The output does not determine the computation

Two machines can compute the same function while retaining different state, exposing different intermediate observations, and paying for different kinds of access. FPRD treats the extensional behavior and its presentation as separate mathematical objects.

$$\text{same input-output map} \;\not\Rightarrow\; \text{same state, schedule, access, or resource demand}.$$

The point is not to multiply descriptions. It is to identify which features survive a change of model and which apparent lower bounds disappear when the object is presented differently. The examples below range from topological sewing to paper-folding geometry and schedule-independent logic gates.

Worked case · finite observations and one model

Compact finite feasibility sews into one presentation

Let $X$ be a Hausdorff behavior space, let $q_N:X\to X_N$ record observations through horizon $N$, and let $S_r\subseteq X$ be the behaviors realizable at resource bound$r$. When the matching cylinders form a neighborhood basis, finite fitting has an exact topological meaning:

$$r\in\bigcap_{N\ge0}F_x(N) \quad\Longleftrightarrow\quad x\in\overline{S_r}.$$

The proof is short and uses classical topology. Its importance to FPRD is organizational: it identifies the precise condition under which the quantifiers $\forall N\,\exists P_N$can be exchanged for $\exists P\,\forall N$. Compactness is sufficient here; it is not automatic for every resource stratum.

Worked case · failure of sewing

When finite fits keep leaving every compact core

Suppose one resource stratum is exhausted by increasing compact cores$S_{r,t}\uparrow S_r$. Its escape modulus $e_x(r,t)$ is the first horizon at which core $t$ can no longer imitate the target.

$$\sup_t e_x(r,t)=\infty \quad\text{but}\quad e_x(r,t)<\infty\ \text{for every fixed }t$$

is exactly the pattern “every horizon can be fitted at resource$r$, but no one compact core contains an exact fit.” This does not prove that the resource coordinate itself must grow; it localizes the failure to escape within a possibly noncompact stratum.

The target is unchanged. Allowing two sequential components makes the stationary presentation tiny; insisting on one component forces both state and canonical height to escape linearly.

Worked case · geometry as a computation model

A folded strip presents a Boolean formula

In the literal-local Fold-and-Look model, input bits open or close prepared feature sites on a strip. A fixed sequence of all-layers folds stacks material, and one final Look asks whether some location is open through every layer. Under the model’s alignment, interiority, and uniform-overlay hypotheses, the geometry has the incidence normal form

$$\operatorname{Look}(w) =\bigvee_{y\in Y_{\mathrm{eff}}} \ \bigwedge_{p\in\Phi(y)}\ell_p(w).$$

Each final fibre becomes one conjunction and the possible through-hole locations become a disjunction. Geometry therefore exposes an exact DNF, letting formula width, term count, and literal occurrences lower-bound layer depth, effective fibres, and feature sites.

Read the model, normal form, sharp bounds, and reflection-line correction →

The previously claimed LIS-based fold lower bound is withdrawn. A legal three-layer fibre can reduce to a width-two term even when its two surviving literal sites have the same orientation. The exact matching-language row count remains proved; the geometric separation is unresolved.

The inequality couples material and precision; it does not prove that material length alone is exponential. The final Look fan-in and input encoding work remain uncharged in this model.

Worked case · abelian computation

Some computations can erase their schedule exactly

Abelian logic gates are a positive control for chronology. Their local operations may occur in different legal orders while the final output is unchanged. One resource question is floor depth: how many nested rounds of addition and integer floors are needed to present an increasing ZILP function?

The finite period-three search found the witness, while the proof rules out every positive-denominator depth-one presentation. Literature and priority comparison remain incomplete; the site does not claim that the separation is externally new.

Worked case · finite semantic obstruction

Even a finite carrier may lack a greatest causal factor

On the five-site cyclic Boolean carrier, a future-row class is exactly an archive mask. A mask is a causal factor of a target when a dihedral motion can place it inside that target. For

$$C_1=\{0,1,2\},\qquad C_2=\{0,1,3\},$$

the edge $E=\{0,1\}$ and diagonal$F=\{0,2\}$ are nonassociated maximal common factors. Neither dominates the other, so no greatest common causal semantic factor exists.

The failure comes from allowing an independent residual frame for each target. Pin one common frame and intersection restores the meet. This is a five-site obstruction in the stated cyclic family, not a theorem about every typed or physical factorization model. Read the finite certificate →

Where the area is moving

Current work looks for exact interfaces: when finite observations sew, when schedule can be erased, when geometry carries a formula, and when semantic factors require a shared coordinate frame. The same discipline applies across machine, algebraic, and physical-looking models.

Working answer to the driving question. A computational feature is intrinsic when it survives every exact presentation change allowed by the comparison. State counts, chronology, geometry, and resource growth are presentation-dependent until a transfer theorem—or a representation-independent obstruction—shows otherwise.