Worked derivation: the paired spatial rank sequence
Prove the nested-counter rank sequence 1,1,2,2,3,3, count the prefix images and establish the precise six-channel lower bound.
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Worked mathematical derivations
Follow eight calculations from declared inputs to their exact results: paired spatial rank, binary readout multiplicities, refinement and clocks, the complex structure of real dynamics, ten symmetric receiver modes, the structural integer 137, the Coulomb ground state, and the relative-speed relation to alpha.
All eight derivations are now available in العربية. Each language version links directly to its counterpart.
Each derivation supplies the state space or operator, definitions, intermediate steps and a checkable conclusion. The SMM/DTT interpretation is placed alongside the mathematical construction, with the remaining physical matching stated precisely. This route makes it possible to inspect why a result follows and which additional assumptions would be needed to enlarge its scope.
8 connected guides
Worked derivation: 64 binary addresses and 27 spatial sites
Calculate every multiplicity, probability and moment in the binary paired readout, then identify the information lost by spatial projection.
Read the guideWorked derivation: refinement dimension and the clock ratio
Prove the dimension of a uniform address geometry and derive its conditional equality to a clock ratio through an independently calibrated encoder.
Read the guideWorked derivation: complex structure from real dynamics
Derive polar phase from an oriented real generator, treat zero modes and time reversal, and fix the sign of positive-energy Schrödinger evolution.
Read the guideWorked derivation: ten normalized symmetric receiver modes
Construct an orthonormal ten-mode pair receiver, prove its universal retention property and calculate the six directions erased by quaternionic products.
Read the guideWorked derivation: the structural integer 137
Derive 137 from a uniquely parsed record grammar, normalize its sampling laws and distinguish dimension, phase degree and physical coupling.
Read the guideWorked derivation: the exact Coulomb ground state
Normalize the hydrogen ground state, prove its global energy bound, calculate radial moments and identify the source dynamics SMM/DTT must recover.
Read the guideWorked derivation: alpha and the relative RMS speed
Derive v_rel,rms/c=α in the Coulomb ground state, prove the virial relation, track reduced mass and explain the noncircular SMM/DTT coupling target.
Read the guideKeep the wider context in view
Start with the foundations, follow the reading paths, or download the printable guides. The open problems explain the mathematical and physical steps that remain in the research programme.