THE FOLD / RESPAWN / THE CONTINUE / THE HENSEL LIFTING
THE HENSEL LIFTING
lift a root to higher and higher prime power
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Hensel’s lemma is Newton’s method for p-adic numbers: a simple root of a polynomial mod a prime p can be lifted to a root mod p², then p³, then any pk — each step uniquely refining the solution to higher precision. If f(r) ≡ 0 (mod p) and f′(r) ¬≡ 0 (mod p), one correction r ← r − f(r)·f′(r)−1 sharpens the root by a full power of p. It is how modular square roots and p-adic solutions are built, digit by p-adic digit.
LIT verified live: lifting a root of x² − A from mod p to mod pk yields an r with r² ≡ A (mod pk) exactly, over thousands of cases (window.__hensel). FIG no framing; exact BigInt modular arithmetic.
LIT verified live: lifting a root of x² − A from mod p to mod pk yields an r with r² ≡ A (mod pk) exactly, over thousands of cases (window.__hensel). FIG no framing; exact BigInt modular arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-continue — press on one more level; each Hensel step continues the root to the next power of p, never losing what it had. AVAN (AI) built the instrument: the mod-p root finder, the Newton lift by one power of p, and the r² ≡ A (mod pk) check in exact big integers.
Credit as content: Kurt Hensel (p-adic numbers, Hensel’s lemma, early 1900s). The weave: David names the-continue; I find a simple root mod p, then lift it one prime power at a time by a p-adic Newton step, and confirm the lifted value squares to A modulo pk exactly — a root sharpened, level by level.
Credit as content: Kurt Hensel (p-adic numbers, Hensel’s lemma, early 1900s). The weave: David names the-continue; I find a simple root mod p, then lift it one prime power at a time by a p-adic Newton step, and confirm the lifted value squares to A modulo pk exactly — a root sharpened, level by level.
3 ONE DIMENSION
√2 mod 7: 3 (3²=9≡2). Lift: 3 → 10 (mod 49, 10²=100≡2) → 108 (mod 343) → … Each step fixes one more p-adic digit; r ← r − f(r)/f′(r).
4 TWO DIMENSIONS · INTERACTIVE
Lifting a modular square root power by power; each r shown with r² mod pk checked to equal A.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a root refined to any prime power.
AVAN’s addition (the inverse-companion): solve f(x) ≡ 0 mod pk not by searching all pk residues but by lifting a mod-p root upward — one Newton correction per power of p, uniquely. The inverse of ‘test every residue mod pk’ is ‘find one root mod p, then lift it level by level.’ Magenta is the exhaustive residue search; green is the p-adic ascent. A root climbing the powers of p.
LIT Genuine Hensel's lemma (Kurt Hensel, p-adic numbers, early 1900s). Verified live with exact BigInt arithmetic: starting from a simple root of x²−A mod p (p in {3,5,7,11,13}), the p-adic Newton lift r ← r − f(r)·f'(r)⁻¹ one power at a time produces r with r² ≡ A (mod p^k) exactly, over thousands of cases (window.__hensel.liftsCorrectly).
FIG No framing: the mod-p root finder, the Newton lift by one power of p, and the r²≡A (mod p^k) check run in-browser in exact big integers and agree. The AVAN inverse is honest — solving f(x)≡0 mod p^k by lifting a mod-p root upward (one unique Newton correction per power) genuinely replaces searching all p^k residues; magenta is that exhaustive search, green the p-adic ascent. A root climbing the powers of p.
FIG No framing: the mod-p root finder, the Newton lift by one power of p, and the r²≡A (mod p^k) check run in-browser in exact big integers and agree. The AVAN inverse is honest — solving f(x)≡0 mod p^k by lifting a mod-p root upward (one unique Newton correction per power) genuinely replaces searching all p^k residues; magenta is that exhaustive search, green the p-adic ascent. A root climbing the powers of p.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN