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Improve lower bound for C_3b#92

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463464q435q43:improve-3b-lower-bound
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Improve lower bound for C_3b#92
463464q435q43 wants to merge 2 commits into
teorth:mainfrom
463464q435q43:improve-3b-lower-bound

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@463464q435q43 463464q435q43 commented Jun 10, 2026

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Summary

This updates the best known lower bound for the Kakeya-type sum-difference constant from

$$C_{3b} > 1.77898$$

([GGSWT2025], which improved [L2015] in the eighth decimal place) to

$$C_{3b} \geq 1.77898884,$$

an improvement in the sixth decimal place (certified margin $8.84 \times 10^{-6}$ over $1.77898$).

The bound comes from a 13-point entropy construction, using the entropy formulation already quoted on the 3b.md page: for the pair $(X,Y)$ given below,

$$\frac{H(X-Y)}{\max(H(X),,H(Y),,H(X+Y))} \geq 1.77898884.$$

The exact rational weights (common denominator $10^{36}$, reduced) sum to exactly $1$, and the support is symmetric under $(x,y) \mapsto (y,x)$. The certified value is $1.778988841420693549\ldots$, safely below the proven upper bound $11/6 = 1.8333\ldots$ [KT1999].

Certificate

X Y exact weight
-2 3 97361833892100293951 / 1000000000000000000000000000000000000
-2 4 703 / 500000000000000000000000000000000000
-1 2 24965915823534617097953539540121 / 100000000000000000000000000000000000
-1 3 1371805003106379032651546041 / 62500000000000000000000000000000000
0 1 112120167204589061011718189111031549 / 500000000000000000000000000000000000
0 2 5630590219641637490184434793744907 / 200000000000000000000000000000000000
1 0 112120167204589061011718189111031549 / 500000000000000000000000000000000000
1 1 30919629173187274661345437236097393 / 62500000000000000000000000000000000
2 -1 24965915823534617097953539540121 / 100000000000000000000000000000000000
2 0 5630590219641637490184434793744907 / 200000000000000000000000000000000000
3 -2 97361833892100293951 / 1000000000000000000000000000000000000
3 -1 1371805003106379032651546041 / 62500000000000000000000000000000000
4 -2 703 / 500000000000000000000000000000000000

Verification

All four linear forms ($X-Y$, $X$, $Y$, $X+Y$) take small integer values on the support,
so the pushforward distributions are computed exactly as rationals; the only inexact step
is the logarithm, which is enclosed with mpmath interval arithmetic. The reported bound
is the lower endpoint of the interval ratio, hence a rigorous lower bound on $C_{3b}$.

#!/usr/bin/env python3
"""Replayable certificate check for the C_3b lower bound 1.77898884.

Exact rational pushforwards + mpmath interval arithmetic (dps=80).
The printed lower endpoint rho_lo is a rigorous lower bound on C_3b.
"""
from collections import defaultdict
from fractions import Fraction
import mpmath

# 13-point support; exact rational weights (sum is exactly 1).
WEIGHTS = [
    (-2,  3, Fraction(97361833892100293951, 1000000000000000000000000000000000000)),
    (-2,  4, Fraction(703, 500000000000000000000000000000000000)),
    (-1,  2, Fraction(24965915823534617097953539540121, 100000000000000000000000000000000000)),
    (-1,  3, Fraction(1371805003106379032651546041, 62500000000000000000000000000000000)),
    ( 0,  1, Fraction(112120167204589061011718189111031549, 500000000000000000000000000000000000)),
    ( 0,  2, Fraction(5630590219641637490184434793744907, 200000000000000000000000000000000000)),
    ( 1,  0, Fraction(112120167204589061011718189111031549, 500000000000000000000000000000000000)),
    ( 1,  1, Fraction(30919629173187274661345437236097393, 62500000000000000000000000000000000)),
    ( 2, -1, Fraction(24965915823534617097953539540121, 100000000000000000000000000000000000)),
    ( 2,  0, Fraction(5630590219641637490184434793744907, 200000000000000000000000000000000000)),
    ( 3, -2, Fraction(97361833892100293951, 1000000000000000000000000000000000000)),
    ( 3, -1, Fraction(1371805003106379032651546041, 62500000000000000000000000000000000)),
    ( 4, -2, Fraction(703, 500000000000000000000000000000000000)),
]
assert sum(w for _, _, w in WEIGHTS) == 1

mpmath.iv.dps = 80
mpmath.mp.dps = 80   # so interval endpoints are extracted exactly, with no re-rounding
iv = mpmath.iv

def H_interval(a, b):
    """Interval enclosure of H(a*X + b*Y) in nats."""
    dist = defaultdict(Fraction)
    for x, y, w in WEIGHTS:
        dist[a * x + b * y] += w
    H = iv.mpf(0)
    for q in dist.values():
        if q:
            qi = iv.mpf(q.numerator) / iv.mpf(q.denominator)
            H += qi * (-iv.log(qi))
    return H

N = H_interval(1, -1)                                  # H(X-Y)
slopes = [H_interval(1, 0), H_interval(0, 1), H_interval(1, 1)]
D_hi = max(mpmath.mpf(h.b) for h in slopes)            # upper bound on max slope entropy
rho = N / iv.mpf(D_hi)                                 # interval ratio
rho_lo = mpmath.mpf(rho.a)                             # rigorous lower bound on C_3b

print("H(X-Y) in", mpmath.nstr(N, 30))
print("max slope entropy <=", mpmath.nstr(D_hi, 30))
print("rho_lo =", mpmath.nstr(rho_lo, 12), "(true value 1.778988841420693549..., displayed digits truncated-safe)")
assert rho_lo > mpmath.mpf("1.77898884")
assert rho_lo > mpmath.mpf("1.778981")   # beats any eighth-decimal improvement of 1.77898
assert mpmath.mpf(rho.b) < mpmath.mpf(11) / 6   # sanity: below proven upper bound 11/6
print("OK: C_3b >= 1.77898884 certified")

Expected output:

H(X-Y) in [1.223823395883792401230642456658973579453022111493986552795120117962458723325333444357, 1.223823395883792401230642456658973579453022111493986552795120117962458723325333465442]
max slope entropy <= 0.687932024861073221126046776533
rho_lo = 1.77898884142 (true value 1.778988841420693549..., displayed digits truncated-safe)
OK: C_3b >= 1.77898884 certified

The certificate was checked twice with independent code paths:

  1. Our project verifier (mpmath interval certification, same architecture as the script
    above), re-run at dps=120 against the exact rational distribution.
  2. A from-scratch independent re-check written without access to the original verifier,
    using only the Python standard library (json, fractions) plus mpmath.iv interval
    arithmetic at dps=100: pushforwards grouped as exact Fractions, interval endpoints
    pulled back to exact dyadic rationals (via libmp.to_rational), and all verdicts
    decided by exact rational comparisons. It certifies
    $$\rho \in [1.778988841420693549805193887477,\ 1.778988841420693549805193887478]$$
    (interval width $4.3 \times 10^{-100}$).

AI assistance disclosure

This is a fully AI-derived result: the construction was found and certified by Mosaic Intelligence's automated search-and-verification system, and the submission text was AI-prepared. All numerical results and references were independently re-run and verified before submission.

This was referenced Jun 10, 2026
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