Internal Spectral Theory (IST)

Internal Spectral Theory (IST) What if space and time are not fundamental? Internal Spectral Theory (IST) is a mathematical research programme investigating whether physical space, locality, and eventually spacetime dynamics can emerge from a deeper spectral and relational structure, rather than being assumed as fundamental ingredients of physics. Its starting point is deliberately pre-geometric. At the fundamental level, IST assumes no physical space, no metric, no spatial dimension, no physical time, and no pre-existing locality. Quantum mechanics is not imposed as a fundamental physical theory at this stage. Instead, IST introduces a minimal relational spectral core consisting of a unital involutive algebra, a self-adjoint spectral element, and an automorphism of order three. This structure admits an exact ternary decomposition without requiring an initial Hilbert-space representation. The resulting spectral relations provide a framework for investigating organisation, non-factorisability, and the possible emergence of stable observable structures. From relational spectral structure to physical readability The first objective is not to construct geometry directly, but to determine which spectral structures can become stable and physically readable. Under suitable C*-admissibility conditions, the relational core admits a universal Hilbert-space representation containing its GNS representations. This provides access to positive states, spectral projections, operator norms, and completely positive maps, without selecting a preferred physical representation in advance. A central result of IST 0 connects the ternary spectral structure to a positive transfer mechanism through the exact identity: Eτ(d²) = d₀² + aa + aa. This second spectral moment determines the normalisation of a completely positive readability channel. Within specified classes of finite weighted chains, IST establishes rigidity conditions connecting bistochasticity, spectral transport, and the gap of an associated weighted Laplacian. These results provide mathematical constraints on spectral stability and organisation. They do not, by themselves, establish an emergent geometry. The intrinsic selection of a stable physical representation remains an open problem. Nucleation and spectral crystallisation IST I investigates the next transition: how a readable spectral system could develop an organised structure capable of supporting spatial interpretation. The proposed mechanism begins with a persistent oriented defect in a non-normal spectral transport operator. Under explicit minimal-boundary and stability assumptions, this defect defines a unilateral spectral structure described using the Wold decomposition. In the primitive semi-bounded class studied, the associated non-compact algebraic closure leads conditionally to a positive discrete series of su(1,1). An additional exact cutoff-coherence condition selects the branch with Bargmann index k = 1. The subsequent crystallisation mechanism is variational. Finite-dimensional crystallised configurations, rather than the original truncated non-compact representations themselves, realise an irreducible su(2) algebra. This distinction is essential: the compact structure is obtained through a separate crystallisation step, not by identifying finite truncations of su(1,1) with su(2). Toward an emergent three-dimensional space The crystallised construction supports a regular tower of finite matrix algebras, with angular representations, inter-shell transport, and radial structure. Under additional shell-completion assumptions, this tower exhibits cubic spectral counting, a characteristic signature compatible with three spatial dimensions. The programme further constructs asymptotically commuting Cartesian observables and examines differential limits compatible, on fixed angular sectors, with the three-dimensional Euclidean Laplacian. However, cubic counting alone is insufficient to identify Euclidean space. Compact geometries can exhibit the same spectral growth. A complete spatial reconstruction therefore requires the simultaneous validation of spectral dimension, local differential behaviour, effective commutativity, isotropy, and convergence across the relevant sectors. IST does not claim that three-dimensional space follows universally from ternary spectrality. It identifies a conditional mathematical reconstruction in which a three-dimensional spatial regime can arise without a fundamental spatial manifold. The global and uniform establishment of Euclidean locality remains open. Time as a distinct emergence problem IST deliberately separates the emergence of spatial structure from the emergence of physical time. Neither spectral depth, ternary order, Wold orientation, nor the numbering of spatial shells is identified with time. Once an effective spatial operator is available, a coherent one-parameter evolution can be considered, including hyperbolic equations compatible with a candidate 3D + t regime. At present, however, the evolution parameter and its physical calibration are introduced rather than derived from the minimal spectral core. The emergence of a physical clock, a universal causal structure, and Lorentzian relativistic behaviour therefore remains an independent research objective. The target is not to assume four-dimensional spacetime from the outset, but to investigate whether space and time can emerge through distinct, ultimately compatible spectral mechanisms. Beyond geometry: matter and interactions The programme also investigates whether internal physical structures can emerge within the spatially readable regime. Potential downstream developments include quantum behaviour, chirality, particle families, mass spectra, gauge sectors, entanglement, and gravitation. These are research directions rather than consequences already established by the minimal core. In particular, ternary spectral organisation is not identified directly with three spatial directions or three particle families. Any such physical interpretation must be supported by additional mathematical constructions, stable invariants, and independently testable consequences. A central constraint is the preservation of the underlying relational non-factorisability while allowing effective spatial and internal structures to become distinguishable. Scientific scope and falsifiability IST is developed as a structured, testable mathematical research programme. Its conclusions distinguish between: Established mathematical statements: exact ternary identities, conditional-expectation relations, channel properties, rigidity results, and algebraic closure theorems within their stated domains. Conditional constructions: positive-sector selection, persistent Wold nucleation, variational su(2) crystallisation, spectral shell completion, and three-dimensional growth. Open problems: intrinsic phase selection, robust global Euclidean locality, absolute physical scale, physical time, relativistic causality, and the emergence of particle sectors. The programme is not presented as a completed theory of fundamental physics. Its purpose is to formulate a sequence of precise mathematical transitions, determine which survive rigorous analysis, and identify the additional conditions required for physical interpretation. The central hypothesis is simple, but its implications are substantial: Geometry may not be a fundamental property of reality. It may instead be a stable, physically readable phase of a deeper relational spectral structure. From spectral relations to organisation. From organisation to nucleation and crystallisation. From crystallisation to candidate space. From spatial structure toward a still-open theory of time and spacetime. Internal Spectral Theory — IST 0 and IST I, October 2026. Christophe Rivière — IST Labs