The Four Sheep Counters
Discrete Laegna is its own system to count positive and negative numbers, their basic limits and unknowns, and the structural scopes that repeat both the properties of infinities and the properties of true and false — from which all of it can be built.
The Counters
Science Resonance
Sheep of Identity
The pure self-referent. The counter that says 'this is'. Atom of presence, the seed of every positive count.
Peano's successor — the +1 axiom. In Laegna, I is not just a number but a witness: the act of counting itself.
{ "counter": "I", "role": "unit_true_one" }Sheep of Origin
The hollow counter. Not absence but the boundary that makes counting possible. The frame around the herd.
Cantor's empty set and the additive identity. In Laegna, O is the boundary-condition every limit collapses into.
{ "counter": "O", "role": "zero_void_null" }Sheep of Affinity
Binds two counters into one composite. The herd-stitcher. Where multiplication and structure emerge.
Boolean ∧ and the tensor product. The constructor that turns a sequence into a lattice.
{ "counter": "A", "role": "and_conjunction" }Sheep of Either
Branches counters into alternatives. The pasture-splitter. Where superposition and possibility live.
Boolean ∨ and the disjoint union. The constructor that turns a sequence into a tree of worlds.
{ "counter": "E", "role": "or_disjunction" }Z · X · Y
Fourier · Gauss · Hilbert
Three octaves form a discrete frame for scale. The same true-and-false relations that pattern I·O·A·E repeat across Z (zero · log), X (center · lin), Y (astro · exp).
log · ∂⁻¹lin · ∂⁰exp · ∂⁺¹Fourier
FREQUENCY COMPOSITIONExtended into octave-frequency composition: the same harmonics repeated across log·lin·exp scales — discrete here as the octave indices Z·X·Y.
Gauss
CURVE ↔ FREQUENCYCurve unit ⇄ frequency unit: a necessary interchange. Discrete counters approximate the accelerated points where both readings agree.
Boole · Peano
LOGIC ↔ ARITHMETICTrue/false and successor/zero meet here as the same gesture: every Laegna counter is both a logical operator and an arithmetic one.
Hilbert
PARTICULAR SOLUTIONSConcrete spaces and operators that resonate cleanly with Laegna's discrete frame — used for verification and bridging.
The herd is counted. The pasture is mapped. Now cross into the continuous infinite — where these same counters dissolve into living structural patterns.
♾ Enter Continuous Laegna →