Proof. For positive \(s\), define \(h_s\) to be affine on the intervals between the following three points, with the indicated values: \[\begin{array}{c|ccc} y&\alpha_{s^{-1}}&\beta_{s^{-1}}&\alpha_{s^{-1}}+1\\ \hline h_s(y)&\beta_s-1&\alpha_s&\beta_s. \end{array}\] Both slopes are positive. Extend by \(h_s(y+1)=h_s(y)+1\), and define \(h_{s^{-1}}=h_s^{-1}\). The complement intervals \([\beta_{s^{-1}}+j,\alpha_{s^{-1}}+j+1)\) map exactly onto \([\alpha_s+j,\beta_s+j)\), proving the set identity for positive letters. Taking complements and inverses proves it for negative letters. At the breakpoints \(|h_s(y)-y|<2\); the difference is affine between breakpoints and periodic. The same displacement bound holds for inverses.

The point \(0\) lies outside all the arc closures. Reading a reduced word from right to left, the first image lies in the interior of the arc of the rightmost letter. At every subsequent step this arc is disjoint from the inverse arc of the next letter, so the image enters the interior of that letter’s arc. This proves (14). ◻

Proof. We may replace \(w\) by its reduced form \(t_1\cdots t_\ell\). Start first at \(j\). Before the first letter the point is outside every arc. If the point before \(t_i\) is outside \(\mathcal A_{t_i}\), its inverse image under \(h_{t_i}\) belongs to \(\mathcal A_{t_i^{-1}}\); this is disjoint from \(\mathcal A_{t_{i+1}}\) because the word is reduced. Thus every indicator \(\mathbf 1_{\mathcal A_{t_i}}\) in (17) is zero, giving the first equality.

Starting instead at \(h_w(0)+j\), the point before \(t_i\) is \(h_{t_i\cdots t_\ell}(0)+j\). By (14) it lies in \(\mathcal A_{t_i}\), so every indicator is one. The sum is \(\ell-n_-=n_+\). ◻

Proof of Proposition 3. Fix a norm \(N\) as in the proposition. We first construct labels with a common nonzero norm and normalize them at the end. For \(k\geq1\), put \[D_k=N(\underbrace{1,-1,\ldots,1,-1}_{k\text{ pairs}}).\] These numbers are positive, and spreading invariance and the triangle inequality give \(D_{k+l}\leq D_k+D_l\).

Step 1: the finite chart chain. Fix an integer \(n\geq1\), and let \[L=8(n+1),\qquad \epsilon=\frac1{10n}.\] Use the homeomorphisms of Lemma 7. The landmarks below record the interval endpoints used by labels and all orbit points needed to follow \(n\) steps. The integer shifts provide a margin around the label window. Let \(\Lambda\) be the following finite subset of \(\mathbb R\), in its usual order: \[ \begin{split} \Lambda={}&\{\alpha_s+j,\beta_s+j:s\in\mathcal S,\ 0\leq j<L\}\\ &\quad\cup \{h_w(0)+j: |w|\leq n,\ -2n\leq j\leq L+2n\}. \end{split} \tag{19}\] All displayed shifts \(j\) are integers, \(|w|\) denotes the length before reduction, the empty word is included, and coincident points are counted only once. For each \(s\) use the partial order-preserving bijection \[T_s=h_s|_{D_s},\qquad D_s=\{a\in\Lambda:h_s(a)\in\Lambda\}.\] Its inverse is \(T_{s^{-1}}\). Proposition 6 gives a finite stationary reversible marked chain on charts; denote its stationary chart law by \(\nu\). Call a transition \((C,s,C')\) good when \[ C'(a)=C(h_s(a))\qquad(a\in D_s). \tag{20}\] Every step is good with probability at least \(1-\epsilon\).

For positive \(s\), define \[ g(C,s,C')=\sum_{j=0}^{L-1} \bigl(b_{C(\alpha_s+j)}-b_{C(\beta_s+j)}\bigr), \tag{21}\] and define negative labels by oddness: \(g(C,s,C')=-g(C',s^{-1},C)\). Strict increase of the charts and spreading invariance imply \[ N(g(C,s,C'))=D_L \tag{22}\] for every transition, good or otherwise.

Step 2: transport along a good trajectory. Consider a trajectory \((C_{i-1},s_i,C_i)_{i=1}^n\) all of whose steps are good, and let \(w=s_1\cdots s_n\) have reduced length \(\ell>0\). Put \[m=\lfloor h_w(0)\rfloor,\qquad z=h_w(0)-m.\] Since every letter moves every point by less than \(2\), \(|h_w(0)|<2n\), so \(-2n\leq m\leq2n-1\); and (14) gives \(0<z<1\). For the integers \(L/4\leq j\leq3L/4\), define \[ r_j=C_0(j),\qquad t_j=C_0(z+j). \tag{23}\] These points belong to \(\Lambda\) by (19), and \(r_j<t_j<r_{j+1}\) whenever the indices occur.

For either starting point \(y_0=j\) or \(y_0=z+j\), follow \(y_i=h_{s_i}^{-1}(y_{i-1})\). For the first family, \[y_i=h_{s_i^{-1}\cdots s_1^{-1}}(0)+j;\] for the second, \[y_i=h_{s_{i+1}\cdots s_n}(0)+j-m.\] The words have length at most \(n\), and the shifts lie in \([-2n,L+2n]\). Thus all these points belong to \(\Lambda\). Moreover, for every \(0\leq i\leq n\), \[ j-2n<y_i<j+1+2n, \tag{24}\] and the whole interval lies in \((0,L)\) for the chosen range of \(j\). Because \(h_{s_i}(y_i)=y_{i-1}\), goodness gives \[ C_i(y_i)=C_{i-1}(y_{i-1})=C_0(y_0). \tag{25}\]

Step 3: cumulative coefficient sums. Let \[G=\sum_{i=1}^n g(C_{i-1},s_i,C_i),\qquad F(k)=\sum_{a=1}^k G_a\quad(0\leq k\leq B).\] Every label has total coefficient sum zero, so \(F(0)=F(B)=0\). For landmarks \(\alpha<\beta\) and \(y\), strict increase of a chart gives \[\sum_{a\leq C(y)}(b_{C(\alpha)}-b_{C(\beta)})_a =\mathbf 1_{[\alpha,\beta)}(y).\] This explains the half-open interval convention. Evaluate \(F\) at a transported rank \(C_0(y_0)\). For a positive step, (21), (25), and \(y_{i-1}\in(0,L)\) therefore give the contribution \(\mathbf 1_{\mathcal A_{s_i}}(y_{i-1})\). For a negative step the label is read in chart \(C_i\). Since \(y_i\in(0,L)\) as well, its contribution is \[-\mathbf 1_{\mathcal A_{s_i^{-1}}}(y_i) =\mathbf 1_{\mathcal A_{s_i}}(y_{i-1})-1.\] Thus in both cases the contribution is \(\chi_{s_i}(y_{i-1})\), and \[ F(C_0(y_0))=c_w(y_0). \tag{26}\] Reduction is applied to this scalar expression; \(G\) retains all \(n\) original edge labels. Let \(n_+,n_-\) be the numbers of positive and negative letters in the reduced word. Lemma 8, applied with the integer shifts \(j\) and \(j-m\), now gives \[ F(r_j)=V:=-n_-,\qquad F(t_j)=U:=n_+, \qquad U-V=\ell. \tag{27}\]

Step 4: converting oscillations to norm. Put \(a=L/4\) and \(b=3L/4\). The selected cuts and their prefix sums determine every consecutive block sum: \[\begin{array}{c|cccccccc} \text{cut}&0&r_a&t_a&r_{a+1}&\cdots&t_{b-1}&r_b&B\\ F&0&V&U&V&\cdots&U&V&0\\ \text{block sum}& &V&\ell&-\ell&\cdots&\ell&-\ell&-V \end{array}\] The last row gives the sum of the block ending at that cut. There are \(b-a=L/2\) inner pairs. Replacing every block by its sum is contractive and yields \[ (V,\underbrace{\ell,-\ell,\ldots,\ell,-\ell}_{L/2\text{ pairs}},-V). \tag{28}\] The last block is nonempty, since the rank \(t_b\) from (23) satisfies \(r_b<t_b\leq B\). We control the endpoint contribution by combining the estimate from all oscillations with the estimate from a single level. Subtracting the endpoint array by the triangle inequality and using \(|V|\leq\ell\) gives \[N(G)\geq\ell D_{L/2}-|V|D_1 \geq\ell(D_{L/2}-D_1).\] Independently, partitioning just at an \(r_j\) or a \(t_j\) gives the arrays \((V,-V)\) and \((U,-U)\). Hence \[N(G)\geq\max\{|V|,U\}D_1\geq\frac\ell2D_1.\] For nonnegative \(A,B\), \(\max\{A-B,B/2\}\geq A/3\). Together with \(D_L\leq2D_{L/2}\), this proves \[ N(G)\geq\frac\ell3D_{L/2}\geq\frac\ell6D_L. \tag{29}\]

Step 5: probability and normalization. The probability that at least one transition fails to be good is at most \(n\epsilon=1/10\), by stationarity and the union bound. Without conditioning on goodness, the letters are independent and uniform in \(\mathcal S\). As letters are appended, the reduced length has conditional expected increment \(1/2\) when positive, since exactly one of the four letters cancels its last letter; at zero the increment is \(1\). Thus \(\mathbb E\ell\geq n/2\). Since \(0\leq\ell\leq n\), \[\frac n2\leq\mathbb E\ell \leq\frac n4+n\mathbb P\{\ell\geq n/4\}, \qquad\text{so}\qquad \mathbb P\{\ell\geq n/4\}\geq\frac14.\] With probability at least \(1/4-1/10=3/20\), all steps are good and \(\ell\geq n/4\). On this event (29) gives \(N(G)\geq nD_L/24\). Every edge label has norm \(D_L>0\) by (22), so dividing all labels by \(D_L\) proves the proposition. ◻

Proof of Lemma 2. By (2), there is a nonreflexive Banach space \(Y\) finitely representable in \(X\). Identify \(Y\) with its canonical image in \(Y^{**}\), and choose \(v\in B_{Y^{**}}\) and \(d>0\) with \(\operatorname{dist}(v,Y)>d\).

The first limit will produce a spreading norm \(N_0\), finitely representable in \(Y\), that controls coefficient prefix sums. A second limit averages increasingly fine subdivisions. It preserves these properties and makes the resulting norm \(N\) invariant under subdivision. Same-sign merging then preserves \(N\), and convexity gives contractive merging for arbitrary adjacent coefficients.

Step 1: finite triangular arrays. For every integer \(M\geq1\), we construct \(y_1,\ldots,y_M\in B_Y\) and \(f_1,\ldots,f_M\in B_{Y^*}\) such that \[ f_i(y_j)=0\quad(j<i),\qquad |f_i(y_j)-d|\leq M^{-1}\quad(j\geq i). \tag{30}\] Suppose \(y_1,\ldots,y_{i-1}\) have been chosen and let \(E=\operatorname{span}\{y_1,\ldots,y_{i-1}\}\). The restriction map \(Y^{**}\to(E^\perp)^*\) is a quotient map with kernel the canonical copy of \(E\): Hahn–Banach extends every bounded functional on \(E^\perp\subseteq Y^*\) to all of \(Y^*\) with the same norm, while finite dimensionality gives \((E^\perp)^\perp=E\). Thus the quotient norm of \(v+E\) equals the norm of its restriction to \(E^\perp\). Hence \[\left\lVert v|_{E^\perp}\right\rVert=\operatorname{dist}(v,E)\geq\operatorname{dist}(v,Y)>d.\] We may therefore choose \(f_i\in B_{Y^*}\cap E^\perp\) with \(v(f_i)=d\), by changing a sign and rescaling if necessary. The finite-dimensional form of Goldstine’s theorem gives \(y_i\in B_Y\) with \(|f_h(y_i)-v(f_h)|\leq M^{-1}\) for \(1\leq h\leq i\). Indeed, \((v(f_1),\ldots,v(f_i))\) belongs to the closure of \(\{(f_1(y),\ldots,f_i(y)):y\in B_Y\}\): otherwise a separating linear functional would give \(v(f)>\sup_{y\in B_Y}f(y)=\|f\|\) for some \(f\in\operatorname{span}\{f_1,\ldots,f_i\}\), contradicting \(\|v\|\leq1\). This proves (30) inductively.

Read the vectors in the reverse order. For any subtuple \(y_{i_1},\ldots,y_{i_m}\) with \(i_1>\cdots>i_m\), the functional \(f_{i_j}\) annihilates the terms after the \(j\)th and approximates \(d\) on the first \(j\) terms. Consequently, \[ d\max_{1\leq j\leq m}\left\lvert \sum_{h=1}^j a_h\right\rvert -\frac1M\sum_{h=1}^m|a_h| \leq\left\lVert \sum_{h=1}^m a_h y_{i_h}\right\rVert \leq\sum_{h=1}^m|a_h|. \tag{31}\]

Step 2: a spreading limit. Fix \(k\). For each \(m\leq k\), choose a finite grid in \([-1,1]^m\) with mesh at most \(k^{-2}\) in the maximum norm. Color every increasing \(m\)-subtuple of the reversed array by the list of the norms of its grid combinations, rounded into intervals of length \(k^{-1}\). The number of colors is finite and independent of \(M\). The finite Ramsey theorem, applied successively for \(m=1,\ldots,k\), shows that, for sufficiently large \(M=M_k\geq k^3\), there is a tuple \((x_1^{(k)},\ldots,x_k^{(k)})\) on which all these colorings are homogeneous. At each arity, thin a previously homogeneous set; homogeneity from earlier arities persists under this thinning. For fixed \(m\leq k\) and \(a\in[-1,1]^m\), the norms of the combinations with coefficients \(a\) on any two increasing \(m\)-subtuples differ by at most \[k^{-1}+2mk^{-2}\leq3k^{-1}.\] The additional error follows from the upper bound in (31).

Taking a diagonal subsequence over all finite rational coefficient tuples, and then using the common \(\ell_1\) Lipschitz bound, gives limits \[N_0(a_1,\ldots,a_m) =\lim_{k\to\infty}\left\lVert \sum_{i=1}^m a_i x_i^{(k)}\right\rVert.\] The limits are compatible with appending zeros. The vanishing subtuple oscillations give (3) for \(N_0\). Passing to the limit in (31) yields \[ d\max_j\left\lvert \sum_{i=1}^j a_i\right\rvert\leq N_0(a)\leq\sum_i|a_i|. \tag{32}\] The norm axioms pass to the limit, and the lower bound gives positive definiteness. On each fixed coefficient cube the convergence is uniform, by the common Lipschitz bound and compactness. Moreover, \[ N_0(a)\geq\frac d2\max_i|a_i|, \tag{33}\] since every coefficient is the difference of two successive prefix sums. Thus the maps \(a\mapsto\sum_i a_i x_i^{(k)}\) give linear embeddings of every fixed coordinate span into \(Y\) with distortions tending to \(1\).

Step 3: averaging subdivisions. For positive integers \(J_1,\ldots,J_m\), let \(F_{\boldsymbol J}a\) be the array obtained by replacing \(a_i\) by \(J_i\) copies of \(a_i/J_i\). Write \(F_J\) when all multiplicities equal \(J\). If \(J_i\leq M_i\) for every \(i\), then \[ N_0(F_{\boldsymbol M}a)\leq N_0(F_{\boldsymbol J}a). \tag{34}\] Indeed, in the \(i\)th block of length \(M_i\), select a uniformly random \(J_i\)-element subset and place \(a_i/J_i\) in those positions, with zeros elsewhere. Every resulting array is obtained from \(F_{\boldsymbol J}a\) by inserting zeros, so has the same \(N_0\) norm. Its mean is \(F_{\boldsymbol M}a\), and convexity proves the claim.

It follows that \[ N(a)=\lim_{J\to\infty}N_0(F_Ja) \tag{35}\] exists. The bounds in (32) survive subdivision, because the original prefix sums still occur at the ends of the blocks. Consequently, for every \(a=(a_1,\ldots,a_m)\), \[ d\max_{1\leq j\leq m}\left\lvert \sum_{i=1}^j a_i\right\rvert \leq N(a)\leq\sum_{i=1}^m|a_i|. \tag{36}\] Spreading invariance and the norm axioms also pass to the limit. The functions \(a\mapsto N_0(F_Ja)\) again have the common \(\ell_1\) Lipschitz bound, so convergence is uniform on every fixed coefficient cube. The lower bound (33) also holds for \(N\). Fix a source dimension \(m\) and \(\eta>0\). First choose \(J\) so that \(F_J\) embeds the \(m\)-dimensional \(N\) span into the \(mJ\)-dimensional \(N_0\) span with distortion less than \(1+\eta\). Indeed, uniform additive convergence on the compact \(N\)-unit sphere, together with (33), gives uniform relative error. Approximate this now fixed \(N_0\) span in \(Y\) with distortion less than \(1+\eta\), and then embed its finite-dimensional image into \(X\) with distortion less than \(1+\eta\). The composite distortion is less than \((1+\eta)^3\); letting \(\eta\downarrow0\) proves (iii).

Step 4: consecutive summation. First, \(N\) is unchanged by every fixed varying-multiplicity subdivision: \[ N(F_{\boldsymbol J}a)=N(a). \tag{37}\] To see this, put \(J_- =\min_iJ_i\) and \(J_+=\max_iJ_i\). The composite \(F_rF_{\boldsymbol J}\) replaces each \(a_i\) by \(rJ_i\) equal pieces. Thus, for every \(r\), \[N_0(F_{rJ_+}a) \leq N_0(F_rF_{\boldsymbol J}a) \leq N_0(F_{rJ_-}a)\] by (34); let \(r\to\infty\).

If adjacent nonzero coefficients have the same sign and rational ratio, subdivide them into equal pieces. Subdividing their sum into the combined number of pieces gives the identical array. Equation (37) shows that merging these coefficients preserves \(N\). Continuity extends this statement to every same-sign pair, and spreading invariance handles zeros.

Fix all other coefficients and fix \(s\ne0\). The function \[t\longmapsto N(\ldots,t,s-t,\ldots)\] is convex and is constant, at the merged norm \(N(\ldots,s,\ldots)\), on the nondegenerate interval with endpoints \(0\) and \(s\). A convex function constant on a nondegenerate interval is at least that constant everywhere: a smaller exterior value, together with the farther endpoint, would force the nearer endpoint below that constant. This proves (4) when \(s\ne0\). Letting \(s\to0\) proves it for \(s=0\) as well. ◻