{"id":"c424dab4-719b-4730-bf6c-1d466ce825f6","arxiv_id":"2506.20507","paper_version":2,"verdict":"UNVERDICTED","confidence":"UNKNOWN","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Theorem A establishes 125 nonzero v_2^32-periodic families in the 2-primary stable stems, 50 new, all vanishing in TMF yet detected by the Atkin-Lehner fixed point spectrum J_0(3).","lead":"The authors prove the existence of 125 infinite periodic families in the 2-primary stable homotopy groups of spheres, 50 of them new, and confirm that they are invisible to ordinary topological modular forms but detected by a fixed point variant. This advances the long-running program of Hopkins and Mahowald and yields new exotic spheres in dimensions congruent to 72, 144, and 168 modulo 192.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A's nonvanishing proof delegates the decisive TMF descent differentials to [CDvN24a, §6]; one incorrect row in Table 3 would invalidate the affected families and their vanishing in TMF.","rationale":"After reading the full manuscript, the single most load-bearing assumption is indeed the one the reader identified: the correctness of the TMF descent differentials quoted from [CDvN24a]. Every nonvanishing claim outside Theorem 5.1 flows through Theorem 5.2, and Theorem 5.2 flows through Table 3. The paper is admirably explicit about this dependence—it points to [CDvN24a, §6] rather than hiding it—but explicit dependence is not verification. The rest of the architecture (synthetic spectra, J0(3), deleting differentials, counting) is internally coherent, and I found no independent error in the periodicity or counting arguments. In particular, the counting in §5.3 is consistent with the table sum of 125, and the paper's own caveats (e.g., Question 5.6) are honest limitations, not contradictions. The concern is therefore not that the result is false but that its correctness cannot yet be assessed from this paper alone; if any Table 3 differential is wrong, the theorem's central claim is compromised. Because this does not provide evidence of error, it leaves the reader's UNVERDICTED verdict unchanged.","tokens_in":32066,"tokens_out":6547,"duration_ms":71431,"concrete_test":"Recompute every differential in Table 3 from the 2-primary descent spectral sequence for TMF using a source independent of [CDvN24a], e.g. the Bruner–Rognes charts [BR21] or a fresh computer-assisted resolution of the E2-page. At minimum verify the d5 of degree 47, d5((2k+1)Δ^2)=2ν¯κΔ, and the d5 of degree 71, d5((4k+1)Δ^3)=±ν¯κΔ^2. If either differential differs in source, target, or filtration r, the corresponding rows of Theorem A are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step of Theorem A is the deleting-differentials argument in §5.2. Corollary 3.3 requires, for each target x in Table 1, that x is hit by exactly the stated d_r in σ(TMF^BP) and that the source is distinguished by the map q-p. These d_r are not proved in this paper; they are quoted from [CDvN24a, §6]. The proof of Theorem 5.2 states: 'references to all of these differentials can be found on their appropriate pages in [CDvN24a, §6]', and Table 3 records only the final differentials. Since [CDvN24a] is a preprint with overlapping authors and is not reproduced or independently checked here, the correctness of every row of Table 3 is an unverified premise. A single wrong row does not just lose one class: e.g., the degree-47 row d5((2k+1)Δ^2)=2ν¯κΔ underlies the families 47a and 47b; if this d5 is absent or has a different source, then [2ν¯κΔ] need not be τ-torsion free in S^BP and need not vanish in π_*TMF. The same structure recurs for each ✓ row. Hence Theorem A's '125 families ... with trivial image in TMF' is conditional on external, unverified differential data. This is a verification gap, not an internal inconsistency, but it is the load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a method based on BP-synthetic spectra and the Atkin–Lehner involution on TMF_0(3) to detect v_2^32-periodic families in the 2-primary stable homotopy groups of spheres that vanish in TMF. The main theorem (Theorem A) lists 125 nonzero v_2^32-periodic families in π_* S_2, all with trivial image in π_* TMF, and recovers/confirms previously known families of Behrens–Hill–Hopkins–Mahowald, Bhattacharya–Bobkova–Quigley, and Bobkova–Quigley. Corollary B derives existence of exotic spheres in dimensions congruent to 72, 144, and 168 modulo 192, and very exotic spheres in dimensions congruent to 143, 145, and 169 modulo 192. The proof has two main ingredients: an analysis of the synthetic Hurewicz image of TMF (Section 4) and a 'deleting differentials' argument using the detection spectrum J_0(3) (Sections 3 and 5).","tokens_in":32319,"tokens_out":8216,"duration_ms":81497,"significance":"If the main theorem is correct, this is a substantial contribution to the computation of v_2-periodic families in the 2-primary stable stems. It provides the first unified and systematic detection of many v_2-periodic families that are invisible to TMF, confirms a number of Hopkins–Mahowald predictions, and yields new exotic sphere existence results. The paper is unusually explicit: it gives detailed tables of generators (Tables 1 and 2), differentials (Table 3), and a careful treatment of the ambiguity in choosing periodic families (Theorem 2.2). The synthetic-spectra framework is used elegantly, and the central detection criterion (Corollary 3.3) is clearly formulated. The main risk is the heavy reliance on external, not-yet-published computations for the TMF descent spectral sequence.","major_comments":[{"comment":"The deleting-differentials argument (Corollary 3.3) is the engine of the proof of Theorem A, and it requires, for each target x in Table 1, that x is hit by exactly the stated differential d_r in σ(TMF^BP) and that the source is distinguished by q-p. These differentials are not proved in this paper; they are quoted from the preprint [CDvN24a, §6]. The proof of Theorem 5.2 states: 'references to all of these differentials can be found on their appropriate pages in [CDvN24a, §6]', and Table 3 records only the final differentials. Since [CDvN24a] is a preprint by overlapping authors and is not reproduced or independently checked here, the correctness of every row of Table 3 is an unverified premise. A single wrong row does not just lose one class: for example, the degree-47 row d5((2k+1)Δ^2)=2ν̄κΔ underlies families 47a and 47b; if this d5 is absent or has a different source, then [2ν̄κΔ] need not be τ-torsion free in S^BP and need not vanish in π_*TMF. The same structure recurs for each ✓ row. Hence Theorem A's '125 families ... with trivial image in TMF' is conditional on external, unverified differential data. I recommend that the authors prove these differentials (or provide a detailed verification) within the paper, or clearly state the theorem as conditional on the acceptance of the companion preprint.","section":"§5.2, Table 3 and Theorem 5.2"},{"comment":"The proof of Theorem 4.2, which establishes the synthetic Hurewicz image of the candidate classes, depends on a number of claims justified by 'inspection of the charts of [IWX22]'. This includes Lemma 4.12 (stem 47), Lemma 4.18–4.20 (stem 71), and Lemma 4.21–4.23 (higher stems). These inspections involve subtle points, such as the τ-power torsion nature of the class l1 and the hidden η-extension from (71,7) to (72,10) in the proof of Lemma 4.18. Since the charts are publicly available, this is not an error, but the paper should provide precise bidegree data (e.g., the exact classes and their coordinates) for each inspection claim so that the reader can verify them without re-deriving the entire charts. This is load-bearing because these classes are the input to the detection argument; a misidentified chart class would invalidate the corresponding family.","section":"§4.2–§4.4"},{"comment":"The application of Corollary 3.3 requires, for each row of Table 3, not only that the stated d_r is deleted, but also that the groups π_{s+1,f-r-i}J_0(3)^BP/τ vanish for i≥1 so that the lifted classes are permanent cycles. The proof asserts this follows from Lemmas 3.12–3.14, but no systematic verification is given for the 14 degree rows. Since the checkerboard pattern and negative-filtration vanishing make this a finite check, I request a table or a clear argument covering all bidegrees involved.","section":"§5.2, proof of Theorem 5.2"}],"minor_comments":[{"comment":"The paper refers to 'Equation (5.1)' and 'Equation (5.2)' when it means Theorem 5.1 and Theorem 5.2, and similarly 'Equation (4.10)' for Proposition 4.10. There are no displayed equation numbers in the text, so these references should be corrected for clarity.","section":"Throughout"},{"comment":"There are typos: 'paralellisable' should be 'parallelisable', and 'prefered' should be 'preferred'.","section":"§1.3 and §3.2.1"},{"comment":"In the phrase 'admits v_k^h-self map v on F', the symbol F is undefined; it should likely be 'on M'.","section":"Definition 2.1"},{"comment":"The row for degree 143 reads 'd5(Δ^6)=η̄κη∆^6', which appears to have a duplicated symbol; please check the notation.","section":"Table 3"},{"comment":"The references to the Adams–Novikov and Adams charts should specify the version and date used, since these are living documents.","section":"References [IWX20a], [IWX22]"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the method is novel. The main concern is the dependence on the companion preprint [CDvN24a], which is by overlapping authors and has not yet been refereed. In my view, this is a fixable issue, but the authors should be required to either include the proofs of the needed differentials or state the theorem as conditional. The papers cites many of the authors' own preprints; this is not circular in a logical sense, but the editor may want to ensure that [CDvN24a] and [CDvN24b] are publicly available and stable before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper you want to know about: Carrick and Davies claim 125 nonzero v2^32-periodic families in the 2-primary stable stems, all vanishing in TMF, with many detected by a height-2 analogue of the image-of-J spectrum. If Theorem A holds, this is the biggest progress on v2-periodic families at the prime 2 in years, and it settles a batch of Hopkins–Mahowald predictions. That's the headline.\n\nWhat is genuinely new: the 50 families not previously in the literature, and the method. The deleting-differentials technique in synthetic spectra, combined with the Atkin–Lehner involution on TMF_0(3), is a clever and portable way to rule out lifts of TMF differentials to the sphere. The paper is unusually explicit: Table 1 and Table 2 give generators, orders, and how the counting works, and §5.3 walks through every row. The recovery of the known families of BHHM20, BBQ24, BQ24 is a sensible sanity check.\n\nWhere I have a soft spot, it's the one the stress-test flags. The nonvanishing of the checked families in Theorem 5.2 depends on the differentials in Table 3, and those are not proved here—they are quoted from [CDvN24a, §6], a preprint by overlapping authors. One wrong row, e.g., the d5 in degree 47, and the affected families disappear. That is a real verification gap, not an internal inconsistency. It doesn't make me think the result is false; the differentials in the TMF DSS are heavily constrained and the authors clearly know them. But for a paper whose main theorem is a list of nonzero classes, I want the referee to spend time on those differentials before accepting.\n\nThere are also smaller places where the proof leans on inspection of the [IWX22] charts, and a few assertions like 'the blue dot indicates...' that are not fully spelled out. That is normal for this literature, but it adds to the self-containedness problem.\n\nBottom line: this is a serious, well-written paper by people who know what they are doing. It deserves a serious referee—two of them, ideally, one of whom checks the TMF DSS differentials against [CDvN24a]. I would not desk-reject it. I would not cite it as established fact until [CDvN24a] is also verified, but I would cite it as a preprint and bring it to reading group.","headline":"A serious, explicitly presented computation of 125 v2-periodic families that deserves referee time, but whose nonvanishing proof leans on differentials imported from an unverified preprint by overlapping authors.","tokens_in":32896,"tokens_out":2519,"would_cite":true,"duration_ms":25843,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55Q45","55T15","55Q51","55P42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the 2-primary stable homotopy groups of spheres contain 125 nonvanishing $v_2^{32}$-periodic families, all with zero image in topological modular forms, and derives exotic spheres in three new congruence classes of…","keywords":["stable homotopy groups of spheres","periodic families","topological modular forms","synthetic spectra","Adams–Novikov spectral sequence","Atkin–Lehner involution","exotic spheres","v2-periodic families"],"falsifier":"Independently compute the 2-primary descent spectral sequence for TMF in the range of Table 3 and check the listed differentials, for example $d_5((4k+1)\\Delta) = \\nu\\bar{\\kappa}$, $d_9(\\eta\\Delta^2) = \\varepsilon\\bar{\\kappa}^2$, and $d_9(\\eta\\Delta^3) = \\bar{\\kappa}^2[\\varepsilon\\Delta]$. If any of these fails, or if the source of a listed differential maps to zero under $q-p$ in the Adams–Novikov spectral sequence for $\\mathrm{TMF}_0(3)$, then the corresponding row of Table 1 need not contribute a nonzero family, and the count of 125 would be too high.","tokens_in":31813,"feed_emoji":"🌀","tokens_out":13809,"duration_ms":120771,"temperature":0.7,"pith_summary":"The paper proves Theorem A: the 2-primary stable homotopy groups of spheres contain 125 nonvanishing $v_2^{32}$-periodic families, distributed over nineteen congruence classes of degrees modulo 192, with orders as listed in Table 1 and with all generators mapping to zero in the homotopy of topological modular forms (TMF). The families are detected instead in the equalizer $J_0(3)$ of the Atkin–Lehner inflation map and the canonical map from TMF to $\\mathrm{TMF}_0(3)$, so they are invisible to TMF yet survive in the sphere. The computation reconfirms and refines previously known families from earlier papers and adds 50 families not previously in the literature. As a corollary, exotic spheres exist in all dimensions congruent to 72, 144, and 168 modulo 192, with very exotic spheres in dimensions 143, 145, and 169 modulo 192.","feed_headline":"125 periodic families hide from topological modular forms","feed_subtitle":"A deleting-differentials trick using the Atkin–Lehner involution on TMF proves they survive in the sphere spectrum.","key_machinery":"The load-bearing mechanism is the deleting-differentials technique in the category of BP-synthetic spectra. Given a fibre sequence of synthetic spectra $F \\to X \\to Y$, if a class $b \\in \\pi_{*,*} X$ has $\\bar{b} \\neq 0$ and every possible source $a$ of a differential $d_r(\\bar{a}) = \\bar{b}$ maps to $f(a) \\neq 0$, then any lift of $b$ to $F$ is $\\tau^{r-1}$-torsion free; applied to $X = \\mathrm{TMF}_{BP}$, $Y = \\mathrm{TMF}_0(3)_{BP}$, and $f = q - p$ with the Atkin–Lehner twist $q = w \\circ p$, this deletes the differentials that would otherwise kill the target classes in $\\pi_* \\mathrm{TMF}$. The equalizer $J_0(3)_{BP}$ of $p$ and $q$ is the detection spectrum, and the paper shows most of Table 1's classes are nonzero in its image.","core_discovery":"The central claim is that Table 1 lists 125 nonvanishing $v_2^{32}$-periodic families in $\\pi_d S_2$ for each degree modulo 192 shown, with the stated orders and with trivial image in $\\pi_* \\mathrm{TMF}$. The proof produces the families by lifting each generator to the synthetic Hurewicz image of $\\mathrm{TMF}_{BP}$, where the known descent-spectral-sequence differentials for TMF kill it, and then using a deleting-differentials argument to show those differentials cannot lift to the sphere; the target instead survives in the equalizer $J_0(3)_{BP}$ of the two maps $\\mathrm{TMF}_{BP} \\to \\mathrm{TMF}_0(3)_{BP}$. A smaller set of families, detected only in the sphere, is handled by a filtration argument using the 1-line of the Adams–Novikov spectral sequence. Corollary B translates the surviving families into exotic spheres in the stated congruence classes.","pith_inferences":["If the same equalizer construction is applied to the level-five and level-seven analogues $\\mathrm{TMF}_0(5)$ at $p=2$ and $\\mathrm{TMF}_0(7)$ at $p=3$, using the level-specific computations referenced in the paper, the deleting-differentials argument could plausibly produce further simple torsion families beyond these 125; the paper leaves this open in Question 5.10.","A systematic computation of the synthetic Hurewicz image of $\\mathrm{TMF}_{BP}$, which the paper does only case-by-case, would likely extend the method to unresolved classes such as $\\bar{\\kappa}[\\nu\\Delta^4]$, $\\bar{\\kappa}[2\\nu\\Delta^5]$, and $\\bar{\\kappa}[\\nu\\Delta^6]$ in degrees 119, 143, and 167, as suggested in Question 5.6.","The fact that several rows of Table 1 are detected only by a filtration argument rather than by $J_0(3)$ suggests that other fixed-point spectra built from TMF by Hecke-type operations could detect additional families that $J_0(3)$ misses, including the 2-torsion family in degree 122 and the 8-torsion family in degree 170 mentioned in Question 5.11."],"forward_implications":["Table 1 yields 125 nonvanishing $v_2^{32}$-periodic families in $\\pi_* S_2$, of which 50 are new and the rest reconfirm known families from [BHHM20], [BBQ24], and [BQ24].","All 125 families vanish in $\\pi_* \\mathrm{TMF}$ but are detected in the equalizer $J_0(3)$, so TMF's classical Hurewicz image is not the only source of $v_2$-periodic phenomena at the prime 2.","Corollary B: exotic spheres exist in all dimensions congruent to 72, 144, and 168 modulo 192, and very exotic spheres in dimensions 143, 145, and 169 modulo 192.","The same techniques reconfirm the families tentatively suggested in the earlier brief report [DFHH14, §15], including the classes in degrees 47 and 48, and confirm the nonvanishing of the family assembled from $\\bar{\\kappa}^6$ in degree 120.","Because TMF and $J_0(3)$ are MU-nilpotent, the proof also shows all listed families have nonzero image in the $K(2)$-local sphere, while the method cannot produce nonzero families in the $T(2)$-local sphere that vanish $K(2)$-locally."],"supporting_citations":[{"why":"Supplies the 2-primary descent spectral sequence for TMF and the differentials (Theorem 5.2, Table 3) that the deleting-differentials argument deletes.","marker":"[CDvN24a]"},{"why":"Establishes the classical 2-primary Hurewicz image of tmf/TMF and the $v_2^{32}$-periodic self-map machinery used to exhibit the generators.","marker":"[BMQ23]"},{"why":"Supplies earlier $v_2^{32}$-periodic families in rows 48 and 120 and the use of $J_0(3)$ detection that Theorem A confirms and extends.","marker":"[BHHM20]"},{"why":"Gives the previously constructed 2-torsion families in congruence classes 23, 47, 71, 74, 95, 119, and 167 that this paper reconfirms.","marker":"[BBQ24]"},{"why":"Provides the η-torsion families in classes 73 and 120 that are reconfirmed here.","marker":"[BQ24]"},{"why":"Constructs the stable Atkin–Lehner involution $w$ on $\\mathrm{TMF}_0(N)$ and the Adams operations that define the equalizer $J_0(3)$.","marker":"[Dav24a]"},{"why":"Computes the action of the Atkin–Lehner map on the $E_2$-page of $\\mathrm{TMF}_0(3)$, used to show sources of deleted differentials do not lift.","marker":"[MR09]"},{"why":"Introduces the category of synthetic spectra whose signature spectral sequences realize the Adams and Adams–Novikov spectral sequences used in the proof.","marker":"[Pst23]"},{"why":"Provides the Adams–Novikov charts for the sphere used to identify lifts and filtrations of the generators in §4.","marker":"[IWX20a]"}],"fun_headline_variants":["125 infinite periodic families hide from topological modular forms","125 periodic families: survive sphere, disappear in TMF","Atkin-Lehner fixed points detect 125 periodic families","New 125 periodic families in height-two stable stems","Exotic spheres in all dimensions congruent to 72, 144, 168 mod 192"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the differentials in the 2-primary descent spectral sequence for TMF listed in [CDvN24a, Section 6] (Theorem 5.2, Table 3) are computed correctly; the paper does not reproduce those computations, and a single misidentified differential would let the corresponding families be killed in the sphere rather than survive.","fun_headline_variants_meta":{"raw":{"variants":["125 infinite periodic families hide from topological modular forms","125 periodic families: survive sphere, disappear in TMF","Atkin-Lehner fixed points detect 125 periodic families","New 125 periodic families in height-two stable stems","Exotic spheres in all dimensions congruent to 72, 144, 168 mod 192"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000908,"raw_usage":{"total_tokens":3863,"prompt_tokens":866,"completion_tokens":2997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":482,"completion_tokens_details":{"reasoning_tokens":2911}},"tokens_in":482,"tokens_out":2997,"duration_ms":25222,"temperature":1.0,"reasoning_tokens":2911,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:47:02.501721+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently compute the 2-primary descent spectral sequence for TMF in the range of Table 3 and check the listed differentials, for example $d_5((4k+1)\\Delta) = \\nu\\bar{\\kappa}$, $d_9(\\eta\\Delta^2) = \\varepsilon\\bar{\\kappa}^2$, and $d_9(\\eta\\Delta^3) = \\bar{\\kappa}^2[\\varepsilon\\Delta]$. If any of these fails, or if the source of a listed differential maps to zero under $q-p$ in the Adams–Novikov spectral sequence for $\\mathrm{TMF}_0(3)$, then the corresponding row of Table 1 need not contribute a nonzero family, and the count of 125 would be too high.","supporting_citations":[],"review_version":1}