{"id":"11af83cc-354a-487e-a693-7d31b53498b0","arxiv_id":"2608.08041","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The exact small-time null-control cost for the 1D heat equation is exp((kappa_* L^2 + o(1))/T) with kappa_* = Gamma(1/4)^4 / (8 pi^3) approximately 0.6966.","lead":"This paper proves that the smallest control energy needed to drive a one-dimensional heat equation to zero in short time T grows like exp(c L squared / T), with the exact constant c now pinned down. The constant closes a gap in the small-time control cost that had been open for about forty years.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lower bound rests on Lemma 2.2's sign-sensitive phase comparison; I rechecked (2.34) and found it consistent, but the real-zero and inner-factor bookkeeping in (2.32) remains the one unverified step that could shift the exponent.","rationale":"I read the paper in good faith and focused on the support of Theorem 1.1. The main constant computations check out: the beta-function optimization in Proposition 3.1 gives B(1/4,1/2)^2/4π² = Γ(1/4)^4/8π³, and the hypergeometric summation in Theorem 3.1 is internally consistent, including the Whipple step and the Dixon evaluation. The reduction from the control problem to the multiplier problem in Proposition 2.3 and the rescaling in Proposition 2.4 are clean. The only place where a hidden sign or factor error could change the exponential rate is Lemma 2.2's phase comparison. I gave special attention to (2.34), since a naive computation with p=1+iζ/μ can suggest a sign error; working with the outer representative 1−iζ/μ (which has the same boundary modulus) confirms the sign in (2.34), and the cancellation in the proof of Lemma 2.2 is formally valid. I could not find a concrete error in the real-zero bookkeeping, but that bookkeeping is exactly the type of step that escapes a quick reading. Because the reader already identifies this lemma as the weakest assumption and recommends an independent check, my assessment does not move the verdict: the result is mathematically plausible, the proof is detailed, and the residual risk is localized and testable. Hence I recommend no change to the ACCEPT verdict with moderate confidence.","tokens_in":21291,"tokens_out":31170,"duration_ms":319393,"concrete_test":"Check the phase identity on a finite model: replace P± by finite products P_N^± over n=2..N, choose T_N and an explicit entire m_N with one prescribed real zero of multiplicity 1 so that Φ_N=P_N^− m_N is a trigonometric polynomial supported in [0,T_N], and compute both sides of (2.32) as distributions by pairing with several C_c^∞ test functions, using (2.31) and (2.34) for the finite Blaschke products. Then compute Ω_N=−log|m_N|, form (HΩ_N)' numerically with a high-accuracy Hilbert-transform routine, and check (2.30) against the same test functions, including one whose support contains the real zero. A mismatch beyond discretization error would locate the sign/factor error; exact agreement would support Lemma 2.2 in a non-asymptotic setting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is proved by matching a lower bound from Proposition 2.4 with the Dardé–Ervedoza upper bound. The lower bound is exactly κ*L^2 only if Lemma 2.2 yields (HΩ)' ≥ −T/2 with no sign or factor error. The delicate step is the distributional identity (2.32), obtained by comparing boundary phases of G=P+m in C− and Ψ=e^{iTζ}G in C+; it must correctly handle the real-zero jump η0 = 2πΣ m_x δ_x and the signs of the modified Hilbert transform in (2.34). I rechecked (2.34) against the outer representative with the same modulus, 1−iζ/μ: its boundary phase is −arctan(x/μ), so 2(H log|1+iζ/μ|)' = −2μ/(μ²+x²) is consistent with the stated sign. The algebra from (2.32)–(2.35) then closes to (HΩ)' + T/2 = ½(η− + eη+ + η0) ≥ 0. What is not settled by my reading is the a.e./distributional treatment at real zeros of G: a missing factor of 2 in η0 or an unaccounted inner-factor phase contribution would change (2.30) and therefore the optimized constant. This is the same step the reader flagged; I found no concrete counter-step, but the comparison is intricate and is the only part of the lower-bound proof that is not independently verifiable from standard references at a glance.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the optimal L^2 null-control cost C_H(T,L) for the one-dimensional heat equation on (0,L), with a Dirichlet boundary control at one endpoint, homogeneous Dirichlet condition at the other, and initial data in H^{-1}(0,L). The main result, Theorem 1.1, asserts that T log C_H(T,L) tends to κ_* L^2 as T tends to 0^+, with κ_* = Γ(1/4)^4/(8π^3) ≈ 0.6966, and that this matches the previously known Dardé–Ervedoza upper-bound constant. The proof combines the moment method with a Paley–Wiener characterization of compact support through two Hardy-space conditions, then compares inner–outer factorizations to derive the one-sided Hilbert-transform inequality (HΩ)' ≥ -T/2 of Lemma 2.2. After a small-time rescaling this becomes a fractional obstacle-type inequality, which is tested against an explicit function ρ_b to obtain the sharp lower bound. The upper bound is taken from the independent result of Dardé and Ervedoza, and Section 3.2 evaluates the series defining their constant in closed form via hypergeometric transformations.","tokens_in":21530,"tokens_out":18911,"duration_ms":204379,"significance":"If Theorem 1.1 stands, it closes a long-standing gap between the earlier lower bound 1/2 and the known upper bound K_0, giving the exact exponential rate of the control cost for a fundamental parabolic model. The lower bound is self-contained and uses no fitted parameters: the constant emerges from an explicit optimization involving the half-Laplacian of the square-root profile q_*. The paper also gives a closed-form evaluation of the Dardé–Ervedoza series, so the constant is identified in a transparent way. I checked the main auxiliary computations, including the sign bookkeeping in Lemma 2.2: for a real zero of multiplicity m, the difference between the upper and lower boundary arguments is -2πm δ, consistent with the factor 2π in η_0 and with the final identity (HΩ)' + T/2 = 1/2(η_- + eη_+ + η_0). The hypergeometric summation in Theorem 3.1 also checks out. This is an important and genuinely quantitative contribution to the control theory of parabolic equations.","major_comments":[],"minor_comments":[{"comment":"The phrase \"obtained in by Lissy\" should read \"obtained by Lissy\".","section":"Abstract"},{"comment":"In the displayed definition of A_T, \"mentire\" should be \"entire\".","section":"§2.2"},{"comment":"The jump relation d arg_+ Ψ_G = T dx + d arg_- G - η_0 is correct, but it would help the reader if the proof stated explicitly that the lower-half-plane boundary argument uses the branch tending to -π on the negative real axis and the upper-half-plane branch uses the branch tending to +π, so that a real zero of multiplicity m contributes exactly 2πm δ to the difference.","section":"§2.3, Eq. (2.32)"},{"comment":"Since only derivatives of the Hilbert transform are used, the modified transform H and the usual transform H differ by a constant and are interchangeable in (HΩ)'; a one-line reminder would prevent sign confusion.","section":"§2.1, Definition 2.2"}],"recommendation":"accept","confidential_remarks":"This is a strong paper that appears to solve a known open problem. The most delicate part is the distributional phase comparison in Lemma 2.2; I read it carefully and found no concrete error, but it would be prudent for the editor to have an independent expert check that lemma before publication. The lower bound is self-contained, and the upper bound rests on an established reference, so there is no circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know about this paper: it proves the long-open exact exponential rate for the null-control cost of the 1D heat equation, C_H(T,L) = exp((κ* L^2 + o(1))/T) with κ* = Γ(1/4)^4/(8π^3) ≈ 0.6966. This closes the gap between Lissy's lower bound 1/2 and Dardé–Ervedoza's upper bound K0.\n\nWhat is genuinely new: the lower bound that matches the upper bound, obtained by a converse of the Beurling–Malliavin phase method. Compact support is converted into the one-sided Hilbert-transform constraint (HΩ)' ≥ −T/2 instead of using phase to construct a multiplier. The paper also gives a closed form for the Dardé–Ervedoza series, S = 2Γ(1/4)/Γ(3/4) − 2πΓ(3/4)/Γ(1/4), which is elegant and correct. I checked the hypergeometric summation and the optimization of J*(b); both are internally consistent. The writing is clear and the organization is sensible.\n\nThe soft spot is Lemma 2.2, specifically the distributional identity (2.32) comparing boundary phases of G and Ψ. This requires careful bookkeeping of real-zero jumps (η0), inner-factor phase derivatives, and the signs in the modified Hilbert transform. I rechecked (2.34) and the algebra leading to (2.35); it is consistent as far as I can see. But the treatment at real zeros is delicate, and this is the one step I could not fully verify at a glance. An independent expert should examine it. If there is an error, it would likely change the constant; but I found no concrete flaw, and the rest of the lower-bound argument—the rescaling, the half-Laplacian obstacle inequality, and the test function—holds up.\n\nThis paper is for control theorists and complex analysts. It deserves a serious referee; the central claim is likely correct, and the method is reusable. Send it to review, with a referee specifically asked to check Lemma 2.2.","headline":"The exact small-time cost constant for the 1D heat equation is settled; the proof is convincing except for one intricate phase-comparison lemma that deserves independent checking.","tokens_in":22133,"tokens_out":3769,"would_cite":true,"duration_ms":33300,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B05","35K05","30H10","42A38","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The optimal small-time cost of one-dimensional heat boundary controls is $\\exp((\\kappa_* L^2+o(1))/T)$ with $\\kappa_*\\approx 0.6966$.","keywords":["heat equation","null controllability","cost of fast controls","moment method","entire functions","Hardy spaces","Hilbert transform","half-Laplacian"],"falsifier":"One concrete check is to test identity (2.32) on a nontrivial entire Hardy function with real zeros, such as $G(\\zeta)=\\sin\\zeta/\\zeta$: both sides must give the same distribution, and the measures $\\eta_-$, $\\eta_+$, and $\\eta_0$ must all be nonnegative as asserted. Alternatively, compute $T\\log C_{\\mathrm H}(T,L)$ numerically for $L=1$ from a truncated moment problem at several small $T$; if the values do not approach $\\kappa_*\\approx 0.6966$ to the expected precision, the claimed asymptotic is wrong.","tokens_in":21013,"feed_emoji":"🔥","tokens_out":9446,"duration_ms":99764,"temperature":0.7,"pith_summary":"The paper establishes the exact small-time asymptotic of the minimal $L^2$ norm of a Dirichlet boundary control that drives the one-dimensional heat equation on $(0,L)$ to zero. The main theorem states that the optimal control cost satisfies $C_{\\mathrm H}(T,L)=\\exp((\\kappa_* L^2+o(1))/T)$ as $T\\to 0^+$, where $\\kappa_* = \\Gamma(1/4)^4/(8\\pi^3) \\approx 0.6966$. This closes a long-standing gap: the best previous lower bound gave the coefficient $1/2$, while the best previous upper bound was a constant $K_0$ given by a convergent series; both are now replaced by the single number $\\kappa_*$. The reason this matters is that the constant quantifies exactly how much harder it is to observe sign-changing solutions than the Gaussian heat-kernel estimate suggests. A sympathetic reader should see this as a first sharp answer for a basic control-theoretic quantity in a model geometry.","feed_headline":"Exact heat-control cost rate: 0.6966 L^2/T","feed_subtitle":"Closes the gap between the 1/2 lower bound and the series upper bound for 1D fast controls.","key_machinery":"The load-bearing mechanism is a phase comparison between inner-outer factorizations in the two half-planes, which converts compact Fourier support into a one-sided inequality on the modified Hilbert transform. The central objects are the canonical product $P_-(\\zeta)=\\prod_{n\\ge 2}(1-i\\zeta/\\mu_n)$, carrying the shifted heat eigenvalues $\\mu_n=\\pi^2(n^2-1)/L^2$, and its reflected twin $P_+$. For an admissible multiplier $m$, Lemma 2.2 produces $(H\\Omega)'\\ge -T/2$ with $\\Omega=-\\log|m|$. After the rescaling $X=T^2\\zeta$, this becomes $|D|u\\ge -1/2$ together with $u\\ge q_T$, the obstacle inequality; $q_T$ converges to $q_*(X)=L\\sqrt{|X|/2}$, a square-root profile. The explicit test function $\\rho_b$ is the Green function of the restricted half-Laplacian on $(-b,b)$ with zero exterior condition, and its half-Laplacian has exactly the sign needed to combine the two constraints and optimize over $b$.","core_discovery":"The core discovery is that the cost of fast boundary controls for the one-dimensional heat equation is governed by a single explicit constant: $C_{\\mathrm H}(T,L)=\\exp((\\kappa_* L^2+o(1))/T)$, with $\\kappa_* = \\Gamma(1/4)^4/(8\\pi^3) \\approx 0.6966$. The paper proves the lower bound by turning the moment problem into an entire interpolation problem, removing the spectral zeros via the canonical product $P_-$, and deriving a necessary one-sided constraint $(H\\Omega)'\\ge -T/2$ on the logarithmic modulus of the multiplier from the fact that the interpolating function has Fourier support in $[0,T]$. After the natural rescaling $X=T^2\\zeta$, this constraint becomes the half-Laplacian inequality $|D|u\\ge -1/2$, paired with the pointwise obstacle $u\\ge q_T(X)$; the limit obstacle is $q_*(X)=L\\sqrt{|X|/2}$. Testing these inequalities against the explicit function $\\rho_b(x)=\\frac1\\pi\\log\\frac{b+\\sqrt{b^2-x^2}}{|x|}$ for $|x|<b$ and optimizing over $b$ yields exactly $\\kappa_* L^2$. The matching upper bound comes from evaluating the previously known upper-bound constant, expressed as a convergent series, in closed form and obtaining the same $\\kappa_*$.","pith_inferences":["The same Hardy-space phase comparison should apply to other one-dimensional equations with quadratic spectra, such as the Schrödinger equation, by replacing the canonical product with the relevant spectral product; the sharp constant would come from the analogous obstacle profile.","Because the sharp constant is extracted from the Green function of the restricted half-Laplacian, the small-time cost problem is probably equivalent to a genuinely variational fractional obstacle problem; the paper does not pursue that duality.","The closed-form value of the series suggests that an explicit optimal multiplier, not just the constant, may exist; constructing it would give a constructive sharp control for the heat equation.","A numerical study of the truncated moment problem at small $T$ would provide an independent check of the rate and could also reveal the size of the $o(1)$ remainder in the exponential."],"forward_implications":["The true small-time rate is $\\kappa_* L^2/T$, so the previous lower bound $1/2$ was not sharp: oscillatory solutions are exponentially harder to observe than the Gaussian heat-kernel barrier alone predicts.","The upper-bound estimate from [7] is optimal: its constant $K_0$, originally a convergent series, evaluates to exactly $\\kappa_*$, so no refined upper construction can improve the exponential rate.","The duality estimate (1.7) now gives a precise small-time stability constant for reconstructing the final state from boundary flux observations, namely $\\exp((\\kappa_* L^2+o(1))/T)$.","The limiting variational data—the obstacle $q_*(X)=L\\sqrt{|X|/2}$ and the constraint $|D|u\\ge -1/2$—provide a canonical profile for the small-time control problem in this geometry.","For control problems that reduce to this one-dimensional spectral geometry, the sharp rate $\\kappa_* L^2/T$ is the benchmark any quantitative estimate has to match."],"supporting_citations":[{"why":"Supplies the upper-bound constant $K_0$ as a convergent series and the small-time observability estimate that yields the matching upper bound.","marker":"[7]"},{"why":"Gives the previous lower bound $\\beta_-\\ge 1/2$, which Theorem 1.1 improves to the sharp constant.","marker":"[24]"},{"why":"Provides the moment method that reduces null controllability of the heat equation to biorthogonal and moment conditions.","marker":"[9, 10]"},{"why":"Provides the Paley-Wiener theorem and $H^p$ boundary-value theory used to express finite support as two Hardy conditions.","marker":"[8]"},{"why":"Supplies inner-outer factorization, Blaschke products, and phase formulas used in the phase comparison of Lemma 2.2.","marker":"[27]"},{"why":"Supplies the classical multiplier phase argument whose direction is reversed in Lemma 2.2 to turn compact support into a one-sided Hilbert-transform constraint.","marker":"[28]"},{"why":"Provides the hypergeometric summation identities used to evaluate the upper-bound series in closed form.","marker":"[31]"}],"fun_headline_variants":["Heat control cost rate proven: 0.6966 L^2/T","Cost of fast heat control: exp(0.6966 L^2/T)","Gap closed: heat control cost constant is 0.6966","Fast heat control cost scales as exp(0.6966 L^2/T)","Exact exponent for fast heat boundary control"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 2.2's claim that comparing the two Hardy-space factorizations yields the one-sided inequality $(H\\Omega)'\\ge -T/2$ with the sign and jump terms exactly as stated; a single reversed sign or a missed jump at a real zero would destroy the lower bound.","fun_headline_variants_meta":{"raw":{"variants":["Heat control cost rate proven: 0.6966 L^2/T","Cost of fast heat control: exp(0.6966 L^2/T)","Gap closed: heat control cost constant is 0.6966","Fast heat control cost scales as exp(0.6966 L^2/T)","Exact exponent for fast heat boundary control"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1897,"prompt_tokens":1046,"completion_tokens":851,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":662,"completion_tokens_details":{"reasoning_tokens":756}},"tokens_in":662,"tokens_out":851,"duration_ms":8837,"temperature":1.0,"reasoning_tokens":756,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:32:25.521866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check is to test identity (2.32) on a nontrivial entire Hardy function with real zeros, such as $G(\\zeta)=\\sin\\zeta/\\zeta$: both sides must give the same distribution, and the measures $\\eta_-$, $\\eta_+$, and $\\eta_0$ must all be nonnegative as asserted. Alternatively, compute $T\\log C_{\\mathrm H}(T,L)$ numerically for $L=1$ from a truncated moment problem at several small $T$; if the values do not approach $\\kappa_*\\approx 0.6966$ to the expected precision, the claimed asymptotic is wrong.","supporting_citations":[{"cited_title":"Dard\\'e and S","cited_arxiv_id":null,"evidence_quote":"Supplies the upper-bound constant $K_0$ as a convergent series and the small-time observability estimate that yields the matching upper bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the previous lower bound $\\beta_-\\ge 1/2$, which Theorem 1.1 improves to the sharp constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Paley-Wiener theorem and $H^p$ boundary-value theory used to express finite support as two Hardy conditions."},{"cited_title":"Mashreghi, Representation Theorems in Hardy Spaces, London Mathematical Society Student Texts, vol","cited_arxiv_id":null,"evidence_quote":"Supplies inner-outer factorization, Blaschke products, and phase formulas used in the phase comparison of Lemma 2.2."},{"cited_title":"Mashreghi, F","cited_arxiv_id":null,"evidence_quote":"Supplies the classical multiplier phase argument whose direction is reversed in Lemma 2.2 to turn compact support into a one-sided Hilbert-transform constraint."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the hypergeometric summation identities used to evaluate the upper-bound series in closed form."}],"review_version":1}