TNFR Arithmetic Pressure — Independence & Completeness Audit (R7)

Status: primality sufficiency PROVED (classical); channel functional independence PROVED (exact witness proof over ℚ); minimality-for-primality NEGATIVE (redundant); structural completeness OPEN / CONJECTURAL (NT-P07). The claim is split into NT-P07a–e (§6). The "minimal and complete" language is downgraded to explicit scope. Arithmetic pressure is a structural descriptor, computed from the factorisation — not a primality-discovery algorithm. Module: src/tnfr/mathematics/arithmetic_pressure.py · Tests: tests/mathematics/test_arithmetic_pressure.py · Benchmark: benchmarks/arithmetic_pressure_audit.py · Audits: TNFR_NUMBER_THEORY.md NT-C01 · Depends on: C5.

1. The three-channel pressure

The selected static arithmetic pressure descriptor (number_theory.py, ArithmeticTNFRFormalism.delta_nfr_value, unit coefficients) is

\[\Delta\mathrm{NFR}(n) = \underbrace{(\Omega(n) - 1)}_{c_1} + \underbrace{(\tau(n) - 2)}_{c_2} + \underbrace{\Big(\tfrac{\sigma(n)}{n} - \big(1 + \tfrac1n\big)\Big)}_{c_3},\]

with Ω the prime-factor count with multiplicity, τ the divisor count, and σ the divisor sum. This module recomputes every channel exactly over ℚ (fractions.Fraction) and agrees with the canonical realisation to 0 mismatch on [2, 300). Three claims that are easy to conflate are audited separately.

2. Primality sufficiency — and redundancy (PROVED / NEGATIVE)

Each channel is non-negative for n ≥ 2 and vanishes exactly at the primes:

\[c_1(n) = 0 \iff \Omega(n) = 1,\quad c_2(n) = 0 \iff \tau(n) = 2,\quad c_3(n) = 0 \iff \sigma(n) = n + 1,\]

and all three conditions are equivalent to "n prime". Verified on [2, 1000] (all_channels_sufficient, channels_nonnegative, pressure_zero_iff_prime).

A direct consequence is that the three-channel set is redundant for primality: a single channel already characterises the primes, so minimal_channels_for_primality = 1 and every ablation still detects the primes (ablation_detects_primes for all six non-empty subsets). Hence the realisation is not minimal for prime detection — the "minimal" claim, read as minimality for primality, is false (a MEASURED negative). The channels are not there to detect primes more than once; they are there to carry distinct structure.

3. Linear independence vs correlation (PROVED + MEASURED)

As real functions on the audited range the channels are linearly independent:

\[\operatorname{rank}[c_1\; c_2\; c_3] = 3,\qquad \operatorname{rank}[c_1\; c_2\; c_3\; \mathbf 1] = 4,\]

so no channel is a linear — or even affine — combination of the others (channel_rank, has_linear_relation == False). Yet they are strongly correlated, all rising with compositeness:

pair	Pearson r on [2, 1000]
c1–c2	0.887
c1–c3	0.877
c2–c3	0.931

Correlation is not linear dependence. High r (≈ 0.9) coexists with full rank. This excludes a fixed linear or affine relation; it does not establish statistical independence or rule out nonlinear dependencies among the channels.

Exact proof (PROVED over ℚ). Beyond the numerical rank, functional independence is proved exactly from three witness points (prove_functional_independence). Suppose a·PΩ + b·Pτ + c·Pσ = 0 for all n ≥ 2. At the prime squares 4 = 2² and 9 = 3² the channels are (1, 1, 1/2) and (1, 1, 1/3); their difference (0, 0, 1/6) forces c = 0 and then a + b = 0. At the semiprime 6 = 2·3 the channels are (1, 2, 5/6), which under c = 0 forces a + 2b = 0, hence b = 0 and a = 0. So a = b = c = 0: the 3×3 witness matrix has exact rank 3 over ℚ (fraction Gaussian elimination). This upgrades independence from MEASURED to PROVED (NT-P07b).

The scalar ΔNFR(n) is only one chosen aggregation (the sum) of these channels; ArithmeticPressureVector / pressure_vector(n) expose them as an independent triple, with .scalar == arithmetic_pressure(n), so the reading is not mistaken for a minimal or unique basis.

4. Class-conditioned pressure (MEASURED)

Conditioning the total pressure on the factor class ([2, 1000]) separates the regimes cleanly — primes at exact zero, the composite classes strictly positive:

class	count	mean ΔNFR	range
prime	168	0.000	[0, 0]
semiprime	288	3.318	[3.07, 3.83]
prime power	25	5.786	[2.03, 17.0]
composite (other)	518	12.515	[6.22, 37.4]

The tight semiprime band and the wide "other" band show the pressure grades compositeness, not merely primality. (pressure_by_class; abundance_class recovers the perfect numbers 6, 28, 496 on [2, 1000].)

5. Structural completeness (OPEN) and the fourth-channel gate

Whether a fourth independent pressure degree is relevant is not proven. completeness_proven() returns False, and a candidate fourth channel is admitted only through the explicit gate admits_fourth_channel, which requires all six conditions: independent structural meaning; not a function of the existing three on the declared domain; derived from the model (not an accuracy search); a documented change in diagnostic capability; preservation of the ΔNFR = 0 prime set; and a full contract with tests. The default (empty) criteria are inadmissible, so the three-channel set is not asserted complete.

6. Honest scope and claim ledger

The implemented channel computation uses the factorisation of n (via Ω, τ, σ), so ΔNFR(n) = 0 is not a fast primality test — it is a structural descriptor read from the factorisation. The C5 circularity audit therefore records uses_factorization_in_features = True (verdict CIRCULAR for discovery): the pressure may not be presented as an independent primality/factoring discovery mechanism. This is a dependency of the implementation, not a lower bound proving that every algorithm for these arithmetic functions must factor. No evolving signed EPI response or law for phase, capacity and support follows from assigning these static descriptors.

Claim	Basis	Status
NT-P07a each channel 0 iff prime; sum 0 iff prime	classical	PROVED
NT-P07b channels functionally (linearly) independent	exact witness proof over ℚ	PROVED
NT-P07c minimal for primality	single channel suffices	NEGATIVE (redundant)
NT-P07d three channels structurally complete	no proof; task-scoped	OPEN
NT-P07e ΔNFR = 0 as a primality algorithm	needs the factorisation	CIRCULAR / no claim
channels correlated but linearly independent	r ≈ 0.9, rank 3	MEASURED
pressure grades compositeness by class	class-conditioned means	MEASURED

The three-channel arithmetic pressure is a linearly-independent set of structural descriptors whose common (and individual) zero set is exactly the primes. It is redundant for primality (not minimal), correlated but linearly independent, and of unproven completeness. The former "minimal and complete" description is superseded; it is not retained as a weaker theorem.