{"id":"e5da5286-ca76-4239-91a3-038744343e15","arxiv_id":"2509.05211","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A new proof technique shows distances and orthogonal projections retain at least half of a planar point's Kolmogorov complexity, improving pinned distance dimension bounds to 3/4 s and generalizing Bourgain's theorem.","lead":"This paper proves that, under mild independence conditions, the distance between two planar points and the projection of a point onto a line each preserve at least half of the point's algorithmic information. This yields an improved lower bound on the Hausdorff dimension of pinned distance sets and a generalization of Bourgain's projection theorem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unjustified sparsity of the candidate sets P_u/P_v breaks the O(1) bound on P_{u,v} and the reconstruction inequalities (19)/(24).","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: the reconstruction argument in Theorem 2 (and Theorem 3) depends on the sparsity of P_u and P_v, but that sparsity is neither proved nor implied by the fact that πu and πv are shortest witnesses for single conditional complexities. A shortest program for one input-output pair can compute an arbitrary enumerable function on other inputs; there is no reason its image should have at most one point per s-dyadic interval. This is not a minor technicality because the O(1) bound on P_{u,v} is what makes the reconstruction of d from the two witnesses cost only O(1) extra bits. Without it, inequality (19) would need an additional term logarithmic in the number of candidate pairs, which could be as large as Θ(r−s) and would destroy the final contradiction. The same issue appears verbatim in Theorem 3. I agree with the reader's verdict: CONDITIONAL, because the proof as written is incomplete at a central point, although the claimed theorems may still be true and repairable. No change from the reader's verdict is needed.","tokens_in":19877,"tokens_out":7130,"duration_ms":77156,"concrete_test":"Formalize the derivation of |P_{u,v}| = O(1) from the actual definition P_u = {U^u(πu, d_u) : d_u ∈ D_s}, without adding any implicit regularity assumption. If the derivation cannot be completed, exhibit a concrete computable function f : D_s → D_r such that f(⌊|d−u|⌋_s) = ⌊|d−u|⌋_r for the specific input but whose image intersects a fixed s-dyadic interval in many points (e.g., let f map all but one input into a single tiny interval). Such an f is a legitimate program for the relevant conditional, demonstrating that the sparsity claim is not a consequence of the definitions. Alternatively, recompute the number of bits needed in the reconstruction step without the O(1) assumption; if the extra term is Ω(r−s), inequality (19) collapses and the proof of Theorem 2 fails as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 2, after selecting d ∈ N_u ∩ N_v, the authors define P_u = {U^u(πu, d_u) : d_u ∈ D_s} and assert that each s-dyadic interval contains at most one element of P_u, and similarly for P_v. This assertion is load-bearing: it is used to conclude that P_{u,v} has cardinality O(1), which in turn is essential for the reconstruction inequality (19), namely K^{u,v}_{r−2t−O(1)}(d) ≤ |πu|+|π_v|+K(⌊d⌋_s)+O(1). The subsequent contradiction relies directly on (19). The same step appears in Theorem 3 with (24). But the assertion does not follow from πu being a shortest witness for the single conditional K^u(⌊|d−u|⌋_r | ⌊|d−u|⌋_s). A program can behave arbitrarily on inputs other than the specific one used in the witness: it may output many distinct values inside one s-dyadic interval when run on different inputs d_u ∈ D_s. Nothing in the definition of prefix Kolmogorov complexity or of a shortest witness imposes any sparsity, Lipschitz property, or consistency across inputs. Thus the set P_u can be dense in an s-interval, making |P_{u,v}| potentially as large as 2^{r−s} rather than O(1). In that case, selecting the correct pair in P_{u,v} requires extra bits, and inequality (19) loses its justification. The same gap invalidates the analogous projection argument. This is the central soft spot in the proof of the main Theorems 2 and 3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops quantitative algorithmic information bounds for distances and orthogonal projections in the plane. It claims that under mild independence conditions the distance |x-y| and a projection coordinate p_e x each retain at least half the finite-precision Kolmogorov complexity of x. The proofs use a combinatorial lemma (surrogate point selection) and prior effective-dimension techniques. From these bounds the paper derives new geometric measure theory results: pinned distance sets of analytic E⊂R^2 with dim_H E ≤ 1 satisfy sup_x dim_H Δ_x E ≥ (3/4) dim_H E, and a generalization of Bourgain's exceptional-sets theorem for projections to sets with optimal Hausdorff oracles.","tokens_in":20259,"tokens_out":17258,"duration_ms":185732,"significance":"If the main inequalities (15) and (21) were established, the paper would deliver a substantial improvement over the best previously known pinned-distance bounds of Shmerkin–Wang and Fiedler–Stull, and a new proof/generalization of Bourgain's theorem. The surrogate-point selection technique and the use of optimal Hausdorff oracles are interesting and potentially reusable. The paper is clearly written and the overall strategy is coherent. However, the proof of both main theorems currently contains an unsupported and in general false assertion about the candidate sets P_u and P_v; until that is repaired, the central claims are not established.","major_comments":[{"comment":"The assertion that each s-dyadic interval contains at most one element of P_u = {U^u(π_u,d_u) : d_u∈D_s} and similarly P_v is not justified and is false for arbitrary witnesses. A program witnessing a single conditional complexity can behave arbitrarily on other inputs and can output many values in the same s-interval; D_s is infinite, so P_u may even be dense. Hence |P_{u,v}| need not be O(1); it can be as large as 2^{r-s}. This O(1) bound is load-bearing for (19): selecting the correct pair would require extra bits and destroy the inequality. The same issue affects (24) in Section 5. A local repair is available by recovering the inputs to π_u,π_v from ⌊d⌋_s and the oracles with O(1) bits, since distance/projection maps are 1-Lipschitz; as written, the proof is incomplete.","section":"Section 4, proof of Theorem 2, Eq. (19)"},{"comment":"The same unsupported sparsity assertion is used to claim that P_{u,v} has cardinality O(1) in the projection argument. This is needed for inequality (24). Since Theorem 3 is used directly to prove Theorem 5 and also (through Theorem 13) Theorem 4, this gap affects the geometric measure theory consequences as well. The steps leading to (24) should be repaired or replaced, for instance by the direct-input-recovery argument sketched above.","section":"Section 5, proof of Theorem 3, Eq. (24)"}],"minor_comments":[{"comment":"The notation D_s is introduced but 's-dyadic interval' is used without definition. It should be stated explicitly that this means an interval of the form [k/2^s, (k+1)/2^s).","section":"Section 2.2"},{"comment":"In the definition of V_1, the existential quantifier over v∈R^2 is non-constructive; the subsequent choice of v_d is not shown to be unique. This is acceptable for an existence argument, but a sentence clarifying that only existence is needed would help.","section":"Section 4"},{"comment":"In the proof of Theorem 13, the inequality labeled (36) is stated as a target and then 'Combining this with (38) and (35)' is slightly confusing because (36) is what is being proved. Reorganize the flow.","section":"Appendix D"},{"comment":"The phrase 'at least half the complexity' is informal; Theorems 2 and 3 contain an εr lower-order term, so the informal summary should be qualified (e.g., 'up to lower-order terms') in the introduction as well as the abstract.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The reader's stress-test concern is correct and lands on the central proof. I nevertheless recommend major revision rather than rejection because the faulty step appears locally repairable: one can sidestep P_u and P_v by observing that the required inputs to the witness programs are determined by ⌊d⌋_s and the relevant oracle up to O(1) candidates. If the authors provide such a repair and verify the constants, the paper's main results may stand. I would ask for a careful rewrite of the reconstruction inequalities (19) and (24) and a proof of the sparsity or its replacement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper is worth a serious look, but it has a real hole in the proof of both main theorems. The headline results—the 3s/4 pinned-distance bound and the Bourgain exceptional-set generalization—are novel and would be significant if the proofs hold. The surrogate point selection and Lemma 1's double counting are genuinely new tools, and the half-information inequalities are strictly stronger than what Shmerkin–Wang and Fiedler–Stull had over the full range. The applications to Hausdorff dimension are correctly derived from Theorems 2 and 3, assuming those theorems stand.\n\nThe soft spot is exactly where the stress test puts it. In Theorem 2, after choosing a high-complexity d, the authors define P_u = {U^u(pi_u, d_u) : d_u in D_s} and assert that each s-dyadic interval contains at most one element of P_u, and similarly for P_v. That assertion is false as stated. A fixed prefix program pi_u, run on different inputs d_u from the same s-dyadic interval, can output completely different values; nothing in the definition of a shortest witness imposes any consistency, Lipschitz property, or sparsity. The program may be total and map each d_u to a distinct value, making P_u dense. The O(1) bound on P_{u,v} is load-bearing: reconstruction inequality (19) (and its analog (24) in Theorem 3) needs that bound to avoid paying r-s bits to specify the correct pair. Without it, the chain of inequalities collapses. This is not a minor oversight; the intuitive claim is simply not entailed by the setup.\n\nThe gap may be repairable—for instance, by using a canonical witness that enforces a single output per interval, or by restructuring the reconstruction argument so it does not require O(1) candidate pairs. But as written, Theorems 2 and 3 are not established.\n\nWho should read it? Researchers in effective dimension and geometric measure theory. The method is promising and the results are important enough that the paper deserves a serious referee, not a desk reject. But the referee should insist on a fix for the reconstruction step before acceptance.\n\nRecommendation: send to peer review, and let the reviewers work on the sparsity claim. If it can be fixed, this is a strong paper; if not, the main results lose their foundation.","headline":"Novel, important bounds, but both main proofs have a load-bearing gap in the reconstruction step: the sparsity of P_u/P_v is asserted, not proven.","tokens_in":20740,"tokens_out":3627,"would_cite":false,"duration_ms":41248,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q30","28A80","03D32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two independent planar points share at least half of their algorithmic information through the distance between them.","keywords":["Kolmogorov complexity","finite-precision complexity","Hausdorff dimension","pinned distance sets","orthogonal projections","effective dimension","point-to-set principle","exceptional sets"],"falsifier":"Fix r,s,x,y satisfying (14) with δ<ε/6·dim^{A,y}(e_{y,x}) and compute both sides of (15); a failure of the inequality at large r would falsify Theorem 2. Concretely, one can probe the proof's load-bearing step by enumerating the outputs U^u(π_u,d_u) over d_u∈D_s for a shortest witness π_u: if the same s-dyadic interval receives two distinct outputs on different inputs, the O(1) bound on P_{u,v} fails and the reconstruction chain (19) would need a different justification.","tokens_in":19803,"feed_emoji":"📏","tokens_out":9142,"duration_ms":89182,"temperature":0.7,"pith_summary":"This paper establishes quantitative half-information theorems for the plane. If two points x and y are sufficiently independent—meaning y's oracle does not lower the finite-precision complexity of x by more than a small linear amount—then the distance |x−y| carries at least half of the algorithmic information content of x, up to lower-order terms (Theorem 2). The same holds for the coordinate of x under orthogonal projection onto a direction e that is sufficiently independent of x (Theorem 3). Because the bounds are quantitative and relativize to arbitrary oracles, the point-to-set principle turns them into measure-theoretic statements: every analytic planar set E of Hausdorff dimension s≤1 has a point whose pinned distance set has dimension at least 3s/4, and the exceptional directions that lose more than half the dimension form a dimension-zero set. A new surrogate-point selection step, driven by a double-counting lemma, is what makes the proofs work.","feed_headline":"Distances keep half of a point's algorithmic information","feed_subtitle":"That bound lifts pinned-distance dimensions to 3/4 and widens projection theorems.","key_machinery":"The load-bearing objects are finite-precision Kolmogorov complexities: K_r(x) is the length of the shortest program that outputs a rational point within 2^{−r} of x, and K^{A,y}_{r,s}(·|·) conditions on coarser approximations and oracles. The proofs run through a combinatorial surrogate-selection lemma: from a large family of candidate points that share x's information profile, a double-counting argument (Cauchy–Schwarz) guarantees two centers u,v and a candidate d such that d is information-rich relative to u,v while the two directions from d to u and v are separated. The distance theorem then reconstructs d from the intersection of two thin annuli, using an annulus-intersection bound that","core_discovery":"The paper's central claim is that a single scalar measurement—the radius of a circle on which x lies, or the signed coordinate of x on a line—retains at least half of the algorithmic information in x whenever the auxiliary center or direction is independent of x. Formally, for all oracles A, points x,y in R^2, and large precisions r,s with K^{A,y}_r(x) ≥ K^A_r(x) − δr and δ < (ε/6) dim^{A,y}(e_{y,x}), the paper proves K^{A,y}_{r,s}(|x−y| | |x−y|) ≥ K^A_{r,s}(x|x)/2 − εr; the projection version replaces y by a direction e and uses δ < ε dim^A(e)/4. From the distance bound and the point-to-set principle, the paper derives Theorem 4: for every analytic E⊆R^2 with dim_H(E)=s≤1, sup_{x∈E} dim_H(∆","pith_inferences":["The surrogate-selection lemma looks transferable to other pairs of scalar measurements whose joint preimage has bounded multiplicity, which would yield half-information bounds for a wider family of geometric queries (not claimed in the paper).","Iterating the argument at several precision scales could plausibly raise the 3/4 constant, though the paper does not pursue this.","In R^n, reconstructing a point from n distances or n projections has finite multiplicity, so an analogous combinatorial lemma may give a family of higher-dimensional bounds; this is an extension, not a paper claim."],"forward_implications":["Pinned distance sets: for every analytic planar E with dim_H(E)=s≤1, some x∈E has dim_H(∆_x E) ≥ 3s/4, improving all previous uniform lower bounds.","Projection exceptional sets: the collection of directions e for which dim_H(p_e E) < dim_H(E)/2 has Hausdorff dimension 0, now for the wider class of sets admitting optimal Hausdorff oracles.","The half-information inequalities are conditional and quantitative, so they can serve as lemmas in multi-scale case analyses across precision intervals.","Both measurements—radial distance and projected coordinate—are information-dense: they discard at most half of the point's algorithmic information."],"supporting_citations":[{"why":"supplies the point-to-set principle that converts effective dimension bounds into Hausdorff dimension statements.","marker":"[14]"},{"why":"provides the annulus-intersection bound used to cover the candidate point set in the distance theorem.","marker":"[37]"},{"why":"supplies the previous pinned-distance lower bound and several lemmas used in the dimension argument.","marker":"[8]"},{"why":"gives the prior best pinned-distance bound that Theorem 4 uniformly improves.","marker":"[33]"},{"why":"is the classical exceptional-set projection theorem that Theorem 5 generalizes.","marker":"[2]"},{"why":"provides the effective-dimension projection machinery and the oracle lemma used in Theorem 13.","marker":"[22]"},{"why":"introduces the pinned-distance effective-dimension framework and observations used in Theorem 13.","marker":"[35]"},{"why":"defines optimal Hausdorff oracles and proves their existence for analytic sets, used in Theorem 5.","marker":"[34]"},{"why":"supplies the radial-projection complexity inequality used in Lemma 12.","marker":"[6]"}],"fun_headline_variants":["Distances and projections retain half a point's complexity","Half the algorithmic info survives in distances and projections","New bound: distance and projection keep 50% of algorithmic information","Pinned distances gain from half-complexity retention","Algorithmic info half-life in geometric measurements"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"In the distance proof (Section 4) the reconstruction step assumes that each shortest-witness program for a conditional complexity emits at most one candidate distance per precision interval, so the pair-candidate set is O(1); if one program could emit several candidates in the same interval on different inputs, the central half-information inequality would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Distances and projections retain half a point's complexity","Half the algorithmic info survives in distances and projections","New bound: distance and projection keep 50% of algorithmic information","Pinned distances gain from half-complexity retention","Algorithmic info half-life in geometric measurements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1174,"prompt_tokens":795,"completion_tokens":379,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":317}},"tokens_in":539,"tokens_out":379,"duration_ms":4215,"temperature":1.0,"reasoning_tokens":317,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:31:45.045399+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix r,s,x,y satisfying (14) with δ<ε/6·dim^{A,y}(e_{y,x}) and compute both sides of (15); a failure of the inequality at large r would falsify Theorem 2. Concretely, one can probe the proof's load-bearing step by enumerating the outputs U^u(π_u,d_u) over d_u∈D_s for a shortest witness π_u: if the same s-dyadic interval receives two distinct outputs on different inputs, the O(1) bound on P_{u,v} fails and the reconstruction chain (19) would need a different justification.","supporting_citations":[{"cited_title":"A Kakeya-type problem for circles.American Journal of Mathematics, 119(5):985–1026, 1997","cited_arxiv_id":null,"evidence_quote":"provides the annulus-intersection bound used to cover the candidate point set in the distance theorem."},{"cited_title":"On the distance sets spanned by sets of dimension $d/2$ in $\\mathbb{R}^d$","cited_arxiv_id":"2112.09044","evidence_quote":"gives the prior best pinned-distance bound that Theorem 4 uniformly improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the effective-dimension projection machinery and the oracle lemma used in Theorem 13."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines optimal Hausdorff oracles and proves their existence for analytic sets, used in Theorem 5."}],"review_version":1}