Recognize Frobenius-truncated prime-field algebra tables

research/frobenius-algebra-recognition/PROOF.md

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+# Recognizing Frobenius-truncated algebras from a multiplication table
+
+This note gives a recognition criterion and an explicit algebra isomorphism
+for a supplied multiplication table over a prime field. It answers a restricted
+part of [Ali Romman's PR5459 handoff](https://github.com/CrocSwap/integer-mult-bounds/pull/5459):
+the inputs and output already refer to one common vector space. It does not
+recognize an arbitrary three-factor tensor up to independent basis changes,
+extract its nucleus, or supply the reductions needed to obtain that table.
+It does not change the selected multiplication bound.
+
+## 1. Input contract and criterion
+
+Let `p` be a prime and let `A` be a nonzero, `n`-dimensional vector space over
+`F_p`, supplied by its structure constants. First check commutativity and
+associativity on a basis. Recover the identity by solving the `n^2` linear
+coordinate equations `u*b_j = b_j` for its `n` unknown coordinates, and check
+the resulting element on both sides of every basis vector. Reject if the
+system is inconsistent or any check fails. Commutativity makes a left unit
+two-sided, and a two-sided unit is unique. These finite checks establish the
+algebra laws for all inputs by bilinearity and trilinearity. A malformed table
+is rejected before the criterion below.
+Primality of `p` is a separate input condition: arithmetic modulo a composite
+integer is not a substitute for `F_p`.
+
+Write `F(a) = a^p`. This is an `F_p`-linear endomorphism because `A` is
+commutative of characteristic `p`. Let `E` be the least nonnegative integer
+such that `p^E >= n`, and set
+
+```
+J = ker(F^E),             Q = J / J^2,
+K_e = ker(F^e),           0 <= e <= E,
+V_e = (K_e + J^2) / J^2,  0 <= e <= E.
+```
+
+Here `F^0` is the identity, so `K_0 = 0`; `J^2` denotes the span of products
+of pairs of elements of `J`. Every `K_e` lies in `J`, so `V_e` is a subspace
+of `Q`. The spaces `V_e` form an increasing flag from `0` to `Q`.
+Define
+
+```
+d_e = dim(V_e) - dim(V_(e-1)),       1 <= e <= E,
+N = p^(sum_(e=1)^E e*d_e).
+```
+
+**Theorem.** The checked table is isomorphic, as a unital `F_p`-algebra, to
+
+```
+B = F_p[x_1,...,x_r] / (x_i^(p^h_i) : 1 <= i <= r),
+```
+
+with positive heights `h_i`, if and only if
+
+1. `J` has codimension one in `A`; and
+2. `N = n`.
+
+The multiset of heights is intrinsic: height `e` occurs `d_e` times. An empty
+list is allowed, in which case `B = F_p` and `n = 1`. Acceptance produces
+generators `g_i` of the prescribed heights and the invertible matrix whose
+columns are their ordered monomials. This matrix, together with its inverse,
+is an explicit transporter between the common-basis multiplication tables.
+
+The word "transporter" here means an algebra isomorphism with the same map
+on both inputs and the output. It is not a general tensor-orbit transporter.
+
+## 2. The Frobenius kernel recovers the nilradical
+
+Let `R` be the set of nilpotent elements of `A`. In a commutative algebra,
+`R` is an ideal: products with arbitrary elements remain nilpotent, and a
+sufficiently large characteristic-power Frobenius kills a sum of nilpotents.
+Choose a vector-space basis `b_1,...,b_s` of `R` and positive integers `t_i`
+with `b_i^t_i = 0`. The ideal generated by these basis elements is `R`.
+Every product of more than `sum_i(t_i-1)` generators contains a factor
+`b_i^t_i`, so `R` is nilpotent.
+
+The chain `R, R^2, ...` decreases strictly until it reaches zero. Indeed, if
+`R^k = R^(k+1)` is nonzero, multiplication by further powers of `R` makes all
+later powers equal, contrary to nilpotence. Since `1` is not nilpotent,
+`dim R <= n-1`, and therefore `R^n = 0`.
+
+Every `a` in `R` consequently satisfies `a^(p^E) = 0`, since `p^E >= n`.
+Conversely, every element killed by this power is nilpotent. Thus
+
+```
+J = ker(F^E) = R,    J^n = 0.
+```
+
+This derivation does not assume that `A` is local. If `J` has codimension
+one, the scalar copy of `F_p` maps isomorphically onto `A/J`. Every element
+outside `J` then has the form `c*1 + j`, where `c != 0` and `j` is nilpotent,
+and is a unit by a finite geometric series. Thus `J` is the unique maximal
+ideal, and `A` is local with residue field exactly `F_p`.
+
+The codimension test matters. An extension field such as `F_4` viewed as a
+two-dimensional `F_2`-algebra has zero nilradical but is outside this family;
+its radical has codimension two. Products of fields are also excluded.
+
+## 3. Necessity and recovery of the heights
+
+Suppose `A` is the displayed algebra `B`. Its monomials
+
+```
+x_1^a_1 ... x_r^a_r,   0 <= a_i < p^h_i,
+```
+
+form a basis, its radical is `(x_1,...,x_r)`, and its tangent quotient
+`J/J^2` has basis the classes of `x_1,...,x_r`. In particular, `J` has
+codimension one and `n = p^(sum_i h_i)`, so every `h_i <= E`.
+
+If `h_i <= e`, the element `x_i` is killed by `F^e`; hence its tangent class
+belongs to `V_e`. For the converse, expand an element of `J` into its linear
+part and terms of total degree at least two. Frobenius raises every monomial
+separately to its `p^e` power. If the linear coefficient of `x_i` is nonzero
+and `h_i > e`, its power contributes a nonzero coefficient to
+`x_i^(p^e)`. No other linear term contributes that monomial. No term of
+degree at least two contributes it either: its surviving power has total
+degree at least `2*p^e`; terms killed by a truncation contribute nothing.
+Thus this coefficient cannot cancel.
+
+It follows that
+
+```
+V_e = span{x_i + J^2 : h_i <= e}.
+```
+
+Therefore `d_e` counts exactly the variables of height `e`, and the dimension
+test `N = n` is necessary. The proof is invariant under an arbitrary common
+change of basis in `A`; the identity and radical need not be coordinate axes
+in the supplied table.
+
+## 4. Sufficiency and an explicit transporter
+
+Assume the two tests pass. For `e = 1,...,E`, extend a basis of `V_(e-1)`
+to a basis of `V_e`, and choose the new representatives directly in `K_e`.
+Linear elimination on a basis of `K_e` modulo `J^2` constructs such lifts.
+Call the resulting elements `g_1,...,g_r` and assign height `h_i = e` to
+each lift added at step `e`. Their classes form a basis of `J/J^2`, and
+
+```
+g_i^(p^h_i) = 0.
+```
+
+The classes are chosen by first entry into the flag. A lift of height `e`
+cannot already lie in `K_(e-1)`, since its class would then belong to
+`V_(e-1)`. No minimal-exponent test beyond the flag is needed.
+
+The substitution `x_i -> g_i` defines a unital homomorphism `phi: B -> A`,
+because all defining power relations hold. It is surjective. One way to see
+this is the finite nilpotent form of Nakayama's lemma. Let `C` be the
+`F_p`-subalgebra generated by `1,g_1,...,g_r`. Tangent spanning gives
+`J = span{g_i} + J^2`. For each `k >= 1`, express every factor of a product
+in `J^k` by this formula. The product of its linear parts lies in `C`, and
+every other term lies in `J^(k+1)`. Hence
+
+```
+J^k is contained in C + J^(k+1).
+```
+
+Starting from `A = F_p*1 + J` and repeating this containment through
+`J^n = 0` gives `A = C`. This is also a direct proof of the particular
+Nakayama implication used here.
+
+The monomial basis of `B` has size `N`, and the dimension test gives
+`dim B = N = n = dim A`. A surjective linear map between these equal finite
+dimensions is invertible. Therefore `phi` is the required isomorphism.
+
+For a concrete certificate, order the exponent tuples in a declared fixed
+order and form the matrix
+
+```
+P[:,a] = g_1^a_1 ... g_r^a_r
+```
+
+in the supplied basis of `A`. Its constant column is the recovered unit.
+The certificate consists of the heights, the generators, `P` and `P^-1`.
+An independent checker can verify invertibility, the unit column, every
+defining generator power and every basis-product identity
+
+```
+mu_A(P*u_a, P*u_b) = P*mu_B(u_a,u_b).
+```
+
+These `n^2` identities imply the identity for all pairs by bilinearity.
+They check the actual transporter rather than trusting the recognition
+path or the inferred heights. Equivalently,
+`mu_B(u,v) = P^-1*mu_A(P*u,P*v)`.
+
+## 5. Why tangent Frobenius alone is insufficient
+
+The filtration uses images of kernels in the algebra before quotienting:
+`V_e = (ker(F^e) + J^2)/J^2`. It must not be replaced by the kernel of the
+induced Frobenius on `J/J^2`. Indeed, `a^p` lies in `J^p`, which is contained
+in `J^2` for every `p >= 2`, so the latter induced map is always zero.
+
+For the explicit example `A = F_2[t]/(t^4)`, the radical is
+`span{t,t^2,t^3}`, with square `span{t^2,t^3}`. Squaring gives
+
+```
+(a*t + b*t^2 + c*t^3)^2 = a*t^2.
+```
+
+Thus `ker F = span{t^2,t^3}`, `V_1 = 0`, and `V_2 = J/J^2`.
+The correct height list is `(2)`, and the predicted dimension is four.
+Frobenius on the tangent quotient alone would give height `(1)` and the
+wrong predicted dimension two.
+
+The dimension test also cannot be dropped. For the square-zero algebra
+`F_2 + J` with `dim J = 3` and `J^2 = 0`, all three tangent generators have
+height one, but `dim A = 4` while the predicted truncated-polynomial
+dimension is `2^3 = 8`. The generators give a surjection with additional
+relations, not an isomorphism to the proposed model.
+
+## 6. Finite algorithm, cost and the one-dimensional edge
+
+All steps are finite linear algebra and arithmetic on the supplied table:
+
+1. Validate commutativity and associativity on basis vectors, recover the
+   unit by solving its linear equations, and check it on both sides.
+2. Compute Frobenius on basis vectors by repeated squaring, its successive
+   kernels through `E`, and a basis of `J^2` from pairwise radical products.
+3. Compute the flag dimensions and adapted lifts by Gaussian elimination;
+   reject unless the codimension and dimension tests pass.
+4. Enumerate the `n` model monomials after acceptance, compute their images,
+   invert the resulting matrix, and check every basis-product identity.
+
+For fixed prime `p`, this is a deterministic polynomial algorithm in `n`.
+There are at most `ceil(log_2 n)` Frobenius levels. Dense table multiplication
+uses at most `O(n^3)` field operations, and the table-law and final
+basis-product checks use polynomially many such operations (a conservative
+bound for a direct implementation is `O(n^6 + n^4 log p)`). No search over
+all algebra elements, all bases or all automorphisms is required. Computing
+the exact candidate dimension is polynomial: its bit length is at most
+`O(n*(log n + log p))`, and is `O(n log n)` for fixed `p`. The implementation
+may compute that exact integer. Optionally, capped multiplication can stop
+as soon as the candidate exceeds `n`; it is not needed for the complexity
+guarantee.
+
+Conditional on a prime input, ordinary field arithmetic and powering are
+also polynomial in `log p`; primality certification must be accounted for
+separately in any variable-characteristic bit-complexity claim. In particular,
+trial division is sufficient for bounded prime fixtures but does not by itself
+establish a polynomial primality algorithm in the bit length of `p`.
+
+When `n = 1`, `E = 0`, `J = ker F^0 = 0`, the flag and height list are
+empty, and the empty product gives `N = 1`. The recovered unit is nonzero and
+forms a basis, so `P` is its single coordinate column. This covers `F_p`
+without introducing a height-zero variable. The zero-dimensional zero ring
+is outside the input contract.
+
+## 7. Scope, credit and remaining work
+
+The criterion establishes the promised algebra class from a checked common-
+basis multiplication table over `F_p`, and constructs an isomorphism to its
+standard monomial model. The written theorem covers every such dimension;
+the package's finite fixtures corroborate it and are not a proof-kernel check.
+
+The immediate source is Ali Romman's
+[PR5459 at `bef3575ddb19c44434a7e2460a353eac7c4d9387`](https://github.com/aliromman/integer-mult-bounds/blob/bef3575ddb19c44434a7e2460a353eac7c4d9387/research/frobenius-tensor-characters/README.md),
+which supplies the unequal-height family, height flag and explicit
+recognition/transporter handoff. His [PR5316](https://github.com/CrocSwap/integer-mult-bounds/pull/5316)
+supplies the truncated-polynomial predecessor. Nick Petalas's
+[PR5252](https://github.com/CrocSwap/integer-mult-bounds/pull/5252) supplies
+the finite-etale comparison, nilpotent follow-up and tensor normalization
+context; David Ganz and the retained atlas lineage retain their credit.
+Nilradicals, tangent generators, [Nakayama's lemma](https://stacks.math.columbia.edu/tag/00DV)
+and truncated-polynomial presentations are classical commutative algebra.
+Section 4 gives the finite nilpotent argument used here directly. No general literature
+priority is claimed for this criterion.
+
+Open next questions are the reduction from an independently based input
+tensor to this checked table, with certified factor maps, and an extension
+to supplied non-prime finite fields with the appropriate semilinear
+Frobenius calculations. Neither follows merely by applying the present
+`F_p`-linear kernel routine. Determinant-character extraction can subsequently
+use the verified heights under its separately pinned theorem. Physical
+coefficient preparation, native realization, precision, movement, supplier
+admission and a new multiplication exponent remain outside this result.
+
+Prepared by apoursh with substantial OpenAI Codex assistance. Apache-2.0.
+No independent human or formal proof-kernel acceptance is claimed.