{"id":"ecff7b48-4803-4d9f-bc10-10e0c270969e","arxiv_id":"1908.08471","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any class of short games with bounded confusion intervals, the boiling point is at most K/2 + J, where K bounds the interval length of the game and J that of its options.","lead":"This paper proves a general upper bound on the temperature of combinatorial games, using the length of confusion intervals of the game and its options. It is the first bound of this kind and yields temperature bounds for subclasses of Domineering and Snort.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 15's existence of thermic versions is load-bearing and the proof's final 'therefore' is unjustified; the main bound depends on it.","rationale":"The reader's weakest assumption identifies exactly the same point: Theorem 15 is asserted in a few lines and the key equality t({G^L|G^R}) = t(G) is not justified. I examined the rest of the proof chain. Theorem 24's segment accounting is sound if a thermic version exists: T_Lvert <= l(~G^L) follows from the right-wall oblique length being at most the confusion interval, and Lemma 23 correctly gives l(~G) = T_Lobl + T_Robl. Theorem 25 then follows directly. The applications are informal in places, but they are examples rather than the central claim, and the path-length and table issues do not threaten Theorem 25 itself. The tightness example also checks out once the ± notation is read as the overheating construction. Thus the single load-bearing risk is the existence lemma. I could not produce a counterexample, and the assertion is plausibly true via a mean argument: if A is an option whose shifted stop reaches m at T, then A has temperature T and mean T+m, so for s < T the game A_s - s has mean m + (T-s) > m and cannot be a left option of the number m in H_s; this would force t(H) = T. But the paper does not supply this argument or anything equivalent. The reader's conditional verdict is therefore appropriate: the main theorem is likely correct but the manuscript should expand the proof of Theorem 15, ideally with a computational check for small games as corroboration.","tokens_in":13481,"tokens_out":42323,"duration_ms":436652,"concrete_test":"Enumerate all short games up to birthday 5 in CGSuite. For each hot game G with T = t(G), compute the options achieving LS(G_T) and RS(G_T); for every such pair (G^L, G^R), form H = {G^L | G^R} and check whether t(H) = T. If any pair yields t(H) < T, Theorem 15 is false and the main bound needs repair. A full pass up to birthday 5 would not prove the theorem but would show no small counterexample exists; it should be accompanied by a rigorous proof that A_T - T, for an option A whose shifted stop achieves m at T, cannot be a left option of the number m at any s < T.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 25 rests on Theorem 24, which rests on Theorem 15: for every hot game G there are options G^L, G^R with t({G^L|G^R}) = t(G). The proof selects options achieving the two stops of G_T at T = t(G), forms H = {G^L|G^R}, observes LS(H_T) = RS(H_T) = m, and concludes t(H) = T. This does not follow. Equal stops at T only give t(H) <= T. H's walls are built only from the two selected options, so they could meet strictly before T, because the other options of G that delay the full mast are absent. The text gives no argument—for example, no use of the mean of the cooled option, no order or dominance property—ruling this out. If such an H had lower temperature, the step T_Lvert <= l(~G^L) in Theorem 24 would not produce the stated bound. I did not find a concrete counterexample, and the claim may be true, but as written this is a genuine gap in the proof chain of the central theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the temperature of short partizan combinatorial games. It defines the confusion interval C(G) of a game G and its length l(G), and claims two main results. First, Theorem 25 states that if S is a class of short games with l(G) <= K and l(G^L), l(G^R) <= J for every G in S, then the boiling point BP(S) is at most K/2 + J; a family of examples is given to show that this bound can be tight. Second, Proposition 27 gives a method to bound l(G) by showing that Right wins in G^L - G - K (with Left moving first), and the method is applied to Domineering snakes in 2 x n grids (temperature at most 3) and to Snort on paths (temperature at most 15/2). The proof route is: Theorem 15 asserts that every hot game has a thermic version with a single Left option and a single Right option; Theorem 24 bounds the temperature of a thermic version in terms of l of the game and of one of its options; Theorem 25 then follows by taking maxima over the class.","tokens_in":13675,"tokens_out":30441,"duration_ms":303078,"significance":"If the main theorem is correct, it is a genuinely general quantitative statement: for the first time, an upper bound on the boiling point of an arbitrary class of short games follows from a simple local parameter, the lengths of confusion intervals. The tightness example shows that the K/2 + J form cannot be improved in general, and the method of bounding l by second-player strategies in G^L - G is elementary and potentially reusable. The paper is self-contained, does not fit parameters to data, and clearly acknowledges the gap between the general bound and the much smaller temperatures known or conjectured for Domineering and Snort. However, the proof of the load-bearing Theorem 15 is not correct as written, so the central theorem is not yet established by this manuscript.","major_comments":[{"comment":"The proof is not valid as written. From the equality of the two stops of G_T and the fact that the selected options realize those stops, one only obtains LS({G^L|G^R}_T) = RS({G^L|G^R}_T) = m, which gives t({G^L|G^R}) <= T; nothing in the argument rules out an earlier meeting of the two walls. The final 'therefore' is actually false for arbitrary options realizing the stops: for G = {3, {5|1} | {1|-3}} one has t(G) = 2 (for t < 2, LS(G_t) = 3 - t and RS(G_t) = 1, while at t = 2 both stops are 1), and both G^L = {5|1} and G^R = {1|-3} realize the stops of G_2, yet H = {{5|1} | {1|-3}} is equal to 1 (H - 1 = {{4|0} | {0|-4}} is a second-player win), so t(H) < 2. Because Theorem 24 and consequently Theorem 25 rely on thermic versions, this gap is load-bearing. The existence claim is probably repairable: one should choose options whose sheared walls form the top segments of the two walls of G's thermograph, and use the fact that the difference of those walls is nonincreasing and strictly positive just below T to conclude t({G^L|G^R}) = T. This repair needs to be written out in full.","section":"Section 3, Theorem 15"},{"comment":"The applications rest on informal strategy arguments. In Proposition 28, the reduction for H = G^L and H = G^R is described only for a representative split, and it is not fully shown that every Left option of H is covered. In Theorem 29, key assertions such as 'Left then has at most three additional non-copiable moves' and 'the influence area is already bounded' are stated without proof, and the induction over end-decorated paths is only alluded to. These are the advertised numerical bounds, so the strategies should be formalized or replaced by a precise inductive argument.","section":"Sections 4.1 and 5.1"}],"minor_comments":[{"comment":"The abstract contains the duplicated phrase 'for for every' which should read 'for every'.","section":"Abstract"},{"comment":"The recursive notation 'P = {P^L|P^L}' should be '{P^L|P^R}'.","section":"Section 2, after Definition 1"},{"comment":"The expressions 'G^L_{t(G)} - t(G)' and 'G^R_{t(G)} + t(G)' need parentheses, e.g. '(G^L)_{t(G)} - t(G)', to avoid ambiguity.","section":"Section 3, Theorem 15 proof"},{"comment":"The hypothesis should read 'for every Left option G^L', and the proof should state that the chosen G^L is the one attaining LS(G).","section":"Section 3, Proposition 27"},{"comment":"The notation '±{9|3}' is not defined, and the example does not explicitly verify that each G_n satisfies l(G_n), l(G_n^L), l(G_n^R) <= 6; a short verification should be added.","section":"Section 3, Example 26"},{"comment":"The sentence 'the length of these oblique segments are at most the distance between the Left and Right stops' is imprecise: the intended inequality is T_Lvert <= l(~G^L), and the intermediate step should be stated more carefully.","section":"Section 3, Theorem 24 proof"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the proof of Theorem 15. My concrete counterexample to the 'therefore' step shows that the gap is substantive, not a missing sentence. I believe the existence claim is true with a different choice of options, so I recommend major revision rather than rejection. The authors should also tighten the strategy proofs in the applications. There is no concern about attribution or scope; the paper fits math.CO."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my quick read. The paper's central contribution is real: Theorem 25 gives the first general upper bound on the boiling point of a class of short games in terms of confusion intervals. That's a useful tool, and the tightness example shows the bound is sharp in a broad class. The applications to Domineering snakes (temperature <= 3) and Snort on paths (<= 7.5) are concrete and new, and Proposition 27's method for bounding confusion intervals via difference games is a nice trick. I'd credit it for being the first general handle on this problem.\n\nThe soft spot is exactly where the stress-test note lands. Theorem 15 claims every hot game has a thermic version: two options G^L, G^R such that t({G^L|G^R}) = t(G). The proof observes that at T = t(G), the stops of the two selected cooled options are equal, and then concludes the new game has temperature T. That doesn't follow. Equal stops at T only give temperature <= T. The new game's thermograph is built only from those two options, so it could plausibly meet strictly earlier, because the other options of G that delay the mast are gone. The paper gives no argument—order, dominance, or mean—to rule this out. I didn't find a counterexample either, but the gap is real and it's load-bearing: Theorem 24 and 25 both rest on it. So the main theorem is not fully proven as written.\n\nTwo smaller things. The segment-length step in Theorem 24 (\"T_L_vert is at most the length of the oblique segments of ~G^L\") is asserted rather than demonstrated; it's plausible but deserves a formal sentence. And the strategy arguments for Domineering and Snort are informal, with diagrams doing heavy lifting; a skeptical reader will want more precise cases. Also careful: there's a small inconsistency in the path-length claim in Section 5.1—the table covers lengths up to 12, but the proof says up to 14. Minor.\n\nWho's this for? Researchers working on temperature theory and partizan games. The main idea is good enough that I'd send it to a serious referee, but the thermic-version lemma needs to be fixed or the theorem restated with an extra hypothesis. I'd want to see that addressed before I cite it.\n\nRecommendation: not a desk reject. Engage it, but ask for a rigorous repair of Theorem 15 and the segment estimates.","headline":"First general boiling-point bound, but the thermic-version lemma has a real gap that needs fixing.","tokens_in":14193,"tokens_out":3292,"would_cite":false,"duration_ms":29875,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A46"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any class of short games, temperature is bounded by K/2 + J, where K and J are the maximum confusion-interval lengths of the games and their options.","keywords":["combinatorial game theory","temperature","boiling point","confusion interval","thermograph","Domineering","Snort","short games"],"falsifier":"A brute-force search over all short games of small game-tree depth looking for a game $G$ with $\\ell(G)\\le 6$, $\\ell(G^L),\\ell(G^R)\\le 6$, and $t(G)>9$ would settle the tightness claim; the paper's own sequence approaches $9$ from below, so a single exceeding value would refute Theorem 25.","tokens_in":13298,"feed_emoji":"♟️","tokens_out":8308,"duration_ms":73658,"temperature":0.7,"pith_summary":"This paper proves the first general upper bound on the temperature of any class of short combinatorial games. It shows that if every game in a class has confusion interval of length at most $K$ and every option has confusion interval of length at most $J$, then every game in the class has temperature at most $K/2+J$. Temperature measures how much a position's advantage can shift, and the boiling point of a class is the largest temperature in it, something previously bounded only for individual games. The paper also proves a way to estimate the confusion interval itself: the number of passing moves Right needs to win in $G^L-G$ gives an upper bound on $\\ell(G)$. It applies both tools to prove that Domineering snakes on a $2\\times n$ board have temperature at most $3$ and that Snort on a path has temperature at most $7.5$.","feed_headline":"Confusion intervals bound game temperature for the first time","feed_subtitle":"If position and option confusion intervals stay short, temperature is at most K/2 + J — and the limit is sharp.","key_machinery":"The argument is carried by the thermograph of a game, the plot of its Left and Right stops as the game is cooled, together with the thermic version $\\tilde G$, a game with a single Left option and a single Right option that has the same temperature as $G$. Decomposing the left wall of the thermograph into vertical and oblique segments below the mast, the paper observes that the vertical segments are paid for by an option's confusion interval while the oblique segments are paid for by the confusion interval of $G$ itself, and the temperature is the sum of both. The length of a confusion interval is $\\ell(G)=LS(G)-RS(G)$, and Proposition 27 turns bounding it into a strategic question: if Right wins $G^L-G-K+\\varepsilon$ moving second, then $\\ell(G)\\le K$. This converts temperature control into checking a difference game, which is how the Domineering and Snort examples are obtained.","core_discovery":"The central discovery is that the temperature of a short game $G$ is controlled by the lengths of its confusion interval and those of its options. Writing $\\ell(G)$ for the distance between Left and Right stops and $\\tilde G$ for a thermic version of $G$ (a game with one Left option and one Right option whose temperature equals that of $G$), the paper proves $t(G)\\le \\ell(H)+\\ell(G)/2$, where $H$ is the option on the side whose thermograph wall has more vertical segment. Consequently, for any class $\\mathscr S$ with uniform bounds $\\ell(G)\\le K$ and $\\ell(G^L),\\ell(G^R)\\le J$, the boiling point satisfies $BP(\\mathscr S)\\le K/2+J$. An explicit sequence of games with $K=J=6$ shows the bound is tight, with temperatures approaching $9$. This is the first boiling-point bound that applies to every class of short games.","pith_inferences":["Beyond the paper's explicit claims, the same strategy should yield sharper constants for classes with few threats: when thermic options can be identified without building the whole thermograph, only their confusion intervals enter the bound, and the paper notes that replacing the constant $5$ by $4+\\uparrow$ in the Snort proof would lower the path bound to $6$.","For games dominated by threats, like $\\{\\{10|1\\}|-1\\}$, the confusion interval can stay small while the temperature is large, so the $K/2+J$ bound is conservative exactly where threats matter; classes of games with mild threats may admit much better bounds.","A testable extension is to apply Proposition 27 to other bounded-degree rulesets, such as Snort on graphs of bounded degree, where the paper's degree conjecture says the temperature is at most the degree of the board."],"forward_implications":["For any class of short games whose confusion intervals and option confusion intervals are uniformly bounded, the boiling point is finite and at most $K/2+J$, with no per-game computation needed.","The bound is sharp in general: for the class of games with $\\ell(G),\\ell(G^L),\\ell(G^R)\\le 6$, the temperatures of the displayed sequence approach $9$, so the boiling point is exactly $9$.","Domineering snakes fitting in a $2\\times n$ grid have temperature at most $3$, proved by showing Right wins $H^L-H-2$ moving second and then applying Theorem 25.","Snort on a path of any length has temperature at most $7.5$, proved by showing Right wins $H^L-H-5$ moving second for paths of length at least $5$.","Since $t(G+H)\\le\\max\\{t(G),t(H)\\}$, these bounds apply to disjunctive sums of such positions, not just to connected boards."],"supporting_citations":[{"why":"Supplies the standard definitions and facts about cooling, temperature, and thermographs, including $t(G+H)\\le\\max\\{t(G),t(H)\\}$, that the proof builds on.","marker":"[17]"},{"why":"Foundational source for thermograph geometry and the intersection-of-walls representation used in the segment decomposition.","marker":"[6]"},{"why":"Prior Domineering temperature computations on rectangular grids that the new general bound complements.","marker":"[3]"},{"why":"Provides values for Snort positions that motivate the path analysis and the degree conjecture.","marker":"[4]"},{"why":"Earlier reduction and temperature results for Domineering snakes that define the subclass examined in Section 4.1.","marker":"[19]"}],"fun_headline_variants":["Confusion intervals set first temperature bound for games","Game temperature tamed by confusion intervals","Short confusion intervals cap game temperatures","First general boiling-point bound via confusion intervals","Confusion interval lengths dictate game temperature caps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that for every hot game one can choose a single Left option and a single Right option so that the two-option game has exactly the same temperature as the original; the paper's justification is brief, and the main bound would not cover all short games if this failed.","fun_headline_variants_meta":{"raw":{"variants":["Confusion intervals set first temperature bound for games","Game temperature tamed by confusion intervals","Short confusion intervals cap game temperatures","First general boiling-point bound via confusion intervals","Confusion interval lengths dictate game temperature caps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000543,"raw_usage":{"total_tokens":2626,"prompt_tokens":998,"completion_tokens":1628,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":1564}},"tokens_in":614,"tokens_out":1628,"duration_ms":12368,"temperature":1.0,"reasoning_tokens":1564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:40:44.006090+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A brute-force search over all short games of small game-tree depth looking for a game $G$ with $\\ell(G)\\le 6$, $\\ell(G^L),\\ell(G^R)\\le 6$, and $t(G)>9$ would settle the tightness claim; the paper's own sequence approaches $9$ from below, so a single exceeding value would refute Theorem 25.","supporting_citations":[{"cited_title":"Combinatorial game theory, volume146of Graduate Studies in Mathematics","cited_arxiv_id":null,"evidence_quote":"Supplies the standard definitions and facts about cooling, temperature, and thermographs, including $t(G+H)\\le\\max\\{t(G),t(H)\\}$, that the proof builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Foundational source for thermograph geometry and the intersection-of-walls representation used in the segment decomposition."},{"cited_title":"Berlekamp","cited_arxiv_id":null,"evidence_quote":"Prior Domineering temperature computations on rectangular grids that the new general bound complements."},{"cited_title":"Berlekamp, J.H","cited_arxiv_id":null,"evidence_quote":"Provides values for Snort positions that motivate the path analysis and the degree conjecture."},{"cited_title":"Snakes in domineering games.Theoret","cited_arxiv_id":null,"evidence_quote":"Earlier reduction and temperature results for Domineering snakes that define the subclass examined in Section 4.1."}],"review_version":1}