{"id":"5d0bd9da-afd6-440d-b212-409100330323","arxiv_id":"1908.04273","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new definition of fractals via porosity-like measure ratio and adjacency conditions is introduced and illustrated on Sierpinski, Pascal, and Koch fractals.","lead":"The paper proposes a new formal definition of fractals, called abstract fractals, built from iterative divisions of a metric space into kept pieces and empty \"complement\" pieces, with conditions controlling the relative measure and boundary contact of the pieces. It argues the definition covers Sierpinski carpet, Pascal triangle modulo 3, and Koch curve, and it asserts a chaos theorem for the associated shift map.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Shift map may not be well-defined: the paper never proves distinct code sequences give distinct points, and its separation condition is too weak to prevent ambiguous addresses.","rationale":"The reader's conditional verdict stressed that the chaos theorem is asserted without proof and that continuity of the shift map is not established. I agree with the conditional assessment, but the deeper problem is well-definedness: phi is defined on code strings, so the coding map from infinite sequences to points of F must be injective, or at least the value of phi must be independent of the chosen representative. The separation condition as written is a weak 'there exists a far-away cylinder' condition, not a strong separation or unique-representation condition. The standard Sierpinski carpet, the paper's first example, illustrates the failure mode: with closed cells, boundary points have multiple addresses and the shift sends the different addresses to different points; with a half-open disjoint convention, the limit-set identity (5) becomes non-obvious and likely false. This does not necessarily destroy the whole framework, since adding closedness plus a true unique-coding hypothesis could repair the theorem, but it means the central chaos claim is at best incomplete. Because a repair is plausible and no definitive counterexample to the abstract definition itself has been constructed, the appropriate verdict remains CONDITIONAL rather than ACCEPT or REJECT.","tokens_in":6683,"tokens_out":16415,"duration_ms":191072,"concrete_test":"Use the paper's Sierpinski carpet. With closed cells, take the point x=(1/3,1/6) in the unit square, which lies in the carpet and has two ternary representations: x=(0.1000...,0.0111...) and x=(0.0222...,0.0111...). Compute phi on both addresses. If the two resulting points differ, phi is not single-valued and Theorem 1 is not valid for the example as stated. If the intended construction uses a disjoint open-cell convention, rerun the check with that convention and verify whether equality (5) still holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is not just a missing proof but the definition of the similarity map itself. In Section 4, phi is defined on points of F by phi(F_{i1i2...}) = F_{i2i3...}. These points are limits of nested sets F_{i1...in}; if two different infinite sequences converge to the same element of X, the notation F_{i1i2...} is ambiguous and phi is not a function. The paper never proves the coding is injective. The 'separation condition' in Section 4 only requires that for each n-cylinder there exists some other n-cylinder at distance at least epsilon0; it does not prevent two cylinders from having intersecting closures or two codes from representing the same point, so it is too weak to guarantee well-definedness. In the standard Sierpinski carpet with closed cells (as the 'common boundary' language in Section 3.1 suggests), boundary points have multiple base-3 addresses; shifting different addresses gives different points, so phi is not well-defined. If the cells are instead taken open or half-open to respect the word 'disjoint', then equality (5) between the construction set F_{i1...in} and the set of limit points is not proved and can fail, because limits can lie in closures rather than in the sets. Thus Theorem 1 rests on an unstated injectivity or closedness hypothesis.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new definition of fractals, called 'abstract fractal,' in a compact metric-measure space. Starting from a set F partitioned into M disjoint subsets (only m of which are retained at each stage), the authors impose four conditions: a ratio condition bounding the measure of retained cells against removed cells (Eq. (1)), an adjacent condition requiring each retained cell to touch some removed cell, an informally stated accumulation condition, and a diameter condition (Eq. (3)). The abstract fractal is then defined as the set of limit points of all infinite sequences of retained indices (Eq. (4)). The paper gives three examples (Sierpinski carpet, Pascal triangle modulo 3, and a Koch-type construction) and, in Section 4, claims that under a separation condition the associated shift map is chaotic in the senses of Poincare, Li-Yorke, and Devaney (Theorem 1), with the proof deferred to the authors' unpublished preprint [1]. Section 5 sketches a relation to iterated function systems and Section 6 discusses the definition's potential as a 'third definition' of fractals.","tokens_in":7004,"tokens_out":8998,"duration_ms":89974,"significance":"If the proposed definition could be made rigorous, it would offer a porosity-based, construction-oriented alternative to the usual IFS and Hausdorff-dimension characterizations of self-similar fractals, potentially covering examples such as the Sierpinski carpet where standard totally disconnected assumptions fail. The paper is honest about the fact that a Hausdorff-dimension theory for abstract fractals has not yet been developed (Section 6). However, as it stands, the significance is limited by (i) the lack of a precise statement and use of the accumulation condition, (ii) the unproved and possibly false coding injectivity needed for the shift map, and (iii) the reliance on an inaccessible preprint [1] for the main theorem. The examples are presented informally and one appears mislabeled. The paper does not yet establish that its definition is a meaningful new criterion rather than a restatement of coding-map dynamics on a subshift.","major_comments":[{"comment":"The construction at the beginning of Section 2 requires F = union_{i=1}^M F_i with the F_i nonempty and disjoint, but the Sierpinski carpet example uses closed sub-squares with common boundaries; closed squares are not disjoint, while open squares would not have union equal to the initial closed square F. This makes it unclear whether the defining conditions are ever satisfied simultaneously, and it also means that Eq. (5), which identifies a subfractal with the set of limit points of infinite sequences, may fail because limit points of open cells can lie outside the cells. The paper needs to specify the topological status of the cells (open, closed, or half-open) and prove the identity in Eq. (5) under that specification.","section":"Section 2, Eq. (4) and Section 3.1"},{"comment":"The accumulation condition is stated only verbally and is not defined mathematically. As a result, it is impossible to verify it in the examples, and indeed Section 3 merely says it is 'clear' or 'valid' without demonstration. Since it is one of the four conditions in the definition, a reader cannot determine whether a given construction qualifies as an abstract fractal. Moreover, the condition is never used in the rest of the paper, including the proof of Theorem 1, so its role in the definition is unclear.","section":"Section 2, accumulation condition"},{"comment":"The similarity map phi is defined by phi(F_{i1 i2 ...}) = F_{i2 i3 ...}; this is a well-defined function only if each infinite index sequence corresponds to a unique point and distinct sequences correspond to distinct points. The paper never proves either statement, and the separation condition stated in Section 4 is far too weak to imply injectivity: it only says that for each n-cylinder there exists some other n-cylinder at distance at least epsilon0, which is compatible with overlapping cylinders and with boundary points admitting multiple addresses. In the standard Sierpinski carpet with closed cells, boundary points do have multiple addresses, so phi is not well-defined on F as given in Eq. (4). Consequently, Theorem 1 is not proven; the asserted transfer of results from the unpublished preprint [1] is not verifiable, and the separation condition is not checked for any of the examples.","section":"Section 4, Theorem 1"},{"comment":"The construction described for the Koch curve retains only two of the three subtriangles at each stage (m=2), yielding a set that is at best a Cantor-type set, not the standard Koch curve, which is generated by four contractions and is connected. Unless a precise definition of 'Koch curve' is supplied that matches this construction, this example does not support the claim that the definition captures the usual self-similar fractals.","section":"Section 3.3"}],"minor_comments":[{"comment":"The abstract Sierpinski carpet is written as F = {F_{i1 i2 ...} | ik = 1,2,...,5}, but the construction uses m=8; this is likely a typo and should be corrected to ik = 1,2,...,8.","section":"Section 3.1"},{"comment":"The ratio condition involves F_{i1...i_{n-1}j} for n=1, where the multi-index i1...i0 is undefined; the condition should be stated for n >= 2 or the base case n=1 defined separately.","section":"Section 2, Eq. (1)"},{"comment":"There is a duplicated word 'with' in 'Start with with an isosceles triangle' and the caption of Figure 4 is missing; the phrase 'Figure illustrats' should read 'Figure illustrates'.","section":"Section 3.3"},{"comment":"Theorem 1 is stated with a double period after 'Devaney..' and the proof is deferred to the unpublished preprint [1]; the authors should either supply a self-contained proof or make [1] available, and the reference should be updated with publication status.","section":"Section 4 and references"},{"comment":"The discussion claims that Cantor sets and Sierpinski fractals 'are also satisfied the Mandelbrot definition,' but the paper has not provided a Hausdorff-dimension computation for abstract fractals and later states that this dimension is 'not yet developed'; this apparent contradiction should be resolved.","section":"Section 6"},{"comment":"The definition of the separation constant in Section 5, min_n inf_{i_n,j_n} d(w_{i_n}(...), w_{j_n}(...)) >= epsilon0, is not precise; the infimum should range over all admissible words of length n, and it is not clear why a single n suffices.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper's central theorem depends on the authors' own unpublished preprint [1], and the examples do not rigorously satisfy the stated definition. The novelty relative to the well-known shift dynamics on a subshift of finite type is not articulated; the authors should clarify what is gained by the porosity/measure conditions. If the authors can resolve the well-definedness issue and provide a precise accumulation condition, the paper could be a useful contribution to a porosity-based perspective on self-similar sets."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one genuinely new thing here is the four-condition definition of an abstract fractal: ratio, adjacent, accumulation, and diameter conditions packaged as a porosity-based abstraction of self-similarity. The examples in Section 3 — Sierpinski carpet, Pascal triangle modulo 3, Koch curve — are worked informally but the pattern is clear and the definition does capture a real family of self-similar sets. That is worth something, and the authors are explicit that this is a formalization rather than a new class of objects. The IFS discussion in Section 5 is mostly a repackaging, but it is honest about the relation to standard IFS theory.\n\nWhere the paper gets shaky is exactly where the reader and stress-test notes put their fingers. Theorem 1 is asserted without proof, and the proof is referred to an unpublished companion preprint [1] that the authors themselves cite. That alone makes the main dynamical result unverifiable from the text. More seriously, the shift map φ(F_{i1i2...}) = F_{i2i3...} is not shown to be well-defined: the paper never proves that distinct infinite code sequences give distinct points. Without that, φ is not a function on the abstract fractal. The separation condition stated in Section 4 only guarantees that some pair of n-cylinders is separated by ε0; it does not prevent two cylinders from sharing a boundary point or two codes from representing the same point. In the Sierpinski carpet with closed cells, boundary points have multiple addresses and shifting different addresses gives different points, so φ fails to be well-defined. If the cells are instead open or half-open, the equality in (5) between the construction sets and the set of limit points is unproved and can fail. This is a load-bearing gap, not a cosmetic one, and it directly undermines the chaos claim.\n\nThere are also smaller issues worth flagging: the accumulation condition in Section 2 is stated in prose and is ambiguous; it is never used in any proof, so the reader is left guessing at its exact content. The Sierpinski carpet section contains a typo — the set F is defined with ik = 1,...,5 instead of 1,...,8. The Pascal triangle example also has some loose notation about which triangles are subdivided. None of these are fatal to the definition itself, but they would need cleaning up in revision.\n\nWho is this for? A reader working on axiomatic definitions of fractals or on porosity-based characterizations of self-similar sets will find the proposal worth engaging with. The definition could be useful even if the chaos theorem is not yet established. I would send it to a serious referee, but the referee should be asked specifically to check well-definedness of the shift map and to report whether the separation condition can be strengthened or replaced. As it stands, the paper is a promising draft, not a finished theorem-bearing manuscript.","headline":"A plausible new formal definition of fractals via porosity, but the central chaos theorem rests on an unpublished preprint and an unproved injectivity of the coding map.","tokens_in":7452,"tokens_out":1297,"would_cite":false,"duration_ms":16643,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","37B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Four subdivision conditions can define a fractal without any dimension calculation.","keywords":["abstract fractal","self-similarity","porosity","chaos","Sierpinski carpet","Koch curve","Pascal triangle","iterated function system"],"falsifier":"Subdivide a square into nine equal subsquares and at every stage keep only the three bottom subsquares. The ratio condition holds with ratio $1/2$, the adjacent condition holds through shared boundaries, and the diameters shrink to zero, so if the accumulation condition is read in the natural way the resulting abstract fractal is a one-dimensional line segment, which is not a fractal in the usual sense. Checking whether this construction actually satisfies the stated conditions decides whether the definition is too broad.","tokens_in":6495,"feed_emoji":"❄️","tokens_out":14896,"duration_ms":148646,"temperature":0.7,"pith_summary":"The paper proposes a new way to define a fractal, based on porosity rather than on self-similarity or on comparing dimensions. An abstract fractal is a set of limit points obtained by recursively subdividing a compact metric space, at each step keeping the same number of pieces while requiring four conditions: a ratio condition on the masses of kept versus discarded pieces, an adjacent condition that each kept piece touches a discarded piece, an accumulation condition about complement points, and a diameter condition that the pieces shrink to points. The authors show that the classic carpet, Pascal-triangle, and Koch-curve constructions fit this scheme, and argue that the definition deserves to be a third criterion of fractalness alongside the usual ones. They further claim that when a separation condition holds, the left-shift map on the abstract fractal is chaotic in the Poincaré, Li-Yorke, and Devaney senses. Accepting the definition would give a porosity-based route into fractal geometry that can cover connected and symbolic sets where dimension calculations are hard.","feed_headline":"Porosity, not dimension, defines fractals in new four-rule scheme","feed_subtitle":"Classic carpet, Pascal-triangle, and Koch sets fit the new conditions; their shift map can be chaotic.","key_machinery":"The machinery is the indexed limit-point construction $F_{i_1i_2\\ldots i_n\\ldots}$: a point of the abstract fractal is determined by an infinite sequence over $m$ symbols, and each finite prefix $i_1\\ldots i_n$ corresponds to a subset $F_{i_1\\ldots i_n}$ of the metric space. The four conditions govern how those subsets are generated: the ratio condition encodes porosity as a bounded mass ratio, the adjacent condition forces the retained structure to touch the pores, the accumulation condition controls complement limits, and the diameter condition guarantees convergence of the nested sets. The same indexing defines the similarity map $\\phi(F_{i_1i_2i_3\\ldots})=F_{i_2i_3\\ldots}$, which turns the abstract fractal into a dynamical system; Theorem 1 asserts this map is chaotic when the separation condition holds.","core_discovery":"The central claim is that the set $F=\\{F_{i_1i_2\\ldots i_n\\ldots}: i_k=1,\\ldots,m\\}$ defined by Eq. (4) is a legitimate mathematical object—an abstract fractal—provided four conditions hold: (1) the ratio of the measure of the $m$ kept pieces to the measure of the discarded pieces stays between fixed positive bounds $r$ and $R$; (2) each kept piece touches at least one discarded complement piece; (3) accumulation points of pairs of complement pieces belong to none of them; and (4) the maximal diameter of the $n$-th level pieces tends to zero. Under these conditions the nested sets converge to points, and the collection of all such points is the abstract fractal. The paper claims that every abstract fractal is an abstract self-similar set in the sense of the companion paper, and that, if a separation condition holds, the similarity map $\\phi(F_{i_1i_2\\ldots})=F_{i_2\\ldots}$ is chaotic in the Poincaré, Li-Yorke, and Devaney senses.","pith_inferences":["The paper leaves implicit that a dimension theory for abstract fractals is still missing; if the ratio constants $r,R$ could be turned into a dimension formula, the abstract definition could be checked against the usual dimension-based one.","A testable extension is to compute the separation constant for the three worked examples; if none of them has a positive separation constant, the chaos theorem applies only to other, possibly symbolic, abstract fractals rather than to the classical ones.","The bottom-row construction—keep only the three lower subsquares of a $3\\times3$ subdivision at every stage—appears to satisfy ratio, adjacent, and diameter conditions while converging to a one-dimensional line segment; if so, the definition as written is broader than the usual notion of fractal and may need an extra condition.","One could randomize the construction, replacing the fixed ratio bounds by almost-sure bounds, which would connect the definition to statistically self-similar porous structures."],"forward_implications":["The classic carpet, Pascal-triangle modulo 3, and Koch-curve constructions each satisfy the four conditions, so all three are abstract fractals with explicit parameters $m$, $M$, and ratio bounds.","Every abstract fractal is an abstract self-similar set in the sense of the companion paper, making self-similarity a special case of the new structure.","If the separation condition holds, the similarity map on $F$ is chaotic in the Poincaré, Li-Yorke, and Devaney senses, so the fractal itself carries complex dynamics.","The new definition gives porosity a role in fractal theory comparable to self-similarity and dimension, and offers a route to a self-similar dimension for abstract fractals.","Because the construction lives in an arbitrary compact metric measure space, the definition applies to porous media and symbolic spaces where exact self-similarity is not readily visible."],"supporting_citations":[{"why":"It supplies the abstract self-similar set definition and the chaos results that Theorem 1 assumes are transferred.","marker":"[1]"},{"why":"It gives the dimension-based definition of a fractal that the paper positions its new criterion against.","marker":"[2]"},{"why":"It defines self-similarity, the property the abstract fractal construction is built to abstract.","marker":"[3]"},{"why":"They provide the iterated function system framework used in Section 5 to realize abstract fractals and to contrast the shift-map approach.","marker":"[4,5]"},{"why":"It supplies the shift dynamical system and the topological-conjugacy proof for totally disconnected IFS fractals that the paper's similarity map is meant to bypass.","marker":"[6]"},{"why":"They introduce unpredictable points and Poincaré chaos, the two notions named in the chaos conclusion of Theorem 1.","marker":"[7, 8]"}],"fun_headline_variants":["Porosity redefines fractals in four-rule abstract scheme","Four rules make fractals abstract, with chaotic shift maps","New fractal definition hinges on porous structure, not dimension","Abstract fractals: self-similarity via four porous conditions","Porous sets define fractals; shift maps turn chaotic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The chain of results depends on the separation condition and on the companion paper's chaos proofs being transferable, neither of which is checked here, and the accumulation condition is stated too vaguely to verify, so the definition's consistency rests on an unstated reading.","fun_headline_variants_meta":{"raw":{"variants":["Porosity redefines fractals in four-rule abstract scheme","Four rules make fractals abstract, with chaotic shift maps","New fractal definition hinges on porous structure, not dimension","Abstract fractals: self-similarity via four porous conditions","Porous sets define fractals; shift maps turn chaotic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000167,"raw_usage":{"total_tokens":1197,"prompt_tokens":827,"completion_tokens":370,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":301}},"tokens_in":443,"tokens_out":370,"duration_ms":4051,"temperature":1.0,"reasoning_tokens":301,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:11:25.917758+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Subdivide a square into nine equal subsquares and at every stage keep only the three bottom subsquares. The ratio condition holds with ratio $1/2$, the adjacent condition holds through shared boundaries, and the diameters shrink to zero, so if the accumulation condition is read in the natural way the resulting abstract fractal is a one-dimensional line segment, which is not a fractal in the usual sense. Checking whether this construction actually satisfies the stated conditions decides whether the definition is too broad.","supporting_citations":[{"cited_title":"and Alejaily E","cited_arxiv_id":null,"evidence_quote":"It supplies the abstract self-similar set definition and the chaos results that Theorem 1 assumes are transferred."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It defines self-similarity, the property the abstract fractal construction is built to abstract."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the shift dynamical system and the topological-conjugacy proof for totally disconnected IFS fractals that the paper's similarity map is meant to bypass."}],"review_version":1}