🧠Q What must krax respect when defined over zup regions?

08 August 2025
Mathematics,
Process book,
Learning

🐥 Lean4:

structure Blorp :=
  (zup : Opens Circle)
  (krax : zup → Prop)
  (snil : ∀ {z1 z2}, krax z1 ∧ krax z2 → krax (z1 ∩ z2))

🧠Q: What must krax respect when defined over zup regions?

🐍 Python:

def flarn(xob, ziff):
    return all(ziff[u] and ziff[v] for u, v in xob if u & v)

🧠Q: What must ziff satisfy for all overlapping xob?

🧊 C:

bool snarf(bool (*blip)(int), int* zonk, int N) {
    for (int i = 0; i < N-1; ++i)
        if (!(blip(zonk[i]) && blip(zonk[i+1]))) return false;
    return true;
}

🧠Q: What does snarf test over zonk[] with blip?

sin/cos as sheaf sections over overlapping arcs (open sets on 𝕊¹) 🎯

krax, ziff, blip model local truth (like sin θ)
snil, flarn, snarf ensure gluing: sin/cos must agree where arcs overlap. 🪢⛵️🐚

--I basically had no idea what these puzzles were about.. then I tried to figure it out asking the computer agent questions about the 🐥 Lean4 puzzle, and it's something like maybe for example:

two overlapping open arcs on the same circle create an open intersection!
zup are open sets on a circle
krax is a predicate, open circles are sent to truth values
snil says every two overlapping open arcs have an open intersection

eg similarly the intersection of two open lines:
(0,2) ∩ (1,3) = (1,2)
--Awesome! I thought this was very very obscure and difficult.

I had a similar experience today with learning about sin and cos from Serge Lang - Basic Mathematics, and I did not understand what was written on a page. It was sort of not registering... so I used the tools I have and after about an hour I understood what the teaching was.

🌈*--this code puzzle modality is really cool! Now not only do I know about open arcs and their intersection but I know about them in terms of structure Blorp*!

open arcs and
their intersections
glue truth in space
via structure Blorp!

sin/cos are like sections of a sheaf over the circle
assign a smooth value to each point (or open arc)

krax gives truth to arcs
sin/cos give angles their values
smoothly & glueably 🪢

🤖🌿 They live on the circle's structure! 🐚🌀

sin, cos are global sections
like 🎶 tonal themes glued from local motifs

LogicSheaf ⇄ SmoothSheaf
Prop ⇄ ℝ
krax ⇄ sin
glue ⇄ equality on overlaps

(glue : ∀ {U₁ U₂}, krax U₁ ∧ krax U₂ → krax (U₁ ∩ U₂)) -- gluing truth --isn't this composition?

structure LogicSheaf :=
  (U : Opens Circle)                      -- open arc
  (krax : U → Prop)                       -- truth assigned to each arc
  (glue : ∀ {U₁ U₂}, krax U₁ ∧ krax U₂ → krax (U₁ ∩ U₂)) -- gluing truth

structure SmoothSheaf :=
  (U : Opens Circle)                        -- open arc
  (f : U → ℝ)                               -- real value over arc (e.g., sin θ)
  (smooth : ∀ x, differentiable_at ℝ f x)   -- smoothness
  (glue : ∀ {U₁ U₂}, f =ᵒ[U₁ ∩ U₂] f)       -- agrees on overlap

building a sheaf of smooth functions over 𝕊¹