Awards recognise answers, never the score.

A peer-reviewed paper that, for at least one pair (n,R) on the list below, strictly improves a published bound on K(n,R), verifiable by direct computation or by a complete proof given in the paper. THE OBJECT. A binary covering code of length n and radius R is a set of binary words of length n such that EVERY one of the 2^n words is within Hamming distance R of at least one codeword. It is written one codeword per line as a string of n characters, each 0 or 1. Checking it is a single sweep over all 2^n words, confirming each is covered. K(n,R) is the smallest possible size, so FEWER CODEWORDS IS BETTER. THE PARAMETERS, each with the published interval for K(n,R) as recorded in Keri's tables of bounds on covering codes. The left number is the best published lower bound, the right the best published upper bound: n=10 R=1 107 .. 120 n=11 R=1 180 .. 192 n=12 R=1 342 .. 380 n=13 R=1 598 .. 704 n=10 R=2 24 .. 30 n=11 R=2 37 .. 44 n=12 R=2 62 .. 78 n=13 R=2 97 .. 128 n=14 R=2 159 .. 248 n=13 R=3 28 .. 42 n=14 R=3 44 .. 64 n=15 R=3 70 .. 112 n=16 R=3 115 .. 192 These are the figures as at the date this bounty was posted. If a bound moves while the bounty is open, the live figure is the one to beat, and the submission must say which figure it is beating and cite it. K(9,1)=62 is settled exactly and is not listed; neither is any other cell whose value is known. WHY THE SPHERE-COVERING BOUND IS NOT THE TARGET. With V = sum over i from 0 to R of C(n,i), every word lies in some ball, so K(n,R) >= ceil(2^n / V). At n=9, R=1 that gives 52 against a true value of 62; at n=13, R=2 it gives 90 against a published lower bound of 97. The bound is correct and useless as a goal. Do not submit a proof that it is unattainable: that is published for every cell here. SCORING. - FULL: a valid covering code strictly smaller than the published upper bound for a listed pair; OR a proof raising the published lower bound for a listed pair; OR a determination of K(n,R) exactly, closing the cell. - PARTIAL: an exact minimum over a named restricted family - the smallest covering code that is linear, or that admits a stated automorphism group - proved exhaustively. Closing a route is a result. - PARTIAL: a reproducible reconstruction of a published record from an explicit recipe, where the published record was reported as a size but not as a rebuildable object. - SCORES NOTHING: a code whose size merely lies inside the published interval without beating its upper end. EVIDENCE REQUIRED. A code must appear in full as a text appendix in the format above, or be reproducible from an explicit finite recipe stated completely in the paper; for a linear code a generator matrix suffices and is preferred, since it is short and a reader can expand it. A size claimed without an exhibited code or an explicit recipe does not qualify. A lower-bound claim must be a complete proof, not a failed search. [certificate: computation]

A peer-reviewed paper exhibiting, for at least one triple (v,k,t) on the list below, an explicit covering design that IMPROVES ON THE BEST PUBLISHED BLOCK COUNT for that cell, verifiable by direct computation from the paper alone. THE OBJECT. A covering design is a family of k-element subsets (blocks) of {0,...,v-1} such that EVERY t-element subset is contained in at least one block. It is written one block per line, as ascending 0-based positions separated by spaces. Checking it is a single sweep: enumerate the t-subsets and confirm each is covered. C(v,k,t) is the smallest possible number of blocks, so FEWER BLOCKS IS BETTER - this is the covering problem, the dual of packing. THE PARAMETERS, each with its Schonheim lower bound L computed from the formula below, not cited from any table: v=16 k=5 t=3 L=61 v=18 k=5 t=3 L=94 v=19 k=5 t=3 L=103 v=20 k=5 t=3 L=116 v=12 k=5 t=4 L=113 v=13 k=5 t=4 L=149 v=14 k=5 t=4 L=219 THE LOWER BOUND, WRITTEN OUT. Start with the value 1 and apply, for i = t-1 down to i = 0 IN THAT ORDER, the step value := ceil( (v-i)/(k-i) * value ). The ceilings nest INWARD; evaluating them in the other order gives a different and incorrect number. Any claimed design with b < L is necessarily invalid and should be reported as a defect in this bounty rather than as a result. THE STANDING RECORD. The cells listed here are those not known to be closed when this bounty was posted. THE SUBMISSION MUST STATE THE BEST PUBLISHED BLOCK COUNT FOR THE CELL IT ATTACKS AND CITE ITS SOURCE. The La Jolla Covering Repository is the standard reference. A block count that merely matches or exceeds a published one is a replication, not a result, however well verified it is. SCORING. - FULL: a valid covering with strictly fewer blocks than the best published count for a listed cell. A covering attaining L is the strongest form of this, since L is a lower bound, and it closes the cell. A complete proof that a listed cell's bound is unattainable, i.e. C(v,k,t) >= L+1, also scores FULL, provided that proof is not already in the literature. - PARTIAL: an exact minimum over a named restricted family, for example the fewest blocks in any covering admitting a stated permutation group as automorphisms, proved exhaustively. Closing a route is a result, not a failure. - PARTIAL: a covering that does not beat the published record but is reproducible from an explicit recipe where the published one is not, so that a reader can rebuild it rather than take it on trust. - SCORES NOTHING: rebuilding a design whose size is already published, including any classical Steiner system. Verification is not novelty. EVIDENCE REQUIRED. The design must appear in full as a text appendix in the format above, or be reproducible from an explicit finite recipe the paper states completely (a permutation group given by generators, plus base blocks). A block count claimed without an exhibited design or an explicit recipe does not qualify. [certificate: computation]

A peer-reviewed paper that, for at least one pair (n,d) on the list below, strictly improves a published bound on A(n,d), verifiable by direct computation or by a complete proof given in the paper. THE OBJECT. A binary code of length n and minimum distance d is a set of distinct binary words of length n, any two of which differ in at least d positions. The words may have any weight. It is written one codeword per line as a string of n characters, each 0 or 1. Checking it is a pairwise scan. A(n,d) is the largest possible size, so MORE CODEWORDS IS BETTER. THE PARAMETERS, each with the published interval for A(n,d) as recorded in Brouwer's tables of bounds for binary codes. The left number is the best published construction, the right the best published upper bound: n=25 d= 8 4096 .. 5421 n=26 d= 8 4104 .. 9275 n=27 d= 8 8192 .. 17099 n=28 d= 8 16384 .. 32151 n=22 d=10 64 .. 84 n=23 d=10 80 .. 150 n=24 d=10 136 .. 268 n=25 d=10 192 .. 466 n=26 d=10 384 .. 836 n=27 d=10 512 .. 1585 n=28 d=10 1024 .. 2817 These are the figures as at the date this bounty was posted. If a bound moves while the bounty is open, the live figure is the one to beat, and the submission must say which figure it is beating and cite it. No cell whose value is known exactly is listed. SCORING. - FULL: a valid code strictly larger than the published construction for a listed pair; OR a proof lowering the published upper bound for a listed pair; OR a determination of A(n,d) exactly, closing the cell. - PARTIAL: an exact maximum over a named restricted family - the largest LINEAR code at these parameters, or the largest admitting a stated automorphism group - proved exhaustively. Closing a route is a result. - PARTIAL: a reproducible reconstruction of a published record from an explicit recipe, where the published record was reported as a size but not as a rebuildable object. - SCORES NOTHING: a code whose size merely lies inside the published interval without beating its lower end. EVIDENCE REQUIRED. The code must appear in full as a text appendix, or be reproducible from an explicit finite recipe stated completely in the paper; at the sizes on this list a recipe is expected, and for a linear code a generator matrix is short and is preferred. A size claimed without an exhibited code or an explicit recipe does not qualify. An upper-bound claim must be a complete proof, not a failed search. [certificate: computation]

A peer-reviewed paper doing at least one of the following for the 1/3-2/3 conjecture. THE STATEMENT. For a finite poset P and elements x,y, let p(x<y) be the fraction of linear extensions of P placing x before y. Let delta(P) be the maximum over incomparable pairs {x,y} of min(p(x<y), p(y<x)). The conjecture is that delta(P) >= 1/3 for every finite poset that is not a total order. THE FRONTIER AS OF POSTING, each item checked against the cited source before this bounty was created: - The best proved general lower bound is delta(P) >= 1/2 - sqrt(5)/10, approximately 0.2764 (Brightwell, Felsner and Trotter, 1995). No proof of the full 1/3 is known. - The conjecture holds for every poset on at most 14 elements. This follows from Gupta 2026 (arXiv:2607.23926), which verifies the Gold Partition Conjecture exhaustively through fourteen elements; the Gold Partition Conjecture implies 1/3-2/3. - It is proved for: posets of width two; posets of height two; semiorders; series-parallel posets; posets with N-free Hasse diagrams; polytrees; and posets in which every element is incomparable to at most six others (the 6-thin posets, Peczarski 2008). SCORING. - FULL: (a) an explicit counterexample - a finite poset with delta(P) < 1/3; or (b) a proof of the conjecture in general; or (c) a proof of a general lower bound strictly greater than 1/2 - sqrt(5)/10. - PARTIAL: exhaustive verification for all posets on 15 or more elements, with the search method and its completeness argument given; a proof for a natural infinite class not implied by those listed above; or an improved constant for a stated restricted class. EVIDENCE REQUIRED FOR A COUNTEREXAMPLE. The poset must be given explicitly, as a list of covering relations on at most 30 labelled elements, so that a reader can compute the exact number of linear extensions by dynamic programming over down-sets. The paper must report: the total number of linear extensions as an exact integer; the pair {x,y} attaining delta(P) and both counts p(x<y) and p(y<x) as exact rationals; and delta(P) as an exact rational. Floating-point values, sampled estimates, and Monte Carlo counts do not qualify on their own - a counterexample to a conjecture about an exact fraction has to be exact. EVIDENCE REQUIRED FOR AN EXHAUSTIVE VERIFICATION. The paper must state how posets were enumerated up to isomorphism, how many were examined at each size, and why the enumeration is complete. A search that reports no counterexample without establishing its own completeness is not a verification and does not score. [certificate: computation]

A peer-reviewed paper exhibiting, for at least one cell (n,d,w) on the list below, an explicit constant-weight binary code together with its size s, verifiable by direct computation from the paper alone. THE OBJECT. A code is a set of w-element subsets of {0,...,n-1}, written one codeword per line as ascending 0-based positions separated by spaces. It is VALID for (n,d,w) if and only if: every codeword has exactly w elements; no codeword is repeated; and every pair of distinct codewords A,B satisfies |A intersect B| <= w - d/2, which is equivalent to Hamming distance at least d. Checking validity is a pairwise scan and needs no theory. THE CELLS, each with its Schonheim upper bound U. Every U here is computed from the formula below, not cited from any table: n=22 d= 8 w=5 lambda=1 U=22 n=26 d= 8 w=5 lambda=1 U=31 n=27 d= 8 w=5 lambda=1 U=32 n=29 d= 8 w=5 lambda=1 U=40 n=30 d= 8 w=5 lambda=1 U=42 n=24 d= 8 w=6 lambda=2 U=92 n=25 d= 8 w=6 lambda=2 U=100 n=26 d=10 w=6 lambda=1 U=21 (lambda = w - d/2 is the maximum allowed pairwise intersection.) THE UPPER BOUND, WRITTEN OUT. Set t = w - d/2 + 1; a valid code has at most one codeword through any t-subset, so it is a packing and the Schonheim bound applies. Evaluate U by starting with the value 1 and applying, for i = t-1 down to i = 0 in that order, the step value := floor( (n-i)/(w-i) * value ). The floors nest INWARD; evaluating them in the other order gives a different and incorrect number. Any claimed code with s > U for its cell is necessarily invalid, and should be reported as a defect in this bounty rather than as a result. THE STANDING RECORD. A submission must state the best published size for the cell it attacks and cite its source; Brouwer's tables of bounds for binary constant weight codes are the standard reference. A size that matches a published one is a replication, however well verified. SCORING. - FULL: a valid code strictly larger than the best published size for a listed cell. A code attaining U is the strongest form of this and closes the cell. A complete proof that A(n,d,w) <= U-1, where that is not already published, also scores FULL. - PARTIAL: an exact maximum over a named restricted family - for example, the largest code admitting a stated permutation group as automorphisms, proved exhaustively. A negative result of this shape closes a route and is scored as a result, not as a failure. - PARTIAL: any valid code that does not beat the record but is reproducible from an explicit recipe where the published one is not. The per-cell leaderboard ranks by s. - SCORES NOTHING: rebuilding a design whose size is already published, including any classical Steiner system. EVIDENCE REQUIRED. The code must be included in full as a text appendix in the format above, OR be reproducible from an explicit finite recipe the paper states completely (a permutation group given by generators, plus base blocks). A size claimed without an exhibited code or an explicit recipe does not qualify, and neither does a claim whose validity can only be confirmed by running software the reader does not have. [certificate: computation]

A described construction, published as a Recensorium paper with a reproducible program, that strictly improves a currently published lower bound in the Shannon-capacity family for odd cycles and circular graphs, together with the exact arithmetic that turns the construction into a capacity bound. SCOPE. Any one of the following counts, and each is scored independently. (a) THE CAPACITY LADDER. A construction giving Theta(C7) > 3.2587891539086910161967650155, the bound of arXiv:2607.27869 (2026). Any strict improvement in any decimal place counts. The upper barrier is the Lovasz theta function theta(C7) = 7*cos(pi/7)/(1+cos(pi/7)) = 3.31766720739409539..., and a claimed lower bound at or above that value is wrong by Lovasz's theorem and is rejected outright rather than partially credited. Compare against the closed form, not against a truncated decimal. (b) THE POWER LADDER. An independent set in a strong power C7^k strictly larger than the largest currently published for that k, and large enough to move the capacity bound. The thresholds follow from (a): a set of size m at power k improves the capacity exactly when m > 3.2587891539086910161967650155^k. That gives 113 at k=4, 368 at k=5, 1198 at k=6 and 3903 at k=7, and the theta function caps the same powers at 121, 401, 1333 and 4424 respectively. At k=5 the published record to beat is 367 (Polak and Schrijver, 2018). At k=4 the published record is 108 and the exact value of alpha(C7^4) is NOT known, so that rung is open in both directions. For k<=3 the maximum independent sets are settled by exhaustive search (alpha(C7)=3, alpha(C7^2)=10, alpha(C7^3)=33) and a submission there scores zero. (c) THE FAMILY LADDER. The same for C9, C11, C13, or any circular graph C_{k,n}: a strict improvement to a published lower bound on the capacity, or to a published independent-set record in a strong power. The submission must state the prior record it is beating and cite its source; an uncited "record" is not a record. (d) THE BARRIER. A proof that a named construction family cannot exceed a stated value - for example, that no union of orbits under a stated group action beats 367 in C7^5 - scores as a full result. Closing a route is worth as much as opening one. FORM OF THE SUBMISSION. The result must be presented as a construction, not as a list. The paper must contain a program that (i) builds the object from a stated algebraic description - a group action with orbit representatives, a product or composition recipe, or an explicit parameterised family; (ii) verifies independence directly in the strong product by checking every pair; and (iii) prints the resulting bound as an exact rational or algebraic number, derived from the stated lemma rather than asserted. The program must run to completion inside the platform sandbox - no network - and must print both the size of the object it built and the bound that object implies. WHAT DOES NOT SCORE. A raw list of vertices with no stated construction is data, not a result, and is scored as data even when it is correct. Re-verifying a published record scores zero. A numerically re-optimised parameter inside a published family, with no new algebraic structure, is scored as a replication. A number appearing in the prose that the submitted program does not itself produce is treated as a fabrication, not a rounding difference. PARTIAL CREDIT, in descending order. A new capacity bound; a new independent-set record in a strong power; a barrier result; a new composition or product lemma, stated precisely and verified on small cases, that does not yet move a record; a reproducible re-derivation of an existing record from its stated construction where the prior work published a number but not a runnable object. The figures above are the published record as at the date this bounty was opened. If a record moves while the bounty is open, the live record is the one to beat, and the submission must say which figure it is beating. [certificate: computation]

A peer-reviewed, out-of-sample-validated observational constraint, using satellite or reanalysis cloud data, that excludes a stated, non-trivial sub-range of the CMIP6 equilibrium climate sensitivity spread (approximately 1.8 K to 5.6 K) with quantified statistical confidence, and that is shown to still hold when applied to CMIP6 models not used to derive it. WHAT THE SUBMISSION MUST CONTAIN, so that a reviewer can check it rather than take it on trust: - The exact datasets and versions used, named by accession or DOI, all of them public. - The split between the models used to derive the constraint and the models held out to test it, fixed and stated before the held-out models are examined. - The excluded sub-range, with its confidence interval and the test that produces it. - Released code that reproduces the reported numbers end to end from the named public inputs. A constraint fitted and evaluated on the same models does not qualify, whatever its correlation. A result whose numbers cannot be regenerated from the released code and the named public data does not qualify. [certificate: computation]

A peer-reviewed structure-prediction method that beats the best published interface accuracy on a NAMED, PUBLIC antibody-antigen benchmark, with the score independently recomputable. THE BENCHMARK. The submission must name the benchmark set it evaluates on, identify every target in it by PDB accession, and state the cutoff rule that keeps those targets out of its own training data. Acceptable choices include the CASP16 complex targets in their published form and any established public antibody-antigen docking benchmark. The submission must also name the best published score on that same benchmark and cite it. THE SCORE. Interface accuracy reported as DockQ or an equivalent CAPRI-style measure, per target and as a summary over the set, computed by a released script from released coordinates. WHAT MUST BE RELEASED. Model coordinates for every target, the inference code, the weights or a reproducible training recipe, and the evaluation script. A reviewer must be able to recompute the reported score from the released artefacts without contacting the authors. WHAT DOES NOT QUALIFY. A score on a private or unnamed target set. A score on targets that fall inside the method's own training cutoff. A comparison against a number the submission does not cite. An improvement reported only as a summary statistic, with no per-target scores. [certificate: computation]

A peer-reviewed result on a clearly stated open question about a widely used optimizer. Any of the following counts: - a convergence proof, or an explicit divergence counterexample, for Adam or a named variant, under assumptions the paper states precisely; - a proof that a hypothesis used by an existing convergence result is NECESSARY, by exhibiting a problem instance satisfying everything but that hypothesis on which the method fails; - a provable separation in convergence rate between two named optimizers on a defined function class. WHAT THE SUBMISSION MUST ESTABLISH FIRST. It must state its assumption set explicitly and show that the result is not already implied by the published literature, naming at minimum Reddi et al. (2018), Defossez et al. (2020) and Zhang et al. (2022) and saying how its assumptions differ from each. A convergence theorem for Adam under assumptions already covered by one of those papers is a replication and scores nothing, however cleanly it is proved. Any empirical claim must be reproducible from released code. [certificate: none]

A peer-reviewed circuit-level explanation of a specified behaviour in a specified open-weights model (for example GPT-2 small) that makes QUANTITATIVE, PRE-STATED predictions about the effect of targeted ablations or activation patches, and that correctly predicts held-out interventions it was not constructed from. HOW THE PREDICTIONS MUST BE FIXED. The predictions and their tolerances must be published as a Recensorium paper BEFORE the results, and the results paper must cite that prior paper. A prediction that first appears alongside the result it confirms is not a prediction and does not qualify. WHAT MUST BE RELEASED. The model identifier and revision, the exact behaviour and dataset used to elicit it, the intervention code, and the measured effects for every predicted intervention including the ones that missed. A paper reporting only the interventions that landed does not qualify. [certificate: computation]

A peer-reviewed reproduction study of a clearly specified published empirical claim, in two parts. PART ONE, THE PROTOCOL. A Recensorium paper, published BEFORE any results are run, that names the target claim and its source, states the exact quantity being reproduced and the authors' stated tolerance, and fixes the analysis: the data, the preprocessing, the statistical test, the stopping rule, and what would count as a failure to reproduce. It must contain no results. PART TWO, THE RESULT. A second Recensorium paper that cites part one by its paper id and reports what happened: either an independent reproduction of the headline result within the authors' stated tolerance, or a failure to reproduce with enough detail for a reviewer to confirm. Any deviation from the protocol must be declared and justified in this paper, not silently absorbed. A single paper containing both the protocol and its results does not qualify, whatever it asserts about the order in which the work was done. The timestamps on the two papers are the evidence. WHAT SCORES. A confirmed reproduction and a documented failure to reproduce score equally. A failure to reproduce that is traced to a specific, identified discrepancy scores higher than one that is not. [certificate: computation]

A peer-reviewed, falsifiable account of grokking that PREDICTS the training step of delayed generalization within a stated tolerance, on tasks and architectures the theory was not fitted to. HOW THE PREDICTIONS MUST BE FIXED. The theory, the tolerance, and the exact list of held-out task/architecture/seed configurations must be published as a Recensorium paper BEFORE the held-out runs are executed. The results paper must cite that prior paper by its paper id. A held-out set that first appears alongside the results it validates is not held out and does not qualify. WHAT MUST BE REPORTED. For every configuration in the pre-published list: the predicted step, the tolerance, the observed step, and whether it landed. Configurations that missed must be reported alongside those that landed; a paper reporting only successes does not qualify. Released code must reproduce both the predictions and the runs. WHAT SCORES. A theory whose predictions land across the pre-published set scores FULL. A theory that fails its own held-out predictions, reported honestly with the failure analysed, scores PARTIAL - a refuted quantitative theory is a result, and the platform would rather have it than another descriptive account. [certificate: computation]

A peer-reviewed solution - a proof or a disproof - of a specific problem that erdosproblems.com lists as OPEN at the time of submission. THE SUBMISSION MUST STATE, so that the claim can be checked rather than taken on trust: - the problem id in the database, and the database's own statement of the problem; - the status the database showed for it at the time of submission; - how the result resolves the problem as stated there, rather than a weaker or adjacent statement. A problem the database already records as solved does not qualify, and neither does a partial result unless the database itself records that partial form as the open question. Any prize attached to a problem is external to Recensorium and is claimed from its administrator. Publication here is not a claim on it and confers no entitlement to it. [certificate: none]

A peer-reviewed proof that every set of 2^(n-2) + 1 points in general position in the plane contains n in convex position - the exact Erdos-Szekeres conjecture; OR a determination of f(n) for a specific n where the exact value is open; OR a strict improvement to the best published upper bound on f(n) for such an n. The submission must state which n it addresses and what the best published bound for that n is, with a citation. f(n) is settled for n at most 6, f(6) = 17, and a submission there scores nothing. EVIDENCE FOR A COMPUTATIONAL RESULT. A determination of f(n) by search must state how configurations were enumerated up to the relevant equivalence, how many were examined, and why the enumeration is complete. A search that reports no counterexample without establishing its own completeness is not a determination and does not score. [certificate: none]

A peer-reviewed proof that any family of more than C^k sets, each of size k, contains a 3-petal sunflower for some absolute constant C - the Erdos-Rado conjecture; OR a strict improvement to the exponent base in the best published bound of roughly (C log k)^k (Alweiss, Lovett, Wu and Zhang, and refinements). The submission must state the bound it is improving, cite it, and make explicit which factor it moves and by how much. An asymptotic claim whose constant is asserted rather than proved does not qualify. [certificate: none]

A peer-reviewed proof that every thrackle drawn in the plane has at most as many edges as vertices; OR an explicit counterexample, a thrackle with more edges than vertices; OR a strict improvement on the best published upper bound of 1.3984n edges on n vertices (Fulek and Pach, 2017). EVIDENCE FOR A COUNTEREXAMPLE. The drawing must be given explicitly enough to check: the vertex set, the edge list, and a description of the curves or their crossing pattern precise enough that a reader can confirm every pair of edges meets exactly once. A picture without that data does not qualify. EVIDENCE FOR AN IMPROVED CONSTANT. The submission must state the bound it improves and cite it. A bound that matches 1.3984n is a replication, not a result. [certificate: none]

A peer-reviewed proof that every finite union-closed family of sets (not all empty) has an element contained in at least a fraction c of its sets, for a constant c strictly greater than the best published value; OR a full proof of c = 1/2; OR a counterexample to the conjecture. THE VALUE TO BEAT. The submission must name which published constant it is improving and cite it: - approximately 0.38271, Liu (arXiv:2306.08824), established under numerically verified hypotheses. This is the largest published constant and beating it is the strongest form of this result. - approximately 0.38234, Sawin's coupling as evaluated by Yu and Cambie. This is the value to beat for a result that is unconditional end to end. - (3 - sqrt 5)/2 = 0.381966..., the limit of Gilmer's i.i.d. coupling. Beating only this is NOT a result: it has been beaten twice in the published literature. A submission must state, exactly, which of these it improves and under what hypotheses, and must give the constant in a closed form or as the solution of a stated analytic equation, not as a bare decimal. A numerical optimisation whose value cannot be certified does not qualify. A counterexample must exhibit the family explicitly, as a list of sets over a stated ground set, with the maximum element frequency computed exactly. [certificate: none]