📐algorithm-mobius-sieve-tests

The groupant beside algorithms/src/mobius_sieve/tests.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

Facts

Components

Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

Model text

descriptant

algorithm-mobius-sieve-tests

Epoch2026-10-30: original implementation and independent fixtures. Prospective relation: Finite executable observation: Return inclusive mu table with mu(0)=0,mu(1)=1. Positive square-divisible k maps0; otherwise (-1)^r for r distinct prime factors. Linear sieve assigns each composite by least prime: p divides i => mu(ip)=0 and stop; else mu(ip)=-mu(i). Reject n>1000000 before allocation. Reference: Hardy and Wright, An Introduction to the Theory of Numbers, chapter XVI, arithmetical functions. Assertions check independent arithmetic oracles, boundaries and identities; passing fixtures do not establish untested behavior.

Predicted cost: O(n) sieve time and storage. Independent trial factorization through10000 has O(N sqrt N) upper bound. Allocation failure follows Vec policy.

Compiled controls are rejected and exact bytes restored. Six file roots contain36 typed views, no fragments. Twice-identical partial syntax binds source; macros remain opaque. No semantic AST, formal proof, approximate surrogate or speed claim.

Formatting follow-up: Rust 2024 style only collapses the final trial-oracle conditional. Previous native evidence was collected at source SHA-256 4503d3db2efbc8ec4d4ea57e6b93f54ec995259c17f5074125b718c75d27dc6c; executable behavior is unchanged. Current binding is 0c965bf9dfb4a78ec5f337a1af8f056de039aa80a2641b8575f0aaad74d1ec2a. Fresh partial syntax is generated twice identically; root integration repeats native checks on the formatted source.

model

algorithm-mobius-sieve-tests

Forward

Finite executable observation: Return inclusive mu table with mu(0)=0,mu(1)=1. Positive square-divisible k maps0; otherwise (-1)^r for r distinct prime factors. Linear sieve assigns each composite by least prime: p divides i => mu(ip)=0 and stop; else mu(ip)=-mu(i). Reject n>1000000 before allocation. Reference: Hardy and Wright, An Introduction to the Theory of Numbers, chapter XVI, arithmetical functions. Assertions check independent arithmetic oracles, boundaries and identities; passing fixtures do not establish untested behavior.

Invariant

Primes ascend; each composite assigned once by least prime. Values in {-1,0,1}; p<=n/i checked before product. Divisible branch stops duplicate generation.

Cost

O(n) sieve time and storage. Independent trial factorization through10000 has O(N sqrt N) upper bound. Allocation failure follows Vec policy.

behavior_complement

Universe is fixture executions and assertion outcomes. A passing actual/expected mismatch is excluded; failure cannot count as success. This finite complement does not cover all API calls.

reverse_reconstruction

Passing summary retains every implementation agreeing on exercised fixtures, including implementations wrong elsewhere. Failure locates an observed disagreement but cannot alone distinguish oracle and implementation defects.

factor_oracle

Trial division and repeated-factor detection independently compare every n=0..10000, without sieve prime discovery.

divisor_identity

sum_{d|n} mu(d)=1 for n=1 and0 for n>1. Tests check n<=512; no n=0 extension.

resource_partition

Zero and one are valid; MAX_N accepted with mu(MAX_N)=0; MAX_N+1 and usize::MAX rejected before allocation.

least_prime

The first prime dividing i stops generation. Changing the other branch sign compiles and fails exhaustive_factor_oracle.

evidence

Evidence

Source SHA-256: 0c965bf9dfb4a78ec5f337a1af8f056de039aa80a2641b8575f0aaad74d1ec2a.

Nine tests total, three per algorithm, pass twice debug and twice release with identical normalized outcomes. Three compiled production controls fail exhaustive_gcd_oracle, exhaustive_factor_oracle and exhaustive_iterative_oracle; exact bytes restored before positives. Strict all-target Clippy and narrow formatting pass. External wave16 numeric bundle retains logs. Finite checks do not establish universal correctness or allocator success.

• debug-1-mobius_sieve.log: a43ee28acac72986c7fd16ecbc4a596e3717dc7b69bfb7422e9c2e14bc3d0ad2
• debug-2-mobius_sieve.log: b9c19a1fe2b12b1ec42cc51c2c800b181cb2d353dea895afefbf7d729c5de96b
• release-1-mobius_sieve.log: 3d751bb23d4c0820235ddce02784b0e24752f5a9266a25ff1ddfdeea674bf6c1
• release-2-mobius_sieve.log: 8044f21edb7c039f930ccab6a3056ca36b8bede55c572fd62239f44f32cda1c6
• mutant-mobius_sieve.log: 4a1888786fe96054fd5e7a379fc033d76390b133291b03d0fdb8b21343cb0f26

Formatting follow-up: Rust 2024 style only collapses the final trial-oracle conditional. Previous native evidence was collected at source SHA-256 4503d3db2efbc8ec4d4ea57e6b93f54ec995259c17f5074125b718c75d27dc6c; executable behavior is unchanged. Current binding is 0c965bf9dfb4a78ec5f337a1af8f056de039aa80a2641b8575f0aaad74d1ec2a. Fresh partial syntax is generated twice identically; root integration repeats native checks on the formatted source.

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