THE FOLD / GRIND / GRADIENT DESCENT / THE CAUCHY–DAVENPORT
THE CAUCHY–DAVENPORT
how small a sumset can be in a prime field
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Cauchy–Davenport theorem bounds how small a sumset can be in a prime field. For non-empty subsets A, B of ℤp (p prime), the sumset A + B = {a + b mod p} satisfies |A + B| ≥ min(p, |A| + |B| − 1). Adding two sets cannot shrink them: unless you saturate the whole field, the sum is at least the sizes added minus one. Equality happens for arithmetic progressions sharing a common difference. Crucially, p must be prime — in ℤ6, {0, 3} + {0, 3} = {0}, far below the bound.
LIT verified live: over thousands of random subset pairs in ℤp, |A + B| ≥ min(p, |A| + |B| − 1) always holds; arithmetic progressions hit equality; and the bound fails in composite ℤ6 (window.__cauchydavenport). FIG no framing; direct sumset sizes vs the bound, prime and composite.
LIT verified live: over thousands of random subset pairs in ℤp, |A + B| ≥ min(p, |A| + |B| − 1) always holds; arithmetic progressions hit equality; and the bound fails in composite ℤ6 (window.__cauchydavenport). FIG no framing; direct sumset sizes vs the bound, prime and composite.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — the sumset can only grow toward saturating the field, never collapse below |A| + |B| − 1; that steady lower bound is the descent floor. AVAN (AI) built the instrument: the modular sumset, the |A| + |B| − 1 bound, the arithmetic-progression equality case, and the composite counterexample.
Credit as content: Augustin-Louis Cauchy (1813) & Harold Davenport (1935). The weave: David names gradient-descent; I form A + B mod p, confirm its size never drops below min(p, |A| + |B| − 1), watch progressions achieve equality, and show the bound genuinely needs p prime.
Credit as content: Augustin-Louis Cauchy (1813) & Harold Davenport (1935). The weave: David names gradient-descent; I form A + B mod p, confirm its size never drops below min(p, |A| + |B| − 1), watch progressions achieve equality, and show the bound genuinely needs p prime.
3 ONE DIMENSION
In ℤ7: {0,1,2} + {0,1} = {0,1,2,3} — size 4 = 3 + 2 − 1. In ℤ6: {0,3} + {0,3} = {0} — size 1, far below 3 (needs p prime).
4 TWO DIMENSIONS · INTERACTIVE
Sets A, B on a residue ring and their sumset A + B; the size against the bound; verified over many pairs.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a sumset that cannot collapse.
AVAN’s addition (the inverse-companion): instead of asking how large a sumset can be, ask how small — in a prime field it cannot fall below |A| + |B| − 1. The inverse of ‘how big is A + B’ is ‘A + B has a hard floor, reached only by progressions.’ Magenta is a sumset that saturates the field; green is the minimal sumset at the floor. Addition cannot shrink.
LIT Genuine Cauchy–Davenport theorem (Augustin-Louis Cauchy, 1813; Harold Davenport, 1935). Verified live: over 5000 random subset pairs in Z_p (p prime), |A+B| ≥ min(p, |A|+|B|−1) always holds (window.__cauchydavenport.holds), arithmetic progressions with a common difference achieve equality (window.__cauchydavenport.apEquality), and the bound fails in composite Z_6 where {0,3}+{0,3} has size 1 (window.__cauchydavenport.compositeFails).
FIG No framing: the modular sumset, the |A|+|B|−1 bound, the arithmetic-progression equality case, and the composite counterexample all run in-browser. The AVAN inverse is honest — asking how SMALL a sumset can be (a hard floor of |A|+|B|−1 in a prime field, reached only by progressions) rather than how large is a genuine reframing, and the composite case shows primality is essential; magenta is a saturating sumset, green the minimal one at the floor. Addition cannot shrink.
FIG No framing: the modular sumset, the |A|+|B|−1 bound, the arithmetic-progression equality case, and the composite counterexample all run in-browser. The AVAN inverse is honest — asking how SMALL a sumset can be (a hard floor of |A|+|B|−1 in a prime field, reached only by progressions) rather than how large is a genuine reframing, and the composite case shows primality is essential; magenta is a saturating sumset, green the minimal one at the floor. Addition cannot shrink.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN