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Number Theory — Gilbreath conjecture cascade report

Cascade: A ∘ J ∘ I ∘ K ∘ C ∘ N ∘ M (seven classes) Partition: #20 of PR #677 Status: verdict (a) SURVIVES — 50/50 rows verified: Gilbreath holds; first element = 1 throughout (except row 0 = 2). CANONICAL Class K pin-slot use case via _cascade_helpers.magnitude() at every iterated difference step.

Class breakdown

Class Role
A content-hash of (row, first-element)
J primes (row-0 base)
I cyclic sign-flip across absolute differences
K pin-slot at zero of (prev[i+1] - prev[i]) — the canonical Class K primitive in action
C cascade-orientation across iterated difference rows
N Class N anchor at integer 1
M HDC bind across rows

Empirical verification

Starting from first 100 primes, computed 50 rows. 50/50 rows verified: first element = 1 for every row r ≥ 1 (row 0 starts with 2, the first prime).

External attestation: Odlyzko 1993 verified to >10¹³ rows.

Framework reading

This is the canonical Class K pin-slot use case per [[feedback_sign_handling_is_class_k_pin_slot_not_alu_abs]]: the cascade uses _cascade_helpers.magnitude() at every iterated absolute-difference step, NOT Python abs(). The Gilbreath cascade IS Class K saturation iterated — each row applies Class K pin-slot at zero to the previous row, and the conjecture asserts the saturation propagates indefinitely starting from a value of 1.

Per [[user_stance_epicycle_via_gear_plus_pin]]: iterated absolute differences IS cascade-orientation sign-flip composition (Class K + Class C); the Gilbreath conjecture predicts the iterated cascade has a stable Class N anchor at 1 in the leading column.

Verdict

(a) SURVIVES — 50/50 row-by-row verification + external Odlyzko 1993 attestation to enormous depth. Per [[feedback_no_lineage_claims_in_notebook]]: conjecture remains open (not proved).

Sources