{"id":"48a0f94a-5e6e-4fe5-b87b-6fd08bbd04d6","arxiv_id":"2507.07051","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At the prime 2, the connective higher real K-theories eo_h are shown to be fp spectra of type h, which implies a divisibility constraint on Euler characteristics and a new obstruction to generalized Moore spectra.","lead":"The authors prove that a family of spectra built from real bordism, one for each chromatic height, are 'fp spectra', the same well-behaved class that includes ko and tmf. Using these, they derive a new necessary condition for the existence of generalized Moore spectra at the prime 2, making progress on a conjecture of Levy.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 1.3 and 1.2 as proved rely on the unpublished Burklund-Levy theorem [12]; Theorem 1.2 can be re-proved without it, but Theorem 1.3 remains conditional on [12].","rationale":"I read the full manuscript. The core structural result Theorem 2.5 is supported by a serious and largely self-contained induction: Section 3 computes the underlying homotopy quotient dimension and shows it is odd, Section 4 develops normed Koszul filtrations, Section 5 upgrades nilpotence to the fp conclusion using equivariant Balmer spectra, and Corollary 6.12 removes the type upper bound by a Euler-characteristic contradiction that does not itself need [12]. I found no internal gap in this chain. The genuinely load-bearing dependency is the transfer from these new Euler characteristics to the standard χ_BP⟨h⟩, via Theorem 6.9. That transfer rests on the unpublished Burklund-Levy theorem [12]. I agree with the reader that this is the weakest point, but with a nuance: Theorem 1.2 can be recovered by applying Theorem 6.8 directly to the given Moore spectrum and using the first half of Proposition 6.11; this route avoids [12]. What cannot be obtained without [12] is the global divisibility theorem Theorem 1.3 and the claimed partial verification of Levy's conjecture. Since the abstract and introduction prominently advertise Theorem 1.3, keeping the reader's CONDITIONAL verdict is appropriate. The mechanical issues noted by the reader (Corollary 3.8 indexing, the typo in Theorem 6.9) do not affect this assessment.","tokens_in":26346,"tokens_out":25147,"duration_ms":277253,"concrete_test":"Obtain the current draft of Burklund-Levy [12] and verify that it proves exactly the rational-generation statement used in Theorem 6.9: χ_BP⟨h⟩ is rationally an isomorphism on K0(Sp^ω_{>h})[1/2] and every generalized Moore spectrum represents a rational generator.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the transfer from the new Euler characteristics to the standard one, χ_BP⟨h⟩. Theorem 6.9 is asserted as a consequence of Burklund-Levy [12, 'To appear'] and says that a single generalized Moore spectrum rationally generates K0(Sp^ω_{>h})[1/2]. This is used in Proposition 6.11 to identify χ_BP^{(G)}⟨m⟩^e with an odd multiple of χ_BP⟨h⟩ as functions on the whole K-group, which is exactly what yields the global divisibility statement Theorem 1.3. If [12] is false, or if it proves a weaker rational-generation statement, then Theorem 1.3 is unsupported. The reader's additional worry about an unstated existence hypothesis for the Moore spectrum is less serious: existence of at least one generalized Moore spectrum at each height follows by iterating the Devinatz-Hopkins-Smith periodicity theorem. Note also that the advertised nonexistence result Theorem 1.2 is less vulnerable: one can evaluate the relation |G|·χ_{R^G} = χ_{R^e} from Theorem 6.8 on the given Moore spectrum and use the direct computation in Proposition 6.11(1), avoiding [12] entirely. The hard dependency is therefore specifically the global divisibility theorem Theorem 1.3 and the partial verification of Levy's conjecture, not the fp/type theorem Theorem 2.5.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the higher truncated Brown-Peterson spectra BP^{(G)}⟨m⟩ and their fixed points eo_h(H), and proves that these are fp spectra in the sense of Mahowald-Rezk, of chromatic type h = m|G|/2. The main structural result, Theorem 2.5, is obtained by an equivariant induction using Koszul-type filtrations and a computation of the underlying homotopy modulo (2,v_1,...,v_h). The paper then uses Levy's Euler characteristic for fp spectra to derive K-theory relations among fixed points, identifies the Euler characteristic of BP^{(G)}⟨m⟩^e with an odd multiple of χ_{BP⟨h⟩}, and concludes that the image of χ_{BP⟨h⟩} is divisible by 2^n (Theorem 1.3). A corollary is the nonexistence criterion for generalized Moore spectra: if S/(2^{i_0},...,v_h^{i_h}) exists, then ν_2(∏ i_k) > ν_2(h) (Theorem 1.2). The final section proves Borel-completeness results for chromatic localizations of MU^{(G)}-modules.","tokens_in":26562,"tokens_out":11561,"duration_ms":121128,"significance":"If the central results hold, this is a substantial contribution to chromatic homotopy theory: it provides new explicit fp spectra at all heights, gives the first 2-primary nonexistence constraint on generalized Moore spectra valid at all heights, and partially verifies Levy's conjecture on the image of χ_{BP⟨h⟩}. The paper is careful and self-contained in its induction in Sections 4-5, and the identification of the Poincaré series and Gaussian binomial coefficients is explicit and falsifiable. The main external risk is the dependence of Section 6 on the unpublished Burklund-Levy theorem [12]; this is acknowledged but must be resolved before the paper can be judged. A second, local but load-bearing issue is the off-by-one indexing in Corollary 3.8, which is easily corrected.","major_comments":[{"comment":"The displayed formula for the dimension of π_e^*BP^{(G)}⟨m⟩/(2,v_1,...,v_h) has an off-by-one indexing error: the product is written over j=0,...,|G|/2−1 of the Gaussian binomial coefficient [jm;m]_2, and by Definition 3.6 the j=0 term [0;m]_2 is zero for m>0. This makes the stated dimension zero and contradicts Theorem 2.6 as well as the asserted oddness. The limiting computation in the proof appears to intend the product over j=1,...,|G|/2 of [jm;m]_2 (equivalently, shifting the index in the displayed product). Because this odd multiplier is used in Proposition 6.11 to identify χ_{BP^{(G)}⟨m⟩^e} with an odd multiple of χ_{BP⟨h⟩}, the indexing error is load-bearing and must be corrected.","section":"Corollary 3.8"},{"comment":"The proof of the global divisibility statement Theorem 1.3 and of the second part of Proposition 6.11 depends essentially on the unpublished Burklund-Levy theorem [12]. Theorem 6.9 is stated as a direct consequence of [12], but the needed assertion is not merely that χ_{BP⟨h⟩} is a rational isomorphism; it is that any generalized Moore spectrum rationally generates K_0(Sp^ω_{>h})[1/2]. As written, the short proof of Theorem 6.9 is valid only if [12] contains exactly this rational-generation statement, and the manuscript does not quote or prove it. If [12] is not yet available in final form, or if it proves a weaker statement, then Theorem 1.3 and the identification of χ_{BP^{(G)}⟨m⟩^e} with an odd multiple of χ_{BP⟨h⟩} are unsupported. The authors should either include a proof of the required rational-generation statement, cite a published version of [12], or explicitly mark Theorem 1.3 and the second part of Proposition 6.11 as conditional. Note also that Theorem 1.2 can be proved without [12] by combining Theorem 6.8 with the first part of Proposition 6.11, so the dependency can be localized.","section":"Theorem 6.9 and Proposition 6.11"}],"minor_comments":[{"comment":"The first line begins with the typo \"Kor all C2 ⊂ H ⊂ G\"; this should read \"For all\".","section":"Lemma 5.3"},{"comment":"The statement contains typographical corruption of the generalized Moore spectrum: the expression \"vih n\" and \"S/(2^{i_0},...,v^{i_h}_n)\" should read S/(2^{i_0},...,v_h^{i_h}).","section":"Theorem 6.9"},{"comment":"In the limiting computation, the displayed quotient (1-x^{2^{jm+i-1}})/(1-x^{2^{i-1}}) should have numerator exponent 2^{(j-1)m+i} in order to match the Gaussian binomial formula after taking the limit; clarifying this would remove ambiguity about the intended indexing.","section":"Corollary 3.8, proof"},{"comment":"The proof would benefit from explicitly stating that F/v is a generalized Moore spectrum of chromatic type h+1, since this is the point at which Proposition 6.11 is applied.","section":"Corollary 6.12"}],"recommendation":"major_revision","confidential_remarks":"The scientific content is well suited to the journal and the main ideas are promising. The primary publication risk is the reliance on the unpublished Burklund-Levy theorem [12] for the global divisibility theorem; the authors should either remove this dependency for Theorem 1.2, as is possible, and clearly mark the remaining conditional statements, or integrate a proof of the needed rational-generation statement. The off-by-one error in Corollary 3.8 is straightforwardly fixable and should not affect the validity of the corrected result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper, worth a serious look. The main new thing is Theorem 2.5: for G = C_{2^n}, the fixed points BP^{(G)}⟨m⟩^H are fp spectra of type m|G|/2 at all heights, with a genuinely new proof via Koszul-style filtrations for equivariant quotients. That gives the first batch of workable fp spectra at every height and, as a byproduct, the first prime-2 criterion valid at all heights restricting generalized Moore spectrum exponent valuations (Theorem 1.2). The K-theory relation |H|·[R^H] ≡ [R^e] modulo torsion (Theorem 6.8) is also a solid new structural statement.\n\nThe proof of Theorem 2.5 is an induction using the nilpotence of norms (Proposition 4.15) and the algebraic finiteness of the underlying homotopy quotient (Theorem 3.1). I cannot machine-check the equivariant argument, but the logic is coherent and the paper is careful about citing the transchromatic input from Beaudry–Hill–Shi–Zeng.\n\nNow the soft spots. Corollary 3.8, the explicit dimension formula, has an indexing error: the product as written runs over j=0 to |G|/2−1 of [jm choose m]_2, so the j=0 factor is [0 choose m]_2 = 0 for m>0, which would make the dimension zero. The intended product almost certainly starts at j=1. The l'Hôpital step in the proof is also written incorrectly. This is fixable and does not threaten oddness of the dimension, which is all that is used downstream.\n\nThe bigger soft spot is the transfer from the Euler characteristic computations to the global divisibility statement Theorem 1.3. That step uses Theorem 6.9, which the authors prove from the Burklund–Levy theorem [12], still 'to appear'. If that result turns out to be false or weaker, Theorem 1.3 (the 2^n-divisibility of χ_BP⟨h⟩) does not follow. The authors are transparent about the dependency, but a referee should insist that the paper either state Theorem 1.3 as conditional or get a written version of [12]. Note that Theorem 1.2 itself does not need [12]: evaluating the relation |G|·χ_{R^G}=χ_{R^e} on the given Moore spectrum and using the direct formula in Proposition 6.11(1) gives the valuation bound directly. So the hard dependency is exactly the global rational-generation statement.\n\nAnother minor issue: Proposition 3.5 appears to have a typo in the exponents of the Poincaré series (x^{2i−1} where x^{2^i−1} is meant for the v_i factors). The subsequent algebra is consistent with the corrected reading.\n\nBottom line: the central theorem is significant and, as far as I can see, sound after the mechanical fixes. The paper deserves a serious referee. I would send it out, asking for correction of Corollary 3.8 and a clear statement about the status of Theorem 1.3's dependence on [12].","headline":"The fp-ness theorem for higher real K-theories is real and new; the paper is referee-ready after fixing a dimension-formula indexing error and pinning down the Burklund–Levy dependency in Theorem 1.3.","tokens_in":27151,"tokens_out":13295,"would_cite":true,"duration_ms":118538,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N22","55P42","55P91"],"pacs":[],"model":"deepseek-v4-flash","headline":"The connective higher real K-theory spectra $eo_h$ are fp spectra of chromatic type $h$ at every height, and this forces any existing generalized Moore spectrum $\\mathbb{S}/(2^{i_0},v_1^{i_1},\\ldots,v_h^{i_h})$ to satisfy $\\nu_2(\\prod…","keywords":["higher real K-theory","fp spectra","chromatic height","generalized Moore spectra","algebraic K-theory of spectra","equivariant stable homotopy theory","truncated Brown-Peterson spectra","Koszul filtration"],"falsifier":"Smash $eo_3 = BP^{(C_2)}\\langle 3\\rangle^{C_2}$ with a known finite type-$3$ complex such as $Z/v_2$ and compute its homotopy groups; the paper predicts infinitely many nonzero $2$-torsion groups in the answer, so finding only finitely many nonzero groups would refute the central fp-type claim.","tokens_in":26066,"feed_emoji":"","tokens_out":20754,"duration_ms":191746,"temperature":0.7,"pith_summary":"This paper tries to establish that the connective higher real $K$-theory spectra $eo_h$---fixed points of equivariant truncated Brown--Peterson spectra $BP^{(G)}\\langle m\\rangle$ with $h=m|G|/2$---are fp spectra of chromatic type $h$ for every height $h$. That means they are bounded below, $2$-complete, have finitely presented cohomology over the Steenrod algebra, and their smash products with finite spectra stay finite exactly above chromatic height $h$. Such spectra are workable, $ko$-like replacements for $BP\\langle h\\rangle$ at heights where almost nothing explicit was known. The proof is carried by an equivariant Koszul filtration, and the same filtration feeds an Euler characteristic computation making $\\chi_{BP\\langle h\\rangle}$ divisible by $2^n$ when $h=2^{n-1}m$. The advertised payoff is a new prime-$2$ obstruction: if a generalized Moore spectrum $\\mathbb{S}/(2^{i_0},v_1^{i_1},\\ldots,v_h^{i_h})$ exists, then $\\nu_2(\\prod i_k)>\\nu_2(h)$. This is a partial verification of a conjecture about the algebraic $K$-theory of finite type-$h$ spectra.","feed_headline":"New obstruction to Moore spectra at every chromatic height","feed_subtitle":"Higher real K-theories are finite and 2-complete, forcing a strict 2-adic bound on Moore-spectrum exponents.","key_machinery":"The load-bearing mechanism is a normed Koszul filtration for equivariant quotients. The paper replaces the non-existent cofiber sequence $BP^{(G)}\\langle m-1\\rangle\\to BP^{(G)}\\langle m\\rangle$ with a filtration whose associated graded pieces are indexed by $C_2$-equivariant functions $f\\colon G\\to\\{0,1\\}$, presenting each layer as an induced norm of a quotient by a proper subgroup's generators; this mirrors the Koszul resolution of a residue field over a polynomial ring. The algebraic input that closes the induction is that $\\pi_*^e BP^{(G)}\\langle m\\rangle/(2,v_1,\\ldots,v_h)$ is a finite $\\mathbb{F}_2$-vector space of odd dimension, computed as $\\prod_{j=0}^{|G|/2-1}\\binom{jm}{m}_2$. On the K-theory side, the same filtration yields the torsion relation between fixed-point classes and identifies $\\chi_{BP^{(G)}\\langle m\\rangle^e}$ with that odd Gaussian-binomial factor times $\\chi_{BP\\langle h\\rangle}$; a cited rational-generation theorem then transfers the divisibility to the Moore-spectrum obstruction. The Euler characteristic itself is $\\chi_E(K)=\\sum_i(-1)^i\\log_2|\\pi_i(E\\otimes K)|$, pairing an fp spectrum $E$ with a finite spectrum $K$ of higher type.","core_discovery":"The central claim, Theorem 2.5, is that for every subgroup $H$ of a cyclic $2$-group $G=C_{2^n}$, the fixed-point spectrum $BP^{(G)}\\langle m\\rangle^H$ is an fp spectrum of type $m|G|/2$; in particular $eo_h(H)$ has fp type $h$. Concretely, these spectra are bounded below, $2$-complete, and finitely presented as modules over the Steenrod algebra, and they detect exactly the chromatic height-$h$ layer: smashing with a finite spectrum of type $h+1$ gives finite homotopy, while smashing with a finite type-$h$ spectrum does not. From this the paper derives the K-theory relation $|H|[BP^{(G)}\\langle m\\rangle^H]\\equiv [BP^{(G)}\\langle m\\rangle^e]$ modulo torsion in $K_0(fp_{\\le h})$, which makes the Euler characteristic $\\chi_{BP\\langle h\\rangle}$ divisible by $2^n$ whenever $h=2^{n-1}m$. Theorem 1.2 follows: existence of $\\mathbb{S}/(2^{i_0},v_1^{i_1},\\ldots,v_h^{i_h})$ forces $\\nu_2(\\prod_{k=0}^h i_k)>\\nu_2(h)$. The proofs proceed by induction on height, using the transchromatic height-shifting layers in the equivariant slice filtration.","pith_inferences":["Beyond the paper's cyclic-group setting, the filtration method is built from the norm functor for cyclic $2$-groups; adapting it to a non-cyclic maximal subgroup such as $Q_8$ would fill the gap at heights $h\\equiv 2\\bmod 4$, where the paper leaves the divisibility question open.","The odd Gaussian-binomial factor separating $\\chi_{BP^{(G)}\\langle m\\rangle^e}$ from $\\chi_{BP\\langle h\\rangle}$ suggests a geometric reading: if these coefficients count degrees of forgetful covers on a moduli stack of formal groups with group actions, the slice layers would acquire a modular interpretation predicting the structure of higher fixed-point spectra.","A natural next step is to compute the minimal $v_h$-periodicity of $eo_h$ from the equivariant slice spectral sequence; if the minimal exponent exceeds the valuation bound, Theorem 1.2 could be upgraded from a divisibility statement to an exponent-by-exponent nonexistence criterion."],"forward_implications":["For every height $h=2^{n-1}m$ with $h\\not\\equiv 2\\bmod 4$, the spectrum $eo_h$ is bounded below, $2$-complete, and finitely presented over the Steenrod algebra, so Adams-spectral-sequence methods previously limited to $ko$ and $tmf$ become available at arbitrary height.","The Euler characteristic $\\chi_{BP\\langle h\\rangle}$ is divisible by $2^n$, confirming the divisibility half of the conjecture that its image is generated by the order of a maximal finite $2$-subgroup of the height-$h$ stabilizer group.","No generalized Moore spectrum $\\mathbb{S}/(2^{i_0},v_1^{i_1},\\ldots,v_h^{i_h})$ can exist unless $\\nu_2(i_0\\cdots i_h)>\\nu_2(h)$, giving the first prime-$2$ nonexistence criterion valid at all heights.","For these spectra, finite-height chromatic localization $L_f^n$ agrees with $L_n$, so the telescope-conjecture condition holds in this class, and the completion map $BP^{(G)}\\langle m\\rangle^H\\to BP^{(G)}\\langle m\\rangle^{hH}$ has bounded-above fiber.","The connective cover of the Borel completion $\\tau_{\\ge0}(BP^{(G)}\\langle m\\rangle^{hH})$ is again an fp spectrum of type $h$."],"supporting_citations":[{"why":"Constructs $BP^{(G)}\\langle m\\rangle$ through twisted monoid rings and supplies the equivariant norm and distributive laws used throughout the Koszul filtration.","marker":"[23]"},{"why":"Gives the recursive formulas for the images of the $v_i$ in $\\pi_*^e BP^{(G)}$ and the $K(h)$-local equivalence between $BP^{(G)}\\langle m\\rangle$ and the height-$h$ Lubin–Tate spectrum $E_h$.","marker":"[6]"},{"why":"Defines fp spectra and the notion of fp type that Theorem 2.5 targets.","marker":"[35]"},{"why":"Introduces the Euler characteristic on fp spectra and the conjecture about the image of $\\chi_{BP\\langle h\\rangle}$ that Theorem 1.3 partially verifies.","marker":"[33]"},{"why":"The to-appear theorem that $\\chi_{BP\\langle h\\rangle}$ is a rational isomorphism and that any existing generalized Moore spectrum rationally generates the source; this is the transfer from the fp theorem to Theorems 1.2 and 1.3.","marker":"[12]"},{"why":"The periodicity theorem producing $v_h$-self maps on generalized Moore spectra, the objects whose exponents Theorem 1.2 constrains.","marker":"[17]"},{"why":"Shows fp spectra of type $h$ have large-degree isomorphisms into their finite-height chromatic localizations, used in Section 7 for the completion statements.","marker":"[21]"},{"why":"Identifies the thick tensor-ideal generated by normed inflations in the equivariant Balmer spectrum, used in Lemma 5.3 to propagate nilpotence.","marker":"[5]"}],"fun_headline_variants":["Higher real K-theories obstruct Moore spectra at every height","2-adic valuation bound rules out Moore spectra at each height","Real K-theories force strict 2-adic exponents for Moore spectra","Chromatic heights impose 2-adic obstruction on Moore spectra","fp spectra of type h block generalized Moore spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument from fp spectra to nonexistence rests on a cited, still-unpublished theorem saying that a certain integer-valued counting invariant rationally detects, up to odd multiples, all finite spectra above height $h$; if that theorem is false or secretly assumes the Moore spectrum already exists, the divisibility and nonexistence results collapse.","fun_headline_variants_meta":{"raw":{"variants":["Higher real K-theories obstruct Moore spectra at every height","2-adic valuation bound rules out Moore spectra at each height","Real K-theories force strict 2-adic exponents for Moore spectra","Chromatic heights impose 2-adic obstruction on Moore spectra","fp spectra of type h block generalized Moore spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1686,"prompt_tokens":976,"completion_tokens":710,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":626}},"tokens_in":592,"tokens_out":710,"duration_ms":8046,"temperature":1.0,"reasoning_tokens":626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:49:26.017345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Smash $eo_3 = BP^{(C_2)}\\langle 3\\rangle^{C_2}$ with a known finite type-$3$ complex such as $Z/v_2$ and compute its homotopy groups; the paper predicts infinitely many nonzero $2$-torsion groups in the answer, so finding only finitely many nonzero groups would refute the central fp-type claim.","supporting_citations":[{"cited_title":"On the non-existence of elements of Kervaire invariant one","cited_arxiv_id":null,"evidence_quote":"Constructs $BP^{(G)}\\langle m\\rangle$ through twisted monoid rings and supplies the equivariant norm and distributive laws used throughout the Koszul filtration."},{"cited_title":"Models of Lubin-Tate spectra via real bordism theory","cited_arxiv_id":null,"evidence_quote":"Gives the recursive formulas for the images of the $v_i$ in $\\pi_*^e BP^{(G)}$ and the $K(h)$-local equivalence between $BP^{(G)}\\langle m\\rangle$ and the height-$h$ Lubin–Tate spectrum $E_h$."},{"cited_title":"Brown-Comenetz duality and the Adams spectral sequence","cited_arxiv_id":null,"evidence_quote":"Defines fp spectra and the notion of fp type that Theorem 2.5 targets."},{"cited_title":"The algebraic K-theory of the K(1)-local sphere via TC","cited_arxiv_id":"2209.05314","evidence_quote":"Introduces the Euler characteristic on fp spectra and the conjecture about the image of $\\chi_{BP\\langle h\\rangle}$ that Theorem 1.3 partially verifies."},{"cited_title":"Burklund and I","cited_arxiv_id":null,"evidence_quote":"The to-appear theorem that $\\chi_{BP\\langle h\\rangle}$ is a rational isomorphism and that any existing generalized Moore spectrum rationally generates the source; this is the transfer from the fp theorem to Theorems 1.2 and 1.3."},{"cited_title":"Nilpotence and Stable Homotopy Theory I","cited_arxiv_id":null,"evidence_quote":"The periodicity theorem producing $v_h$-self maps on generalized Moore spectra, the objects whose exponents Theorem 1.2 constrains."},{"cited_title":"RedshiftandmultiplicationfortruncatedBrown–Petersonspectra","cited_arxiv_id":null,"evidence_quote":"Shows fp spectra of type $h$ have large-degree isomorphisms into their finite-height chromatic localizations, used in Section 7 for the completion statements."}],"review_version":1}