Coordinates and conventions

Raw height is the largest absolute weight in a presentation. Canonical height is measured after trimming and pushing the weight of a fixed shortest access word into each state. Both coordinates minimize over machines that agree with the target, including its undefined words, through a finite horizon.

The superscript $1$ means deterministic or sequential realization. Thus $A_f^1(N)$ is minimum state demand, $h_f^1(N)$ is minimum raw height, and $H_f^1(N)$ is minimum canonical height. Exact realization includes rejection of every word on which the partial target is undefined.

A correction is a separate target computed by a bounded-state sequential machine and subtracted before measuring the residual. This is broader than a global length gauge, but narrower than arbitrary changes to the presentation. The correction family is part of every theorem statement below; changing it changes the result.

FPRD-T75 · constructive theorem

Finite-horizon output variation controls raw height

Let $f:\Sigma^*\rightharpoonup G$ have nonempty domain $D$ , where $G$ has a fixed finite symmetric generating set and word norm. Put $D_N=D\cap\Sigma^{\le N}$ . For $u,v\in D_N$ , set

\Lambda_f(N)=\max_{u,v} \frac{|f(u)^{-1}f(v)|}{d_{\rm pref}(u,v)+2}.

Let $\beta_N$ be the maximum, over $p\in\operatorname{Pref}(D_N)$ , of the shortest $|z|$ with $pz\in D_N$ , and choose a fixed shortest domain word $w_*$ . Then

\Lambda_f(N)\le h_f^1(N)\le \max\!\left\{|f(w_*)|,(2\beta_N+3)\Lambda_f(N)\right\}.

The lower bound follows by telescoping a common-prefix computation. For the upper bound, build the full horizon trie and anchor each prefix at a nearby completed domain word. Adjacent anchors are at prefix distance at most $2\beta_N+1$ , so their output difference fits on one trie edge.

Always $\beta_N\le N$ . If the domain has bounded completion radius $B$ , the coefficient becomes independent of the horizon. In particular, a trim $q$ -state NFA gives $B\le q-1$ .

There is also a completion-free uniform bound for every domain:

h_f^1(N)\le |f(w_*)|+24(|w_*|+1)\Lambda_f(N)+1.

Its additional ingredient is Basso and Sidler's Lipschitz extension theorem for tree-valued maps on spaces of Nagata dimension zero. The prefix tree gives extension constant eight; the comparison with $\Lambda_f(N)$ contributes a factor three, and rounding back to a Cayley-graph vertex contributes the final one.

The construction is allowed exponentially many states. It controls raw height with state count free; it is not a state-complexity bound. If $D_N=\varnothing$ , the variation and completion radius are zero and an everywhere-undefined height-zero machine handles that horizon. The coefficient $2\beta_N+3$ is exact for the displayed anchor construction, not proved optimal over all machines.

FPRD-T76 · exact state count and height windows

Two arms transfer group geometry into machine resources

For $N\ge1$ , take the partial target with exact domain $a^*\#\cup a^*\$$ , undefined elsewhere, and values

f(a^m\#)=x^m,\qquad f(a^m\$)=y^m,

write

d_m=x^{-m}y^m,\quad D(N)=\max_{m<N}|d_m|,\quad E(N)=\max_{m<N}\max\{|x^m|,|y^m|\}.

If $d_0,\ldots,d_{N-1}$ are distinct, continuation delays force an exact spine of $N$ nonfinal states plus a final state:

A_f^1(N)=N+1,

\frac{D(N)}4\le h_f^1(N)\le \max\{\min(|x|,|y|),D(N)\},

\frac{E(N)}2\le H_f^1(N)\le E(N).

Raw height sees the delay between the two exits. Canonical height instead sees the absolute values pushed from the unique access spine onto those exits. Delay distinctness is not used in the raw-height window; it is load-bearing for the exact state count and canonical lower bound.

A fixed three-state unambiguous nondeterministic weighted automaton realizes the entire target. The growing value $A_f^1(N)=N+1$ is therefore a deterministic finite-horizon demand, not a representation lower bound across all ambiguity classes.

If $x=y$ , all delays coincide and a two-state sequential machine works at every horizon, so the state and canonical lower bounds require distinct delays. Over $\mathbb Z$ with $x=1,y=0$ , the exact canonical height is $\lceil(N-1)/2\rceil$ , showing that the factor-two lower window can be sharp.

FPRD-T77 · asymptotic separation

Two incompatible drifts defeat one global gauge

In the discrete Heisenberg group $H_3(\mathbb Z)=\langle X,Y,Z\mid Z=[X,Y],\ [X,Z]=[Y,Z]=1\rangle$ , use the exact partial domain $\{0a^m\#0,0a^m\$0,1a^m\#1,1a^m\$1:m\ge0\}$ with values

X^m,\ X^mZ^m,\ Y^m,\ Y^mZ^m.

Under a global length-and-constant gauge $L_{r,c}f(w)=r^{|w|}f(w)c$ , where one pair $(r,c)$ may be chosen for each horizon but must act on every branch, equal-length branch delays remain conjugates of $Z^m$ , whose norm is $\Theta(\sqrt m)$ . Comparing the X- and Y-components in abelianization cancels the common gauge and leaves $m(-1,1)$ . The central estimate is elementary: a word of length $L$ encloses only $O(L^2)$ central area, while rectangular commutators realize $Z^m$ in $O(\sqrt m)$ letters. Therefore, for $N\ge2$ ,

A_f^1(N)=2N,\qquad \widehat h_f(N)=\Theta(\sqrt N),\qquad \widehat H_f(N)=\Theta(N).

This is not invariant under every reasonable correction. A two-mode gauge selected by the first marker, using $(X^{-1},X^2)$ and $(Y^{-1},Y^2)$ , sends both components to the pair $1,Z^m$ and collapses canonical height to square-root scale. An unrestricted letter-emitting transducer makes the proposed coordinate trivial. The theorem is exactly about one global gauge.

FPRD-T78 · robust separation

Every fixed-state sequential correction leaves linear canonical cost

Let $\langle n\rangle$ be the delimiter-terminated binary code of $n$ . On the exact regular support

D=b^+ca^*\#\,\mathrm{Code}\,\mathrm{Code},

define, for $i\ge1$ and $m,r,s\ge0$ ,

f(b^ica^m\#\langle r\rangle\langle s\rangle) =\left\lfloor\frac{m}{i+1}\right\rfloor +\mathbf 1_{(r,s)=(i,m)},

and leave $f$ undefined elsewhere. A horizon machine stores $(i,m)$ , emits one whenever the current $a$ -depth is divisible by $i+1$ , and uses the final bit for the equality test. All weights are zero or one, so $h_f^1(N)\le1$ ; presentation state count is unrestricted.

A permitted baseline is any deterministic sequential integer-weighted machine with at most $k$ states, defined on the target support through the horizon. Its topology and unrestricted weights may be chosen separately for each horizon. Residual raw and canonical demand take separate infima over all such baselines and all exact presentations. For every permitted baseline, two of the first $k+1$ slope addresses reach the same correction state. Their residual values separate linearly, while the challenge rows allow only $O(k^2)$ possible access-word collisions. A linearly large residual value therefore remains on a forced shortest access word:

R^h_{f,k}(N)\le1,\qquad R^H_{f,k}(N)=\Theta_k(N).

The target is finitely described and its support is regular, but it is not max-plus rational at any ambiguity level. Indeed, fixing the nonmatching suffix $\#\langle0\rangle\langle0\rangle$ produces unary slopes $1/(i+1)$ with unbounded reduced denominators, whereas an $n$ -state integer max-plus automaton obtains unary asymptotic slopes from cycle means with denominator at most $n$ . The classical twins theorem also rules out this robust separation for trim unambiguous rational targets with sublinear raw demand. The corresponding finitely ambiguous rational question remains open and requires a different target.

FPRD-T79 · reusable lower-bound criterion

Full-dimensional gradient regions force ambiguity

A $k$ -ambiguous automaton has at most $k$ successful runs on each word. Restrict such a max-plus automaton to the letter-bounded language $a_1^*\cdots a_d^*$ . Cut an accepting run at the block boundaries. On a sufficiently fine eventual product-congruence cell—every block length above one threshold and fixed modulo one common period—each block contributes an affine unary sequence and each boundary-state tuple contributes one affine function of the block lengths.

Distinct active tuples give distinct runs, so at most $k$ affine functions remain. Consequently, if a single cell contains $m$ full-dimensional regions with $m$ different gradients, then the intrinsic ambiguity degree is at least $m$ .

Gaubert's unary rational-series theorem supplies the eventual residue-affine form; finite experiments only corroborate it. The common period need not be bounded by state count: disjoint unary cycles of lengths two and three use five states but have least support period six.

FPRD-T80 · exact intrinsic ambiguity

A three-gradient series has ambiguity degree exactly three

Define

f_{\triangle}(w)=\max\{-3|w|_a,-|w|,-3|w|_b\}.

On $a^x b^y$ , the three open cones $y>2x$ , $x/2<y<2x$ , and $x>2y$ carry gradients $(-3,0)$ , $(-1,-1)$ , and $(0,-3)$ . The gradient criterion excludes every two-ambiguous realization. Three one-state linear components realize the displayed maximum, hence

\operatorname{ambdeg}(f_{\triangle})=3.

The proof uses boundary-state factorization. An earlier path-scheme argument conflated schemes with and without cycle multiplicities and was withdrawn.

Only the three open cones are used in the lower bound; axes and tie rays do not matter. The matching upper bound is the disjoint union of three one-state components with letter weights realizing $-3|w|_a$ , $-|w|$ , and $-3|w|_b$ .

What remains open

Does a finitely ambiguous rational series have bounded raw height but unbounded canonical height after every fixed-state sequential correction?
Which restricted correction families yield a meaningful hierarchy rather than trivializing the target?
Can the gradient criterion produce exact ambiguity degrees for broader natural families?
How do the two-arm height windows align with quantitative twinning and subgroup-distortion results already in the literature?

Sources

No documented external or specialist review of these FPRD results is recorded. Basso–Sidler supplies the metric-extension theorem; Mohri supplies classical normalization context; and Osin is a broader distortion comparator. The finite-horizon bounds, two-arm formulas, and Heisenberg witness are FPRD deductions. Gaubert supplies the unary residue-affine import used by the gradient theorem. Klimann and coauthors supply the unambiguous rational boundary; Bell and Bell–Semukhin are nearby bounded-language results over probabilistic automata, not sources of the gradient criterion.