The Mathematical Anchor in the NSk/𝜓 Programme: Coarse-Graining Stabilisation Certificate and Spectral Representation

This preprint presents the current self-contained version of the Nowak–Stachowiak mathematical anchor as the central module NSk--Anchor of the NSk/ψ programme. Its purpose is to establish the anchor as an independent mathematical theorem, prior to geometry, prior to physical interpretation, and prior to downstream realizations. The module is organised in three internal layers. The first layer, PRE-PURE, provides the finite combinatorial basis of the construction: the lattice of partitions, the refinement relation, partition entropy, and the entropic loss generated by coarse-graining. This layer is entirely non-geometric and non-physical. In particular, the anchor is not derived from fitted physics, but from a purely mathematical refinement structure. The second layer, PURE, establishes the anchor itself. Starting from the REAL principles G1–G5, the paper derives a finite admissible system, a descent potential, a critical threshold α₍crit₎, a descent certificate, and a global telescoping law. This yields a unique minimal state Ω and the corresponding anchor witness AnchorWit = (α, Fα, (Dα)ₖ, Ω). At this stage the anchor is a fully mathematical object: it does not presuppose geometry, spectrum, or physical interpretability. The physicality gate ΦGate appears only as a later criterion selecting physically interpretable realizations; it does not participate in the definition or existence of the mathematical anchor itself. The third layer, CORE (Level I), constructs an abstract spectral representation of the anchor. For every spectral triple (H, Γmax, WNSk) satisfying the minimal package (SP1)–(SP3) together with the window assumptions (W1)–(W7), the paper defines a spectral energy function Fspec and proves that it is continuous, strictly increasing on its active range, and therefore invertible on the admissible interval of spectral energies. As a consequence, for every admissible density ρ there exists a unique distinguished spectral scale Γeff* and the associated anchoring length and anchoring energy, l₀ = 1 / Γeff*,E₀ = ħc Γeff*,E₀ l₀ = ħc. This is a purely mathematical result: it does not depend on bounded geometry, on any concrete realization of H as a geometric operator, or on physical interpretability of the tags or weights. A key feature of the present version is the strict separation between the anchor itself and its realizations. The module proves the existence of the mathematical anchor and of its abstract spectral representation, but does not develop concrete geometric realizations, Weyl-type realizational packages, or EXEC-layer constructions inside the same document. In this sense, NSk--Anchor should be read as a theorem-bearing foundation: a stable source of results about the mathematical anchor, not as a realization paper for any one chosen class of manifolds or physical models. The module is therefore intended to serve as a frozen reference layer for later NSk/ψ developments, including vacuum selection, global gap constructions, toroidal models, entropy-based modules, and future geometric or physical realizations.