{"id":"f28c4dcf-f200-49db-8e1e-5f8792fbd76a","arxiv_id":"2505.22372","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A set-theoretic proof that posets projecting to wide forcings, and filter-based Mathias forcings, exhibit the full range of nonamalgamable generic extension patterns.","lead":"This paper shows that many forcing extensions of a model of set theory, including extensions by Cohen-style and Mathias-style forcings, can be arranged to be incompatible in the multiverse: no single larger forcing extension can contain them all. It broadens a known 'blockchain' phenomenon to a wider class of forcing notions, which matters for understanding how different mathematical universes relate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 19's proof starts from a projection π_i:Q_i→B_i, but Proposition 9 only supplies a projection into a cone B_i↾p_i; this gap must be repaired before the main theorem is proven.","rationale":"The reader's verdict CONDITIONAL is appropriate: the paper has a number of compressed arguments. The most clear-cut load-bearing problem is the projection-to-full-Boolean-completion assertion, because it is a false reading of Proposition 9 and affects the entire tagged-condition framework of the main theorem. The reader's weakest_assumption concerned decoding signal ambiguity; that is also relevant, but in the Mathias case an implicit induction (coding points are increasing and all intermediate points are bounded between consecutive coding points) makes the assertion true, and in the Cohen case Claim 18 proves noncoding extensions. The cone issue is not a matter of omitted detail: the stated projection does not follow from the cited proposition. The fix via replacing P_i by the cone P_i↾p_i is natural and likely works, so this does not warrant rejection, but it must be added for the proof to be complete. I therefore recommend keeping the CONDITIONAL verdict, pending the cone repair and a fuller decoding verification.","tokens_in":16684,"tokens_out":46923,"duration_ms":458753,"concrete_test":"Check whether every wide poset P has cone-homogeneous Boolean completion B(P) (for any nonzero p, B(P)↾p ≅ B(P)). If not, as expected, take the specific wide posets used in the motivating examples (e.g., Hechler, amoeba, specializing Aronszajn tree) and verify that each Q_i's projection from Proposition 9 can be replaced by the cone P_i↾p_i while preserving the hypotheses of Theorem 19; then re-run the proof of Theorem 19 with B_i replaced by B_i↾p_i and confirm every tagged-condition and decoding step remains valid. If the cone version fails at any step, Theorem 19 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 19, the line 'By Proposition 9 we have, for each i∈I, a projection π_i:Q_i→B_i' is not justified. Proposition 9 concludes there are p_i∈B(P_i) and π_i:Q_i→B_i↾p_i. Removing the cone restriction requires cone-homogeneity of B(P_i), which is true for Cohen forcing (Corollary 10) but not established for arbitrary wide posets; wide does not imply homogeneity (e.g., a κ-branching tree can have a rigid Boolean completion). All subsequent tagged conditions, coding points c_n, and antichain probes live in B_I, so the proof as written depends on an unjustified projection. The gap is repairable: since P_i is wide of size κ, the cone P_i↾p_i is also wide of size κ, and Q_i still adds a generic for it; replacing P_i by this cone preserves the hypotheses and the construction should go through. But that repair must be stated; without it the main theorem lacks a proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the nonamalgamability results of HHK+19 in two directions. Theorem 17 proves the finite-obstacle pattern for families of posets that add Cohen reals, using projection tags and a coding scheme into rows of 1s. Theorem 19 relaxes the hypothesis from wide posets to posets that project onto wide posets. Section 4 introduces coding via Todorčević oscillation for filter-based Mathias forcing, proving nonamalgamability for a pair of generics (Proposition 22), for finite-obstacle families with a fixed filter (Theorem 23), and for families of filters linearly ordered by inclusion (Theorem 24). The overarching method is the by-now standard one of coding a catastrophic real z into the projected generics while preserving genericity and the intersection property.","tokens_in":16913,"tokens_out":11102,"duration_ms":114374,"significance":"If the technical gaps are repaired, the paper makes a solid contribution: it broadens the class of forcings known to exhibit arbitrary finite-obstacle nonamalgamability patterns, and it provides a genuinely different coding mechanism for Mathias-type forcings, including mixed Cohen/Mathias examples. The paper is careful in setting up the projection terminology and the tagged-condition framework, and it correctly identifies the cone-homogeneity issue for Cohen forcing as a special feature. The authors also honestly discuss limitations such as the open random-real question. However, the manuscript currently contains several load-bearing claims that are not adequately proved, so the strength of the contribution cannot be fully assessed until those are fixed.","major_comments":[{"comment":"The proof begins 'By Proposition 9 we have, for each i∈I, a projection π_i:Q_i→B_i'. Proposition 9 only yields a projection into a cone B_i↾p_i for some p_i∈B(P_i). Since wide posets are not in general cone-homogeneous, the step from a cone to the whole Boolean completion is unjustified; the argument in Corollary 10 relies on the special homogeneity of Cohen forcing. This gap is load-bearing because the entire tagged-condition construction takes tags in B_I. The gap is repairable: because P_i is wide of size κ, any cone below a condition is also wide of size κ (the maximal antichain of size κ from the width property lies inside the cone), and Q_i still adds a generic for P_i↾p_i. Replacing each P_i by such a cone, and each B_i by B_i↾p_i, preserves the hypotheses and lets the proof go through. This replacement must be stated explicitly.","section":"§3.2, Theorem 19, first paragraph of the proof"},{"comment":"The assertion 'Consequently, if A∈A then p′≤p is a noncoding extension' is not justified. If B∈B is an obstacle not contained in A, then p may already have a row of 1s on B\\A, and p′ may add 1s on B∩A at that same height; then p′ has a row of 1s across B that p lacked, even though p′\\p has no 1s on I\\A. The bullet points in the claim only control where new 1s are added, not whether pre-existing partial rows are completed. This is load-bearing because the decoding argument in Theorem 17 relies on the claim that the only rows of 1s across an obstacle are the ones added in the obstacle case. The construction needs an additional invariant, or a different choice of p′, to ensure that noncoding steps never complete a partial row across an obstacle.","section":"§3.1, Claim 18"},{"comment":"The proof of Proposition 22 states that 'our construction ensured that the only points in the intersection of the two generic reals are the coding points' without giving the necessary induction. One must verify at each step that the new coding point lies above both current stems and that the subsequent one-point extensions are made from upper parts that exclude all points of the other stem; otherwise extraneous common elements could appear and corrupt the decoding of z. The same invariant is needed in Theorem 23 for intersections over finite obstacles; the sentence 'Inspecting the construction...' is not a full proof. Since nonamalgamability in this section is obtained precisely by decoding z from these intersections, this gap affects the central claim. Theorem 24, whose proof transfers verbatim, inherits the same concern.","section":"§4, Proposition 22 and Theorem 23"}],"minor_comments":[{"comment":"There is a typo in the cone-density verification: 'π(¯q≤p)' should read 'π(¯q)≤p'.","section":"§2.2, proof of Proposition 9"},{"comment":"The phrase 'build >p_{n+1}≤c_n' should read 'build p_{n+1}≤c_n'.","section":"§3.2, proof of Theorem 19, near the end of Case 2.2.2"},{"comment":"In the decoding paragraph, the sentence 'the only time we could have added such an index ℓ was in the obstacle case' depends on the noncoding-extension property; once Claim 18 is fixed, this sentence should be made into an explicit reference to the invariant.","section":"§3.1, proof of Theorem 17"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in spirit and the main theorems are probably salvageable, but the gaps are not purely cosmetic: Claim 18 appears to be false as stated, and the projection issue in Theorem 19 needs an explicit cone replacement. The Mathias decoding also requires a formal induction. If the authors can supply the missing invariant and details, the paper would be a good fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does two real things: it generalizes the HHK+19 nonamalgamability theorem from wide posets to posets that project onto wide posets (Theorem 19), and it proves analogues for filter-based Mathias forcing (Theorems 23 and 24). The tagged-condition machinery and the Mathias oscillation coding are genuinely new, and the paper carefully preserves the intersection property M[G_{A0}]∩M[G_{A1}] = M[G_{A0∩A1}]. No circularity; the use of HHK+19 Lemma 2.9 as a black box is appropriate.\n\nThe main flaw is in Theorem 19. The proof starts 'By Proposition 9 we have, for each i, a projection π_i:Q_i→B_i', but Proposition 9 only gives a projection into a cone B_i↾p_i. Wide posets are not automatically cone-homogeneous, so this is not justified. The repair is easy: replace each P_i by its cone below p_i, which is still wide and the same size, and then run the same argument. But the paper must state this; as written the proof overreaches.\n\nThe Mathias constructions have a softer issue. Proposition 22 relies on the claim that the only points in the intersection of the two generic reals are the coding points. That is asserted but not demonstrated; extensions chosen to meet dense sets could in principle pick a point already in the other real's stem. I suspect it's true with careful choices, but the text should argue it. In Theorem 23 they allow finitely many accidental points and recover a tail of z. That step is actually fine—finite initial segments of z are in M, so a tail of a catastrophic real is again catastrophic—but the finite-accident claim itself is only sketched. There are also compressed steps in Theorem 19's case analysis (e.g., Case 2.2.2) where the existence of an extension with the required projection bound is asserted rather than shown.\n\nThis is a solid contribution for people working on the generic multiverse and forcing projections. It deserves a serious referee. My recommendation is accept with revision: fix the cone gap in Theorem 19, spell out the coding-point arguments in the Mathias section, and expand the compressed steps.","headline":"Solid extension of the HHK+19 blockchain results to wide projections and filter-based Mathias forcing; Theorem 19 has a repairable cone-projection gap and the Mathias coding arguments need more detail.","tokens_in":17449,"tokens_out":17680,"would_cite":true,"duration_ms":174411,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonamalgamable forcing patterns extend to any poset projecting onto a wide poset.","keywords":["nonamalgamability","generic multiverse","forcing projections","wide posets","filter-based Mathias forcing","catastrophic real","finite obstacles","amalgamation"],"falsifier":"Run the recursive construction of Theorem 23 in a concrete countable model M with a specified catastrophic real and with F the cofinite filter (so M_F is Cohen forcing). If at any stage a point enters the intersection of the stems for some i∈B outside the designated coding steps — say because a coordinate is first mentioned — then the decoded bits of z will be corrupted. A reader could check whether such an accidental intersection arises in the first few steps of the recursion.","tokens_in":16502,"feed_emoji":"♾️","tokens_out":5280,"duration_ms":51354,"temperature":0.7,"pith_summary":"This paper shows that the full nonamalgamability patterns previously known for wide forcing notions also hold for any forcing notion that projects onto a wide poset, and for filter-based Mathias forcing. The authors construct generic filters for a family of posets indexed by a set I that realize exactly the amalgamation pattern determined by a prescribed family of finite obstacles: subfamilies the obstacles allow amalgamate, while every forbidden subfamily fails to amalgamate, and the intersection of extensions for two allowed subfamilies is exactly the extension for their intersection. The mechanism is a coding of a catastrophic real for the ground model into the generic objects; any single model containing all the generics for a forbidden set would have to contain that real, which is impossible. This broadens the reach of nonamalgamability to many common ccc forcings, such as Hechler or amoeba forcing, and introduces a new coding method for Mathias-type forcings.","feed_headline":"Nonamalgamable forcing patterns extend to wide-projection posets","feed_subtitle":"Any poset projecting onto a wide one can host full nonamalgamability patterns; filter-based Mathias too.","key_machinery":"The central mechanism is the concept of a projection between forcing notions, used to form 'tagged conditions' (q,p) where q is a condition in the target forcing and p is a condition in the projected forcing below π(q). By maintaining a descending sequence of tagged conditions that is infinitely often strong in the sense of the ≤* relation, the authors arrange that the filter generated by the working parts q projects exactly onto the filter generated by the tags p, allowing independent control of the original generics and the projected generics. The projected generics carry the coding information about the catastrophic real. For the Mathias case, the coding uses the oscillation of the stems: after each common point of the two generic reals, the ordering of the next elements encodes one bit of the catastrophic real.","core_discovery":"The central claim is that, given a family of subsets of an index set defined by finite obstacles, and a family of posets each of which adds a generic for a common wide poset of sufficient size, one can find generic filters for the original posets that realize the three-part pattern: generic existence for every allowed subfamily, nonamalgamability for every forbidden subfamily, and exact intersection of extensions of allowed subfamilies. The same three-part pattern is established for filter-based Mathias forcing, using a different coding scheme based on oscillation points in the stems. The construction simultaneously builds the generics and codes a ground-model-catastrophic real into the projected generics (or the stems of the Mathias reals), so that any model containing all the generic objects for a forbidden index set would contain the catastrophic real and therefore could not lie in the generic multiverse.","pith_inferences":["The same method might apply to any family of posets that all project onto a common wide poset without requiring equal cardinality, by using a more flexible coding scheme.","The robustness of the coding could be tested by attempting to replace the fragile 'no accidental row of 1s' or 'only coding points in the intersection' conditions with an error-correcting code, which might make nonamalgamability easier to achieve in other settings.","The open question about random real forcing (Question 20) could be attacked by trying to find a projection from random forcing to a wide forcing, or by proving that no such projection exists, following the approach of this paper.","The exact intersection property suggests a connection to the structure of grounds in the generic multiverse; one could investigate whether such patterns correspond to a lattice of models generated by the allowed subfamilies."],"forward_implications":["If Theorem 19 holds, nonamalgamability is not a special feature of wide posets themselves; any forcing that adds a generic for a wide poset of appropriate size exhibits the same patterns.","Filter-based Mathias forcing, which is σ-centered and therefore ccc, realizes the full nonamalgamability pattern, adding a new class of examples outside the wide-poset framework.","The construction allows mixing Cohen-type forcings with filter-based Mathias forcing, giving nonamalgamable pairs such as one Cohen real and one Mathias real.","The intersection property (3) provides a strong form of exactness: the common elements of any two allowed extensions are exactly the elements of the extension for the intersection of the index sets."],"supporting_citations":[{"why":"Provides the base theorem for wide posets (Theorem 4) and Lemma 6, which is used to defeat potential counterexamples to the intersection property.","marker":"[HHK+19]"},{"why":"Establishes that wide posets exhibit nonamalgamability and observes that the core of properties (1) and (2) can be obtained by quotient forcing, which the paper extends.","marker":"[Ham16]"},{"why":"Introduces the notion of oscillation used in the coding scheme for the Mathias forcing construction.","marker":"[Tod88]"},{"why":"Supplies the fact that filter-based Mathias forcing adds a Cohen real if and only if the filter is not Ramsey, used to place the Mathias construction in context.","marker":"[JS91]"}],"fun_headline_variants":["Wide-projection posets host nonamalgamability patterns","Filter-based Mathias yields nonamalgamable extensions","Nonamalgamability via wide projections and Mathias coding","Coding catastrophic reals for nonamalgamable generics","Extended nonamalgamability for wide posets and Mathias"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction relies on the claim that no extraneous coding signals appear in the generic objects — for the Cohen/tag constructions, that noncoding extensions never create a new row of 1s across an obstacle, and for the Mathias construction, that the only points common to the generic reals are the coding points.","fun_headline_variants_meta":{"raw":{"variants":["Wide-projection posets host nonamalgamability patterns","Filter-based Mathias yields nonamalgamable extensions","Nonamalgamability via wide projections and Mathias coding","Coding catastrophic reals for nonamalgamable generics","Extended nonamalgamability for wide posets and Mathias"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000466,"raw_usage":{"total_tokens":2223,"prompt_tokens":740,"completion_tokens":1483,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":356,"completion_tokens_details":{"reasoning_tokens":1395}},"tokens_in":356,"tokens_out":1483,"duration_ms":10142,"temperature":1.0,"reasoning_tokens":1395,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:11:49.494501+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the recursive construction of Theorem 23 in a concrete countable model M with a specified catastrophic real and with F the cofinite filter (so M_F is Cohen forcing). If at any stage a point enters the intersection of the stems for some i∈B outside the designated coding steps — say because a coordinate is first mentioned — then the decoded bits of z will be corrupted. A reader could check whether such an accidental intersection arises in the first few steps of the recursion.","supporting_citations":[],"review_version":1}