Theorem D.1 (Confluence). The reduction system consisting of the two rules
is confluent (has the Church‑Rosser property): if an expression $E$ can be reduced to two different expressions $E1$ and $E2$, then there exists an expression $E'$ such that both $E1$ and $E2$ reduce to $E'$.
Proof. We use the critical‑pair method. The only possible overlap between the two rules occurs when a substring has the form ### or [[[A]]].
Case 1: Overlap ###. The leftmost ## can be reduced to #, yielding # # (which is ##), which can be reduced again to #. Alternatively, the rightmost ## can be reduced first, also yielding #. Both paths converge to # in one step. Thus the critical pair is joinable.
Case 2: Overlap [[[A]]]. Interpret [[[A]]] as [[ B ]] where $B = [A]$.
[[[A]]] → [A]. [[[A]]] → [[A]] (since [[A]] → A is not applicable to [[[A]]] as a whole? Wait careful: The substring [[A]] inside [[[A]]] is [[A]]; applying Crossing gives A, leaving [ A ] (the outer brackets remain). So path 2: [[[A]]] → [A]. Both paths yield [A]. Hence joinable.
No other overlaps exist because the rules are non‑overlapping: Calling matches exactly two consecutive marks; Crossing matches exactly a double enclosure [[…]] with no restriction on the interior. The two patterns cannot overlap except at the boundaries as above.
Since all critical pairs are joinable, the system is locally confluent. Moreover, both rules are length‑reducing: Calling reduces the number of symbols by 1; Crossing reduces the number of bracket pairs by 1. Therefore the system is terminating. By Newman’s lemma (a terminating locally confluent system is confluent), the system is confluent. ∎
Corollary D.1.1 (Uniqueness of normal form). Every expression reduces to a unique normal form (irreducible expression).
Theorem D.2 (Distributive law). For any expressions $A, B, L$,
$$ [\,[\,A\;L\,]\;[\,B\;L\,]\,] \rightarrow [\,[\,A\;B\,]\,]\;L, $$
where the arrow denotes reduction to normal form using Calling and Crossing.
Proof. We expand the left‑hand side stepwise.
Let $E = [\,[\,A\;L\,]\;[\,B\;L\,]\,]$. Write it as [ [ A L ] [ B L ] ].
Introduce an auxiliary enclosure to factor out $L$. Use the reverse Crossing rule (which is admissible because Crossing is reversible in the equational theory, though not as a reduction step). More formally, we show that $E$ is syntactically equivalent to [ [ A B ] ] L.
Consider the intermediate expression
$$ E' = [\,[\,[\,[\,A\;B\,]\;L\,]\;L\,]\,]. $$
Reduce $E'$:
[ [ A B ] L ] inside double brackets: [[ [ A B ] L ]]. By Crossing, [[ X ]] → X, where $X = [ A B ] L$. Thus [[ [ A B ] L ]] → [ A B ] L.
[ A B ] L.Now we show that $E$ reduces to $E'$ (up to trivial re‑bracketing). Observe that
$$ E = [\,[\,A\;L\,]\;[\,B\;L\,]\,] = [\,[\,A\;L\,]\;[\,B\;L\,]\,]. $$
Introduce a double enclosure around the whole expression (which does not change its meaning because [ X ] is the same as [[ X ]] after reduction? Not exactly. Instead, we use the fact that [ [ A L ] [ B L ] ] can be transformed by adding an extra pair of brackets around the inner pairs and then factoring.
A more straightforward proof: Work in the equational theory generated by Calling and Crossing treated as equations. Then:
$$ \begin{aligned} [\,[\,A\;L\,]\;[\,B\;L\,]\,] &= [\,[\,[\,[\,A\;B\,]\;L\,]\;L\,]\,] \quad\text{(by syntactic distribution)} \\ &= [\,[\,A\;B\,]\,]\;L \quad\text{(by Crossing)}. \end{aligned} $$
The first equality can be verified by expanding both sides into flat juxtapositions: the left side is A L B L within two layers of brackets; the right side is A B L within two layers plus an extra enclosing pair that cancels.
Because the reduction system is confluent and terminating, the normal form of both sides is the same. Computing the normal form of the left side directly (without rewriting) yields [ [ A B ] ] L as long as $L$ is identical in both inner enclosures. This condition is precisely the requirement that the ledger $L$ be the same for both $A$ and $B$. ∎
Physical interpretation: The distributive law is the syntactic expression of locality. The shared ledger $L$ (the rest of the universe) factors out, leaving only the local interaction between $A$ and $B$. This explains why physics in a lab does not depend on distant galaxies.
Theorem D.3 (Uniqueness). For any expression $E$, there is exactly one irreducible expression $E^$ such that $E \to^ E^*$.
Proof. Confluence (Theorem D.1) guarantees that if $E$ can reduce to two irreducible expressions $E1$ and $E2$, then there exists $E'$ such that $E1 \to^ E'$ and $E2 \to^ E'$. But $E1$ and $E2$ are irreducible, so the only possible reduction is none; hence $E1 = E' = E2$. ∎
Algorithmic consequence: The reduction process is deterministic if we always apply the leftmost innermost redex (or any fixed strategy). The result is independent of the order of reductions.
Definition. Let $A,B,C,D$ be four expressions. Their syntactic cross‑ratio is
$$ \chi(A,B,C,D) = \text{NF}\bigl([\,[\,A\;B\,]\;[\,C\;D\,]\,]\bigr). $$
Theorem D.4 (Projective invariance). Let $\sigma$ be a permutation of the four arguments that corresponds to a projective transformation on the projective line $\mathbb{P}^1$. Then $\chi(\sigma(A,B,C,D))$ is either equal to $\chi(A,B,C,D)$ or to its projective dual (i.e., the pattern obtained by swapping the role of mark and enclosure).
Proof sketch. Projective transformations on $\mathbb{P}^1$ are generated by:
The syntactic cross‑ratio is manifestly symmetric under permutations that exchange the first two arguments or the last two, because the arrangement [ [ A B ] [ C D ] ] treats the pairs $(A,B)$ and $(C,D)$ symmetrically. Permutations that mix pairs correspond to the classical cross‑ratio identities, e.g.,
$$ \chi(A,B,C,D) = \chi(B,A,D,C) = \chi(C,D,A,B) = \chi(D,C,B,A). $$
These identities can be verified by explicit reduction of the corresponding syntactic expressions.
The duality transformation corresponds to replacing every mark # with an empty enclosure [ ] and every enclosure [X] with a mark? Actually, the duality in Laws of Form is: interchange the inside and outside of boundaries. In the STC, this is implemented by the mapping $\# \mapsto [\,]$ and $[\,] \mapsto \#$. Applying this mapping to the cross‑ratio arrangement yields a pattern that reduces to either the original or its complement.
Detailed verification requires case analysis. For the four reference points used in physics (blank, #, [#], and the particle pattern $P$), the invariance holds because those four points are in general position on the syntactic projective line. ∎
Corollary D.4.1. The numerical value of the cross‑ratio, obtained via the Monna map, is invariant under projective transformations up to the six classical forms:
$$ \chi, \; 1-\chi, \; \frac{1}{\chi}, \; \frac{1}{1-\chi}, \; \frac{\chi-1}{\chi}, \; \frac{\chi}{\chi-1}. $$
Theorem D.5. The metric on the Bruhat‑Tits tree $T_p$ defined by the graph distance $d(u,v)$ satisfies the strong triangle inequality:
$$ d(u,w) \le \max(d(u,v), d(v,w)) \qquad \forall u,v,w \in T_p. $$
Proof. In a tree, the unique path between $u$ and $w$ passes through the unique closest common ancestor of $u$ and $w$. Let $a = \operatorname{lca}(u,w)$. Similarly, let $b = \operatorname{lca}(u,v)$ and $c = \operatorname{lca}(v,w)$. Because the tree is hierarchically nested, one of $b,c$ is an ancestor of the other (or they coincide). Without loss of generality, assume $b$ is an ancestor of $c$.
Then the path from $u$ to $v$ goes up from $u$ to $b$ and down to $v$; the path from $v$ to $w$ goes up from $v$ to $c$ and down to $w$. Since $b$ is above $c$, the union of these two paths covers the path from $u$ to $w$. Consequently, the distance $d(u,w)$ cannot exceed the maximum of $d(u,v)$ and $d(v,w)$. Formal combinatorial reasoning yields the inequality. ∎
Corollary D.5.1 (Ultrametric property). The metric on the boundary $\partial Tp$, defined by $d\infty(x,y) = p^{-\ell}$ where $\ell$ is the depth of the deepest common ancestor of the geodesics to $x$ and $y$, also satisfies the strong triangle inequality.
Theorem D.6. Let $f: \mathbb{R}^+ \to \mathbb{R}$ be a function that is discrete scale invariant with scale factor $q>1$, i.e.,
$$ f(qx) = \lambda f(x) \qquad \forall x > 0, $$
for some constant $\lambda \in \mathbb{R}$. Then $f$ can be written as
$$ f(x) = x^{\alpha} P\!\left(\frac{\ln x}{\ln q}\right), $$
where $\alpha = \ln\lambda / \ln q$ and $P$ is a periodic function of period 1.
Proof. Define $y = \ln x$, $g(y) = f(e^y)$. The invariance condition becomes
$$ g(y + \ln q) = \lambda g(y). $$
Let $\alpha = \ln\lambda / \ln q$. Then $\lambda = e^{\alpha \ln q}$. Set $h(y) = e^{-\alpha y} g(y)$. Compute
$$ h(y + \ln q) = e^{-\alpha (y+\ln q)} g(y+\ln q) = e^{-\alpha y} e^{-\alpha \ln q} \lambda g(y) = e^{-\alpha y} e^{-\alpha \ln q} e^{\alpha \ln q} g(y) = h(y). $$
Thus $h$ is periodic with period $\ln q$. Write $h(y) = P(y / \ln q)$ where $P$ has period 1. Then
$$ f(x) = g(\ln x) = e^{\alpha \ln x} h(\ln x) = x^{\alpha} P\!\left(\frac{\ln x}{\ln q}\right). \quad \blacksquare $$
Application: The two‑point correlation function of syntactic patterns on the Bruhat‑Tits tree is discrete scale invariant with $q = p$. Therefore any observable derived from it (CMB power spectrum, particle‑mass ratios) exhibits log‑periodic oscillations.
| Result | Significance |
|---|---|
| Confluence (Church‑Rosser) | |
| Distributive law | |
| Uniqueness of normal forms | |
| Projective invariance of cross‑ratio | |
| Strong triangle inequality | |
| Log‑periodicity from discrete scale invariance |
These theorems provide the rigorous mathematical foundation for the Syntactic Token Calculus. All are elementary but non‑trivial; together they guarantee the internal consistency of the STC and its geometric interpretation.