The division algebras

The entry floor — floor 1, one doubling, adjoining a square root of −1 — is the only floor that reads the channel it stands on. It is the Gaussian decality: Z[i] over the seven channels has exactly ten prime components, three verdicts across the seven. At p ≡ 1 (mod 4) the doubling splits into F_p × F_p — 5, 13, 17. At p ≡ 3 (mod 4) it is the field F_p², division intact — 3, 7, 11. At 2 it ramifies: x² + 1 = (x + 1)², and the doubling is a local ring — one maximal ideal, everything outside it a unit — carrying a nilpotent, so division is gone there too.

Floor 2 reads nothing. The quaternion doubling (−1, −1 | F_p) is M₂(F_p) at every odd channel, by explicit isomorphism constructed per channel; Wedderburn and Chevalley–Warning are what stand behind it. Channel 2 degenerates instead: the doubling is commutative there, the 16-element local ring F₂[C₂ × C₂] with 8 units and (1 + i)² = 0. That degeneration is a fact about the presentation and not about the place, which designed ramification separates. Three channels arrive at floor 2 still holding division and none carries it out, against the ladder over R, where division reaches floor 3 and dies only at floor 4. Across channels the floor lifts by construction, matrix rings commuting with products:

Scope. Isomorphisms constructed and checked — relations and spanning — at every odd channel of Z/510510; channel 2's commutativity exhaustive over all 256 pairs. The splitting law itself is classical.

verifier: explore_division_ladder.py

At floor 2 the norm is the determinant, and det is a polynomial in the entries — hence channel-local by the criterion in its n-variable form. It is a multiplicative size-analogue living inside the channel-local class, where integer size is the information wall (the hiding lemma). It is the multiplicative half of size and no more: a field of characteristic p carries no order, so nothing here compares two elements.

The norm's fibers are exactly flat over the units — p³ − p matrices at every nonzero value — and its zero set is the cone, where the algebra's zero divisors live: p³ + p² − p elements at floor 2 and p⁷ + p⁴ − p³ at floor 3. Invertibility is per-channel cone avoidance, so a matrix over the whole rung is a unit iff its determinant misses every channel's cone: |GL₂(Z/6)| = 288.

Scope. Multiplicativity exhaustive at p = 2, 3 and sampled at Z/6; fiber and cone counts exhaustive at every odd channel of Z/510510, the floor-3 cone by dynamic programming over the same channels.

verifier: explore_division_ladder.py

A squarefree rung is a meadow: every element a has a pseudo-inverse a′ with a·a′·a = a and 0′ = 0, defined for all of the ring at once, which is what makes division total there (the wall-breaker). Doubling reinstates what squarefreeness banished — at channel 2 already on the entry floor, and everywhere on the one above it: p² − 1 nonzero nilpotents per channel at floor 2, elements with a zero power — the matrix unit E₁₂, a single 1 above the diagonal, squares to 0 — now unavoidable over a field rather than a symptom of a repeated prime. Five candidates follow them down, and each fails in a different place.

• Powering, a^m+1 = a: dead for every m, since E₁₂ has no nonzero power.
• Any polynomial in A with scalar coefficients: dead — a commuting inner inverse at a nilpotent forces the algebra to zero.
• The adjugate map adj(A)·det(A)^2p−3 — adj being the transposed cofactor matrix, so adj(A)·A = det(A)·I identically — is total, polynomial and exact on units, but it collapses the entire cone to 0 and the meadow laws die there.
• Von Neumann regularity, some X with AXA = A: total, but choiceful — no canonical X without further structure.
• The Moore–Penrose inverse — the choice-free patch, pinning the candidate G by four equations in A, G and the transpose — exists exactly where Pearl's rank criterion rank(AA^T) = rank(A^TA) = rank(A) holds, and is total at exactly the p ≡ 3 (mod 4) channels.

The failures are exactly the rank-1 matrices whose row or column space is isotropic — 4p(p − 1) of them at each p ≡ 1 (mod 4) channel, so 80, 624 and 1088 at 5, 13 and 17, and five of channel 2's sixteen matrices, where (1, 1) is isotropic for the same form. So the floor-1 norm x² + y² governs the floor-2 meadow: the congruence deciding whether the entry floor splits is the same one deciding whether the floor above it has a canonical inverse. That coupling is 2×2-specific and dies immediately above. At 3×3, x² + y² + z² is isotropic over every F_p by Chevalley–Warning, so a rank-1 witness built on an isotropic vector breaks Pearl's criterion at every channel of the rung and totality fails everywhere.

Scope. Nilpotent counts and the adjugate and von Neumann candidates exhaustive at p ≤ 5, the scalar-polynomial one at p ≤ 3; the powering tier dies algebraically at any m. Moore–Penrose existence exhaustive with uniqueness at p = 2, 3 and exhaustive at p = 5, the failure-set equality checked per channel across Z/510510. The 3×3 failure is criterion-independent at p = 2, 3 — none of the p⁹ candidate inverses satisfies the four equations — and rests on Chevalley–Warning at the rest.

verifiers: explore_division_ladder.py, explore_designed_ramification.py

Recoding a channel by a discrete logarithm turns × into + — Napier's move made exact over a residue window — and gives the rung a second coordinate system, the index transform ind (Number systems). At floor 2 there is no such coordinate at all: GL₂(F_p) is nonabelian. What survives is its abelianization, and the whole of it is the norm. The commutator subgroup is exactly SL₂ — the commutators generate all p³ − p matrices of determinant 1 (Dieudonné) — so every homomorphism to an abelian group factors through det, and ind ∘ det is the whole of Napier's move at floor 2: what is left of the logarithm is exactly the channel-computable size. Channel 2 is exceptional again, and in the opposite direction: GL₂(F₂) ≅ S₃ has abelianization C₂, which det cannot see, since F₂^* is trivial.

Scope. Commutator generation computed at every odd channel of Z/510510, the generated subgroup coming out at full order p³ − p with every element of determinant 1. The factor-through-det consequence is algebraic.

verifier: explore_division_ladder.py

The locality criterion's proof never used commutativity — a matrix function is four functions of four scalar variables — so channel-local = compatible transfers verbatim to M₂(Z/N) and the cross-channel wall keeps its teeth. Within one channel the polynomial class saturates instead: every function M₂(F_p) → M₂(F_p) is a generalized polynomial, one permitting coefficients on both sides of the variable. The construction is explicit — the matrix units multiply as E_1r·X·E_c1 = x_rc·E₁₁, extracting an entry into a corner where entries multiply like scalars, and Lagrange does the rest; the classical shadow is A ⊗ A^op ≅ End(A) for central simple A. And the CRT glue collapses to one line, e₂·f₂(X) + e₃·f₃(X), because the central idempotents are now coefficients rather than degreewise bookkeeping.

Scope. Circuits built and verified pointwise at p = 2 (all 16 inputs) and p = 3 (all 81); the CRT glue at M₂(Z/6), both channels exact on 150 random points.

verifier: explore_division_ladder.py

Division dies at floor 2. Composition — N(xy) = N(x)N(y), the norm being a homomorphism of the multiplication — outlives it by two. Floor 3 is no longer associative but is alternative and still composes, and its cone is exactly its zero divisors: x·conj(x) = 0 at norm 0, with conj the conjugation each doubling carries up. Floor 4 loses both composition and alternativity, by explicit witnesses over F₃. That is Hurwitz's dimensions 1, 2, 4, 8 verbatim over the tower — composition algebras exist in those dimensions over any field and the ladder ends where it ends everywhere. Over R the two deaths coincide, division and composition both reaching floor 3 and failing at floor 4; the gap of two is what a channel opens. What the tower changes is not where the shadow stops but what it falls on: the surviving norm is channel-local, and the size it half-replaces is the deleted window.

Scope. Nonassociativity, alternativity and composition at floor 3 by witness plus 3000-pair samples at p = 3, 5; the floor-4 failures by explicit witness at p = 3. The dimension classification is classical.

verifier: explore_division_ladder.py

A quaternion algebra (a, b | Q) is ramified at a place when it stays a division algebra there and split when it becomes a matrix algebra; Hilbert reciprocity makes the ramification set finite and of even size across all places, and every such set is realized by some algebra. So which channels ramify is chosen, not found. Four algebras across the seven channels give 28 verdicts, and all 28 hold as predicted: the ramified channels are exactly the designed finite primes — {3, 5} for (5, 3), {7, 11} for (77, −1), {2} for (−1, −1), {3} for (−3, −1) — and channel 2 splits in each of the three designs that leave it out.

Separating design from presentation needs a maximal order: a subring that is a lattice of full rank in the algebra and maximal among such, certified here by its reduced discriminant — the invariant that equals the algebra's own exactly when the order is maximal — coming out as the product of the designed finite primes. Such an order reduces to M₂(F_p) at exactly the undesigned channels, channel 2 included whenever it is undesigned, so the naive order's mod-2 collapse was the artifact and what survives it is real ramification.

A designed channel does not vanish, it fattens. The order's reduction there is local; its radical — the nilpotent ideal divided out to reach a field — has dimension 2, and the field left is F_p², the quadratic extension, which is the Deuring–Eichler shape. And the visible parity comes out odd: (−3, −1) ramifies at {3, ∞}, so the set the channels can see has size one. Evenness is conserved only across all places and the tower deleted one, so the finite half of a reciprocity constraint is free to be odd — a design rule whose partner term sits at the missing window and is read off the channels that remain.

Scope. All 28 verdicts fixed in advance of the run, four algebras × seven channels, with the reading's refutation riding on channel 2's four. Maximality certified by reduced discriminant; Hilbert symbols computed at every relevant place and reciprocity checked; the ramified shape certified per designed channel — local, radical of dimension 2, residue field verified a field of order p².

verifier: explore_designed_ramification.py