The flat page is finished. Not abandoned — saturated. The 1D Collapse
Theorem settles why: under the strict axiom a one-line string can express only commutative
operations and symmetric relations, so the notation is necessarily two-dimensional, and the plane
offers exactly two lawful carriers of asymmetry — the vertical axis, and adhesion.
Both are already occupied: Stk, Up, Dn and Frac
carry vertical order; Adh carries the above/below gluing. A thirteenth flat constructor
could add abbreviation; a third place for asymmetric content lay beyond it. The twelve are not
a list that happened to stop. They are a closure of the plane.
That makes the direction of growth non-arbitrary. If the notation is to say more, it needs more places to put asymmetry — which means a surface with more structure than the plane. What follows proposes constructors for the folded carriers and puts each through the volume's own Layer-4 machine check. Every animation below runs that check: the strokes of a term reflect across the axis and must land on the strokes of its reflection.
Up/Dn. That is not the volume's result and it is not right. The strict
axiom fixes every atom under reflection, so no glyph is chiral at all — the sign isotype is
precisely the retired chiral layer. Asymmetry is carried by position, and the
1D Collapse Theorem counts the positions: two. Adhesion is the second, and the earlier draft omitted
it.The check, running
Press Reflect. The black strokes are the drawn term; they sweep across the mirror axis. The green strokes are the drawing of the reflected term, computed independently from the grammar. The term passes iff the two coincide — that is the machine check verbatim, at 0.05pt tolerance. Two hundred and seventy-nine terms were checked; all pass.
A balance with distinct sides. Its reflection is the argument-reversed drawing — a different picture asserting the same proposition. This is the Mirror theorem in its ordinary case.
Row admits only commutative content (WF3), which is exactly what makes argument reversal harmless.
Vertical orientation is the page’s one lawful asymmetry: reflection carries Up pointwise, and the drawing is fixed.
Enclosure open above: indeterminate extent.
The dual OBox never had: open below. Where OBox leaves the extent of a term indeterminate, UBox leaves its ORIGIN so — which is what an antiderivative has: no distinguished basepoint, only the family. The classical +C is that indeterminacy, and here it is a formed enclosure saying so, rather than a slot left to stand for it. Mirror-fixed, so lawful on the page.
The pair {t, ρt} as ONE presence across the ridge. The machine check refused ρ(Fold t)=Fold(ρt) and forced reflection to FIX a fold — a pair is unchanged by exchanging its members. Fold is a chirality quotient.
The argument is achiral, the two faces coincide, and the fold lies flat. The branch point stated as a formation condition, in the grammar’s own terms.
Three faces at a common apex. The reading group acts on them as the full symmetric group S₃, so only TOTALLY SYMMETRIC content is writable here — which is to say, elementary symmetric functions. The corner’s grammar forces Vieta.
What the check corrected
Twice the check refused a declared design, and both refusals were instructive.
The fold was declared to reflect pointwise, ρ(Fold t) = Fold(ρt), like every other unary constructor. It passed on atoms and stopped exactly at those terms whose argument was itself argument-reversing. The only action making the renderer equivariant is that reflection fixes a fold — which is the construction's own content, since a pair is unchanged by exchanging its members. Fold is a chirality quotient: a chiral argument yields an achiral fold, verified on every test term.
With reflection fixing each fold, Fold(t) and Fold(ρt) still drew as mirror images of one another — so the fold was carrying an order it claimed not to have. A fold presents an unordered pair, so it must present its canonically lesser face first. This is a clause of exactly the kind WF5 already imposes on numerals, and it was forced rather than invented.
A fold lies flat exactly when t = ρt. Degeneracy of the fold is
achirality of the argument. The branch point is thereby stated as a formation condition: a fold held closed, read
directly off the drawing.
A proposal the axiom refuses
The same discipline that retires → and < retires a proposal
made here in an earlier draft. A rotation-sense mark — a three-armed pinwheel distinguishing
clockwise from counter-clockwise on the corner — was offered as the corner's new orientation atom. It
is chiral: reflection carries it to its opposite. Tested against the strict axiom, it fails, exactly
as every glyph of the retired chiral layer fails.
Every glyph of this notation is fixed under reflection, on every carrier. The strict axiom fixes every atom under its reading group's reflection, so a chiral atom is unwritable by construction. What a folded carrier supplies is not chiral marks but additional lawful positions in which asymmetric content may be placed — which is what the 1D Collapse Theorem counts on the plane, and what the count would have to be redone for on the tent, the corner and the cone.
The forms that survive are the positional ones. UBox is mirror-fixed and asks only
for a lawful place to be open. Fold is fixed by reflection and carries its asymmetry in
the ridge, not in any mark. Tri places its arguments at three positions about an apex
and carries reflection-fixed glyphs throughout.
The carrier tower
Generalise WF3 rather than the constructor: a lawful operator must be invariant under the argument permutation induced by its reading group. Each carrier then dictates its own algebra.
| Carrier | Reading group | Induced permutation | What is writable |
|---|---|---|---|
| page | ℤ/2 | the swap | commutative binary operations (WF3) |
| tent | C2v | face exchange | a third position for asymmetry: which side of the ridge |
| corner | C3v | all of S3 | totally symmetric ternary operations |
| cone | O(2) | rotation and reversal | bracelets; asymmetry graded by winding |
Expressive power trades alphabet for chirality
An atom is a stroke-set fixed by the reading group, hence a union of orbits, hence the alphabet has 2orbits members — and orbit counts come from Burnside's lemma, the same lemma governing the right-folding tower. On a 5×5 stroke lattice the page admits 2160 lawful atoms and the tent only 288. Folding costs roughly 272 of alphabet and buys a new place to put asymmetry. Richer carriers are poorer in atoms: the notation becomes more expressive by becoming more constrained.
Status
The saturation claim is the volume's own 1D Collapse Theorem, quoted rather than re-derived: two lawful carriers of asymmetry on the plane, both occupied. The constructors are stipulations: each is declared with its reflection action and has passed involutivity, well-formedness preservation, and renderer equivariance on the terms exhibited — which earns them a hearing, not a place in the codex. WF7 and the fold's reflection action are derived. The alphabet computation is exact for the stated lattice and model-dependent otherwise, though the direction of the trade stands open.