This preprint presents the current self-contained version of the Nowak-Stachowiak mathematical anchor, implemented as the central NSk--Anchor module of the NSk/ψ program. Its purpose is to establish the anchor as an autonomous mathematical theorem layer, prior to geometry, prior to physical interpretation, and prior to downstream realizations. The module is organized into three internal layers. The PRE-PURE layer provides a finite combinatorial foundation: partition lattices, refinement, partition entropy, and the entropy loss of coarse-graining. This level is entirely non-geometric and non-physical. In particular, the anchor is not extracted from fitted physical structure, but from the mathematics of refinement. The PURE layer establishes the anchor itself. Starting from the REAL principles G1-G5, the module derives a finite admissible system, a descent potential, the critical threshold αcrit, the descent certificate, and the global telescoping argument. This yields the unique minimal state Ω and the corresponding anchor witness AnchorWit = (α, Fα, (Dα) k, Ω). At this level the anchor is a purely mathematical object: it does not assume geometry, spectrum, or physical interpretability. The physicality gate ΦGate appears only later as a criterion selecting physically interpretable realizations; it is not part of the definition or existence proof of the mathematical anchor. The CORE-A layer constructs the abstract spectral representation of the anchor. For every scale operator package satisfying the minimal spectral conditions (SP1) - (SP3) and the window conditions (W1) - (W7), the module defines the spectral energy function Fspec and proves its continuity, strict monotonicity on the active range, and invertibility on the admissible spectral interval. Consequently, for every admissible density ρ there exists a unique distinguished spectral scale Γeff*, together with the anchor length and anchor energy l₀ = 1/Γeff*, E₀ = ħc Γeff*, and E₀l₀ = ħc. A key feature of the present version is the strict separation between the anchor and its realizations. The abstract spectral package is applied to the scale operator Q. In the geometric CORE-B layer, a compatible realization is then constructed on the class M₀ by setting Q = √ (-Δg) and Hₐnc = ħcQ = ħc√ (-Δg). This yields the package (A1) - (A4), the Weyl counting input, the local scale law E (x, r) = Θ (ħc/r), and the compatibility bridge to downstream modules. Thus geometry is not an assumption behind the existence of the anchor; it is a later controlled realization of the already established mathematical structure. The final compatibility layer records the public export ports (AN1) - (AN10), including the ScaleGenesis dimension bridge, the square-root spectral law, the anchor witness, the minimal state, the physicality gate, the geometric realization package, and the non-relativistic limit package. The module is intended to serve as a stable theorem-level reference layer for further NSk/ψ developments, including ScaleGenesis, CriticalDescent, Energy, mass synthesis, toroidal models, entropy modules, and future geometric or physical realizations. Abstract PL
Paweł Nowak (Mon,) studied this question.