The complete singular squared-distance Laplacian conjecture

The single target is the complete two-clause concluding conjecture identified by D5/L/QuadraticForms/dasmohanty2024inverse. The Library note owns the bibliography, equation (3.6), omitted-list locators and source qualifications. The exact proposition is DasMohantySingularSquaredDistance.completeConjecture; the theorem DasMohantySingularSquaredDistance.result has that proposition as its type. Issue https://github.com/the-omega-institute/trureturing/issues/25619 and immutable preregistration comment 6574192179 identify the external event.

For every finite number t of parts with t at least two, and every positive integer part-size function n on Fin t, let N be the sum of the sizes, s the number of parts of size at least two and h the number of singleton parts. Vertices are pairs consisting of a part and a vertex in that part. The real squared-distance matrix delta has diagonal zero, same-part off-diagonal four and between-part one. Assume delta is singular and the sum of all its cofactors is nonzero. The latter is the sum of all adjugate entries, not a principal cofactor.

The functions phi, psi and theta, their single and double omissions, and all three entry cases of sourceL are exactly the author-defined omitted-list functions and equation (3.6). Empty products are one and empty sums zero. No expanded hatted-Psi display is substituted for the omitted-list definition.

The matrix sourceL is Hermitian. Its entire positive real characteristic-root multiset equals N-t copies of 1/4 together with h-1 copies of one, with algebraic multiplicity and no other positive roots. Its positive, zero and negative indices are N-s-1, one and s, respectively. These are a conjunction on the same actual matrix and domain. All finite part counts, repeated sizes and zero multiplicities are included; no additional matrix identity, genericity, classification or spectral-transfer premise is assumed.

The singular case requires the actual source matrix, rather than an inverse of singular delta. Inertia alone cannot specify the positive values and their exact multiplicities. The two source clauses therefore require both a rank-one inertia relation and exhaustion of the positive characteristic roots.

The complete theorem discharges both source clauses on the original domain. No source-matrix realization or spectral-transfer assumption remains in the conclusion. Three source-owner information-escape audits remain unfinished under the linked-issue exception in CLAUDE.md section 3.9; they are distinct from the mathematical theorem. Publisher text, license and correction checks, complete forward citations and worldwide priority remain outside the verified source scope.

Write delta as the block matrix with same-part coefficient three and diagonal subtraction four, plus the all-ones matrix. Positive integer sizes make each factor 3n(i)-4 nonzero. Its explicit block inverse and the determinant lemma give ratio=-1 in the singular case, hence psi=-phi and nonzero source denominators. Substitution in the literal three sourceL entry cases yields the pivot-one rank-one downdate of the negative block inverse A.

An invertible constant-and-difference basis on each actual part diagonalizes A. Exactly s constant columns are negative; the remaining N-s columns are positive. The existing structured rank-one inertia theorem gives sourceL inertia (N-s-1,1,s). Vectors with zero sum in each part form an (N-t)-dimensional quarter eigenspace. Singleton-supported vectors with zero total sum supply at least h-1 dimensions in the one eigenspace. Singularity forces a singleton part, so these two algebraic multiplicity lower bounds sum to the full positive index. Multiset inclusion and equal cardinality then prove exact exhaustion.

A mathematical counterexample would require positive integer part sizes with t at least two, singular literal delta and nonzero sum of all cofactors, for which either the exact positive-root multiset or the specified inertia fails for the same equation-(3.6) sourceL. A discrepancy caused by replacing omitted-list definitions with a malformed expanded display is a different target. A lower multiplicity bound, inertia alone or one finite graph does not establish the complete event.

The frozen owning module contains the exact source functions and the theorem result : completeConjecture. The owning Blueprint explains singular_source_downdate, singular_source_reduction and result, with typed attribution to the single Library note. The proof uses existing finite-dimensional determinant, inertia, characteristic-root and dimension suppliers. The result has proof_shape: bind-only and admission_basis: open-problem-resolution; auxiliary theorems have genuine proof consumers. No escape witness or computational-content admission is claimed.

The three public audit targets are singular_source_downdate, singular_source_reduction and result. None has a validated four-slot registration, and no Reg source is retained. The first two lack a completed zero-readout rejection and selector bridge: the unsuccessful attempt left False at lines 54/57 and 59/62 and selector syntax errors at 125/159. The result audit lacks the rational-cofactor cast at line 63, adjugate-sum transport at line 65 and actual root-multiset observational dependence at line 74. Lawful variation/sensitivity and current source selection/binding are unvalidated. These concrete gaps are disclosed at https://github.com/the-omega-institute/trureturing/issues/25619 under CLAUDE.md section 3.9; the issue is not a registration certificate.

Proved: on the complete original domain, the literal source matrix is the pivot-one downdate described above, has inertia (N-s-1,1,s), and has exactly the positive-root multiset specified in the conjecture. The decisive structure is the common finite vertex realization together with the partwise and singleton zero-sum eigenspaces; their multiplicities exhaust the positive index. The singular target is distinct from the source's nonsingular inverse theorem and the inertia of delta itself. The theorem resolves both concluding clauses without modifying those previously stated source results. The quoted source identifies no subsequent theorem conditional on this concluding conjecture.

Unfinished: the three source-bound audits and the source/literature boundaries below. This dossier asserts no further candidate, generalization, global novelty or priority result.

The author-defined omission convention is the target. The inconsistent expanded hatted-Psi products are not certified. The arXiv record's nonexclusive distribution license does not provide a publisher reuse license. Publisher full text, license and erratum, complete forward citations, worldwide nonresolution, hidden repository aliases and global priority are unverified. The bounded source screen found no complete resolution in its inspected scope; that finding is not exhaustive literature evidence. The three information-escape audits remain unfinished, independently of the frozen mathematical proof.