See the calculus

Nine instruments, in order. Each one shows one thing, and each opens by itself in the browser — without installation, without network, without an account.

They are not illustrations of what a text already said. They are the place where the statement can be verified with a ruler: when two things have the same length, they have the same length on the screen, and you measure.

The order

  1. The pair becomes a point
    Two number lines become a graph
    Everyone knows that a function is a machine: enter x, get y.
  2. The Angle
    in degrees, in radians, and where π comes from
    An angle does not have its own size.
  3. The triangle
    opposite, adjacent, and hypotenuse depend on which angle
    Opposite and adjacent seem like names of sides.
  4. Distance
    where a circle originates
    A circle is not drawn: it is generated by a condition — all points that are the same distance from a center.
  5. The Circle
    Sine, cosine, and tangent as lengths
    Sine, cosine, and tangent are not calculator buttons: they are lengths, measured from a floor that someone chose.
  6. The Unrolling
    from the circle to the graph
    The circle's arc leaves a question in the air: I know the sine as a wave — what does it have to do with this circle?
  7. The rate
    the function described by the change
    Slope is how much you rise when you go one.
  8. The Intersection
    The slope is a tangent
    The word tangent names two different things: a length measured in the circle and the line that touches a curve.
  9. The family
    Read the curve and say what machine made it
    You recognize a parabola by eye.

Why this order, and not another

One dimension only has position — there is no place to curve. It is the second that unlocks the shape, and that is why the pair becomes a point is the root and not a beginning among others: even an angle needs two dimensions to exist. After the root, the tree branches into two that only meet at the end.

   READING ORDER — one line

   pair-to-point → the angle → the triangle → the distance → the circle
                 → the unwinding → the rate → the meeting


   DEPENDENCY — two branches, and only here does the tree exist

                    pair-to-point
              ┌───────────┴───────────┐
           the rate    the angle → the triangle → the distance → the circle
              │                  → the unwinding
              └───────────┬───────────┘
                     the meeting

   The rate does NOT DEPEND on trigonometry: slope exists without angle. But it
   spends the word "tangent", and that word has to be disambiguated first.
   That is why the reading line places it late, and the branch stays true.

Dependence and order of reading are not the same thing, and it's worth separating them. The rate does not depend on trigonometry — slope exists without any angle. But it uses the word tangent, which names two things, and this ambiguity needs to be resolved first. That's why the queue puts it in seventh, although the graph of dependence keeps it in its own branch.

This wasn't theory: with the rate in second, it was the only slice with declared debt at the check — spending sine, cosine, and π before anyone was defined. Descending to seventh, the debt zeroed out. The gate saw what the argument didn't.

How this is verified

Each slice declares, in a machine-readable manifesto, its order and the symbols it responds to. The `auditar_fatias.py` checks: no symbol used without coming from the slice itself or a previous one; no borrowing without someone repaying it later; no idle declaration; no network request. It runs with negative control — injects defects and demands they be caught, because zero findings only mean something if the check proves it can disprove.

This index is also generated: it reads the order and the description of the slices themselves. Editing the generated HTML is the flaw, not the shortcut.

PT