What Channels Cannot See

The CRT decomposition has exact limits, and they are as structural as the decomposition itself. This page charts them at the k = 7 demo rung, Z/510,510, where every element is a 7-tuple of remainders mod 2, 3, 5, 7, 11, 13, 17 — small enough that every claim below is proved or checked exhaustively over a stated range. The chart: what no channel reads and what buying it back costs, what a second coordinate system buys and what it taxes, why a third does not exist, what kind of size a channel can compute, what replaces the ordering of the integers once size is gone, and how far — and at what price — the replacement can be read.

The size wall

When the modulus is a product of distinct primes — every rung of the tower — a function is computable channel-by-channel exactly when it is a polynomial — some mix of ring additions and multiplications (proved). Every question of size — sign, comparison, overflow — falls outside: none of it is channel-computable at any rung with two or more channels, and no per-channel recoding brings it into view. Any coordinate system that keeps addition parallel hits the same wall.

The hiding is exact, not approximate. Fix six of the seven residues of an unknown element and leave mod 17 open: 17 candidates remain, spaced 30,030 apart — an even ladder across the whole ring, splitting the two halves 9 to 8. Leave the mod-2 channel open instead and the two candidates split the halves exactly. Six channels of seven, and the size question is still essentially a coin flip. Positional digits carry size; residues provably cannot. And the hiding is not the tower's doing: in every cyclic ring — prime-power moduli included — knowing an element only by its remainder mod a proper divisor of the modulus leaves candidates from both halves, and the split is exactly even whenever the forgotten factor is even (checked for every modulus to 240; the 9-to-8 versus exact-even contrast above is that rule in action); what the rungs specifically buy is where this section opened — channel-by-channel computable = polynomial.

The price of size

The wall is exact, but it is a price, not a prohibition. Every known exact route to size pays a stated cost — an exact fraction summed at the ring's full width, or a digit-by-digit conversion whose digits chain, depth seven here — and the third buys comparison back not by recoding the channels, which the wall forbids, but by adding one. The quotient sum ⌊x/2⌋ + ⌊x/3⌋ + … + ⌊x/17⌋ never decreases as x climbs, and — the useful surprise — it is computable channel by channel modulo a fresh modulus: 716,167, the sum of the seven cofactors 510,510/p, which is coprime to the ring automatically (proved, for every set of distinct primes). Size becomes one bolted-on extra channel. The sort it induces has ties — the sum stays flat from x to x + 1 exactly when x + 1 has no zero residue, 92,160 adjacent pairs here — and the mod-2 residue, already among the stored residues, breaks every one: quotient sum and parity together make a key that orders the ring strictly (checked exhaustively at this rung). Sign is then one comparison of the key against a fixed constant; overflow, one comparison of the wrapped sum's key against an operand's.

The price is the new channel's size. Its modulus is the ring's times the sum of the channel primes' reciprocals — 1.4 times 510,510 here — so the one channel carrying size is wider than the seven it serves combined. Buying size back beats rebuilding the integer exactly when that reciprocal sum stays below 1, and on the tower it never does: 1/2 + 1/3 + 1/5 already exceeds 1, so every rung from the third up keeps its wall. A ring designed instead from a few large primes flips the verdict — eight channels of 16 to 32 bits put the comparison key about a tenth narrower than full reconstruction, with every channel still a prime field. The deleted window can be bought back exactly; on the tower's own rungs the price always exceeds the ring.

The second coordinate system

Multiplication, on the other hand, has a second exact coordinate system. Within each channel the nonzero residues are the powers of one base — mod 7 the powers of 3 run 1, 3, 2, 6, 4, 5 — so rewriting residues as exponents turns multiplication into addition, the slide-rule move: 2 × 6 becomes exponents 2 + 3 = 5, and 3⁵ ≡ 5 ≡ 2 × 6 (mod 7). Composed across channels — for the invertible elements, the 92,160 of 510,510 with no zero residue — one multiplication is seven small exponent additions, powering is exponent scaling, and the multiplicative order of an element is a per-channel gcd read (verified exhaustively at k = 7). The price flips: in exponent coordinates addition is the operation that needs a lookup table — one per channel, 44 entries in all here (Zech's logarithm, the classical device).

Each pair of demands has its coordinate system; the three together have none. Residues run + and × in parallel but hide size. Exponents make × and powering linear but tax +. Positional digits show size but chain every carry. No coordinate system hosts all three — at Z/30, the small rung where it can be checked completely, the census of coordinate systems is exhaustive.

The ladder ends at two rungs

The two moves do not extend to a third. The residue split turns multiplication into channel-by-channel work, at the price of size; the exponent rewrite turns multiplication into addition, at the price of the lookup table; a third move — a coordinate system in which powering becomes addition — generically does not exist. Powering is not a ring operation to begin with: exponents live in a smaller companion ring — here, raising any element to the power a + 240 gives the same result as the power a (for exponents ≥ 1) — so the question restarts one ring down. There, channel by channel, the move exists only when the channel's prime p has 8 not dividing p − 1: three quarters of primes pass, one in four is blocked, and at k = 7 the mod-17 channel already fails. And the operation itself runs out of values: power towers — x to the x to the x, floor on floor — reduced mod 510,510 stop changing once four floors are up, for any base x ≥ 2 (computed by a recursion checked against exact integer towers), because the chain of companion rings 510,510 → 240 → 4 → 2 → 1 bottoms out. A third coordinate would have almost nothing left to coordinate. A mirror world isolates what the first wall is made of: in polynomial arithmetic, where multiplication has no carries, size is exactly as invisible to channels — but the workarounds that read size back, silent-failure-prone over the integers, either answer exactly or fail in plain sight there (checked completely at a small ring). Carries are where silent size errors live. Multiplication can be rewritten all the way down to addition; powering, here, cannot: the ladder ends at two rungs.

Three walls, one question

The three walls look unrelated — size is unreadable, the exponent table is uncompressible, the third coordinate is missing — but they are one statement with one parameter: through which residue window can the operation be read? Size: through none — the reading it lived in is the one the construction deletes, and no channel table of any size substitutes (an information obstruction). The exponent table: addition, rewritten into exponents, cannot be read through any window one ring down — within each channel, the lookup table is incompatible with every nontrivial proper quotient of that channel's exponent range p − 1 (proved, for every prime p ≥ 5 and every such quotient, composite ones included), so it resists every compression (a structure obstruction). The third coordinate: blocked one ring down by the window at 2 — a channel's exponent range is not a clean prime field once 8 divides p − 1, and exactly that thickness forbids the move (an existence obstruction). Three failure modes, one question, asked at level zero, then twice one level down.

The size a channel can see

One floor up, the wall has a counterweight. Take 2×2 matrices with entries in the ring: the determinant is a polynomial in the entries, so by the criterion above it is channel-computable — a genuinely size-like, multiplicative quantity (it composes, it gates invertibility) living inside the locality class where integer size is the wall. It is the multiplicative half of size only — no remainder arithmetic recovers an ordering — but at every odd channel it is the whole of what survives: every structure-preserving reading of an odd channel's invertible matrices that lands in commutative territory factors through the determinant (its kernel is exactly the commutator subgroup — verified at every odd channel of the demo rung; the group fact is classical). The mod-2 channel is the exception here as elsewhere: its matrix group is too small for the determinant to say anything.

The same floor shows the deleted window from the other side. A quaternion layer over the ring can be designed: choose which channels go strange — instead of splitting into 2×2 matrices, a chosen channel fattens into a field of p² elements — and the choice is free: {3, 5}, {7, 11}, even a single channel. A classical conservation law says the set of strange places always has even size — but it counts all the readings of a number, the deleted one included. A visible strange-set of size one is therefore possible: its partner sits at the deleted window (computed: four designed algebras, every verdict at all 7 channels as predicted, 28 of 28; the bookkeeping behind the conservation law is classical, the channel pattern is the tower-side reading).

What replaces order

With size deleted, the ordering of the integers is gone — and the loss is total, not partial. Test any relation by asking whether it equals the AND of its per-channel readings: less-than fails maximally — every channel's reading saturates, so the channels jointly retain nothing of the order (exhaustive at Z/210) — and arranging the ring on a circle instead of a line buys nothing, since the circular relation carries the line's order inside it. The test has a communication reading. Split the channels into one versus the rest and lay the relation out as a table — rows for everything the chosen channel sees of the two elements, columns for everything the other channels see. A relation reads channel-by-channel exactly when every such table is one solid rectangle, some set of rows times some set of columns (proved) — in communication terms, the shape of a question that needs no conversation. Less-than's tables never are: not at any split of Z/30 (measured whole), and at the two-channel ring Z/6, where the whole table is 4 by 9, covering its yes-entries takes four rectangles — the most a table that size can need (computed exactly). Deciding order across a channel split is a conversation between the channels, and the ring runs no wire between them. What passes the test is divisibility, and it reads exactly one bit per channel: zero here, or not. The zero pattern — which channels an element has died in — is the surviving skeleton: where you live, not how big you are.

That skeleton supports a logic. Write AND as ab, OR as a + b − ab, NOT as 1 − a: ring polynomials, so all three run channel-by-channel. On elements whose every residue is 0 or 1 — the zero patterns themselves — they are exact Boolean logic; every other element is a graded truth value, a residue that is neither 0 nor 1 being a channel only partly true. Two measurements read the grade, and both are ring operations: raising to the power 240 — the universal exponent the ladder section met — collapses each residue to 0 or 1 and answers “true to some degree?” channel by channel (mod 7, the graded residue 3 collapses to 3²⁴⁰ ≡ 1), while the mirrored map 1 − (1 − a)²⁴⁰ answers “fully true?” (the same 3 reads 0). Each is the other's reflection through NOT, exactly, and the pair behaves as the possibility and necessity of the modal logic S5 — its laws, taken in equation form (the graded elements carry no order for the textbook form to live in), hold exactly here — quantifying the degree of truth inside a channel. The other quantifier — across the channels: compress “is this element zero?” into a single 0-or-1 answer inside the ring — does not exist: no polynomial computes it over any composite modulus (proved), while a single prime field has it as Fermat's 1 − a^p−1. Degree quantifiers come free; one bit quantified across the channels costs the deleted window. It is the size wall's question again, asked of logic.

The pair turns measurement into certificates. Take any formula built from AND, OR and NOT over graded inputs, pushing the NOTs down onto the inputs first (De Morgan holds identically here, so the value does not change). Its honest verdict, channel by channel, is what the two measurements say of the formula evaluated in the ring. Now measure each input instead — two bits per input — and evaluate the formula twice as plain Boolean arithmetic: once generously, reading every plain input by its some-degree bit and every negated one as “not fully true”, and once strictly (fully-true, “not true to any degree”). The verdict's two bits land between the two evaluations in every channel (proved): a generous 0 certifies the formula false there, a strict 1 certifies it fully true, and where the two agree the verdict is settled outright — from two bits per input, without revisiting the elements. The slack between them lives only in channels where some input was already graded; channels where every input is classical are always certified. And one bit is not enough: the some-degree bit alone can certify only falsity, the fully-true bit alone only truth — read the some-degree bit as the whole answer and it errs at a nonzero rate, enumerated exactly. The two-sided decision needs the pair.

The two measurements are the ends of a family. For each divisor m of 240, raise to the power m and ask “fully true?” of the result: the answer, channel by channel, is one bit — does the residue's multiplicative order divide m? — a zero residue, which has no order, reading no (proved). At m = 1 this is the fully-true bit; at m = 240, where every order divides, the some-degree bit; between them sit the rest — one gate per divisor, twenty in all here, all distinct (computed), ordered as the divisors are and combining as gcds do: the AND of two gates is exactly the gate at their gcd (proved). Repetition is this family in disguise: “fully true?” of a AND a AND … AND a, m copies evaluated in the ring, is the order-divides-m gate, so what repeating an input adds to its two bits is its order and nothing else. And the family decides what the pair cannot. The formula a AND NOT a, evaluated in the ring, is fully true exactly at residues satisfying x − x² = 1 — primitive sixth roots of unity wherever a channel has them (mod 7 and mod 13 here), plus one degenerate point in the mod-3 channel — while its strict evaluation from the two bits per input is identically zero: the pair can never certify it. The order gates decide it in every channel with zero errors (proved): order divides 6 and neither 2 nor 3 — except in the mod-3 channel, where the equation's one truth point is the channel's only graded residue, read straight off the family's two ends: some degree, not fully.

With two inputs the gates read words. Take any product of powers of a and b — ab, a²b, ab⁻¹ — and ask the order gates of it. Within each channel, the full profile of answers measures the two positions exactly up to one shared rescaling: a second pair of residues answers every gate of every word identically precisely when it is (a^u, b^u) for a single invertible exponent u applied to both (proved — a fact about cyclic groups, and not special to two inputs); across channels the rescalings are independent, one per window, as everything here is. What a second input genuinely adds is relative position: where a generates a channel's nonzero residues, which power of a the residue of b is — pinned exactly. The reach then has a clean boundary: a yes/no question about the pair is decidable by words precisely when its answer is constant under that shared rescaling (proved). Every multiplicative relation aⁱb^j = 1 passes. Additive questions are blind — with a crystallographic exception. Ask “does a + b = c?”: at c = 0, where b = −a is multiplication in disguise, the question is decided outright; on every other line the words see only points of multiplicative order 1, 2, 3 or 6 — c = ±2 keeps (±1, ±1), c = 1 keeps the primitive sixth roots, c = −1 the cube roots, and every remaining line is invisible in full (proved, with census across the primes to 100). Orders 1, 2, 3, 4, 6 — counting c = 0's quarter-turn points — is the same short list a plane crystal's rotations obey, by the same arithmetic. One word deserves singling out: the gates of ab⁻¹ grade equality — channel by channel, for invertible residues, whether a and b sit at the same position around the multiplicative cycle at each of the twenty resolutions, exact equality at the bottom (proved) — a graded “how equal” in a ring with no “how close”.

Every wall in this section ends at one door. The reads so far draw on a fixed alphabet — powers of the inputs and of their NOTs. But this ring divides totally: every nonzero residue inverts channel by channel, the inverse of 0 is 0, division never refuses. Let that inverse nest freely with NOT and the walls fall in one line: NOT(a⁻¹) · (NOT a)⁻¹ · a works out to the constant −1 at every residue except 0 and 1, and to 0 at those two (proved) — a constant, conjured from a variable. From −1, NOT and multiplication reach every other constant, and with constants in hand “is x equal to c?” is one order gate of xc⁻¹: every residue pinned outright, 0 and 1 being the original pair's own bits. Identification, not measurement. Each wall above — the pair's, the order gates', the words' — is a statement about what the alphabet withholds: total division buys back exact identification.

The reach of measurement

Short of that door, the words' reach has an exact map, and the crystal list of the lines is one row of it. Ask the same question of any curve — any polynomial condition on the pair, not just a line: which of its points do the words see? By the boundary above, exactly those whose whole rescaling orbit stays on the curve, and one criterion sorts them (proved). Take a point whose two coordinates are invertible — the residues that have orders. Each term of the curve, evaluated there, lands somewhere on one cycle of length d — d the least common multiple of those two orders — and collecting coefficients slot by slot turns the curve into a word of length d. The point is seen exactly when that word either is zero outright — the curve contains the pair's whole power cycle — or is a codeword of a classical error-correcting code: the cyclic code that the d-th cyclotomic polynomial generates. Coding theory then writes the menu. That code's minimum weight is exactly the least prime factor of d (proved, in every characteristic), so a curve written in M terms can hold seen points of order d — short of containing their cycles whole — only if the least prime factor of d is at most M. Lines are the three-term row, their unit coefficients and constant term cutting “least prime factor at most 3” down to the short list above; each prime number of terms seats a species that fewer terms cannot; and the first arrival past the lines is order five, needing five terms, where a conic's budget of six first fits it: the pentagon conic y² + xy + x + y + 1 = 0 holds exactly one seen orbit of order-5 pairs at every prime p ≡ 1 (mod 5), and none elsewhere (proved; censused at p = 11, 31, 41, 61) — a degree-2 curve seeing torsion no line can.

And reading has a price list. Keep the alphabet of the words — products of powers, inverses included, NOT not yet — and count the gates a decision needs. Every “does the order divide m?” is one gate by construction, the graded equality reads included; pinning an order exactly costs more. “Is the order of x exactly d?” takes one gate per distinct prime of d, plus one (proved, both directions: each prime of d forces a gate of its own, no gate serves two primes, and one more must accept d itself) — except at the top, where “does x generate the channel?” drops the extra gate. Two currencies now price one locus: the pentagon orbit, five terms to cut out, is decided by exactly three gates (proved; at p = 11 and 31 every one- and two-gate program fails, exhaustively) — deciding a locus and writing its equation are different costs, neither convertible into the other. The questions words cannot decide are the table's infinite rows: a gate's bit never separates two pairs linked by the shared rescaling, so a question that splits an orbit — “does a + b = 3?” in the mod-7 channel — has no finite gate count at all. The walls of the previous section sit in the same table, each priced in a different currency — the alphabet a read requires rather than the gates it spends. The door is on the table too: with NOT joining the alphabet, total division decides every question at exactly one gate, and the whole price moves into the formula handed to the gate — where it cannot be hidden. Only so many formulas exist under any given operation count (counted exactly), so as the channel prime grows, the costliest residue's identification formula grows without bound: size, deleted as data, returns as cost, and which residue you pin carries a tax with no ceiling. The six-operation formula above that conjures −1 is optimal — at p = 5 and p = 7 no shorter formula reaches any constant beyond the pair's own 0 and 1 (computed exactly) — and a single ring gate runs all seven channels in parallel (by construction), so every per-channel price here is already the ring's.

The cycle that survives

One order-like structure does survive the deletion: the ring's own cycle — 0, 1, 2, … around the circle and back to 0. It survives as an object; the question is how much of it the channels can read, and the answer is: not even its direction. Pick three distinct positions and ask whether they run clockwise or counterclockwise around the cycle. Reveal any proper subset of the channels — six of the seven here, or five, or one — and each position is pinned only to an even ladder of candidates spanning the whole ring, and wherever the visible residues admit a distinct triple at all, they admit it running in both directions (proved). The determined fraction of the direction is exactly zero — one bit of cyclic orientation costs the entire complement. And this is not a special power of the tower: it is a fact about remainder arithmetic in every cyclic ring, for every proper divisor of the modulus. The tower inherits it at every rung; what the tower adds comes next.

What every channel does see of the cycle is the sieve. Step through the ring by any size coprime to the modulus and the mod-p residue returns to zero exactly every p steps: the pattern of zero-crossings around the cycle is the Eratosthenes pattern, and every cycle through the ring shares it. The cycles differ only in the finer braid — which channel sits where between crossings. The skeleton all the orders have in common is the sieve itself.

Among all those cycles, stepping by 1 is still singled out — and here the primes do the work. Step by t and the hand on channel p advances t mod p of its p marks per tick; step by 1 and the hands sort by channel — mod 2 fastest, half a turn per tick, then mod 3, down to the largest prime. Stepping by 1 is the only step size that sorts the speeds this way precisely when 2 is among the primes and no prime in the set is more than double the one before it (proved, both directions). Every rung of the tower passes: 2 is there by construction, and prime gaps never double, by Bertrand's postulate. Build a ring from {3, 7} instead — no 2, and 7 more than doubles 3 — and six different step sizes sort the speeds equally well.

The deepest consequence is what happens to induction. Over the integers, “keep descending, you must stop” powers everything from Euclid's algorithm on up — and descent there can run unboundedly long before it stops. In the ring, descent through the zero patterns halts in at most k steps at rung k — the depth is a constant of the ring, not a function of the element — so every descent loop unrolls into a fixed-depth circuit. Euclid splits in half: the gcd itself is a one-step read of the joint zero pattern (checked exhaustively at small rungs), while Euclid's loop — divide, take the remainder, repeat — has no analogue at all: a remainder is a size comparison, and the sizeless substitute just cycles (explicit witness). Division-with-remainder died with the size window; only its purpose survives, and it survives bounded.