Algebra changes character with its coordinates

An algebraic operation may be elementary in the coordinates that generate it and highly irregular in the coordinates in which it is observed. Conversely, a canonical-looking algebra of observations may be much larger than the finite controls that actually produce the sequence. FPRD studies both presentations and the map between them.

This is not a claim that algebra is subjective. The group law, formal series, or transformation action is fixed. What changes is which structure is local, which generators are finite, and which regularity survives transport back to the visible carrier.

Worked case · one operation, two roles

A group product that is also its distance

A metric-group product on the nonnegative half-line is one binary operation $\star$ that makes$[0,\infty)$ a group and defines its metric by$d(x,y)=x\star y$. The group axioms force identity zero, exponent two, and commutativity. The numerical condition left to prove is

$$x\star y\le x+y.$$

Thus one operation must simultaneously encode symmetric difference in an abstract Boolean group and obey the triangle inequality in ordinary real coordinates.

Worked case · transport through a presentation

Nested digits make the construction explicit

For $c>0$, define$h_c(t)=t/(c+t)$ and read an infinite digit stream by

$$E_c(a_0,a_1,\ldots) =a_0+h_c\!\left(a_1+h_c\!\left(a_2+\cdots\right)\right).$$

For $c\ge1$, this readout is a continuous bijection from$\mathbb N^{\mathbb N}$ to the half-line. Writing $D_c=E_c^{-1}$, coordinatewise bitwise XOR transports to

$$x\star_c y=E_c\bigl(D_c(x)\mathbin{\oplus}D_c(y)\bigr).$$

Worked case · intrinsic versus ordinary topology

Computable digit dynamics can look discontinuous on the line

With digit names, the $c=2$ product is computed by XOR followed by the nested readout. In ordinary real coordinates, however, its discontinuities are exactly

$$\bigl(\mathbb Q_{>0}\times(0,\infty)\bigr) \cup \bigl((0,\infty)\times\mathbb Q_{>0}\bigr).$$

The product is still Baire class 1 and Riemann integrable on compact rectangles, and each translation$T_a(x)=a\star_2x$ is cadlag. But no metric-group product on the half-line can be separately continuous in the ordinary topology. The same algebra is therefore computable in its intrinsic names and noncomputable from ordinary Cauchy names.

This is a clean representation result: the operation is unchanged, while the available names determine whether its local rule is visible. The decoder itself has a weaker measurability classification than the transported binary product, so regularity is not silently passed back through the coordinates.

Worked case · discrete rigidity

On the integers, the metric inequality forces XOR

The proof grows the operation table through consecutive binary blocks; injectivity and the numerical bound leave exactly the XOR value at each step. This is rigidity on the discrete carrier. It does not classify metric-group products on the real half-line.

Worked case · visible algebra versus control algebra

A finite presentation can cast an infinite algebraic shadow

Let a polynomial/Hadamard automaton over$\{a,b\}$ give both letters the control updates

$$X\longmapsto XY,\qquad Y\longmapsto Y+1, \qquad F(X)=F(Y)=1.$$

Its controls recognize$f(w)=|w|!$ and$g(w)=|w|+1$. Both depend only on word length, so this is already a finite presentation by commutative controls.

The example blocks a tempting normalization test: finite generation of the visible derivative algebra is not necessary for a finite positive presentation. The derivative-algebra failure already appears over a unary alphabet, where commutativity itself is automatic; the two-letter form shows the same obstruction in the nontrivial permutation setting. It does not settle whether every commutative output has some presentation with commutative controls.

Open frontier · regular actions

Better ordinary regularity requires a different presentation

The nested-XOR family contains translations of infinite total variation on $[0,1]$. A metric-group product whose every translation is locally of bounded variation would instead be equivalent to a regular elementary abelian 2-group action by locally BV permutations satisfying

$$|h_a(x)-x|\le a\qquad(a,x\ge0).$$

FPRD-T124 is an equivalence, not an existence theorem. Several natural coordinate doctrines are ruled out, but a different regular locally-BV action is neither constructed nor refuted.

Integrated program · quotients and receipts

The missing datum is often an obstruction class

The Quotient Arsenal now supplies one reusable chain of exact interfaces. Matrix invariants leave a Jordan fibre receipt; a cochain quotient produces a connecting class; graph covers retain monodromy; factorization lifts retain a transported product cokernel; anonymous graph cards retain an equivariant matching defect. In each case the quotient is useful precisely because its loss is named and tested.

$$\text{local lifts sew} \quad\Longleftrightarrow\quad [\text{defect}]=0,$$

The form of the bracket changes—cohomology, cokernel, monodromy, or orbit module—but the proof obligation does not. A vanishing witness must supply an actual correction, and the corrected object must still lie in the intended admissible class.

The matrix calibration makes the distinction concrete. Conjugation invariants remember the characteristic polynomial and hence the eigenvalues with multiplicity. They forget the Jordan partition. That missing partition is recovered exactly from the increments of $\dim\ker(A-\lambda I)^k$. A different question—whether two states have identical outputs $CA^kv$ for every future time—has the universal quotient $V/\bigcap_{k=0}^{n-1}\ker(CA^k)$. These are two legitimate quotients with different declared observations; neither should be mistaken for the other.

In the abelian cochain case, this is the classical connecting homomorphism made constructive. Lift a closed quotient probe, take its differential in the null subcomplex, and test whether that cocycle is a coboundary. A primitive is the correction that produces a closed lift. For nested quotients the second obstruction is a coset, because changing the intermediate lift changes its raw representative.

A finite Čech totalization extends this test from one quotient map to a cover of local presentations. It keeps internal closedness and overlap agreement on separate axes. Strict descent requires a correction in Čech degree zero; a higher-degree primitive supplies coherence data but not necessarily one global section. The same choice issue appears in a cross-axis obstruction staircase: the raw second defect depends on the first primitive, while its quotient class is canonical. For reduction in stages, direct products have zero extension curvature, whereas the real Heisenberg group supplies an explicit nonzero-curvature boundary.

Read the matrix quotient, Jordan receipt, and observation boundary →

Read the cohomological sewing theorem and its exact boundaries →

Read the local-to-global totalization and stage receipts →

Where the area is moving

The area now has two complementary directions: transport explicit algebra through representations while tracking the regularity it loses, and recover finite algebraic presentations from families of observable derivatives, or prove why a proposed recovery doctrine is too small. Rational series, transformation actions, and polynomial loop invariants provide further test cases.

Working answer to the driving question. Computational models generate algebra through their update and observation maps. The intrinsic structure is what survives exact transport; finite generation, continuity, and computability may belong only to one side of that transport and must be proved separately.