M36d The Doorway Programme - D00-D16 Papers about HyperSymmetric Core (v0.9)

M36d is a colleciton of D00-D16 Papers about HyperSymmetric Core and related objects. The Doorway Programme develops the HyperSymmetric Core (HSC) after its introduction in M36c. It begins from the h-etage at rank R = 3, generated by freezing the HyperCore Triad member Cpow, and studies the resulting operation system through its algebra, complex domain, rooms, base dependence, TrigCore, star ranks, cascade dynamics, physical interpretation, and final classification as a correction layer. The central generator is GC (t) = exp sqrt (e (1+t) ln (1+t) ), with normalised Abel coordinate AHSC (y) = exp W ( (ln y) ²/e) - 1, where W is the Lambert W function. The Room 1 operation is C-Multiplication, a odotC b = GC (AHSC (a) + AHSC (b) ), so that AHSC (a odotC b) = AHSC (a) + AHSC (b). D00 -- Introduction D01 -- Spectral Gap Main object: C-Multiplication versus ordinary multiplication. D02 -- Complex Domain and Hermit Unit Main object: Complex inverse structure of C-Multiplication. D03 -- HSC Room 2 Main object: Room 2 of the HSC, named C-Apow. D04 -- Halbzeug of the HSC Main object: HSC half-object / Halbzeug fountain. D05 -- Base Dependence Main object: HSC at arbitrary base B. D06 -- Extended Triads Main object: The Mult, C-Mult dual and interpolation family. Extended Nuclear Quad. D07 -- HSC TrigCore Main object: HSC phase structure on the unit circle. D08 -- SC TrigCore Contrast Main object: Comparison of SC and HSC TrigCore behaviour. D09 -- Cascade of the HSC Main object: HSC cascade map as a complex dynamical system. D10a -- Star Ranks of the Symmetric Core (recap) Main object: Negative Rooms / alive SC basement. D10b -- Star Ranks of the HSC Main object: HSC negative Rooms and h-LSE. D10c -- Complex Room Numbers Main object: Complex HSC Rooms via Schroeder iteration. D10d -- SC--HSC Dual Main object: Mature SC--HSC comparison. D11 -- Fountains of the HSC Main object: Identity and Halbzeug fountains. D12 -- Eta4 Science and the W-Backbone Main object: eta₄, tetration criticality, Lambert branch backbone. D12b -- h-Triad at Room 1 Main object: Cross-Abel h-Triad. D13 -- Groups, Phantom, Hyperphantom Main object: Phantom / Hyperphantom (HSC version of the unit i) taxonomy and group behaviour. HSC orbit: near-fourfold but not closed. Collatz conjecture relevance (a balance problem). D14 -- HSC and the Millennium Problems Main object: HSC relevance classification Core results: HSC coordinate: AHSC (y) ~= (ln y) ²/e Therefore HSC naturally targets balance problems, Yang--Mills: structural shadow only, belongs beyond direct HSC at R = 7/2. Navier--Stokes target: vortex stretching versus viscous damping. HSC enstrophy: OmegaH (t) = integral AHSC (1 + |omega (x, t) |²) dx Large-vorticity asymptotic: AHSC (1+|omega|²) ~= (4/e) (ln|omega|) ² Global controlled bound: OmegaH (T) <=~ OmegaH (0) + T E₀/nu Status: does not solve NS regularity; narrows the gap between doubly-log enstrophy and ordinary enstrophy. D15a -- HSC as Double-Log Sector Main object: Renormalization interpretation. HSC physics: double-log Sudakov / correction sector HSC represents higher-order correction structure, not bare leading exponents. D15c -- Electroweak Running Main object: HSC correction parameter in electroweak running. D15d -- Proton Decay and QCD--GUT Entanglement Main object: Whether HSC supplies leading proton-decay scale. D16 -- Unified Answer Main object: Final classification of the HSC. Core result: HSC = operational structure of higher-order corrections. Physics reading: SC = single-log leading sector / tree-level boundary data HSC = double-log correction sector / running machinery Main HSC operations: C-Mult C-Apow h-LSE CatHSC