OBSERVATION CHAMBER / ACTIVE

P-adic Lift Tree

Lift polynomial roots from mod p through powers of p, and watch each branch continue uniquely, split, die, or fail to begin.

structure Operational

01 · Define the problem

Lift controls

Ready

Preset

Customize polynomial and lift conditions

Integer coefficients a₀, a₁, …

Constant term first; commas separate coefficients. The 200-digit limit counts the canonical absolute value, excluding sign and leading zeroes. f(x) = x² − 2

Prime p

Target depth

Initial residues optional Enter comma-separated residues to follow selected seeds only.

Current process

mod p

Ready to enumerate the initial roots modulo p.

Next: test f(r) ≡ 0 (mod p)

Advanced limits

Maximum displayed nodes

Computation limit: depth 18 and 4096 materialized nodes. A cap is a safe-stop, never a partial proof.

02 · Observe the branches

Lift flow

pending unique split dead end depth complete

Choose LIFT ONE LEVEL to build the first real layer.

Scroll vertically for deeper levels; wider split levels scroll sideways. Each node shows its residue, new p-adic digit, and verified congruence.

03 · Inspect a branch

Lift result

Not computed

Select a computed node to highlight one branch.

Selected residue: —
Added digit: —
Derivative mod p: —
Root condition: —

Selected branch, from the initial residue onward
Level	Residue	Digit	Outcome
No branch selected.

Compare two integers

Ordinary vs p-adic distance

Ordinary distance |x − y|: —
Valuation v_p(x − y): —
p-adic distance p^−v: —

Values are calculated exactly from the BigInt difference.

Keep the computed evidence

Report & image

Exports are enabled only for the current, non-stale input revision.

Model limits

This is finite-depth, exact modular enumeration. It observes compatible residues modulo powers of p; it is not a proof of every statement about a p-adic limit. Invalid input, composite p, a depth or node cap, Worker failure, cancellation, and stale replies produce no certified branch or report.

Current input revision

P-adic Lift Tree report

No report is available.

SYSTEM NOTE

For a root a with f(a) ≡ 0 (mod p^n), enumerate child = a + t p^n for t in {0,…,p−1} and keep exactly the candidates with f(child) ≡ 0 (mod p^(n+1)). For simple roots, compare the measured child with the Hensel correction. The distance panel reports v_p(x−y) and |x−y|_p = p^(−v_p(x−y)).

OBSERVATION POLICY

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Route

This chamber belongs to Structures & Symmetries.

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