The cost belongs to a presentation

A reduction arrow, a weighted series, or a tensor decomposition can summarize what is computed while hiding how the computation is represented. FPRD keeps a resource receipt beside the extensional object. Depending on the problem, that receipt may record states, weight height, dependency depth, live storage, fan-out, coefficient growth, or precision.

$$\mathcal R=(\text{state},\text{height},\text{depth}, \text{storage},\text{fan-out},\text{precision},\ldots).$$

The tuple is not a universal definition; it is a discipline. Every bound must say which coordinates it controls, which model measures them, and which coordinates it leaves open. The cases below show why that bookkeeping changes the mathematical question rather than merely lengthening its statement.

Worked case · exact reduction budgets

A frontier is not yet an instantiated reduction

In the retained GapCVP reduction, field size, support, and moment horizon are written as powers$q\asymp N^Q$,$K=N^\kappa$, and$T=N^\tau$. Keeping those coordinates separate reduces all strict exponent obligations to the decisive inequality

$$(1-28c)Q>9+14c.$$

A frontier still leaves a realization obligation. The fully checked point uses

$$c=\frac1{30},\qquad (Q,\kappa,\tau)=(242,34,172).$$

FPRD-T114 carries this tuple through the field, support, reconstruction, radius, parity, and gap-transfer conditions. Thus the area page distinguishes the sharp symbolic boundary from the one concrete end-to-end certificate currently exposed and executable.

Worked case · behavior versus access

Many possible futures must also be reachable

For a binary language, the suffix ball through length$N$ has$V_N=2^{N+1}-1$ words. A prefix can therefore expose at most$2^{V_N}$ different Boolean future rows. One decidable language realizes every one of those rows at every horizon.

Counting behaviors and locating them are different resources. A model may have the full finite behavior space in principle while forcing an exponentially long address before every behavior becomes observable.

Worked case · compact value versus decision

A short powering circuit need not have a short integer output

For an explicitly given rational recurrence and a binary index, the zero question can be normalized to an integer recurrence and then to one selected entry of a binary matrix power. Repeated squaring gives a polynomial-size division-free circuit even though the represented integer may have exponentially many bits.

These are upper bounds for the explicit variable-order input model, not a deterministic polynomial-time algorithm and not a reduction from arbitrary EquSLP. The fixed-order bounded-Skolem literature uses the same circuit-zero subroutine but solves a different quantified problem.

Worked case · normalization cost

Small transition weights can hide large canonical values

Raw height measures the largest weight in a chosen weighted-automaton presentation. Canonical height first trims the machine and pushes the value of a shortest access word into each state. The same finite target can look inexpensive in the first coordinate and expensive in the second.

With state count unrestricted, finite-horizon output variation controls raw height on every partial group-valued target. A completion-radius construction gives an elementary domain-sensitive coefficient, while the Basso–Sidler extension theorem gives a coefficient independent of the domain's completion radius. A separate two-arm family shows why raw and canonical geometry diverge: branch delays control raw height, absolute power growth controls canonical height, and incompatible Heisenberg drifts resist one global gauge but not a marker-dependent two-mode correction. Read the variation and two-arm theorems →

The boundary is already visible in two smaller exact families. A sparse partial parity-comb target has repaired raw height one but repaired canonical height linear in its forced minimum-state comb, for the proved horizons $k\ge3$. Inside the trim unambiguous max-plus class, the classical twins theorem yields a strict finite-horizon height dichotomy: sequential series have bounded demand, while nonsequential series have raw and canonical height $\Theta(N)$. The min-count series shows the hypothesis is sharp: ambiguity degree two has linear state demand but height one from horizon two onward. These are claims about height; nonsequentiality by itself is not used as a state-demand lower bound. Read the repaired gap and ambiguity boundary →

The theorem survives every fixed-state sequential correction in the declared family in canonical height; the raw upper bound uses the zero correction. It is not a theorem about arbitrary gauges. The target is also not max-plus rational at any ambiguity level; whether a finitely ambiguous rational series exhibits the same separation remains open.

A separate letter-bounded theorem turns geometry into an ambiguity lower bound. On each sufficiently fine eventual residue cell, a$k$-ambiguous max-plus automaton is the maximum of at most $k$ affine functions. Therefore $m$full-dimensional regions with different gradients force ambiguity at least $m$. The three-gradient series $\max(-3|w|_a,-|w|,-3|w|_b)$attains the bound exactly. Read the gradient criterion →

Worked case · memory before a late query

Pathwise width does not pay for query-oblivious placement

A future selector can choose one of $2^r$addresses even though every completed computation demands only one of them. If one-way tracks must be placed before the selector appears and the targets are spaced $r+1$apart, a single track can cover at most one target.

The result is fixed-cut and model-specific. Its converse starts from a supplied snapshot and track layout; it does not construct them online or make optimal layouts compatible across multiple cuts.

Worked case · algebraic certificate to executable DAG

One Strassen formula supports different machines

The standard $2\times2$ Strassen certificate fixes seven nonscalar products and eighteen additions or subtractions. It does not fix a schedule. In the declared one-shot scalar-register model, the same formulas admit both

$$(\text{depth},\text{peak live storage})=(5,5) \quad\text{and}\quad(4,6).$$

The faster schedule needs another live register. This small example explains the larger program: rank and arithmetic count do not determine the executable dependency graph.

The same issue persists under composition. For a gadget ideal$I$ and a compatible context ideal$J$, exact live storage is recovered from their two frontiers and the boundary values with pending use on either side. A min–max path computation on these compatible states is therefore a complete sewing law for the declared one-shot model. Strong bisimulation gives one sound quotient; completed storage-observation traces give a strictly coarser sound receipt in the untrimmed case. The completion marker is essential, and the known strictness witness uses a dead gate; the corresponding trimmed-DAG question remains open. See the theorem chain and its model boundary →

This work does not prove $\omega=2$, does not make the compiled bases practical, and does not identify live scalar storage with communication or memory-hierarchy I/O. Its result is a resource-preserving realization of supplied algebraic certificates.

Where the area is moving

Current work asks which coordinates can be controlled simultaneously: reduction parameters without hidden asymptotic debt, residual diversity together with access radius, weighted state together with canonical height and ambiguity, and fast arithmetic together with executable scheduling and precision.

Working answer to the driving question. An algorithm’s cost becomes meaningful only after the representation, permitted normalization, update schedule, and observation interface are fixed. FPRD tries to preserve the strongest theorem that survives that full receipt—and to label every coordinate the theorem does not control.