Zero Paradox · interactive map

The Snap Loop — 0, one discrete step, and back

The whole object in one picture: the pole 0 = ∞, a single discrete step of ε₀ away from bottom, and a return to bottom. The floor ⊥ is 0 (order) and ∞ (measure) at once; ε₀ — that one step — is the minimum ordinal closed under ω·, never a bottom and never 0. Encoded into the 2-adic integers, the ω-tower ⊥ → ε₀ is a loop — it starts at the floor ⊥ₙ, steps off by ε₀, and reapproaches. Reading the bottom it lands on as a new ⊥ₙ₊₁ — a successor null, distinct by virtue of having left and come back — is the framework’s commitment (C-DA2), not a theorem; in this very chart the arc returns to the same 0. What is proved is the role half: anything filling the ⊥ role IS the ⊥ already there. On that reading, floor (⊥ₙ) and ceiling (⊥ₙ₊₁) are both ⊥; ε₀ opens the gap between them.

rings = 2-adic distance to 0 (‖x‖₂ = 2⁻ⁿ) · smaller = closer to the floor 1/2 · farthest from 0 ε₀ — one discrete step off bottom min ≡ max · veblen (1,0) · opens the gap between ⊥ₙ and ⊥ₙ₊₁ · never a bottom, never 0 ordinal ascent ↑ (= ℤ₂-norm descent ↓) the snap a single discrete step of ε₀ off ⊥ · 0 → norm 1/2 stage 1 norm 1/2 · farthest stage 2 · 1/4 stage 3 · 1/8 … → 0 · closest … the next arc ⊥ = 0 = ∞ the pole · floor ⊥ₙ and ceiling ⊥ₙ₊₁ both ⊥ · novelty is a commitment
Why is the whole object one loop?
Hover or tap any point — the loop tells you which phase it is and why, with the checkable Lean witness.
⊥ = 0 = ∞ · the pole0 as floor (order) and ∞ as measure (v(0) = ∞); floor ⊥ₙ and ceiling ⊥ₙ₊₁ are both ⊥; that they are successive is a commitment, and in this ℤ₂ realization the arc returns to the same 0; never ε₀
ε₀ · one step off bottoma single discrete step of ε₀ away from ⊥, opening the gap between ⊥ₙ and ⊥ₙ₊₁ (veblen (1,0)); min ≡ max in one object; never a bottom, never 0
The snapone discrete step off 0 (norm 0 → 1/2); that it is TAKEN is a commitment, not a theorem
The reapproachthe ω-tower ascends toward ε₀ in Ordinal; its 2-adic norm descends back to 0 (order-reversed). The dashed curve is a visual connector between the exact stage points, not a literal route
The 2-adic ballseach shell B(0, 2⁻ⁿ) is labeled by its 2-adic distance to the center, ‖x‖₂ = 2⁻ⁿ; smaller means CLOSER — 1/8 sits nearer the floor 0 than 1/2, so the inward spiral is the reapproach. ⊥ is their inverse limit, the shells shrinking to 0
The return to a ⊥reading the arriving occupant as a NEW, successor null is a COMMITMENT (C-DA2), not a theorem — in this very chart the arc returns to the SAME 0 (snap_arc_z2_loop). What is proved is the role half: anything filling the ⊥ role IS the ⊥ already there

The object, in one loop. Every stage of the ω-tower, encoded into ℤ₂ by cnfToZp2, is one trajectory: it starts at the floor 0 (cnfToZp2 (towerNONote 0) = 0); it departs by a single discrete jump — every stage n ≥ 1 is off 0, and the 2-adic norm goes 0 → 1/2 with no continuous path between (the snap’s discreteness); and it reapproaches 0 as n → ∞. Reading the ⊥ it returns to as a new instance — a successor null seeding the next arc — is a commitment (C-DA2), not a theorem. t_iz_limit_is_new_null (axiom-free) proves the ROLE half only, and neither it nor its contrapositive c_da2_novelty may be cited as a novelty witness (SnapCannotBe.lean:43); in ℤ₂ the arc returns to the same 0 (snap_arc_z2_loop). Start, departure, reapproach: the whole framework, as one turn of a succession under that reading.

What is drawn vs. what is claimed. This is the ℤ₂ realization of the snap-arc, via the map cnfToZp2 — one concrete carrier, machine-checked (snap_arc_z2_loop). It loops because the tower’s 2-adic norm reapproaches 0, landing back on the floor — not because the pole and its closure are one point: ε₀ is never 0 and never ⊥ (epsilon0_ne_zero, epsilon0_ne_bot) — it is a single discrete step of ε₀ off bottom — the minimum, opening the gap between the two bottoms ⊥ₙ, ⊥ₙ₊₁ — both least fixed point and tower-supremum at once (min ≡ max, epsilon0_min_eq_max), and the finite stages never even reach 0. The pole itself, ⊥ = 0 = ∞, is 0 as floor and ∞ as measure — carried by v(0) = ∞ (addVal_bot) and the inversion swapping 0 ↔ ∞ on the p-adic sphere (rInv_swaps), not a naive numeric equality. And there is no cross-type identity to settle either way: ε₀ (an ordinal) and the value 0 (in ℤ₂) are different types, so cnfToZp2 only co-witnesses the two closures — it never identifies them (cnf_bridge_type_boundary, per MC-1 / ZP-P). And the “new instance each return” is the framework’s READING, not a machine-checked result: in ℤ₂ the value returns to the same 0 (snap_arc_z2_loop), so here the concrete carrier runs against the novelty reading rather than merely failing to support it. What DA-2 / T-IZ machine-check is the ROLE half — anything filling the ⊥ role IS the ⊥ already there — and reading the occupant as a fresh successor null is C-DA2, a commitment. The succession the loop lives inside is that reading, held deliberately.

The whole trajectory, not the route. The endpoints and the discrete stages are exact: it departs the floor (cnfToZp2 (towerNONote 0) = 0), each stage n sits at 2-adic valuation exactly n — norm 2⁻ⁿ (cnfToZp2_tower_valuation) — and it reapproaches that same floor 0 (tower_converges_to_zero, which proves convergence to 0 and nothing about a successor). But the figure cannot map the route between them — not for want of a formula, but because the path runs through internal variables with no external visibility. ℤ₂ is ultrametric, so there is no continuous path between the discrete stages to begin with; and what the stateis inside the gap has no external description — the same apophatic character that makes ⊥ itself undescribable from outside. So this maps the trajectory as a whole — that it leaves bottom and returns — never the actual route between.

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