The division algebras

The entry floor — one doubling, adjoining a square root of −1 — is the only floor whose division verdict 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 — the repeated-factor sense: 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's division verdict reads no channel. 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 — division gone either way. The commutative shape is a fact about the presentation and not about the place — every naive order collapses mod 2 whatever algebra it presents — while whether the place ramifies is the algebra's own affair: (−1, −1) genuinely ramifies at 2, and designed ramification separates the two. 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, channel-computable, by the criterion in its n-variable form. It is a multiplicative size-analogue living inside the channel-local class, where integer size leaks almost nothing to any proper subset of the channels (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. The shape measured here is the floor-2 matrix ring M₂(F_p) at every channel: the doubling itself at the odd ones, and at channel 2 — whose own doubling degenerates to the commutative ring above — the matrix ring taken in its own right. It carries p² − 1 nonzero nilpotents per channel, 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 candidate 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 its identification of channel-local with compatible, meaning a ≡ b (mod p) forces f(a) ≡ f(b) (mod p), 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 — at the p ≡ 3 (mod 4) channels, the last three to hold it; the other four lose it on the entry floor and see a gap of three floors, so the exact two is the surviving channels' and the ladder's. 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 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 sat at the deleted place.

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 over the completion there — the p-adic field at a finite place, R at the archimedean one — and split when it becomes a matrix algebra over it; 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. Division lives over the completion, and a residue window is too shallow to keep it; what the window shows of a ramified place is the order's reduction, and that reduction 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, channel 2's four being the ones that separate design from presentation. 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

What deleting a place buys at a global constraint is exactly the deleted term's own local group — the group that term ranged over — no more, and both factors of it, torsion and rank. The torsion factor is read three times below — by reciprocity, the Brauer sum and the product formula. Reciprocity's term is a sign, so the budget is one parity bit: the odd visible sets above. The Brauer sum generalizes it — a quaternion algebra carries one invariant per place, ½ where it ramifies and 0 where it splits, and reciprocity is the law that the invariants sum to 0; the real term's group has order 2 — Frobenius leaves R, C, H as the real division algebras, and the invariants count the two with center R, so that group is {R, H} — so the budget is one bit and 2-torsion's alone: for the odd-degree siblings of the quaternion construction (cyclic algebras, invariants in (1/n)Z/Z) the visible invariants still sum to zero exactly (Albert–Brauer–Hasse–Noether). And the product formula ∏_v |x|_v = 1 — over all places, with |x|_p = p^−v_p(x) and v_p(x) the exponent of p in x — loses an archimedean term whose values range over everything, so the visible constraint is void: every finitely supported vector of exponents is realized. The control is F₂(t), the rational functions over the two-element field, with the place that reads a fraction's degree deleted — non-archimedean, and the same total freedom — so the law is deletion's, not archimedean-ness's.

The rank factor is Dirichlet's S-unit theorem read as the same budget. An S-unit is an element invertible once denominators are allowed at the finite places in a set S; the S-unit group has rank |S| − 1, counting the archimedean places in S. As a budget: each deleted term buys the rank of its own value group — 1, always — so the rank added to the unit group's own is the count of deleted terms, while the one lost rank is the product formula's single relation. That is why Z's unit group is {±1}: its S is the one archimedean place alone, and the relation eats the one rank. Which prime is deleted enters only through its splitting: over Q(√−5), deleting the places over p buys rank 1, 2, 1, 2 at p = 2, 3, 11, 29, as x² + 5 ramifies, splits, stays irreducible, splits — the count of terms, never the name of the prime. What the rank cannot see — which place — returns one floor down: the valuation image sits at finite index in the integer lattice with one coordinate per deleted finite place, and that index is the order of the subgroup of the class group — the finite group of ideal classes, trivial exactly when every ideal is principal — generated by the deleted places' classes. Over Q the class group is trivial and the budget is paid with no correction. The witness pair: p = 2 and p = 11 each delete one term — equal rank, index 2 against 1. A torsion budget again, at the lattice the rank forgot.

Volumes price the same way. Map an S-unit to the logarithms of its absolute values at the places of S: the image is a full lattice in the hyperplane where the coordinates sum to zero — the sum is the product formula — and that lattice's covolume — the volume of one fundamental cell, computed as the absolute determinant of a basis's log vectors with one coordinate dropped, a number independent of both choices — is the regulator. With nothing deleted it is the field's own unit regulator, the covolume the unit group already had. Measured over real quadratic fields, each deleted term multiplies it by the covolume of its own local log-value group — log NP, where NP counts the residues at the place, so a place whose residue field has p² elements contributes log p², not log p — and the one correction, applied once and not per term, is the same class-group index. Nothing else enters: no cross-term residual at sixteen S-sets over Q(√2) and Q(√10). Deletion buys volume multiplicatively. And the witness pair returns in volumes: over Q(√10) the ramified place over 2 against a split place over 41 — equal rank, covolume ratio 2 against 1, the correction once more the class of the deleted ideal.

The product-formula, rank and regulator faces are one theorem seen three ways. The classical object is the idele class group: one coordinate per place of the field, each ranging over that completion's nonzero elements, almost all of them local units — absolute value 1 at their place — modulo the field's own elements placed diagonally. An idele's norm is the product of its coordinates' absolute values; the product formula is the statement that the field's elements land in the norm-one part, and that is what makes the quotient there exist: norm-one ideles modulo the field, and this quotient is compact (Fujisaki's lemma, with the product formula as its one entry). The equivalence runs both directions: this one compactness is exactly class-group finiteness plus the unit theorem. The class group is the compact quotient's discrete image, finite because compact; the rank statement is its projection onto the sum-zero log hyperplane, full because the hyperplane's quotient by the span of the log image is again a vector space, and a compact vector space is zero; the covolume is its invariant volume — under the natural normalization, class number × regulator — per-term because an invariant volume on a product of local factors is a product of local volumes, which is why volumes multiplied above. The parity and Brauer points read the same object one level up: reciprocity is a character of the ideles — a homomorphism to the unit circle — trivial on the field's elements, which is what lets it descend to the quotient, and the Brauer sum reads the same quotient's degree-2 cohomology, the home of the invariants above, compactness entering as the finiteness input. Five faces, one object at two levels — and one which-place correction because it is one object being read.

A sixth face deletes less, and what refuses is its correction. The class group is itself a quotient of the idele class group — by the subgroup whose finite coordinates are all local units, archimedean coordinates free. Delete not a place but a layer of one: at a chosen set of finite places, cut the local units down to a congruence depth — one residue layer at every modulus measured here — instead of removing anything. The places stay, ramification there is newly visible, and what is bought is a ray class group — ideal classes counted with a congruence condition at a modulus m, a product of finite places — since every quotient of the idele class group by an open subgroup is one. Its order is the class number times the count of residues invertible at m, divided by one correction: the image of the global units among those residues. The local factor is per-term outright — residues factor place by place — but the correction is not: the unit image is a single subgroup of the whole product, generated by at most unit-rank + 1 elements, and its failure to factor into per-place projections is exactly the excess of the true ray class number over the per-term prediction. Measured over Q(√2): excess 2 at one two-place modulus against 1 at another, so the entanglement is the modulus's property, not the construction's — and largest at conjugate pairs, the two places over one prime, where the fundamental unit's norm ties one residue to the other's negated inverse, holding the joint image at one place's own size while the two-place product is that size's square: 16 against 256 at the places over 17. Over Q(√−5), where the units are ±1, the excess at s odd unramified places is 2^s−1 — unbounded in the place count, forced by counting generators. So both deletions price the same way — per-term local factors, then one global corrector — and the two finitenesses the compactness bundles are the budget's two correctors: the class group when a place is deleted whole, the unit image when only a layer is. Neither corrector factors place by place — the class-group index at the two places over 3 in Q(√10) is 2 where the two classes' separate orders multiply to 4 — and what the layer deletion adds is a corrector whose entanglement is measurable, and unbounded.

The threshold track prices the other failure shape: below an unmet hypothesis, what survives is local data deeper than a residue window. The square ladder is the specimen. Every residue at every channel is a sum of two squares, so the classical two- and three-square laws over Z are channel-invisible: two squares needs the parity of the exponent of each prime ≡ 3 (mod 4) — a valuation, depth below the window; three squares the 2-adic condition n ≠ 4^a(8b + 7) — strip the 4s, then read mod 8: depth, then precision; four squares nothing. Witnessed by pairs congruent modulo 510510 with opposite verdicts: 3 and 2552553 for two squares, 7 and 510517 for three, all four verdicts by direct search. Both tracks price an import's failure in the same coin — the local coordinates the deletion removed or the window coarsened.

Scope. The parity point is the designed-ramification rule above; the Brauer point is a property standing on Frobenius and the classical exact sequence; the product formula and the function-field control are verified in exact arithmetic. The rank face is a rule at its measured scope — over Q and an imaginary quadratic field, where the unit rank is 0 and the S-unit rank is the term count outright: ranks and indices measured over enumerated S-units at |S| ≤ 7 over Q and at eight S-sets over Q(√−5), in exact arithmetic throughout; that the rank attains |S| − 1 in general is Dirichlet's theorem. The regulator face is a rule at its measured scope — covolumes and lattice indices measured over enumerated S-units at sixteen S-sets across Q(√2) and Q(√10), the fundamental units coming out at 1 + √2 and 3 + √10; the product form in general is the classical S-regulator formula. The compactness statement and its equivalence with the two finitenesses are classical (Fujisaki's lemma), as is the volume normalization; they are cited, not re-proved. The ramification face is a rule at its measured scope — unit images and ray class numbers computed by exact subgroup closure at two- and three-place moduli over Q(√2) and Q(√−5), all 21 place-pairs over Q(√2) scanned, the extension over class number 2 confirmed by ideal witnesses; that every open-subgroup quotient is a ray class group is classical. The square ladder is exhaustive to 20000, the pair verdicts by direct search.

verifiers: explore_deleted_budget.py, explore_rank_face.py, explore_regulator_face.py, explore_ramification_face.py