Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- A family of real numbers indexed by the integer lattice is summed along the cubes of side 2N+1 centred at the origin; it is cube-summable when these finite sums converge, and its lattice sum is then their limit.
The Trilinear Estimate for Cyclic Gradients of Differences of Unitary Laws
lemmalem:nc-unitary-cyclic-gradient-trilinear-2026aAnalysisProbabilityFor two unitary laws with gauge difference x, the difference of the laws evaluated on the squared truncated cyclic gradients of x is bounded by a constant times the gauge norm of x times the square of its length-weighted gauge, uniformly in the truncation.Admissible Hamiltonians on Unitary Laws
definitiondef:nc-unitary-admissible-hamiltonian-2026bAnalysisPDEA Hamiltonian on unitary laws times the gauge space is admissible if it is jointly continuous and depends on the momentum only through its pairing with two-sided tangent vectors.The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance
definitiondef:nc-unitary-law-embedding-2026aAnalysisProbabilityUnitary laws embed into the gauge space by restriction to cyclically reduced words; this defines the gauge distance between laws.The Free Unitary Heat Generator Points into the Space of Unitary Laws
lemmalem:nc-free-unitary-heat-inward-2026aAnalysisProbabilityThe free unitary heat direction is an inward tangent vector at every unitary law, realised by a Cayley-transform curve built from a free semicircular family.Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones
lemmalem:nc-unitary-tangent-extrema-2026aAnalysisProbabilityAt a relative minimum on the unitary law space, the gradient of a test function is nonnegative on inward and zero on two-sided tangent vectors; the reverse holds at a relative maximum.Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws
definitiondef:nc-unitary-tangent-vectors-2026aAnalysisProbabilityDefines inward and two-sided tangent vectors to the space of unitary laws as velocities, in the gauge space, of one-sided and two-sided curves of laws.The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge
lemmalem:nc-free-unitary-heat-generator-estimate-2026aAnalysisProbabilityThe free unitary heat generator is bounded by |w|^2 on each word, continuous into the gauge space, and strictly dissipative: its differences pair with differences of laws to at most minus one quarter of the squared length-weighted gauge.The Free Unitary Heat Generator on Unitary Laws
definitiondef:nc-free-unitary-heat-generator-2026aAnalysisProbabilityDefines the free unitary heat generator on unitary laws: a diagonal term -|w|/2 lambda(w) plus signed products of lambda over pairs of letters with the same generator.The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It
lemmalem:nc-unitary-word-gauge-2026aAnalysisProbabilityThe gauge space is a real Hilbert space, and the gauge distance is a metric on unitary laws whose convergence is convergence on every word and for which the unitary laws are compact.The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge
definitiondef:nc-unitary-word-gauge-2026aAnalysisProbabilityDefines the weights theta^|w| with theta = 1/(6144d), summable word functions, the gauge space of weighted square-summable functions on cyclically reduced words with its norm and inner product, and the length-weighted gauge.Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family
lemmalem:nc-unitary-law-realisation-2026aAnalysisProbabilityTraces of words in unitaries of a tracial W*-probability space form a unitary law, and every unitary law is realised this way together with a semicircular family obeying the free second-moment relations.The Closed-Score Hypothesis Appears to Fail for the Free Entropy of One Variable: a Sketched Example, Conditional on the Tangent Inequality
remarkrem:nc-closed-score-fails-free-entropy-2026aAnalysisProbabilityA sketched one-variable example, conditional on the tangent inequality for minus the free entropy. Densities oscillating on a vanishing scale keep the free Fisher information bounded while their conjugate variables oscillate at unit size, and a coupling that reads the fine positi…Projected Closedness of Conjugate Variables along Square-Integrable Realisations with Bounded Free Fisher Information
lemmalem:nc-conjugate-variables-projected-closedness-2026aAnalysisProbabilityIf laws with conjugate variables are realised next to positions converging in L2 to a bounded law, and the realised conjugate variables stay bounded, then the limit law has conjugate variables with the same Fisher bound, and the realised conjugate variables converge weakly agains…The Barycentric Decomposition of the Noncommutative Wasserstein Distance
theoremthm:nc-barycentric-decomposition-wasserstein-2026aAnalysisProbabilityFor a coupling of two noncommutative laws, projecting the second variables onto the tracial algebra of the first gives a barycentric law and coupling. The cost splits orthogonally, the squared distance is at most the squared distance to the barycentric law plus the drop in second…Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum
lemmalem:nc-field-realisation-2026aAnalysisProbabilityA square-integrable tuple which, together with a copy of the positions of a bounded law, has the joint law of the positions and a given field of the GNS space is uniquely determined by the positions, and its joint law with the momentum of any bounded plan is that of the plan with…Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing
definitiondef:nc-plan-couplings-2026aAnalysisProbabilityThe couplings of a square-integrable plan with a law are the three-variable laws whose first two variables have the plan as law and whose third variable has the given law; they always exist, and every three-variable law has a displacement and a momentum pairing that do not depend…Hamiltonians on Phase-Space Noncommutative Laws that are Bounded at Zero Momentum at Bounded Positions
definitiondef:nc-hamiltonian-bounded-zero-momentum-2026aAnalysisProbabilityA Hamiltonian on phase-space laws is bounded at zero momentum at bounded positions if its values at zero momentum are bounded uniformly over all positions whose law has a given norm bound.Wall-Confined Free Energies with Regular Plan Subgradients
definitiondef:nc-regular-plan-subgradients-2026aAnalysisProbabilityA wall-confined free energy has regular plan subgradients if every bounded plan in the plan subjet of a function touching the free energy from below at a local minimum point of their difference forces that point into the score domain and coincides with its score plan.- A wall-confined free energy has closed score if, whenever realisations of score plans have strongly converging positions and bounded scores, the limit law lies in the score domain and the scores converge weakly to a realisation of its score plan.