A computational claim you cannot recompute is hearsay. We ask what a certified answer must be so that anyone can recompute it and refuse anything that does not match, and we answer that it must be a canonical form: a representative stable under the ways the answer can be re-expressed. We build three and show each survives a different transport. An optimum survives change of algorithm: one integer reached by exhaustive search below a size wall, by exact tropical variable elimination past that wall (bounded by a graph's width, not by its number of variables), and by the published literature. A quantum code's number of logical qubits survives change of proof: it is the block length minus two matrix ranks certified by a proof kernel, and its distance is either enumerated exactly or witnessed by an exhibited logical operator proven genuine by a rank increase. A certified answer survives change of language: its content hash agrees byte-for-byte across two independent implementations and refuses any value, such as a floating-point number, with no language-independent form. The consequence is operational: a reader re-derives the answer and serves it only if it matches. We place a live 2024 quantum-error-correction result on the classical side of this discipline, certifying the code parameters its own published matrices imply without disputing the demonstration. The through-line is old and exact: truth is invariant under change of notation, and a certificate is that invariant made computable. Published by The Institute for Applied Ontological Mathematics.
A computational claim you cannot recompute is hearsay. We ask what a certified answer must be so that anyone can recompute it and refuse anything that does not match, and we answer that it must be a canonical form: a representative stable under the ways the answer can be re-expressed. We build three and show each survives a different transport. An optimum survives change of algorithm: one integer reached by exhaustive search below a size wall, by exact tropical variable elimination past that wall (bounded by a graph's width, not by its number of variables), and by the published literature. A quantum code's number of logical qubits survives change of proof: it is the block length minus two matrix ranks certified by a proof kernel, and its distance is either enumerated exactly or witnessed by an exhibited logical operator proven genuine by a rank increase. A certified answer survives change of language: its content hash agrees byte-for-byte across two independent implementations and refuses any value, such as a floating-point number, with no language-independent form. The consequence is operational: a reader re-derives the answer and serves it only if it matches. We place a live 2024 quantum-error-correction result on the classical side of this discipline, certifying the code parameters its own published matrices imply without disputing the demonstration. The through-line is old and exact: truth is invariant under change of notation, and a certificate is that invariant made computable. Published by The Institute for Applied Ontological Mathematics.
Quantum error correction protects quantum information by encoding it in a code described by three numbers: how many physical qubits it uses, how many logical qubits it protects, and the smallest error it cannot detect. The last is an optimum over exponentially many candidates, so it is read off a solver rather than checked. Here we show that code parameters can be certified by a short certificate, a list or a pairing or a symbolic instance, whose correctness the kernel of a proof assistant decides by computation, and whose size is itself a theorem rather than an estimate. Two published code families have distances settled here: the bivariate bicycle code's [[144,12,12]], certified, and the affine-permutation codes printed at [[1152,580,≤12]] and [[2304,1156,≤14]], settled at twelve and fourteen. Eleven code families and thirty-nine parameter sets follow, with nothing beyond three standard axioms trusted and the pipeline's specification step preset to twenty columns.
Four questions sit at the base of fault-tolerant quantum computing: whether a code with given parameters can exist, whether two circuits denote the same operation, how much entropy a measurement irreversibly produces, and whether a compiled gate still means what it should. They are normally answered by four disjoint techniques. We show they are one technique. In each case the answer is an invariant—the object a governing transformation leaves fixed—and that invariant can be held as a certificate a proof kernel checks and an operator-native substrate computes. The instruments are the quantum weight enumerator, whose shadow variant yields what is, to our knowledge, the first machine-checked nonexistence certificate for a quantum error-correcting code (no pure [[4, 1, 3]] stabilizer code, reduced to an integer impossibility); the spider normal form, unique both as a linear map and as a content-addressed tree, at the minimal axiomatic footprint of a proof assistant; the measurement boundary defect, whose logical entropy equals a kernel-counted cohomology dimension; and the reduction receipt, round-tripping a certified gate word to bare metal on two independent substrates. Everything downstream of the explicitly labelled modeling step in each domain is exact integer or rational arithmetic. The common shape—find the invariant the transformation fixes, and read it off as a certificate—turns four hard quantum questions into arithmetic a machine can hold. Published by The Institute for Applied Ontological Mathematics.
Computing the minimum distance of a quantum LDPC code is easy in one direction and expensive in the other: exhibiting a light logical operator is a search, while proving that no lighter one exists is what actually decides the parameter. This note proves an orbit reduction for the second half. If a group of code automorphisms acts on the qubits with r orbits, the minimum distance is the smallest of r restricted problems, one per orbit representative, so the number of solver calls is the number of orbits and not one. We give the a priori condition under which restricting to a single qubit is legitimate, namely transitivity on all qubits, and a counterexample showing what happens without it: on a hypergraph product code with n = 27 and d = 3, the widely used tool dist-m4ri prints a pair of equal bounds, which its own documentation calls a confirmed exact distance, and that value is 6. The reduction is coupled to a proof producing pipeline in which every nonexistence claim is re-verified by an independent checker, drat-trim or, for the largest closure, cake_lpr, whose soundness is proved inside CakeML. This closes with replayable certificates the distances of the five smaller bivariate bicycle codes of Bravyi et al., up to and including the code with n = 288 and d = 18, whose exactness was published with no replayable artifact and is established here at both ends. For the two larger codes it yields the first lower bounds a third party can re-check from the deposited files: d at least 18 for the code with n = 360, against a published upper bound of 24 and against d at least 11 for the best previously re-executable method, and a first lower bound of any kind for the code with n = 756. A separate and much cheaper ingredient, a sweep over logical operators supported on unions of cosets of a subgroup, tightens two upper bounds printed in a recent table of two block group algebra codes. The code printed there as [[140, 8, at most 16]] carries a logical operator of weight 14; the code printed as [[144, 10, at most 16]] has distance exactly 12, established at both ends and so closed four below the published bound. Both witnesses are re-checked outside any solver by two rank computations over the field with two elements, so neither of these two upper bounds rests on a solver at all. Both entries are printed with the inequality sign in that reference, which states no method for the tables they sit in; nothing it proves is contradicted here, and no error of its authors is corrected. What is improved are two bounds they offered as bounds. Finally we show that the two instance form of the reduction is a consequence of commutativity and not of the two block structure. On two block group algebra codes over non-abelian groups, taken from the published catalogue, the translation subgroup has two orbits for one code and twelve for another, so the number of solver calls has to be computed from the group before any search rather than assumed. On three such codes we certify the minimum distance. The argument that makes one orientation enough in the abelian case does not apply there, and we check that it does not, so both orientations are certified separately rather than assumed equal; on these three codes the two turn out to coincide, which is a measured fact about each code and not a consequence of that argument. The published values for these codes come from randomised search, and a certificate of nonexistence a reader can replay is, to our knowledge, new for non-abelian codes of this family. What this work does not claim is stated explicitly in the note: the restriction itself is not our idea, it is an option of dist-m4ri present since 2020; a machine checkable chain covering a symmetry breaking step is not a first; our proofs are checked by an independent checker and not by the kernel of a proof assistant; and the encoding step is not formally verified. A public question asking whether the orbit reduction is already in the literature was posted on MathOverflow on 28 August 2026 and carried no answer at the time of deposit. Every table of numbers in the note is generated from the deposited artifacts by a script that refuses to emit a row it cannot source. 27 pages. Verification scripts and certificate digests included as separate archives. MSC 2020: 81P73, 94B05, 68Q25, 20C05.
A classical bit fails one way: it flips. A qubit fails three ways—X, Z, or both at once—and that single change of arithmetic, from one error per site to three, is the whole difference between the classical theory of error-correcting codes and the quantum one. It has a precise home. The MacWilliams identity of 1963 is the exact law relating a linear code's weight distribution to its dual's; its quantum analogue, due to Shor and Laflamme, carries the same law across the stabilizer formalism with one coefficient changed—a +1 becomes a +3. We prove that quantum identity in the kernel of a proof assistant, and we prove it the honest way: by reusing, unchanged, the finite character-orthogonality argument that certifies the classical identity, isolating the +3 to a single new lemma about the four Pauli operators at one qubit. The classical coefficient is not merely inapplicable to quantum codes; we exhibit a code on which the +1 law is false and its failure is itself a theorem. From the certified identity the entire Shor–Laflamme weight spectrum of a code follows—the count of stabilizers and of logical operators at every weight—and from the spectrum two numbers the field cares about: the code distance (the minimum weight of a logical operator, recovered as the first place the spectra part) and the sub-threshold error-rate prefactor (the count of minimum-weight logical representatives, which the distance alone cannot see). We certify both for the five-qubit perfect code and the Steane code, exhibit the exact spectra, and place the result beside a companion certificate that reaches the distance by a single witness: the witness is a local probe, the enumerator a global census, and a code is fully accounted for only when both have spoken. A public instrument reproduces every number by independent enumeration, trusting neither our kernel nor our word. Published by The Institute for Applied Ontological Mathematics.