Finite-Sector Block-Primitive Criterion

• Finite-Sector Block-Primitive Criterion is a methodological pattern that uses finite tests on structured blocks to certify domain-specific primitive properties.
• It unifies diverse approaches across control theory, finite fields, symbolic dynamics, permutation groups, and cryptography by replacing infinite searches with finite criteria.
• The criterion converts complex, infinite evaluative processes into tractable tests, enabling precise stability, primitivity, and security proofs in various applied fields.

to=functions.exec_command ＿色json 彩票主管":"/bin/bash","command":"python - <<'PY'\nimport urllib.request, urllib.parse\nurl='https://export.arxiv.org/api/query?search_query=id:([2511.20995](/papers/2511.20995))&start=0&max_results=1'\ntry:\n print(urllib.request.urlopen(url, timeout=20).read()[:300].decode('utf-8','ignore'))\nexcept Exception as e:\n print('ERR',e)\nPY"} 天铭ెంట్ to=functions.exec_command 天天乐彩票 񹚊ppassistant to=functions.exec_command _天天啪 彩彩票娱乐/json {"result":"ERR <urlopen error timed out>"} The expression Finite-Sector Block-Primitive Criterion denotes, in the cited literature, a family of finite certification principles in which a structured “sector” or “block” decomposition is used to prove a notion of primitivity. In one explicit terminology alignment, it corresponds directly to the exact, finite-dimensional representation of full-block circle-criterion multipliers for discrete-time feedback systems with non-repeated, sector-bounded nonlinearities (Biertümpfel et al., 26 Nov 2025). In broader usage across adjacent areas, closely analogous criteria certify the existence of primitive elements in finite fields, the primitivity of permutation-group actions on blocks, the finiteness of block maps between substitution subshifts, the primitivity of words in free groups, and the primitivity of round-function groups in block ciphers (Bagger et al., 29 Jul 2025).

1. Domain, terminology, and recurring structure

The phrase is not tied to a single mathematical discipline. Rather, the cited sources use the same vocabulary around three recurrent ingredients: a sectorized or block-structured ambient object, a finite criterion, and a primitive target property. This suggests a family resemblance across fields rather than a single canonical theorem.

Domain	Primitive object	Finite criterion
Control theory	Full-block circle-criterion multiplier class	Finite copositivity constraints
Finite fields	Primitive elements or Singer cycles	Sieve inequalities or characteristic-polynomial tests
Finite groups and designs	Primitive block action or $p$-solvability obstruction	Triangle obstruction or maximal stabilizer
Symbolic dynamics and free groups	Block maps or primitive words	Finite template sets or graph-distance tests
Cryptography	Primitive round-function group	SPN reduction or non-type-preserving mixing

In all of these settings, the term primitive is domain-specific. A primitive element of $\mathbb{F}_{q^r}$ generates $\mathbb{F}_{q^r}^{\times}$; a primitive permutation group admits no nontrivial invariant partition; a primitive substitution is one whose incidence matrix is eventually positive; and a primitive word in $F_k$ belongs to some free basis. The unifying motif is that a potentially infinite or hard-to-check condition is replaced by a finite, structurally parameterized test.

2. Exact finite-dimensional full-block criterion in control

The most direct control-theoretic instantiation studies a discrete-time LTI system in feedback with a non-repeated, sector-bounded nonlinearity $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$, where each scalar component satisfies

$$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$

For a set of input/output pairs $G \subset \mathbb{R}^{2m}$, a symmetric matrix $M \in S^{2m}$ defines a QC if $z^T M z \ge 0$ for all $z \in G$. The classical diagonal sector multipliers form

$\mathbb{F}_{q^r}$0

but the complete static multiplier class is the full-block family

$\mathbb{F}_{q^r}$1

This class is necessary and sufficient for the union of graphs $\mathbb{F}_{q^r}$2, but in raw form it is defined by an uncountable family of LMIs and is therefore not computationally tractable (Biertümpfel et al., 26 Nov 2025).

The key reduction uses the piecewise-linear “wedge” function

$\mathbb{F}_{q^r}$3

applied elementwise as $\mathbb{F}_{q^r}$4. The paper proves the set equivalence

$\mathbb{F}_{q^r}$5

where $\mathbb{F}_{q^r}$6 is the incremental graph of $\mathbb{F}_{q^r}$7. This converts “for all $\mathbb{F}_{q^r}$8” into “for all incremental pairs of a single wedge nonlinearity.”

With $\mathbb{F}_{q^r}$9 and $\mathbb{F}_{q^r}^{\times}$0, the finite class is

$\mathbb{F}_{q^r}^{\times}$1

which imposes $\mathbb{F}_{q^r}^{\times}$2 copositivity constraints on $\mathbb{F}_{q^r}^{\times}$3 matrices. The central theorem states that

$\mathbb{F}_{q^r}^{\times}$4

and hence

$\mathbb{F}_{q^r}^{\times}$5

This is the exact finite-dimensional characterization: the complete full-block multiplier class is recovered from a finite family of copositivity constraints. For $\mathbb{F}_{q^r}^{\times}$6, copositivity is exact via Diananda’s theorem, since every copositive matrix in dimensions $\mathbb{F}_{q^r}^{\times}$7 is representable as PSD plus componentwise nonnegative; for $\mathbb{F}_{q^r}^{\times}$8, one typically uses outer approximations (Biertümpfel et al., 26 Nov 2025).

The multiplier enters a dissipation inequality through the affine map $\mathbb{F}_{q^r}^{\times}$9. Under $F_k$0, if there exist $F_k$1 and $F_k$2 such that

$F_k$3

then the interconnection is well-posed, internally stable, and has finite induced $F_k$4-gain $F_k$5 for every non-repeated $F_k$6. The resulting SDP,

$F_k$7

yields the least conservative static-multiplier certificates. In the illustrative $F_k$8 example, the certified gains were $F_k$9 for $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$0, $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$1 for the vertex-relaxed full-block class $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$2, and $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$3 for the exact finite class, with the exact formulation also attaining the largest certified sector bound $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$4 versus $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$5 and $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$6 (Biertümpfel et al., 26 Nov 2025).

3. Finite-field primitive elements, sieves, and block companions

In finite-field theory, a closely related criterion arises from the existence problem for primitive elements inside structured subsets $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$7. Writing $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$8, an element $\Phi(v) = [\phi_1(v_1),\dots,\phi_m(v_m)]^T$9 is primitive when $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$0. For multiplicative characters $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$1, the relevant sums are

$$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$2

Using the Vinogradov indicator for $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$3-free elements, the counting function

$$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$4

is obtained, where $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$5 and $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$6. Under the uniform hypothesis $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$7 for all nontrivial $$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$8, the modified prime sieve yields

$$(\phi_i(\xi)-\alpha \xi)(\beta \xi-\phi_i(\xi)) \ge 0.$$9

after decomposing

$G \subset \mathbb{R}^{2m}$0

If $G \subset \mathbb{R}^{2m}$1 and

$G \subset \mathbb{R}^{2m}$2

then $G \subset \mathbb{R}^{2m}$3 contains a primitive element. The paper emphasizes that the sieve depends only on a uniform character-sum bound and is therefore flexible for structured subsets (Bagger et al., 29 Jul 2025).

A particularly important application is the complement $G \subset \mathbb{R}^{2m}$4 of $G \subset \mathbb{R}^{2m}$5 affine $G \subset \mathbb{R}^{2m}$6-hyperplanes in general position, for which $G \subset \mathbb{R}^{2m}$7. Combining the prime-sieve corollary with the Grzywaczyk–Winterhof bound

$G \subset \mathbb{R}^{2m}$8

and, for even $G \subset \mathbb{R}^{2m}$9, the Cheng–Winterhof refinement

$M \in S^{2m}$0

the paper proves explicit sufficient conditions ensuring that $M \in S^{2m}$1 contains a primitive element, thereby improving earlier unsieved results of Fernandes and Reis (Bagger et al., 29 Jul 2025).

The same source formulates a block version directly: if $M \in S^{2m}$2 and each block satisfies $M \in S^{2m}$3, then $M \in S^{2m}$4, so the modified sieve applies with $M \in S^{2m}$5. This is the most literal finite-sector/block extension in the finite-field setting.

A second finite-field criterion concerns consecutive primitive elements in $M \in S^{2m}$6. For odd $M \in S^{2m}$7, Jarso and Trudgian prove that four consecutive primitive elements always exist when $M \in S^{2m}$8. Their sieve introduces

$M \in S^{2m}$9

for block length $z^T M z \ge 0$0, where $z^T M z \ge 0$1 are the primes dividing $z^T M z \ge 0$2 but not a chosen divisor $z^T M z \ge 0$3, and gives explicit thresholds implying $z^T M z \ge 0$4, hence an $z^T M z \ge 0$5-block of consecutive primitive elements (Jarso et al., 2021).

A matrix-theoretic analogue appears in the theory of block companion Singer cycles. An $z^T M z \ge 0$6-block companion matrix $z^T M z \ge 0$7 is a Singer cycle iff its characteristic polynomial $z^T M z \ge 0$8 is primitive of degree $z^T M z \ge 0$9. This is the basic block-primitive criterion for recursive vector sequences and word-oriented LFSRs. The paper further conjectures the exact count

$z \in G$0

and proves the corresponding fiber formula for $z \in G$1 (Ghorpade et al., 2011).

4. Block graphs, permutation-group primitivity, and invariant designs

In finite-group representation theory, the relevant object is the block graph $z \in G$2, whose vertices are the primes dividing $z \in G$3, with an edge $z \in G$4 when the principal $z \in G$5- and $z \in G$6-blocks have a nontrivial common irreducible character. The central criterion is a triangle obstruction: if $z \in G$7 has no triangle containing a prime $z \in G$8, then $z \in G$9 is $\mathbb{F}_{q^r}$00-solvable. Specializing to $\mathbb{F}_{q^r}$01, $\mathbb{F}_{q^r}$02 is solvable iff $\mathbb{F}_{q^r}$03 has no triangle containing $\mathbb{F}_{q^r}$04, where $\mathbb{F}_{q^r}$05 is the solvable radical. The same framework also characterizes nilpotency by the absence of edges, and shows that for finite nonabelian simple groups the block graph is complete except for $\mathbb{F}_{q^r}$06, missing the edge $\mathbb{F}_{q^r}$07, and $\mathbb{F}_{q^r}$08, missing $\mathbb{F}_{q^r}$09 (Brough et al., 2017).

For finite simple groups of Lie type, a further block-theoretic criterion governs the Steinberg character. If $\mathbb{F}_{q^r}$10 is defined over $\mathbb{F}_{q^r}$11 with defining characteristic $\mathbb{F}_{q^r}$12, and $\mathbb{F}_{q^r}$13, then the Steinberg character lies in the principal $\mathbb{F}_{q^r}$14-block iff $\mathbb{F}_{q^r}$15 is a regular number of $\mathbb{F}_{q^r}$16. This criterion is used to certify block-graph edges and, in turn, triangles (Brough et al., 2017).

In design theory, block-primitive has its classical permutation-group meaning. For a $\mathbb{F}_{q^r}$17-invariant design $\mathbb{F}_{q^r}$18 with $\mathbb{F}_{q^r}$19 transitive on $\mathbb{F}_{q^r}$20, the action on blocks is primitive iff the block stabilizer $\mathbb{F}_{q^r}$21 is maximal in $\mathbb{F}_{q^r}$22, equivalently iff there is no nontrivial $\mathbb{F}_{q^r}$23-invariant partition of $\mathbb{F}_{q^r}$24. The cited construction starts with a primitive action $\mathbb{F}_{q^r}$25, a maximal subgroup $\mathbb{F}_{q^r}$26, and an $\mathbb{F}_{q^r}$27-orbit $\mathbb{F}_{q^r}$28, then defines

$\mathbb{F}_{q^r}$29

This yields a $\mathbb{F}_{q^r}$30-invariant $\mathbb{F}_{q^r}$31-design with

$\mathbb{F}_{q^r}$32

and if $\mathbb{F}_{q^r}$33 is $\mathbb{F}_{q^r}$34-transitive, a $\mathbb{F}_{q^r}$35-design with the usual

$\mathbb{F}_{q^r}$36

The paper proves both the forward construction and the converse statement that every point- and block-primitive $\mathbb{F}_{q^r}$37-invariant design arises by “merging” $\mathbb{F}_{q^r}$38-orbits for a maximal block stabilizer (Saeidi, 2024).

5. Primitive substitutions, block maps, and primitive words

In symbolic dynamics, the criterion concerns the finiteness of block maps between substitution subshifts. Let $\mathbb{F}_{q^r}$39 and $\mathbb{F}_{q^r}$40 be primitive aperiodic substitutions that are either uniform or Pisot, with equal Perron–Frobenius eigenvalues

$\mathbb{F}_{q^r}$41

Then there exists a finite set $\mathbb{F}_{q^r}$42 of block maps $\mathbb{F}_{q^r}$43 such that every block map $\mathbb{F}_{q^r}$44 is of the form

$\mathbb{F}_{q^r}$45

for some $\mathbb{F}_{q^r}$46 and $\mathbb{F}_{q^r}$47. The mechanism uses Mossé recognizability, almost inverses in the category of dill maps, and balanced-growth invariants $\mathbb{F}_{q^r}$48 and $\mathbb{F}_{q^r}$49. Under the conjugation scheme

$\mathbb{F}_{q^r}$50

the equality $\mathbb{F}_{q^r}$51 keeps $\mathbb{F}_{q^r}$52, while the composition laws

$\mathbb{F}_{q^r}$53

force bounded radius and bounded deviation, producing only finitely many possible templates modulo shift (Salo et al., 2013).

In free groups, a graph-theoretic block criterion is given via Stallings core graphs. If $\mathbb{F}_{q^r}$54 are finitely generated and $\mathbb{F}_{q^r}$55 is a quotient of $\mathbb{F}_{q^r}$56, then

$\mathbb{F}_{q^r}$57

where $\mathbb{F}_{q^r}$58 is the shortest directed-path length in the DAG of immediate quotients. For a word $\mathbb{F}_{q^r}$59, primitivity is the special case $\mathbb{F}_{q^r}$60, $\mathbb{F}_{q^r}$61. The same paper studies the measure-preserving criterion: primitive implies measure preserving on every finite group, and the converse is proved for $\mathbb{F}_{q^r}$62 and, more generally, for subgroups $\mathbb{F}_{q^r}$63 with $\mathbb{F}_{q^r}$64 (Puder, 2011).

These two settings share a common structural feature: an infinite search over local rules or basis changes is reduced to a finite region of the relevant parameter space. In substitution systems the finite region is defined by bounded $\mathbb{F}_{q^r}$65, $\mathbb{F}_{q^r}$66, and effective radius; in free groups it is the finite fringe of the core graph.

6. Primitive round-function groups in block ciphers

In symmetric cryptography, block-primitivity is a permutation-group property used to exclude imprimitivity attacks. For Lai–Massey schemes on $\mathbb{F}_{q^r}$67, with round core $\mathbb{F}_{q^r}$68, linear map $\mathbb{F}_{q^r}$69, and round permutations

$\mathbb{F}_{q^r}$70

the main result reduces the primitivity analysis of the Lai–Massey group

$\mathbb{F}_{q^r}$71

to that of the associated SPN group

$\mathbb{F}_{q^r}$72

If $\mathbb{F}_{q^r}$73 is primitive on $\mathbb{F}_{q^r}$74, then $\mathbb{F}_{q^r}$75 is primitive on $\mathbb{F}_{q^r}$76. A standard sufficient SPN-side criterion assumes $\mathbb{F}_{q^r}$77, $\mathbb{F}_{q^r}$78, with $\mathbb{F}_{q^r}$79 a bricklayer permutation and $\mathbb{F}_{q^r}$80 strongly proper, while the S-box $\mathbb{F}_{q^r}$81 is non-affine and strongly anti-invariant. Under these hypotheses, the Lai–Massey cipher resists imprimitivity attacks under independent round keys (Aragona et al., 2020).

A related criterion for SPNmod and generalized GOST-like ciphers uses non-type-preserving mixing layers. With $\mathbb{F}_{q^r}$82, $\mathbb{F}_{q^r}$83, and block matrix $\mathbb{F}_{q^r}$84, the type of a subgroup of $\mathbb{F}_{q^r}$85 is recorded as $\mathbb{F}_{q^r}$86, corresponding to white, ruled, and black boxes. The matrix is type-preserving precisely when it preserves all such types, and non-type-preserving otherwise. A simple sufficient test is: $\mathbb{F}_{q^r}$87 which implies that $\mathbb{F}_{q^r}$88 is non-type-preserving. If the round function is $\mathbb{F}_{q^r}$89 with parallel invertible S-box $\mathbb{F}_{q^r}$90, then the group $\mathbb{F}_{q^r}$91 is primitive whenever $\mathbb{F}_{q^r}$92 is non-type-preserving. The paper verifies this property for the mixing layers of AES, PRESENT, and GOST-like rotations in the stated parameter ranges (Aragona et al., 2018).

7. Scope, limitations, and comparative interpretation

The criteria surveyed here are exact within their own ambient categories, but their hypotheses are sharply domain-dependent. The control-theoretic finite full-block characterization is exact only after passing to copositivity, and exact conic implementation is presently restricted to $\mathbb{F}_{q^r}$93; for $\mathbb{F}_{q^r}$94, ordinary PSD-plus-nonnegative relaxations are generally outer approximations (Biertümpfel et al., 26 Nov 2025). The finite-field sieve requires a uniform character-sum bound $\mathbb{F}_{q^r}$95, and its sharpness depends on how well that bound reflects the structure of $\mathbb{F}_{q^r}$96; order-dependent or average character information is not yet incorporated in the stated form (Bagger et al., 29 Jul 2025). The consecutive-primitive-element sieve becomes computationally difficult for longer blocks, as shown by the unresolved $\mathbb{F}_{q^r}$97 regime (Jarso et al., 2021).

In group- and design-theoretic settings, maximality and orbit enumeration are decisive but may be computationally burdensome for large groups. The design-construction framework is theoretically complete, yet the paper explicitly notes that enumerating all designs for larger groups may be limited by computational complexity (Saeidi, 2024). In free groups, the Stallings-core criterion is algorithmic and complete, whereas the measure-preserving characterization is proved in the paper only for $\mathbb{F}_{q^r}$98 and for the range $\mathbb{F}_{q^r}$99 (Puder, 2011). In symbolic dynamics, the finiteness theorem is tied to the uniform-or-Pisot balanced-growth regime and the eigenvalue alignment $\mathbb{F}_{q^r}^{\times}$00; non-Pisot primitive substitutions need not satisfy the same bounded-deviation mechanism (Salo et al., 2013).

Taken together, these results show that a finite-sector or block-primitive criterion typically has three features: a structural decomposition into sectors, blocks, or orbit pieces; an exact or explicit finite test replacing an infinite search; and a primitive conclusion that excludes hidden decompositions. The phrase therefore names not a single theorem, but a recurrent methodological pattern linking control-theoretic multipliers, finite-field sieves, permutation-group actions, symbolic block codes, free-group primitivity, and block-cipher security.