51. Numerical evidence¶

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Data sources: f:/Primes/twin_stats_21_27.csv, f:/Primes/twin_stats_28_detail.csv (originals, never edited) and the harnesses tools/*_harness.py with protocols tools/RESULTS_*.md. This chapter has no Lean counterpart: it is data, not claims. The numbers are given here as a reference point and a stress test of the laws before formalisation — not as a proof.

What we check, and why¶

The programme rests on a series of exact Diophantine laws (the carrier of two, the sign law, height drop) and on one target balance inequality B₅ = N₀₀ − N₃₃ > 0 (there are no fewer twins than "double composites"). The formal proofs of these laws live in Engine/*; here we show that on real data they hold — and with what margin.

All checks run in exact integer arithmetic and pointwise, on every block and node, not on average. A law that holds on average is of no use to us: we need a law that is never violated even once.

The numerical evidence never crosses the red line (no probabilities, no distribution of primes). It measures already-defined integer quantities rather than postulating their density.

1. The balance `N₀₀ − N₃₃` and four-corner (blocks `N = 2^k`)¶

Centres 6m±1 are classified by the pair of side ranks, coarsened to {0,3} (prime side / richly composite side): N₀₀ (twin), N₃₃ (both richly composite), N₀₃, N₃₀ (mixed). The target is B₅ = N₀₀ − N₃₃ > 0; the four-corner ratio R_fc = N₀₀N₃₃/(N₀₃N₃₀) < 1 encodes the negative correlation of the side ranks.

k	N₀₀	N₃₃	N₃₃/N₀₀	R_fc = N₀₀N₃₃/(N₀₃N₃₀)	Γ/M
21	59 382	745	0.0125	0.876	0.671
22	109 168	1 213	0.0111	0.918	0.675
23	202 492	2 816	0.0139	0.920	0.682
24	375 236	4 796	0.0128	0.976	0.682
25	698 496	9 596	0.0137	0.951	0.687
26	1 302 736	18 721	0.0144	0.962	0.688
27	2 435 911	33 386	0.0137	0.972	0.686
28	4 566 323	72 230	0.0158	0.977	0.695

(The columns N₀₀,N₃₃,Γ/M come from the GLOBAL rows of the CSV; R_fc from tools/RESULTS_fourcorner.md, exact sieve, old-free version. Γ = N₃₀ + 4N₃₃, M is the mass scale.)

What we see:

B₅ = N₀₀ − N₃₃ > 0 with a wide margin. N₃₃ is only ~1.1–1.6 % of N₀₀ and never comes close to it. The ratio is roughly stable rather than falling to zero (it is important not to confuse N₃₃ with the neighbouring N₃₁,N₃₂). This is direct numerical support for 16. By contradiction.
R_fc < 1 on every block (0.876 → 0.977). The four-corner negative association holds pointwise, exactly as required by 12. Four-corner, 14. Remainder decomposition.
Γ/M ≈ 0.67 → 0.70 — a stable structural constant (carrier dispersion of ~70 % of the mass). There is no "clock monopolist", consistent with the exclusivity forcing argument of 06. No way back.

Note (the parity wall is visible in the numbers). R_fc rises monotonically toward 1 with the block, i.e. the margin 1 − R_fc → 0⁺. This is precisely the "margin-to-zero wall" of 16. By contradiction: the inequality holds on all the data, yet its margin melts away, which is why the estimate must be uniform rather than per-block. This is exactly what makes the twin node a distributional one, kin to the estimate of L(x) in 32. Rank-parity node.

2. Where exactly the symmetry breaks (an exploratory margin estimate)¶

The exploratory harness symmetry_break_harness.py (this is not a proof step but a "what-if" estimate) fits the margin series 1 − R_fc at κ = 4.0 with a power law:

\[1 - R_{\mathrm{fc}} \;\sim\; 12.12 \cdot A^{-1.161}.\]

The exponent α = 1.161 > 0 means margin → 0⁺ as A → ∞, but smoothly, without vanishing.

The harness's conclusion: there is no finite size at which the symmetry suddenly breaks. For the twins to run out, the margin would have to jump off the smooth law and change sign — and nothing of the kind is observed in the data. This is indirect support for the wall being a knife-edge rather than a cliff (see 16. By contradiction).

3. Old-peel: three laws at 100 % (3000 real catches, `A = 200`)¶

The harness oldpeel_harness.py on 3000 real rank-1 catches (6n+ε = a prime > A, 6n−ε composite, caught by a small p ≤ A, unfolding 6n−ε = p(6t+δ)):

law	result
sign law `δ = −π·ε` (`p ≡ π mod 6`)	3000 / 3000 = 1.0000
height drop `t < n/5`	3000 / 3000 = 1.0000
regeneration `t > 0` (the flow continues)	3000 / 3000 = 1.0000

Both algebraic laws (old_peel_sign, old_peel_height_drop) and the closure core (infinite strict descent ⟹ False) are proven in Lean (19. Old-peel). The numbers confirm that the mechanism is real at every node checked, not hypothetical. The observed descent t < n/5 is even sharper than the proven soft bound t < n.

4. The clean graph and the global absorber (fan-in)¶

The numbers for the clean graph and the global absorber (tools/RESULTS_clean_graph.md, tools/RESULTS_global_absorber.md):

The "sink-is-clean" gap. 178/300 = 59.3 % of clean centres have all their active edges in the boundary. This is precisely the fact that the clean graph is not self-sufficient and requires boundary regeneration (23. Clean graph): 59 % of the apparent "stops" are in fact boundary exits.
Fan-in is real. A single absorber centre collects up to 570 lineages (570/2000 = 28.5 % of the endpoint shares); small centres (≤ 50) absorb the mass of the descent. This is the numerical form of the single remaining node GlobalOldAbsorption (24. Boundary decomposition, 29. The last link): the fan-in 570 → 1 exists, but turning it into an engine — a pigeonhole over the fibre (Hall capacity) — is exactly the irreducible gap.

Note. The numbers here measure the open node itself, not its closure. They show that the absorber and the fan-in are not an artefact but a real structure; yet the reality of the fan-in is not the same as its reducibility to an engine. We record this honestly: the data confirm the existence of the wall to which the whole programme has been reduced, and say nothing about its conquest.

5. Rank is bounded and squeezed (product-core)¶

The harnesses rank2_harness.py, cov_harness.py (k=24, κ=4):

The rank range 1 ≤ r ≤ 4. On the legal layer the product of factors > A is pinched between (A+1)⁵ and 6X_A+1 < A⁵, which gives r ≤ 4 (factor_rank_le_four, 28. MkNode); the data confirm this.
The singular spectrum of the rank-2 covariance: [764953, 733, 224, 98], sv₂/sv₁ = 0.001 — rank 1 explains 99.9999 % of the energy. In other words, the structure of the centres is almost one-dimensional: a single clean mode dominates, and higher-rank corrections are negligible. This supports the cubic squeeze and the shortness of repeat trains (07. The short train, 08. The bounded cycle).

6. The separating scale is reachable with exponential headroom¶

A numerical run (tools/RESULTS_separating_scale.md): the condition P_A > 6X_A + 1 (with X_A ≈ A^{4.5}) already holds for small A and only strengthens further, since log P_A ~ A/ln 10 (Chebyshev) outruns 4.5·log₁₀ A:

A	log₁₀ P_A	log₁₀(6·A^{4.5})	`P_A > 6X_A`?
50	17.0	8.4	✓
200	81.1	11.1	✓
1000	414.5	14.3	✓

This is the numerical evidence beneath legal_lift_lt_primorial / no_productHall (26. Separating scale): on the legal domain a < P_A, so the coarsened passport a mod P_A is already exact and injective — the node ProductHall is closed constructively, without Classical.choice.

7. The Riemann branch: where the numbers are, and where only theory so far¶

The identity λ(n) = (−1)^{Ω(n)} = (−1)^{rank(n)} and the flip λ(p·m) = −λ(m) are proven in Lean (31. Riemann via Liouville) — this is not empirics but a consequence of liouville_apply/cardFactors_mul from mathlib.

The estimate |L(x)| = O(√x · x^ε) (LiouvilleBound) remains an arithmetic input. We have no separate numerical harness for L(x)/√x yet, and we do not hide this.

The only numerical support for the Riemann branch at present is indirect. L(x) is the rank-parity imbalance — the same object as N₀₀−N₃₃ in §1 — and §1 shows that the balance of ranks holds with a margin (see 32. Rank-parity node).

What matters¶

Numerical stability does not close any of the open nodes. Both the target twin balance (16. By contradiction, §1) and the estimate of L(x) (31. Riemann via Liouville, §7) are needed uniformly and asymptotically, not at specific k ≤ 28 / A ≤ 1000.

The data are consistent with the hypothesis and confirm that the mechanisms (old-peel, fan-in, rank ≤ 4, separating scale) are real; but confirmation on a finite segment is a reference point, not a conclusion. The single irreducible node — GlobalOldAbsorption (§4) — is made visible by the data, but not closed.