Correct format ceiling displays and stage endpoint checks

research/format-limits/MANIFEST.sha256

@@ -1,7 +1,7 @@
 31f6e8be64f321cc58730c4fbbdba0d2f7f8b704fb3d5da23303e219d754c47d  DEPENDS.json
 c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4  LICENSE
 d16991367957b1de0cb3c9d4b61667482c3d4985903f15846051c65812c81611  NOTICE.md
-0fa0d6b2df96ddc06b876c41c5fa8d2335b97f36bc8f17d1371296d23199c263  README.md
+1bcd80c1cd61fddae9f3a7f00fc9d1c796410408251e576600e94c19b16b1e5e  README.md
 ecdc9cb66e0631cd311ffae9c742dd1173efb99803a537778c76cdd4ea0bda1e  SOURCES.json
 810e9c5c3222811ee5d4b52b3429f7700ab578836f76418bada521bea98b1001  data/ceiling/gap568-inputs.json
 5445f842587ab968c5001491e6479e7cc691b5114ab234652ce25b9272e2092f  data/census/anchors.json
@@ -18,8 +18,8 @@ cde64e210720e0cafbea5a242ee3747d3f34447bfdbc53c62114d6c34c71feeb  data/vendor/pr
 6f99cffd73076a768d39c21bf97ced42b6a5d3228b1dcef79d6f072dd35c0ee5  data/vendor/pr569/width7-eight-FULL-WORD.json.gz
 ee0fecf86bddcc5f4e675c60fbd5c572bd48269368a37a38c070346eefa3dd4a  data/vendor/pr571/combined-eight-local-FULL-WORD.json.gz
 6a080563ee39cc2c1c527bb6fb9c822253a7ee7202b7c5e53030e5526f973817  requirements.txt
-d763514fc92933875b8cef8535b881cae4536937918f2a3a5d8dc87dda207dff  scripts/a1_ceiling_roots.py
-1628325b8c15519a6fb5850461344b90a88d950948d8c47d8d19c00a95c17a1d  scripts/a2_ceiling_sup.py
+7ef64e261cbcc5ff69c67653264d0448d1c004bf6b879314fc2814f45274d331  scripts/a1_ceiling_roots.py
+3a6ab0ad504b6e306ade332974d9e0a6226b39ee3cc378eda642367e9ce9214f  scripts/a2_ceiling_sup.py
 1dc82914bf5a6ee5d1f27ea5cfbe404b83279c327cac2f56142a47a443a87159  scripts/a3_axioms.py
 b2e90cbb329bfebd954166637ed04488479dd7acca32e0c6102d4a3631c347f4  scripts/a4_gap.py
 6479c189696f53cc505febb2c81c4b69a5bf54f211ee4ae64ff841041244b060  scripts/a5_scope_multipliers.py

research/format-limits/README.md

@@ -117,7 +117,7 @@ is increasing with fixed point s0, so κ ≤ X/(1 + X) with X = min(X_bit*, X_cx
 
 Roots are rational-interval brackets of width < 10⁻¹⁵; the upper end is the certified value.
 
-- **κ ≤ 5.1594938·10⁻³ < 5.1595·10⁻³**; the complex side binds. The bit-only corollary is c/(1 + c) ≤ 5.9647·10⁻³.
+- **κ ≤ 5.1594939·10⁻³ < 5.1595·10⁻³**; the complex side binds. The bit-only corollary is c/(1 + c) ≤ 5.964704·10⁻³.
 - **The supremum over s is at s = 5** on both units. For odd s = 7..401, the certified D_s/Cost_s stays below X(5);
   the largest values are 5.2311·10⁻³ (bit) and 4.5292·10⁻³ (complex), both at s = 7. For s ≥ 403 a tail bound applies:
   Cost_s/v ≥ 2Σ_{p<K} 2pa·ln(m/(2pa)) ≥ 4a·Σ_{p<K} p ln(K/p) (using m > 2Ka), and

research/format-limits/scripts/a1_ceiling_roots.py

@@ -33,8 +33,9 @@
     'cx-120/72': {3: 3.5307429201300307e-3, 5: 4.535607180902823e-3, 7: 4.059761375276634e-3,
                   9: 3.6334780884322328e-3, 11: 3.2929429181725836e-3, 13: 3.0181922014469375e-3},
 }
-KAPPA_CEILING = 5.1595e-3          # rev. 2, complex binds
-KAPPA_BIT_ONLY = 5.9647e-3          # bit-only corollary c/(1+c)
+KAPPA_CEILING = Fr('0.0051595')       # rev. 2 headline, complex binds
+KAPPA_FINE = Fr('0.0051594939')       # outward display of the complex ceiling
+KAPPA_BIT_ONLY = Fr('0.005964704')    # outward display of the bit-only corollary c/(1+c)
 KAPPA_REV1 = 4.5151e-3              # superseded rev. 1 value (x = 3/5)
 FRONTIER = {'bit': 1.733052823330162e-3, 'cx': 0.00159868506668458107128}   # #555 c (s = 9), #568 b (s = 9)
 HEADROOM_S9 = {'bit': 2.658, 'cx': 2.499}
@@ -56,9 +57,9 @@
 xb, xc = tab['bit-111/63'][5]['hi'], tab['cx-112/64'][5]['hi']
 Xmin = min(xb, xc)
 kap = Xmin / (1 + Xmin)
-check('A1.kappa-ceiling', xc < xb and near(kap, KAPPA_CEILING, 7),
-      'X_bit* %.5e, X_cx* %.5e (complex binds); kappa <= X/(1+X) = %.7e' % (xb, xc, float(kap)))
-check('A1.kappa-bit-only', near(xb / (1 + xb), KAPPA_BIT_ONLY, 7), 'c/(1+c) <= %.7e' % float(xb / (1 + xb)))
+check('A1.kappa-ceiling', xc < xb and kap <= KAPPA_FINE < KAPPA_CEILING,
+      'X_bit* %.5e, X_cx* %.5e (complex binds); kappa <= %s' % (xb, xc, KAPPA_FINE))
+check('A1.kappa-bit-only', xb / (1 + xb) <= KAPPA_BIT_ONLY, 'c/(1+c) <= %s' % KAPPA_BIT_ONLY)
 x1 = min(tab['bit-120/72'][5]['hi'], tab['cx-120/72'][5]['hi'])
 check('A1.rev1-superseded', near(x1 / (1 + x1), KAPPA_REV1, 7),
       'the x = 3/5 scope gives %.7e; the punctured units have x = 63/111, 64/112 < 3/5' % float(x1 / (1 + x1)))

research/format-limits/scripts/a2_ceiling_sup.py

@@ -20,13 +20,15 @@
 
 for name, (h, cst, v) in UNITS.items():
     x = Fr(cst, v)
-    x5 = X(5, h, 1, x)['hi']
+    x5_box = X(5, h, 1, x)
+    x5 = x5_box['hi']
+    x5_lo = x5_box['lo']
     worst, sworst, allbelow = Fr(0), None, True
     for s in range(7, 402, 2):
         prof, m, D = profile(s, h, 1, x)
         clo, _ = cost_iv(prof, m)
         ub = D / clo
-        allbelow &= ub < x5
+        allbelow &= ub < x5_lo
         if ub > worst:
             worst, sworst = ub, s
     ew, es = EXPECTED_WORST[name]
@@ -36,7 +38,7 @@
     K = 201
     s = 2 * K + 1
     tail = float(1 - x) * s / (4 * a * (K * K / 4 - K / math.e - K / 2))
-    check('A2.tail.' + name, tail < float(x5) and near(tail, EXPECTED_TAIL[name], 7),
+    check('A2.tail.' + name, tail < float(x5_lo) and near(tail, EXPECTED_TAIL[name], 7),
           'analytic tail bound at s = 403: %.3e (decreasing in s)' % tail)
 bad = [K for K in range(2, 5001) if sum(p * math.log(K / p) for p in range(1, K)) < K * K / 4 - K / math.e - K / 2]
 check('A2.tail-inequality', not bad, 'sum_{p<K} p ln(K/p) >= K^2/4 - K/e - K/2 for K = 2..5000')