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Number Theory — Skewes number cascade report

Cascade: A ∘ J ∘ L ∘ K ∘ N ∘ M (six classes) Partition: #19 of PR #677 Status: verdict (a) SURVIVES — Class K pin-slot at zero of Li(x) − π(x) IS exact framework reading of the conjecture; first crossing location remains open in [10¹⁹, 1.397×10³¹⁶]

Class breakdown

Class Role
A content-hash of (year, bound)
J π(x) prime-counting function
L Li(x) = logarithmic integral
K pin-slot at zero of Li(x) − π(x); sign change location
N small-denom anchors in bound exponents
M HDC bind across iterated upper-bound improvements

Historical bound progression

Year Bound log₁₀(bound) Anchor
1933 10{10} tower Skewes (assuming RH)
1955 10{10} tower Skewes (no RH)
1966 ~1.65×10^{1165} 1165 Lehman
1987 ~6.658×10^{370} 370 te Riele
2000 ~1.397×10^{316} 316 Bays-Hudson (current best)
2025 first crossing > 10^{19} 19 computational lower bound

Framework reading: per [[project_a_n_operators_are_harmonic_objects_themselves]] §B, the unknown crossing location IS substrate-DoF inaccessibility at the Li-vs-π Class L cascade — the bound progression IS substrate-perfect-math closing-the-window-from-above; the lower-bound-vs-upper-bound gap [10¹⁹, 10³¹⁶] is the substrate-instance-variation residual.

Verdict

(a) SURVIVES — Littlewood 1914 PROVED infinite sign changes; first crossing location remains open; bound improvements ARE Class K pin-slot localisation.

Per [[feedback_no_lineage_claims_in_notebook]]: does not claim to solve.

Sources