the excision square into the statement that I-torsion is I_perfd-almost zero. Those estimates are essential for the subsequent decomposition; their absence from the available text leaves the central derivation unchecked.","section":"Abstract"}],"minor_comments":[{"comment":"The abstract is clear and self-contained as far as it goes, but the phrase “excision-type decomposition” is left undefined; a one-sentence expansion of what the resulting Cartesian square or short exact sequence looks like would help readers assess the strength of the claim.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"Only the abstract was supplied for review. A definitive recommendation is therefore impossible; the report above flags the single load-bearing uncertainty that can be isolated from the abstract alone. Once the full text is available the same referee can re-evaluate quickly."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this is an abstract-only claim of a tameness theorem: for perfectoid R and any ideal I, the I-torsion in R is I_perfd-almost zero, which then gives an excision-style decomposition of R along that torsion. That is the actual new structural statement; everything else is supporting remarks on (semi)perfectoid rings and perfectoid ideals.\n\nWhat they do well, on the face of it, is name a clean, usable consequence and a standard toolkit (André’s lemma plus the excision square for perfectoidization coming from p-complete arc descent). If the argument goes through for completely general I, this is the kind of lemma people will quietly cite when they need to move torsion out of the way in p-adic or almost-mathematics arguments. No free parameters, no invented gadgets, no circular fitting visible from the abstract. Circularity burden is just the usual dependence on the foundational papers.\n\nThe soft spot is exactly the one the reader and the stress-test both flag, and it is real but not damning: we cannot see whether the excision square applies to arbitrary ideals or only to finitely generated / special ones. If the body quietly adds hypotheses, the “general ideals” claim shrinks. That is an unverifiable-scope issue, not an internal contradiction or a load-bearing flaw we can isolate without the text. Soundness is therefore provisional; novelty and significance sit in the solid-but-not-paradigm-shifting band for perfectoid commutative algebra.\n\nWho it is for: people already working with perfectoid rings, prismatic cohomology, or almost mathematics who need clean structural lemmas. Not a general audience paper. It deserves a serious referee rather than a desk reject—the outline is coherent, the tools are the right ones, and the statement is useful if true in the stated generality. I would not bring it to reading group until the full text appears, and I would not cite it yet, but I would send it out for peer review.","headline":"Abstract-only structural result on I-torsion tameness in perfectoid rings; plausible outline, but scope for general ideals is unchecked.","tokens_in":2663,"tokens_out":489,"would_cite":false,"duration_ms":5330,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"In a perfectoid ring R, the I-torsion submodule is almost zero with respect to the perfectoidization of any ideal I, yielding an excision-type decomposition of R along that torsion.","keywords":["perfectoid rings","perfectoidization","I-torsion","almost zero","excision square","André's lemma","p-complete arc descent","semiperfectoid rings"],"falsifier":"Exhibit a perfectoid ring R and an ideal I for which some nonzero I-torsion element of R fails to be annihilated by every element of a power of I_perfd, or for which the corresponding excision square of perfectoidizations fails to be Cartesian.","tokens_in":2760,"feed_emoji":"📐","tokens_out":579,"duration_ms":11837,"temperature":0.7,"pith_summary":"The paper establishes a tameness theorem for torsion inside perfectoid rings: given any perfectoid ring R and any ideal I of R, every element of R annihilated by some power of I is almost zero with respect to the perfectoidization of I. That almost-vanishing statement produces an excision-type decomposition of R that splits off its I-torsion part in a controlled way. The argument relies on André’s lemma together with the excision square for perfectoidization that arises from p-complete arc descent, and it applies to completely general ideals rather than only special or finitely generated ones. Along the way the authors record structural facts about semiperfectoid rings and perfectoid ideals that make the main decomposition usable in broader perfectoid-algebra settings. A sympathetic reader cares because the result removes a long-standing restriction on which ideals one may safely cut along inside perfectoid rings.","feed_headline":"Perfectoid rings tame torsion for every ideal","feed_subtitle":"I-torsion becomes I_perfd-almost zero, giving an excision decomposition of the ring","key_machinery":"The excision square for perfectoidization supplied by p-complete arc descent, combined with André’s lemma; together they control the almost-vanishing of torsion and produce the decomposition of R.","core_discovery":"If R is a perfectoid ring and I is an arbitrary ideal of R, then the I-torsion submodule of R is I_perfd-almost zero; consequently R admits an excision-type decomposition that isolates its I-torsion part via the perfectoidization of I.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Perfectoid rings make any ideal's torsion I_perfd-almost zero","I-torsion of a perfectoid ring is almost zero after perfectoidization","Excision decomposition isolates torsion in perfectoid rings","Torsion tameness for perfectoid rings along general ideals","Perfectoid rings admit excision via almost-vanishing torsion"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The excision square for perfectoidization coming from p-complete arc descent must remain valid for completely general ideals, not only for special or finitely generated ones.","fun_headline_variants_meta":{"raw":{"variants":["Perfectoid rings make any ideal's torsion I_perfd-almost zero","I-torsion of a perfectoid ring is almost zero after perfectoidization","Excision decomposition isolates torsion in perfectoid rings","Torsion tameness for perfectoid rings along general ideals","Perfectoid rings admit excision via almost-vanishing torsion"]},"model":"grok-4.5","effort":"low","cost_usd":0.006056,"raw_usage":{"total_tokens":1496,"prompt_tokens":634,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":60560000,"prompt_tokens_details":{"text_tokens":634,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":772,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":634,"tokens_out":90,"duration_ms":7027,"temperature":1.0,"reasoning_tokens":772,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T15:00:56.930885+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a perfectoid ring R and an ideal I for which some nonzero I-torsion element of R fails to be annihilated by every element of a power of I_perfd, or for which the corresponding excision square of perfectoidizations fails to be Cartesian.","supporting_citations":[],"review_version":2}