Count regular rank-one projector triple classes (no κ)

research/regular-projector-triple-orbits/PROOF.md

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+# Regular fully coupled rank-one triples
+
+This proof was produced by Nick Petalas. Apache-2.0. It is a scoped written
+linear-algebra proof. Finite corroboration is separate from its universal
+quantifiers. Independent human and formal proof-kernel review are unclaimed.
+
+## Contract and result
+
+Let F be any field, n>=3, and let P1,P2,P3 be an ordered triple of rank-one
+idempotents on V=F^n. Put S=P1+P2+P3. Require
+
+1. rank(S)=rank(S^2)=3;
+2. p=tr(P1 P2), q=tr(P1 P3), r=tr(P2 P3) are all nonzero.
+
+Two triples are equivalent exactly when one invertible linear map
+simultaneously conjugates all three, preserving their order. No transpose,
+projector permutation or field automorphism is included in this equivalence.
+Set u=tr(P1 P2 P3). Then u is nonzero. The complete invariant is
+
+    (F,n,p,q,r,u).
+
+The admissible quadruples are exactly p,q,r,u in F* such that
+
+    Delta = 1-p-q-r+u+pqr/u != 0.                       (1)
+
+A unique representative on the three active coordinates is obtained from
+
+    G = [ 1  1    1  ]
+        [ p  1    u/q]
+        [ q  rq/u 1  ]                                (2)
+
+by setting Pi=e_i times row i of G, and adjoining n-3 zero coordinates.
+Over a finite field of order Q, the exact number of these conjugacy classes
+is
+
+    N(Q)=Q^4-5Q^3+9Q^2-9Q+6.                          (3)
+
+This is one fixed algebraic state class, not a classification of arbitrary
+triples or a bound on a complete physical register search.
+
+## Rank-one factorization and the active space
+
+Write Pi=x_i y_i with x_i nonzero and y_i a row covector. Idempotence says
+y_i x_i=1. Put C=[x_1 x_2 x_3], D=[y_1;y_2;y_3] and G=DC. Then S=CD.
+The first rank condition implies rank C=rank D=3. Thus C is injective and
+D is surjective, so rank(S^2)=rank(C G D)=rank G. The second rank equality
+therefore makes G invertible.
+
+Consequently W=im C and N=ker D have dimensions 3 and n-3 and intersect
+only at zero. Thus V=W direct-sum N. All Pi preserve W and kill N. In the
+basis x_1,x_2,x_3 of W, Pi has exactly row i of G in its ith row and zero
+in the other rows. Conversely any invertible unit-diagonal G constructs
+rank-one idempotents with rank(S)=rank(S^2)=3, including after adjoining zero
+spectators.
+
+## Completeness of the four traces
+
+The factorization Pi=x_i y_i is unique up to x_i -> d_i x_i,
+y_i -> d_i^(-1) y_i, with d_i in F*. Its Gram matrix therefore changes as
+
+    G -> diag(d)^(-1) G diag(d).                       (4)
+
+Ambient simultaneous conjugation preserves G when the factors are carried
+along. Conversely, equal G gives an intertwiner by mapping the three image
+basis vectors to their counterparts and extending by any isomorphism between
+the common kernels. Thus simultaneous conjugacy is exactly diagonal similarity
+of these Grams, together with equal ambient n and field F. This argument is
+direct and does not assume a predecessor's two-projector classification.
+
+Write g_ij=y_i x_j. Rank-one multiplication gives
+
+    p=g_12 g_21, q=g_13 g_31, r=g_23 g_32,
+    u=g_12 g_23 g_31,
+    v=tr(P1 P3 P2)=g_13 g_32 g_21,
+    uv=pqr.                                           (5)
+
+The nonzero pair traces imply that all six off-diagonal g_ij are nonzero,
+and hence u is nonzero. Apply (4) with
+d_1=1, d_2=1/g_12, d_3=1/g_13. The first row becomes (1,1,1). Equations
+(5) then force exactly (2). Any further diagonal similarity preserving this
+first row has d_1=d_2=d_3 and acts trivially. This proves uniqueness and
+completeness over every field. Expanding the determinant of (2) gives (1),
+which proves that there are no further realizability restrictions.
+
+The ambient centralizer of a representative is F* on W times GL(N).
+Indeed a commuting map preserves each image line and each kernel, so it
+preserves W and N. On W it is diagonal; the six nonzero couplings force
+its three diagonal entries to agree. Conversely these maps commute.
+Over a finite field each ambient class thus has size
+
+    |GL_n(F)| / ((Q-1)|GL_(n-3)(F)|).                  (6)
+
+Use |GL_0|=1. Formula (6) counts actual ordered triples in one class, not
+additional conjugacy classes.
+
+## Exact count over every finite field
+
+There are (Q-1)^4 candidate quadruples. Count those with Delta=0. For fixed
+p,q,u, rewrite that equation as
+
+    A+B r=0,  A=1-p-q+u, B=pq/u-1.                    (7)
+
+If u!=pq, then B!=0, so exactly one r solves (7). This r is nonzero exactly
+when A!=0. Before excluding A=0 there are (Q-1)^2(Q-2) choices of p,q,u.
+For A=0 we must have u=p+q-1, nonzero and unequal to pq. The inequality
+u!=pq is equivalent to (p-1)(q-1)!=0. There are (Q-2)^2 pairs with
+p,q outside {0,1}, of which Q-2 have p+q=1. Therefore the number to remove
+is (Q-2)(Q-3). This case contributes
+
+    (Q-1)^2(Q-2) - (Q-2)(Q-3).
+
+If u=pq, then B=0 and A=(1-p)(1-q). Thus Delta=0 exactly when p=1 or
+q=1, independently of the nonzero r. This contributes
+
+    (2Q-3)(Q-1).
+
+Adding the two contributions gives Q^3-3Q^2+5Q-5. Subtracting this from
+(Q-1)^4 proves (3). The argument uses only field arithmetic and applies to
+every prime power, including characteristic two. At Q=2 the intermediate
+products containing Q-2 are zero, so the same counting argument remains valid.
+
+## Why pair data alone is insufficient
+
+Over F5 take (p,q,r)=(1,1,1). The choices u=2 and u=3 give the invertible
+Grams
+
+    [1 1 1]           [1 1 1]
+    [1 1 2]     and   [1 1 3].
+    [1 3 1]           [1 2 1]
+
+Both determinants equal 3. They have the same pair traces. Their oriented
+triple traces differ, so they are not simultaneously conjugate. This also
+shows why transposition or reversing a word cannot silently be included in
+the stated equivalence. The witness is a distinction between states, not a
+paid-call lower bound.
+
+## Finite evidence and scope
+
+The checker enumerates every unit-diagonal Gram with nonzero off-diagonal
+entries over F2,F3,F4,F5,F7. Its diagonal orbits are compared as complete sets
+with a second enumeration that starts from four trace parameters. A six-term
+determinant and row reduction separately check invertibility. Actual matrix
+multiplication checks idempotence and the projector traces. F4 uses
+F2[t]/(t^2+t+1), rather than integers modulo four.
+
+For every canonical ternary class, another check enumerates every GL3(F3)
+basis change, computes all three conjugated matrices and their traces, and
+counts the distinct endpoints. Zero spectators of dimensions one and two,
+ten invalid-domain cases and the pair-trace ambiguity witness exercise the
+boundaries. These finite checks corroborate the proof; they do not replace
+its all-field argument or establish a physical caller. Singular active
+systems, missing pair couplings, unordered equivalence, native implementation,
+dirty-helper programs and a complete recursive invoice are outside scope.

research/regular-projector-triple-orbits/README.md

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+# Exact orbits of regular fully coupled projector triples
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+For an ordered rank-one idempotent triple P1,P2,P3 over any field, put
+S=P1+P2+P3. Assume rank(S)=rank(S^2)=3 and all three pair traces are nonzero.
+Then the ambient dimension and
+
+    p=tr(P1P2), q=tr(P1P3), r=tr(P2P3), u=tr(P1P2P3)
+
+are a complete invariant under simultaneous GL conjugacy. The admissible
+quadruples are exactly the nonzero p,q,r,u with
+
+    1-p-q-r+u+pqr/u != 0.
+
+Over every finite field of order Q there are exactly
+
+    Q^4-5Q^3+9Q^2-9Q+6
+
+such classes, in every fixed ambient dimension n>=3. [PROOF.md](PROOF.md)
+gives the all-field proof, explicit representative and centralizer. Two
+triples over F5 give different classes with identical
+pair traces. Thus a descriptor retaining only the three pair traces loses
+information even in this regular class.
+
+Claim type: scoped written algebraic proof with exact finite corroboration.
+The new result is the triple normal form and closed class count. It does not
+claim global literature priority for rank-one factorizations, diagonal
+similarity or cycle products, which are classical methods.
+
+## Evidence and reproduction
+
+From the repository root, with CPython 3.12 and its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/regular-projector-triple-orbits/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/regular-projector-triple-orbits/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/regular-projector-triple-orbits
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/regular-projector-triple-orbits --check-source-history
+```
+
+Another machine can substitute its CPython 3.12 executable. Direct replay
+also works from a Git-free export of the six package files. The verifier
+writes nothing. It prints a generated receipt with the six file hashes and
+`PASS_REGULAR_PROJECTOR_TRIPLE_ORBITS`.
+
+The full six-entry Gram domains contain 1, 64, 729, 4,096 and 46,656 matrices
+over F2,F3,F4,F5,F7. They give 0, 6, 50, 186 and 1,070 classes respectively.
+Every accepted Gram is compared with its four-trace representative. A second
+enumeration expands every representative's complete diagonal orbit, and the
+two full sets must agree. Matrix multiplication checks the projector traces;
+row reduction separately checks invertibility. F4 uses t^2+t+1.
+
+For every ternary representative, all 11,232 GL3(F3) basis changes are checked,
+giving 67,392 conjugated triples and 33,696 distinct endpoints. This agrees
+with the proved scalar centralizer. Spectator dimensions zero, one and two,
+ten domain rejection controls and an explicit F5 pair-trace ambiguity fixture
+also pass. The universal quantifiers come from the proof, rather than an
+extrapolation of these finite domains. No predecessor scientific checker is
+imported or run.
+
+## Scope, provenance and credit
+
+This is an original state-classification side question, with a changed
+`bit-new-register-model` scope recorded in [TASK.json](TASK.json). It concerns
+full simultaneous conjugacy on a three-dimensional active direct summand.
+The separately claimed width-two monomial/reversal quotient concerns
+target-preserving word symmetries and a different equivalence relation.
+
+Credit MatShep's [PR4762](https://github.com/CrocSwap/integer-mult-bounds/pull/4762),
+head `3b4e0449262b0b1e07b80e009ef71cc0ca5bf816`, for the related two-idempotent
+factor descriptor, and codyw912's
+[PR4735](https://github.com/CrocSwap/integer-mult-bounds/pull/4735), head
+`4afbea22e79e19d1d65702eb4090d454ee39d967`, for its inclusion/observation
+framework. Those results are context, not proof premises or audit targets.
+The proof here derives the three-factor reduction directly. Credit the
+eumemic/woklerx scalar-projector register lineage and the retained native,
+frame, helper, compiler and outer contributors for the application context.
+No predecessor files or credit lines are changed.
+
+Canonical base is `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8,
+selected PR6030. [SOURCES.json](SOURCES.json) pins task provenance. There are
+no external scientific runtime inputs. The
+[advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19373460)
+records the scope. Live claims, local reservations, recent account work,
+main README searches and all-state general/formula PR searches were checked.
+The broader cache covered 556 open PRs and 6,605 Discussion records with
+complete remote pagination; sparse local coverage was 137 of 8,917 notes.
+Arbitrary historical diffs and unpublished work remain limits.
+
+Singular active systems and zero pair traces are excluded. Projector order,
+field and ambient dimension are retained. Transpose, reversal, field
+automorphisms and projector permutations are not part of this equivalence.
+This is not a complete word-state invariant. A search must still restrict
+conjugations to its target stabilizer and preserve all other state. Physical
+basis changes, native maps, input access, arbitrary dirty restoration,
+precision/tape/stock, compiler closure and a complete paid/target invoice
+are unpriced. No supplier, paid-call saving, kappa or selected-frontier change
+is claimed. All five selected framework assumptions remain.
+
+No selected scientific cold replay, Lean, Docker, independent human review
+or formal proof-kernel verification is claimed.
+
+## Open next questions
+
+- Classify a singular or partially uncoupled triple. Begin with one fixed
+  singular Gram and retain its factor kernel and observation image. The
+  regular direct-sum reduction used here is unavailable there.
+- Bind this descriptor to one target-preserving search and charge every
+  basis change. Begin with an explicit alphabet and target stabilizer.
+  A concrete caller and its full paid invoice are still missing.
+
+Structured first steps and blockers are in TASK.json. No construction
+expansion follows from this algebraic count.
+
+## Verifier constraints
+
+All function bodies have fewer than 60 physical lines. There is no recursion,
+network, subprocess, dynamic code execution or output-file creation. Public
+matrix and field boundaries validate their finite domains. Internal calls
+use those validated objects. Loop bounds are Q<=7, matrix dimension<=5,
+six Gram entries, four trace parameters, and the 3^9 ternary basis candidates.
+Mathematical guards are explicit exceptions and remain active under -O.
+
+Power-of-ten rule 3 deviation: CPython allocates managed integers, tuples,
+sets and dictionaries during checking. At most 46,656 raw Gram candidates,
+38,520 accepted Grams per set, 1,312 class keys and 33,696 distinct ternary
+endpoints are retained in fixed bounded phases. File reads are limited to
+one MiB plus one byte for each of six fixed files. The normal and optimized
+registered commands exercise these bounds. A fixed-allocation embedded
+implementation would require a different runtime. Warning-strict execution
+and source compilation are the available diagnostics; no separate static
+linter or repository-wide build configuration is introduced.

research/regular-projector-triple-orbits/SOURCES.json

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