The FPRD view: expose the coordinates

The construction is best understood as a chain of presentations: a real is decoded into an infinite digit stream, streams combine by coordinatewise nim-XOR, and a nested readout returns the result to the half-line. The operation is simple in digit coordinates and necessarily irregular in ordinary Euclidean coordinates. Keeping both presentations visible explains the theorem and its boundary at once.

FPRD-D35 · definition

One operation must do two jobs

A metric-group product is a binary operation $\star$ on $[0,\infty)$ such that the same operation makes a group and defines a metric by $d(x,y)=x\star y$ . The axioms force identity zero, every element to be self-inverse, and the group to be commutative. The remaining numerical condition is

x\star y\le x+y.

So the problem becomes concrete: present a Boolean group law on the half-line whose group distance never exceeds ordinary addition.

FPRD-T115 · explicit construction

Nested digits transport XOR to the half-line

For $c>0$ , put $h_c(t)=t/(c+t)$ and read a digit stream by the infinite nesting

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

For every $c\ge1$ , this is a continuous bijection from $\mathbb N^{\mathbb N}$ onto the half-line. If $D_c=E_c^{-1}$ , define

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

where $\oplus$ is coordinatewise bitwise XOR. Subadditivity of the readout gives $x\star_c y\le x+y$ , hence a metric-group product. The endpoint matters: the truncation tail is $1/(N+1)$ at $c=1$ , while for $c>1$ it is $(c-1)/(c^{N+1}-1)$ .

For $0<c<1$ , the readout is discontinuous and misses $(0,1-c]$ . Thus $c=1$ is sharp for this family, not a universal boundary for all possible constructions.

FPRD-T116 · ordinary regularity

The c=2 product is tame, but not continuous

In ordinary Euclidean coordinates, the binary product is Baire class 1 and Riemann integrable on every compact rectangle. Its discontinuities are exactly

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

Both axes are continuous. Every translation $T_a(x)=a\star_2x$ is cadlag. This is essentially optimal: no metric-group product on the half-line can be separately continuous in the ordinary topology.

The stream decoder is $\Sigma^0_2$ measurable, or first Lebesgue class, but it is not standard Baire class 1. That distinction prevents a false regularity transfer from the product back to its coordinate decoder.

FPRD-T117 · geometry and computation

Intrinsic coordinates are computable; ordinary ones are not

With its intrinsic metric, the c=2 half-line is complete, separable, zero-dimensional, homogeneous, isosceles-free, and has every nonnegative real as a distance. Digit names make the product computable: XOR the names and evaluate the nested readout.

Ordinary Cauchy names tell a different story. A total Type-2 computable operation on those names would be ordinarily continuous, contradicting the forced discontinuity theorem. The same mismatch rules out an o-minimal definition. The object has not changed; the information made persistent by its representation has.

FPRD-T118 · universal bound

Every length-a window detects a jump

For any metric-group product and any $a>0$ , each closed interval $[r,r+a]$ contains a discontinuity of $T_a$ . The length is sharp. Translation profiles also recover the intrinsic distance exactly:

\sup_{b\ge0}|T_b(x)-T_b(y)|=x\star y.

Thus the common continuity and equicontinuity sets are exactly the continuity set of the identity map from ordinary coordinates to the intrinsic metric. The profile identity is classical in origin as specialized here; no novelty claim is attached to it.

FPRD-T119 · exact family classification

The digit support determines every translation jump

Let $\mathrm{Fin}$ be the finitely supported digit streams and $B_c^+=E_c(\mathrm{Fin})\setminus\{0\}$ . For every $c\ge1$ , all translations are cadlag and their discontinuity sets together fill the same countable dense set $B_c^+$ .

The zero parameter has no jumps.
An infinite-support parameter jumps at every point of $B_c^+$ .
A finite nonzero parameter ending at coordinate s has a closed scattered jump set of Cantor–Bendixson height s+1.

If its final digit is $A$ and $r=v_2(A)$ , its largest jump-free interval has length $h_c^s(2^r)$ . The common continuity set is therefore co-countable. For positive integer c, $B_c=\mathbb Q_{\ge0}$ ; that arithmetic identity is not asserted for arbitrary real c.

FPRD-T120 · discrete rigidity

On the nonnegative integers, the metric bound forces XOR

Suppose $\circ$ is any Boolean group law on $\mathbb N_0$ with identity zero and $p\circ q\le p+q$ . Then $\circ$ must be bitwise XOR.

The proof grows the table one binary block at a time. Assume XOR is already forced on $H_k=\{0,\ldots,2^k-1\}$ and put $b=2^k$ . Translation by b injects $H_k$ outside itself, while the metric bound restricts its ith value to at most b+i. The only unused value is b+i. Associativity and exponent two then force the complete XOR table on $H_{k+1}$ .

This proves uniqueness on the discrete carrier, not on the real half-line. Elliott's 1953 Boolean-algebra theorem is nearby prior art under different lattice-valued hypotheses; the bounded search does not establish novelty.

FPRD-T121 · variation boundary

Cadlag does not mean bounded total variation

Every finite-support translator in the nested family is locally of bounded variation. Only finitely many cylinder levels are affected, the tail maps are monotone inside each cylinder, and the jump intervals have bounded overlap.

But one can place nonzero digits sparsely enough that infinitely many disjoint boundary levels each contribute a fixed positive amount of jump mass. The resulting infinite-support translation already has infinite total variation on $[0,1]$ .

This refutes simultaneous local BV for the nested family. It does not classify every infinite-support stream and does not rule out a different metric-group presentation whose translations are all locally BV.

FPRD-T122 · universal dense sections

Irregular sections occur densely in parameter space

For a translation $T_a(x)=a\star x$ , let $D_a$ be its ordinary discontinuity set. For every nonempty ordinary-open interval U, the parameter set

A_U=\{a:D_a\cap U\ne\varnothing\}

is open and dense in the intrinsic metric. Openness follows because positive oscillation survives a small change in the exact uniform profile distance. Density inserts a small translator's forced discontinuity into a compact interval where the current translation is a homeomorphism.

If the intrinsic metric space is Baire, intersecting these sets over a countable ordinary interval base gives an intrinsic dense $G_\delta$ of parameters whose discontinuities are dense on the whole ray.

Without any Baire assumption, the exact conclusion is smaller but still strong:

\forall\eta>0\;\exists a>0\;\exists\epsilon\in(0,\eta) \quad D_a\text{ is dense in }(0,\epsilon).

The same discontinuity set is somewhere dense, nonclosed, non-scattered, and not locally finite. Both a and epsilon may depend on eta; one whole-ray-dense section without intrinsic Baireness remains open.

FPRD-T123 · atomic-score obstruction

Positive coordinate weights cannot label the whole ray uniquely

Let a Boolean support subgroup contain every singleton and give each coordinate a positive weight $\lambda_i$ . If every allowed support A has finite additive score

S(A)=\sum_{i\in A}\lambda_i,

then S cannot be a bijection onto the full half-line. Assuming bijectivity, toggling a singleton of weight u forces its membership at score x to equal $\lfloor x/u\rfloor\bmod2$ . Distinct weights must separate by factors of two, and surjectivity forces the full chain $\{2^n u:n\in\mathbb Z\}$ . The unique supports for u/3 and 2u/3 are alternating binary tails; their symmetric difference then has score u and collides with the singleton of weight u.

This rules out positive coordinate-additive scores with all singletons. Signed scores, interaction terms, non-atomic encodings, and non-coordinate actions remain outside the theorem.

FPRD-T124 · exact open frontier

Simultaneous local BV becomes a regular-action problem

A metric-group product whose every translation is locally BV exists if and only if the half-line admits a regular elementary abelian 2-group H of locally BV permutations. If $h_a(0)=a$ , the selected action must satisfy

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

Evaluation at zero transports such an action to the required Boolean law; the displacement bound gives metric domination. This is an equivalence, not an existence theorem.

Two natural doctrines fail. A locally finite interval-exchange action would make every discontinuity set locally finite, contradicting FPRD-T122. In the ordinary binary architecture, every repair confined to ambiguous dyadic names gives infinite variation to a translator with infinitely many adjacent bit changes. At prefix depth n, each adjacent transition contributes more than one half to the exact finite-prefix variation formula.

Infinite support alone is not enough—the all-one tail admits a finite-variation endpoint repair. Dense jump sets can also have summable jump sizes. A regular locally-BV Boolean action using another ordering or non-coordinate presentation remains neither constructed nor refuted.

Evidence and limits

The written proofs were reconstructed and adversarially audited across the construction, ordinary regularity, universal translation bounds, exact finite/infinite-support classification, integer rigidity, and the bounded-variation frontier. Executable checks replay finite XOR-subadditivity, rational round trips, threshold probes, 15,500 translation-boundary cases, finite XOR uniqueness tables, and the binary variation formula. Those checks are regression evidence, not a mechanical proof of the infinite theorems.

The excursion does not claim the original abstract existence result, external specialist review, or a settled novelty determination for the combined nested-XOR construction.

Metric-group products and ordinary regularity