For any modulus $m\ge 2$, define the coprime density

$$\delta (m)=\frac{\phi (m)}{m},$$

the fraction of residues mod $m$ that are coprime to $m$. For a prime $p$, this equals $\delta (p)=1-\frac{1}{p}$.

Assuming independent sampling, the probability of a composite $m$ passing the test is approximately

$$Pr(\text{pass}\mid m)\approx {\delta }_S(m{)}^k.$$

This yields the following key behavior:

• For even composites ($q=2$): $Pr(\text{pass})\le (1/2{)}^k$
• For $q=3$: $Pr(\text{pass})\le (2/3{)}^k$
• For semiprimes with large factors ($q\gg 1$): the bound weakens, requiring larger $k$

Thus the heuristic strongly rejects composites with small factors, and becomes probabilistically weaker only for products of large primes.

function is_prime_candidate(m, k, slice="half", 
                           small_primes=[2,3,5,7,11,13]):
    // (1) Deterministic prefilter by small primes
    for q in small_primes:
        if m % q == 0 and m != q:
            return False
    
    // (2) Choose slice
    if slice == "half":
        S = {1, 2, ..., floor(m/2)}
    else if slice == "quarter":
        S = {1, 2, ..., floor(m/4)}
    else:
        S = {1, 2, ..., m-1}
    
    // (3) Random sampling
    for i in 1..k:
        r = random_choice(S)
        if gcd(r, m) != 1:
            return False
    
    return True  // passes fractional-slice test

The parameter $k$ governs tradeoff between false-positive rate and computational cost. For a target false positive probability $\epsilon$ and smallest divisor threshold $Q$, choose

$$k\ge \frac{\log \epsilon }{\log (1-1/Q)}.$$

The fractional-slice heuristic can be visualized in two complementary ways: on the unit circle, and as a 2D plane of concentric rings.

Map each residue $r\in \{1,2,\dots ,m-1\}$ to the point

$$z_r=e^{2\pi ir/m}.$$

Coprime residues lie on "open channels" that avoid blocked directions determined by the factors of $m$.

Blocked Farey channels appear as gaps along the arc where $gcd(r,m)>1$. Prime moduli maximize the fraction of occupied (coprime) points in every slice; composites produce visible voids aligned with their divisors.

The unit circle can be extended to a 2D plane of concentric rings to capture the nested structure of residues for multiple moduli.

• Let each ring correspond to a modulus $m$.
• Place the points $z_r=e^{2\pi ir/m}$ on the ring of radius proportional to $m$.
• Coprime residues are marked (filled dots), blocked residues are shown differently.

Beyond fractional-slice sampling, we can analyze the geometric uniformity of coprime residues by measuring chord lengths between consecutive points on the unit circle. This provides an independent geometric signature for primality.

1. Input: modulus n
2. Compute coprime residues R_n = {r : gcd(r,n)=1, 1≤r<n}
3. For each consecutive pair (r_i, r_{i+1}):
     L_i = 2·sin(π·(r_{i+1}-r_i)/n)
4. Compute statistics:
     μ_L = mean(L_i)
     σ_L = std_dev(L_i)
     CV = σ_L / μ_L
     gap_ratio = max(L_i) / min(L_i)
5. Decision thresholds (range-dependent):
     For n ≤ 100:    CV < 0.22 → likely prime
     For n ≤ 500:    CV < 0.15 → likely prime
     For n ≤ 1000:   CV < 0.12 → likely prime
     For n > 1000:   CV < 0.30/√(n/100) → likely prime
     Universal:      CV > 0.27 → likely composite
     Ambiguous zone: thresholds provide classification confidence
6. Return CV, gap_ratio, uniformity_score = 1/CV, verdict
            

• Semiprime leakage: Composites with large prime factors can pass the test.
• Carmichael-like pseudoprimes: These may evade simple gcd-based filtering.
• Slice bias: Improper slice selection can undercount blocked channels.

The heuristic is therefore best used as a rapid prefilter prior to a deterministic or probabilistic primality test (e.g., Miller–Rabin).

Fractional-slice coprimality sampling provides a geometric and probabilistic bridge between Nested Farey Channel theory and practical number testing. It captures the essential property that prime moduli maintain maximal channel openness even when viewed through restricted arcs of the unit circle, while composites introduce blocked Farey channels visible through partial sampling.

This heuristic can serve both as:

1. A fast preliminary screen for prime candidates within modular sieves.
2. A geometric diagnostic tool for visualizing the distribution of coprime residues across modular arcs.

Further work includes: quantifying slice bias functions ${\beta }_S(m)$, linking partial-channel openness to residue equidistribution, and integrating this fractional sampling into the modular sieve hierarchy for scalable prime detection.