Classical structure

The split is not one of domain. The primes determine the shifted primes and the shifted primes determine the primes, so every quantity named below is a function of the same prime set and saying one "depends on {p_i}" rather than on {p_i−1} says nothing. What the split says is that no geometric quantity which varies along the tower factors through λ. Geometric: curvature, the degree, the chord gap, the shell distribution, and the time a random walk on the neighbour graph takes to mix — each reads a channel's size. Dynamical: λ, orbit lengths, the orders an element can have, transparency itself — each reads the multiplicative structure of p − 1, which the size of p does not determine.

The witness is on the trajectory and needs no designed tower. The primes 31, 37, 41 and 43 are all transparent, so λ stands at 55,440 across k = 10 through k = 14 — five rungs — while the degree runs 119 to 267 and every geometric quantity above moves with it. No function of λ can do that, so none of them is one.

The word varies is load-bearing, and what it excludes is a triviality rather than a result. The Hamming gap — the smallest Hamming distance between two distinct elements — is 1 on every rung by construction, since changing one channel is always available. A constant is a function of anything, so that is the one geometric quantity no constant-λ witness can separate, and there is nothing in it to separate.

What the tower buys with it is a two-speed growth: capacity at every rung — a dimension, degree, φ — and complexity only at jumps, which thin out as the transparency density approaches 1.

Scope. Each quantity's dependence is by construction; the separating witness is computed rung by rung for k = 3..14, and it separates only the quantities named.

verifier: explore_tower_geometry.py

Give the neighbour graph its Ollivier–Ricci curvature — the coarse curvature that measures how far a step from one point can be transported onto a step from its neighbour. Each channel contributes κ_i = (p_i − 2)/degree, so the total is

The 2-channel is exactly flat — a two-point circle has no room to curve — the largest channel is the most curved at 5/17, and the k = 7 total is 44/51. The degree grows like k² ln k / 2 while the subtracted term's numerator is only k, so the curvature climbs toward 1, reaching 0.948 by k = 14: the torus gets rounder as it grows.

Curvature therefore runs opposite to the chord gap, which shrinks as the largest circle grows. The rounder the rung, the less its elements are separated by the metric that embeds them — one fact read twice, since a large prime is a large circle, which is locally flat and finely divided.

The Gauss–Bonnet count over the same channels, χ = N(1 − k + ∑1/p_i), is not a new invariant twice over. It is the Euler characteristic of the complex built on this graph below, and −χ is the integer whose prime divisors are the absent primes a sub-ring names — so the naming thread and the curvature thread are counting the same number.

Scope. The per-channel formula and the k = 7 figures are verified exhaustively; the totals are computed k = 3..14, and the limit follows from p_k → ∞.

verifier: explore_tower_geometry.py

Fix a point and call the elements at Hamming distance d from it the shell at d. The channels are independent, so the generating function of the shell sizes factors,

and the coefficient of x^d is the number of elements at distance d. Two coefficients are already familiar: e₁ is the degree, and e_k — the elements differing in every channel — is φ(N), the units. The distribution is not centred. Its mean distance is k − ∑1/p_i, growing as k − O(log log k), and the largest shell is the next-to-last: 39% of Z/510510 sits at distance 6 of a possible 7.

So the torus is almost uniformly far from any reference point. The fraction of elements at distance k/2 or more passes 97% at k = 6, 99% at k = 8 and 99.9% at k = 10.

Scope. The factorization is exact on every squarefree rung; the concentration figures are computed k = 3..14.

verifier: explore_tower_geometry.py

The j-th Betti number of the torus — the rank of its j-th homology, counting independent j-dimensional holes — is β_j(T^k) = C(k, j), one for each way of choosing j of the k circles. The rung's idempotents — the elements with e² = e — are indexed the same way, since idempotence is decided channel by channel and each channel is a field: one idempotent per subset of channels. So β_j counts the idempotents whose subset has size j, and the Betti numbers sum to 2^k, the number of idempotents. At k = 7 they run (1, 7, 21, 35, 35, 21, 7, 1) and sum to 128.

Poincaré duality, C(k, j) = C(k, k−j), is then the complement involution on channel subsets: an idempotent e and its complement 1 − e sit at dual Betti numbers. Both structures are graded by the same lattice of channel subsets, and the grading is the one the CRT decomposition already fixes.

Scope. Both counts are C(k, j) by construction, computed k = 3..14. What is claimed is the shared index set and nothing more: no map between homology and the ring is asserted, and a reader who finds the coincidence forced — both families are indexed by subsets of a k-set — is reading it correctly.

verifier: explore_tower_geometry.py

The neighbour graph is a Cartesian product of complete graphs, one per channel, and the spectrum of such a product is indexed by exactly the lattice the last two claims share: one eigenvalue per subset S of channels. A complete graph on p vertices has eigenvalue p−1 once and −1 with multiplicity p−1, so taking the off-eigenvalue on S and the top one elsewhere gives

One letter is doing two jobs from here on, and the argument separates them: bare λ, or λ of a rung, is the universal period above, while λ carrying a channel subset is an eigenvalue of the neighbour graph.

The eigenvalue therefore depends on S only through ∑_i∈Sp_i: the spectrum is the set of subset sums of the channel primes, reflected through the degree. At k = 7 that turns a question about a graph on 510,510 vertices into an arithmetic one about seven small numbers. The extremes are the degree 51 itself, at S = ∅, and −k = −7 when every channel is off. That second one has multiplicity ∏(p_i−1) = φ(N) = 92,160 — multiplicity being how many independent eigenvectors share the value — because the eigenvectors there are exactly those non-trivial in every channel. The shell at distance k, which is the units, has that same size, and the agreement is worth naming rather than leaning on: both count one non-zero choice per channel, once over eigenvectors and once over residues. They are equal, not identical.

And the count comes out short of the 128 subsets. The distinct eigenvalues number 53, not 128, because distinct subsets can share a sum — {2, 5} and {7} both give 7 — and 53 is not an accident either. The primes total 58, so 59 integers are candidate sums, and exactly six of them are unreachable: 1, 4, 6 and 52, 54, 57. That set is symmetric about 29 for a structural reason rather than a numerical one — a subset and its complement sum to 58, so a sum is reachable if and only if its reflection is. Only the three small gaps carry information, and each is simply not a sum of distinct members of the set: 1 falls below the smallest prime, while 4 and 6 would need 2+2 and 3+3. 59 − 6 = 53.

And none of that is a fact about k = 7. At every rung from the fourth on, the reachable sums are the whole range from 0 to the prime total minus exactly those six values — the same three small gaps and their three reflections — so the distinct eigenvalues always number the total minus 5. The deficit never grows, however many primes are added. The reason is that a new prime replaces the reachable set by its union with a shifted copy of itself: past the fifth rung the shift is short enough that the two overlap, so the interior can only stay full, while the three small gaps survive every step by being smaller than any prime that ever arrives.

Scope. The eigenvalue formula holds for any Cartesian product of complete graphs, so for every squarefree rung. The count of 53 and the missing set are exhaustive over the 128 subsets at k = 7. The k-independence is proved from the fifth rung on, the overlap condition holding there directly and above it by Bertrand's postulate, with the base rung and the fourth checked directly and every rung through the thirtieth verified by exhaustion. The complement symmetry is proved for every rung.

verifier: explore_torus_geometry.py, explore_subset_sum_weld.py

Fill in every complete subgraph of the neighbour graph as a simplex — the graph's clique complex. Two elements are adjacent when they differ in exactly one channel, so if u and v differ in channel i and v and w in a different channel j, then u and w differ in both and are not adjacent. Every clique therefore has all its differences in one channel: the maximal cliques are exactly the lines — all channels but one held fixed — of which there are L = ∑N/p_i, any two meeting in at most one element. The complex is a union of simplices glued at isolated points, so take the lines in an order where each after the first already meets the union of the earlier ones — the graph is connected, so one exists — and attach them in turn. Attaching a simplex along c ≥ 1 points already present adds c − 1 circles up to homotopy, and summing c over every line counts incidences minus elements, kN − N, the first line contributing none. Each of the L − 1 attachments therefore yields one circle fewer than the points it lands on, leaving a wedge of circles,

with b_j = 0 for every j ≥ 2. What b₁ counts is the excess independent edges the lines supply beyond what the elements can hold: a line of size p_i offers p_i − 1 independent edges, so the lines jointly offer kN − L, while a spanning tree on the N elements absorbs only N − 1.

And it is one more than the Euler characteristic the curvature claim above already carries. Each line is contractible and the lines meet only at elements, each of which lies in exactly k of them, so χ = L − (k−1)N, which is the N(1 − k + ∑1/p_i) above; a wedge of circles has no homology past the first for that count to hide, so b₁ = 1 − χ, which is 2,346,894 at k = 7.

The subset lattice is entirely absent here. The Betti numbers of the torus and the neighbour spectrum are both indexed by subsets of channels; this complex sees only the lines, and its b₁ is a single number rather than a graded family — the two Betti counts on this page share a word and nothing else. Primality is unused as well, the p_i entering only as complete-graph sizes, so sizes 8 and 9 give 56 by the same formula. At two channels b₁ collapses to (p−1)(q−1), which is why a product form fits the pairs and nothing above them: the true form is a difference of three terms, and no product of per-channel factors gives the middle one, ∑N/p_i. The one rung whose b₁ equals N itself is Z/30, and it is the only multiset of sizes that can do it — the condition rearranges to ∑1/p_i = k − 2 + 1/N, which is out of reach from k = 4 up and impossible at k ≤ 2. That leaves the triples whose reciprocals exceed 1, and they are not a finite list: besides {2, 3, 3}, {2, 3, 4} and {2, 3, 5} there is the whole family {2, 2, x}, closed by its own form rather than by enumeration — b₁ = 4x − 3 against N = 4x, short by three at every x. Of the three that remain only {2, 3, 5} lands exactly. So the rung that made this count look like it counted the ring is the one rung where it could have.

Scope. Proved for every k and all channel sizes at least 2, primality unused; the uniqueness of Z/30 likewise. Checked against independent homology computations — boundary-map ranks over the field of two elements, the cliques enumerated from common neighbours rather than from the lines — on 20 cases including repeated and non-prime sizes, with the second Betti number computed on all 20 and the third on the 18 carrying simplices of that dimension, and the maximal cliques enumerated directly on five. That field is the instrument's choice and costs nothing here: a wedge of circles has no torsion for it to see, which is what the proof establishes and the check is testing.

verifier: explore_clique_betti.py

Let T be a subset of the channels of a rung S, and N_T, N_S their products. Hold the channels of S outside T at one fixed value: the elements agreeing there span an induced subcomplex isomorphic to the whole complex of T, and there are c = N_S/N_T such slices, disjoint and covering every element. First homology reads them independently — one slice's circles inject into the large complex at rank exactly b₁(T), and the c slices are jointly independent at rank exactly c·b₁(T). The counts assemble as

the copy count multiplying the sub-ring's own homology, and the dropped channels paying an additive term that does not read T at all. The identity itself is arithmetic — substitute the formula above on both sides and it rearranges — so what the homology adds is that the first term is a genuine subspace count: the c copies really are independent circles of the large complex and not merely a number that fits.

Scope. The identity is proved for every nested pair, from the formula above. The independence is computed — boundary ranks over the field of two elements — on 28 complexes with at most 210 elements, the image and joint ranks landing on b₁(T) and c·b₁(T) in every case — including the single-channel T, where both are 0 and the check is empty, a one-channel complex being a simplex with no circles in it. The counting identity itself was checked on all 1,932 nested pairs through k = 7. Free first homology is what carries the field computation back to the integers.

verifier: explore_naming_complex.py

An absent prime s names a sub-ring exactly when it divides −χ, which is b₁ − 1 by the count above. For a connected graph an s-fold covering multiplies the Euler characteristic by s, so its first Betti number satisfies b₁ − 1 = s(b₁′ − 1) for the graph below it. Naming is therefore a statement about shape: s names S iff the complex of S is homotopy equivalent to a connected s-fold covering of a graph, one with (b₁(S) − 1)/s + 1 circles.

No symmetry of the complex realizes it. The graph is a product of one complete graph per channel, those factors have pairwise distinct sizes and none of them factors further, so by Sabidussi's theorem every automorphism acts one channel at a time: it permutes each channel's residues among themselves and does nothing else, confirmed by direct enumeration on the smallest case. An element of order s is then a tuple of permutations each of order 1 or s, s being prime; and a permutation of order s on p_i points leaves p_i − s·(number of s-cycles) points fixed — nonzero, because s is absent and so divides no p_i. Fixing a point in every channel fixes an element. So for every absent s, every order-s automorphism has a fixed element, no order-s action on the complex is free, and the covering the divisibility asks for exists only up to homotopy. It is not a covering by a symmetry: the graph beneath it is this complex modulo a symmetry the complex does not have.

Scope. Proved for every rung and every absent s. The channel sizes enter only as pairwise distinct integers none of which s divides. The homotopy equivalence to a wedge of circles is the theorem above; the automorphism group is Sabidussi's, its order confirmed at 12 by brute force on the two-channel rung Z/6. What is proved is that no free action of order s exists, not that no covering does.

verifier: explore_naming_complex.py

−χ(S) is b₁(S) − 1: the integer the operator factors is the circle count minus one. So the operator depends on the complex of S only through its homotopy type — the coordinates never mattered — and, since b₁ classifies wedges of circles up to homotopy, it descends to a well-defined map on homotopy types. Death — an orbit reaching the empty set — reads as contractibility: {2, 3} apart, a seed no orbit can reach and the one set that dies at b₁ = 2, every death runs through a singleton, whose complex is one simplex, b₁ = 0.

The descent is not vacuous. The 2047 subsets of the first eleven primes take only 2020 distinct −χ values: 17 collision classes, the eleven singletons at −χ = −1 being one and the other 16 holding sets of two or more primes, 8 mixing sizes and 8 same-size ({2, 13} with {3, 7} at −χ = 11), so distinct rings ride one homotopy type into one future. {2, 3, 5} shares −χ = 29 with {2, 31} — one wedge of 30 circles, one image — and the largest class, −χ = 71, holds {2, 3, 11}, {5, 19} and {7, 13}.

And the map exists only at that level. Consecutive orbit values are coprime — one step's value has the next step's set as its support, and no member prime divides its own set's −χ — while an s-fold covering multiplies the Euler characteristic by s, and a ring map needs one modulus dividing the other, where here a set and its image share no prime at all. So neither a covering map nor a ring map connects the complexes at consecutive steps of an orbit still running, in either direction: where naming asked for a covering that exists up to homotopy, the claim above, iteration supplies no covering at all.

Scope. The descent and the coprimality are proved; the collision census is a rule, exhaustive over the subsets of the first eleven primes; the no-realization check ran on 10,039 consecutive pairs across the sweeps, 0 divisibility steps. The b₁ wiring was controlled first against independent homology — boundary-map ranks over the field of two elements — printing b₁ = 0, 2, 30 at {2}, {2, 3}, {2, 3, 5}.

verifier: explore_seed_flower_complexes.py (11 checks)

Through k = 9 — channels up to 23 — the 36 channel pairs carry 36 torus knots and all 36 signatures are negative: in this orientation the tower is left-handed throughout. The values are small and highly degenerate, T(3,5) and T(3,7) both landing on −8, the signature of the E8 lattice.

The 84 channel triples carry 84 Brieskorn homology spheres. Every one of the 84 Milnor-fiber signatures is negative and divisible by 8 — the fiber is spin with unimodular intersection form — so by Neumann–Wahl each triple has an integer Casson invariant σ/8. Those run from −1, the Poincaré sphere Σ(2, 3, 5), to −308 at Σ(17, 19, 23), and their totals by rung k = 3..9 are −1, −8, −47, −187, −615, −1625, −3873.

Channel 2 is the suspension coordinate — the one whose exponent turns a pair's singularity into a triple's — and it costs the construction nothing: σ(2, q, r) = σ(T(q, r)) exactly, since a squared third variable is how a knot signature arises from a Milnor fiber at all. A triple containing 2 therefore adds no signature and no Casson data beyond its own pair, and the genuinely new three-dimensional objects are the 56 odd triples. The 2-channel behaving as a degenerate direction is the shape the naming thread reports too, where 2 is never named by a sub-ring that omits it.

One numerical coincidence does not survive. The crossing numbers of the pairs sum, over the rungs k = 2..9, to 3, 18, 67, 210, 509, 1104, 2073 and 3660, and the 210 at rung 5 is the modulus of rung 4 — but no later total equals any earlier modulus.

Scope. Exhaustive over the primes through 23, so k ≤ 9. Signatures are the standard Brieskorn–Hirzebruch lattice counts, anchored two ways: σ(T(2,3)) = −2 at the trefoil, and the three-variable count reproducing the knot signature at r = 2 for all 28 odd pairs. Divisibility by 8 and the Casson identification are known background, verified here rather than proved; the crossing-total statement is asserted over the same range.

verifier: explore_torus_knots.py