{
 "title": "Numbers read one place at a time",
 "pages": [
  {
   "url": "/claims/",
   "title": "Every claim and its tier",
   "kind": "page",
   "parent": "/"
  },
  {
   "url": "/code/",
   "title": "Code: every program on the site",
   "kind": "page",
   "parent": "/"
  },
  {
   "url": "/digits/",
   "title": "Digit readers: the real place",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/",
   "title": "Numbers read one place at a time",
   "kind": "page",
   "parent": null
  },
  {
   "url": "/instruments/",
   "title": "Exact truth",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/key/",
   "title": "How to read a claim",
   "kind": "page",
   "parent": "/"
  },
  {
   "url": "/local/",
   "title": "One place up close",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/machines/",
   "title": "Machines and learning",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/object/",
   "title": "The object: every finite place at once",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/residues/",
   "title": "Residue readers: the finite places",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/terms/",
   "title": "Every term",
   "kind": "page",
   "parent": "/"
  },
  {
   "url": "/tools/",
   "title": "Tools: programs to call on your own input",
   "kind": "page",
   "parent": "/"
  },
  {
   "url": "/walks/",
   "title": "Growth: places priced and compared",
   "kind": "section",
   "parent": "/"
  },
  {
   "url": "/binding/",
   "title": "BINDING — what multiplication keeps of a code, and what a sum costs",
   "kind": "doc",
   "parent": "/residues/",
   "source": "/binding.md",
   "claims": [
    {
     "heading": "The dictionary criterion",
     "anchor": "the-dictionary-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "binding.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The additive twin",
     "anchor": "the-additive-twin",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "binding.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The superposition wall",
     "anchor": "the-superposition-wall",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "binding.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The mixing bound",
     "anchor": "the-mixing-bound",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "binding.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The characteristic cap",
     "anchor": "the-characteristic-cap",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "binding.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The capacity sweep",
     "anchor": "the-capacity-sweep",
     "tier": "pattern",
     "shown": "pattern",
     "range": "toy scale: D = 1,000, N = 100, 200 trials per m",
     "scripts": [
      "binding.py"
     ],
     "proof": false,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/bridges/",
   "title": "BRIDGES — where growth meets open number theory",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/bridges.md",
   "claims": [
    {
     "heading": "The seating law",
     "anchor": "the-seating-law",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "bridges.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The phoenix",
     "anchor": "the-phoenix",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "bridges.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The atom criterion",
     "anchor": "the-atom-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": "proved both ways; every odd h ≤ 999 realized by a built modulus",
     "scripts": [
      "bridges.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The least unrealizable odd number",
     "anchor": "the-least-unrealizable-odd-number",
     "tier": "rule",
     "shown": "rule",
     "range": "proved; the search to b ≤ 1000 an observation; the other survivors settled and the nine open primes, the published record's",
     "scripts": [
      "bridges.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Transparent rungs are transparent moves",
     "anchor": "transparent-rungs-are-transparent-moves",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "bridges.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/budget/",
   "title": "BUDGET — what omitting a place frees at a global constraint",
   "kind": "doc",
   "parent": "/object/",
   "source": "/budget.md",
   "claims": [
    {
     "heading": "The budget theorem",
     "anchor": "the-budget-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [],
     "proof": true,
     "source": null
    },
    {
     "heading": "The constraints over Q",
     "anchor": "the-constraints-over-q",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "Serre, A Course in Arithmetic (1973), ch. III, the product formula for Hilbert symbols and the existence theorem; Albert, Brauer, Hasse and Noether (1932) for the Brauer sum."
    },
    {
     "heading": "The degree control",
     "anchor": "the-degree-control",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "budget.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The rank face",
     "anchor": "the-rank-face",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "budget.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The volume face",
     "anchor": "the-volume-face",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "budget.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The layer face",
     "anchor": "the-layer-face",
     "tier": "rule",
     "shown": "rule",
     "range": "the formula known; the excess proved when the units are ±1, and its bound at a conjugate pair proved; the census verified at the 21 pairs of places over 3, 7, 17, 23 of Q(√2",
     "scripts": [
      "budget.py"
     ],
     "proof": false,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/cascade/",
   "title": "CASCADE — whether the greedy walk over a number ring must lock",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/cascade.md",
   "claims": [
    {
     "heading": "The escape is a conjunction",
     "anchor": "the-escape-is-a-conjunction",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "cascade.py",
      "module_law.py"
     ],
     "proof": true,
     "source": "H. Heilbronn, On the class-number in imaginary quadratic fields, Quart. J. Math. 5 (1934), 150–160."
    },
    {
     "heading": "The budget inequality",
     "anchor": "the-budget-inequality",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the rise through a carrier the walk opens is one level at odd p and at most two at 2",
     "scripts": [
      "cascade.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The characteristic sweep",
     "anchor": "the-characteristic-sweep",
     "tier": "rule",
     "shown": "rule",
     "range": "verified every odd p < 1000",
     "scripts": [
      "cascade.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Dead levels recur at eleven characteristics",
     "anchor": "dead-levels-recur-at-eleven-characteristics",
     "tier": "rule",
     "shown": "rule",
     "range": "proved by covering at 2 and at ten odd p; the walks verified from every start to 300 at 2, 3 and 5",
     "scripts": [
      "cascade.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The residual is inhabited at every bound",
     "anchor": "the-residual-is-inhabited-at-every-bound",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "cascade.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The shape of a dead level",
     "anchor": "the-shape-of-a-dead-level",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the identity and the one-class count, for every odd p and level",
     "scripts": [
      "cascade.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The full-price opening at odd p is a dead level over Z",
     "anchor": "the-full-price-opening-at-odd-p-is-a-dead-level-over-z",
     "tier": "criterion",
     "shown": "criterion; rule",
     "range": "proved both ways",
     "scripts": [
      "cascade.py",
      "growth.py"
     ],
     "proof": false,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/clock/",
   "title": "CLOCK — what a place charges a growth walk",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/clock.md",
   "claims": [
    {
     "heading": "The ladder is ψ's orbit, save a head at the bend",
     "anchor": "the-ladder-is-ψs-orbit-save-a-head-at-the-bend",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clock.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The head criterion",
     "anchor": "the-head-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "clock.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The width",
     "anchor": "the-width",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clock.py",
      "tame.py",
      "weld.py",
      "wild.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A ramified quadratic place over 2 reads its width off δ mod 8",
     "anchor": "a-ramified-quadratic-place-over-2-reads-its-width-off-δ-mod-8",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [
      "carrier.py"
     ],
     "proof": false,
     "source": "A. Ranum, The group of classes of congruent quadratic integers with respect to a composite ideal modulus, Trans. Amer. Math. Soc. 11 (1910) 172–198, types 5, 6 and 7 of its section 35."
    },
    {
     "heading": "The two numbers",
     "anchor": "the-two-numbers",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clock.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The walk stops at its high-water price",
     "anchor": "the-walk-stops-at-its-high-water-price",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "stop.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Under a per-item clock the sup sets the stop",
     "anchor": "under-a-per-item-clock-the-sup-sets-the-stop",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "stop.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Under the global clock the sup is a floor",
     "anchor": "under-the-global-clock-the-sup-is-a-floor",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "stop.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A block clock keeps at most one runaway per block",
     "anchor": "a-block-clock-keeps-at-most-one-runaway-per-block",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "dial.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The admission census is the high-water lemma, item by item",
     "anchor": "the-admission-census-is-the-high-water-lemma-item-by-item",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "census.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Two blocks of a number ring never both run away",
     "anchor": "two-blocks-of-a-number-ring-never-both-run-away",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "dial.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A ring's door is one valuation and its clock a count",
     "anchor": "a-rings-door-is-one-valuation-and-its-clock-a-count",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "door.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The staircase column",
     "anchor": "the-staircase-column",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "column.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A norm does not fix a column",
     "anchor": "a-norm-does-not-fix-a-column",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "column.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A move's price is an index",
     "anchor": "a-moves-price-is-an-index",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "column.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "A ring's block keeps one runaway, in counts",
     "anchor": "a-rings-block-keeps-one-runaway-in-counts",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "door.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A carrier bears on a place down to a ladder member",
     "anchor": "a-carrier-bears-on-a-place-down-to-a-ladder-member",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "carrier.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Which place a ring's block runs",
     "anchor": "which-place-a-rings-block-runs",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "race.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The void tie crosses kinds and the tie-break chooses the place that runs",
     "anchor": "the-void-tie-crosses-kinds-and-the-tie-break-chooses-the-place-that-runs",
     "tier": "rule",
     "shown": "rule",
     "range": "proved for every cubic ring meeting either paragraph's hypotheses; walked at six",
     "scripts": [
      "race.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "A void walk pays its winner's lone path",
     "anchor": "a-void-walk-pays-its-winners-lone-path",
     "tier": "theorem",
     "shown": "theorem; criterion",
     "range": "the rows, the cell",
     "scripts": [
      "winner.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A lidded block locks the walk for good",
     "anchor": "a-lidded-block-locks-the-walk-for-good",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "the lid, the certificate",
     "scripts": [
      "lid.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/collide/",
   "title": "COLLIDE — when two products of weighted sets are one polynomial",
   "kind": "doc",
   "parent": "/machines/",
   "source": "/collide.md",
   "claims": [
    {
     "heading": "The frame",
     "anchor": "the-frame",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Non-uniqueness needs a seed",
     "anchor": "non-uniqueness-needs-a-seed",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A witness-free tie is a regrouping or passes a seed",
     "anchor": "a-witness-free-tie-is-a-regrouping-or-passes-a-seed",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py",
      "quarantine.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Term counts multiply",
     "anchor": "term-counts-multiply",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Size 2 closes in the elements",
     "anchor": "size-2-closes-in-the-elements",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Size 3 closes in the exponent vectors",
     "anchor": "size-3-closes-in-the-exponent-vectors",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Ljunggren's theorem on trinomials",
     "anchor": "ljunggrens-theorem-on-trinomials",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "W. Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand. 8 (1960) 65–70, Theorem 1 and the corollary extending it to x^n + ε₁x^m + ε₂."
    },
    {
     "heading": "Menu size is the grading",
     "anchor": "menu-size-is-the-grading",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Factorization by number of terms",
     "anchor": "factorization-by-number-of-terms",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "C. E. van de Woestijne, Factors of disconnected graphs and polynomials with nonnegative integer coefficients, arXiv:1103.0709 (2011), Lemmas 3.3, 3.4 and 3.10, Theorems 3.16 to 3.20."
    },
    {
     "heading": "The escape at (2,5)",
     "anchor": "the-escape-at-25",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The descent dimension",
     "anchor": "the-descent-dimension",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collide.py",
      "faces.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Two factorizations no proper face separates",
     "anchor": "two-factorizations-no-proper-face-separates",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "faces.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The sign scan",
     "anchor": "the-sign-scan",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "faces.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The face law",
     "anchor": "the-face-law",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "faces.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The cycle lemma",
     "anchor": "the-cycle-lemma",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "mixing.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The mixing theorem",
     "anchor": "the-mixing-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "mixing.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The family mixes at every e",
     "anchor": "the-family-mixes-at-every-e",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "mixing.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The binomial-pair criterion",
     "anchor": "the-binomial-pair-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "mixing.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Yost's indecomposability bound",
     "anchor": "yosts-indecomposability-bound",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "D. Yost, Irreducible convex sets, Mathematika 38 (1991) 134–155, Proposition 6, as restated in K. Przesławski and D. Yost, More indecomposable polyhedra, Extracta Math. 31 (2016) 169–188 (arXiv:1607.00643), whose Theorem 9 there adds the equality case, first proved in M. Kallay's unpublished 1979 thesis."
    },
    {
     "heading": "The half-size law",
     "anchor": "the-half-size-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "halfsize.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The torsion split",
     "anchor": "the-torsion-split",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "rooted.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The least free degree",
     "anchor": "the-least-free-degree",
     "tier": "rule",
     "shown": "rule",
     "range": "verified exhaustively through degree 5",
     "scripts": [
      "rooted.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Mills's theorem on quadrinomials",
     "anchor": "millss-theorem-on-quadrinomials",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "W. H. Mills, The factorization of certain quadrinomials, Math. Scand. 57 (1985) 44–50, Theorem 2."
    },
    {
     "heading": "The cofactor theorem",
     "anchor": "the-cofactor-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "quadrinomial.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The size-4 closure",
     "anchor": "the-size-4-closure",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "quadrinomial.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The row lemma",
     "anchor": "the-row-lemma",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "pentanomial.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The least free size off a line",
     "anchor": "the-least-free-size-off-a-line",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "pentanomial.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Mann's bound",
     "anchor": "manns-bound",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [
      "pentanomial.py"
     ],
     "proof": false,
     "source": "H. B. Mann, On linear relations between roots of unity, Mathematika 12 (1965) 107–117, Theorem 1, as restated in J.-H. Evertse, The number of solutions of linear equations in roots of unity (preprint, Leiden), section 2."
    },
    {
     "heading": "The Filaseta–Solan question",
     "anchor": "the-filasetasolan-question",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "M. Filaseta and J. Solan, An extension of a theorem of Ljunggren, Math. Scand. 84 (1999) 5–10, Theorem 1 and the remark after it."
    },
    {
     "heading": "The size-5 reduction",
     "anchor": "the-size-5-reduction",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "pentanomial.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/coordinates/",
   "title": "COORDINATES — which coordinate system reads which operation",
   "kind": "doc",
   "parent": "/residues/",
   "source": "/coordinates.md",
   "claims": [
    {
     "heading": "The two-readings criterion",
     "anchor": "the-two-readings-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The carry wall",
     "anchor": "the-carry-wall",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Addition and multiplication each decide",
     "anchor": "addition-and-multiplication-each-decide",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "Presburger 1929; Mostowski 1952; Putnam 1957; Villemaire 1992; Korec 1993 and 1995; Bès 1997; all as given in Bès, A survey of arithmetical definability (Proposition 5, Theorems 23 and 36, the notes on Pascal triangles); J. Robinson, J. Symbolic Logic 14 (1949) 98."
    },
    {
     "heading": "The suffix and floor reads",
     "anchor": "the-suffix-and-floor-reads",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The Internet checksum",
     "anchor": "the-internet-checksum",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "checksum.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The valuation row",
     "anchor": "the-valuation-row",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The index coordinates",
     "anchor": "the-index-coordinates",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The Zech wall",
     "anchor": "the-zech-wall",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The support log",
     "anchor": "the-support-log",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The exponent wall",
     "anchor": "the-exponent-wall",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The rigidity dichotomy",
     "anchor": "the-rigidity-dichotomy",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The bearers",
     "anchor": "the-bearers",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "coordinates.py",
      "size.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/dual/",
   "title": "DUAL — the leading end of base b, read against the trailing end",
   "kind": "doc",
   "parent": "/digits/",
   "source": "/dual.md",
   "claims": [
    {
     "heading": "The leading-tile lemma",
     "anchor": "the-leading-tile-lemma",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The two-ends law",
     "anchor": "the-two-ends-law",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The crown exchange",
     "anchor": "the-crown-exchange",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The coset rigidity",
     "anchor": "the-coset-rigidity",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The affine class theorem",
     "anchor": "the-affine-class-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The least-lookahead law",
     "anchor": "the-least-lookahead-law",
     "tier": "theorem",
     "shown": "theorem; pattern",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The two-ends exchange",
     "anchor": "the-two-ends-exchange",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "dual.py",
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The structural split",
     "anchor": "the-structural-split",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The redundant reader",
     "anchor": "the-redundant-reader",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The hedge",
     "anchor": "the-hedge",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "What the purchase spends",
     "anchor": "what-the-purchase-spends",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The redundancy leak",
     "anchor": "the-redundancy-leak",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The period law",
     "anchor": "the-period-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The dual Haar measure",
     "anchor": "the-dual-haar-measure",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Each end prices in its own metric",
     "anchor": "each-end-prices-in-its-own-metric",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "dual.py",
      "reading.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/element/",
   "title": "ELEMENT — the walk that must stay principal",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/element.md",
   "claims": [
    {
     "heading": "The ring is a supply matrix",
     "anchor": "the-ring-is-a-supply-matrix",
     "tier": "rule",
     "shown": "rule; theorem",
     "range": "verified at six rings: every colour to degree 9 against a brute count of places; the counts to degree 400 computed by an exact recursion, whole at every degree",
     "scripts": [
      "element.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The element reading sees only classes",
     "anchor": "the-element-reading-sees-only-classes",
     "tier": "criterion",
     "shown": "criterion",
     "range": "proved in any Dedekind domain with finite class group; brute force at three rings",
     "scripts": [
      "sensor.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The class gap between principal and all ideals",
     "anchor": "the-class-gap-between-principal-and-all-ideals",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the identity, by character orthogonality",
     "scripts": [
      "sensor.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The coarsest colouring",
     "anchor": "the-coarsest-colouring",
     "tier": "theorem",
     "shown": "theorem",
     "range": "the witnesses built at six rings for every pair of colours of degree ≤ 6",
     "scripts": [
      "element.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The minimal rider is unique",
     "anchor": "the-minimal-rider-is-unique",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "element.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The bare door",
     "anchor": "the-bare-door",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "element.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Greed pays no avoidable rider",
     "anchor": "greed-pays-no-avoidable-rider",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "element.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The margin step",
     "anchor": "the-margin-step",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "element.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The eventual chain",
     "anchor": "the-eventual-chain",
     "tier": "rule",
     "shown": "rule",
     "range": "verified over every branch of six rings, an 8-move stretch continued to 300 moves; the argument below lacks its induction's base",
     "scripts": [
      "element.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Where the chain changes hands",
     "anchor": "where-the-chain-changes-hands",
     "tier": "rule",
     "shown": "rule",
     "range": "verified over every branch of six rings, an 8-move stretch continued to 300 moves: 2, 2, 2, 4, 20 and 120 branches",
     "scripts": [
      "element.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The orbit law and the rider recursion",
     "anchor": "the-orbit-law-and-the-rider-recursion",
     "tier": "theorem",
     "shown": "theorem",
     "range": "past the change range; verified over every branch of six rings, an 8-move stretch continued to 300 moves",
     "scripts": [
      "element.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The element limit",
     "anchor": "the-element-limit",
     "tier": "rule",
     "shown": "rule",
     "range": "verified over every branch of six rings, an 8-move stretch continued to 300 moves; unbounded and stopped read by the orbit law's recursion from each branch's seed, a limit past the eventual chain's missing base",
     "scripts": [
      "element.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The number rings price in (min, ×)",
     "anchor": "the-number-rings-price-in-min-",
     "tier": "rule",
     "shown": "rule",
     "range": "verified at Z[√−5] and Q(√−23",
     "scripts": [
      "element_ring.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The lock carries its rider",
     "anchor": "the-lock-carries-its-rider",
     "tier": "rule",
     "shown": "rule",
     "range": "verified at the 53 seeds, the void and every principal ideal of norm ≤ 40, the last 20 of 60 moves; each lock's permanence proved",
     "scripts": [
      "element_ring.py",
      "module_law.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The element lid",
     "anchor": "the-element-lid",
     "tier": "theorem",
     "shown": "theorem",
     "range": "the lid and the four statements over any number ring; the lock's one prime over an imaginary quadratic field",
     "scripts": [
      "rider.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/fingerprint/",
   "title": "FINGERPRINT — one real number reads a residue tuple up to sign",
   "kind": "doc",
   "parent": "/object/",
   "source": "/fingerprint.md",
   "claims": [
    {
     "heading": "The fingerprint",
     "anchor": "the-fingerprint",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "fingerprint.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The fingerprint theorem",
     "anchor": "the-fingerprint-theorem",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "fingerprint.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/flatten/",
   "title": "FLATTEN — the least height flattened to a given depth",
   "kind": "doc",
   "parent": "/instruments/",
   "source": "/flatten.md",
   "claims": [
    {
     "heading": "Flattening is annihilation",
     "anchor": "flattening-is-annihilation",
     "tier": "property",
     "shown": "property",
     "range": "proved",
     "scripts": [],
     "proof": true,
     "source": null
    },
    {
     "heading": "The least height is a lattice minimum",
     "anchor": "the-least-height-is-a-lattice-minimum",
     "tier": "property",
     "shown": "property",
     "range": "the radius proved; that the floating-point enumeration misses nothing rests on its slack and is not proved; the route agrees with exhaustion at the 12 cells of width at most 6, and on whether h = 1 at 33",
     "scripts": [
      "flatten.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "At rank 1 the least height is the central binomial",
     "anchor": "at-rank-1-the-least-height-is-the-central-binomial",
     "tier": "property",
     "shown": "property",
     "range": "proved; read by the lattice route at all 28 rank-1 cells of the chart",
     "scripts": [
      "flatten.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "In the corner the least height is a pure product's",
     "anchor": "in-the-corner-the-least-height-is-a-pure-products",
     "tier": "rule",
     "shown": "rule",
     "range": "exhaustive at the 63 corner cells, h read by the lattice route, whose minimality rests on the enumeration's slack",
     "scripts": [
      "flatten.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "On the chart the products fail more often the deeper",
     "anchor": "on-the-chart-the-products-fail-more-often-the-deeper",
     "tier": "rule",
     "shown": "rule",
     "range": "exhaustive over the 695 chart cells, h read by the lattice route, whose minimality rests on the enumeration's slack",
     "scripts": [
      "flatten.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Through its table the inverse dial is a pure-product degree",
     "anchor": "through-its-table-the-inverse-dial-is-a-pure-product-degree",
     "tier": "rule",
     "shown": "rule",
     "range": "exhaustive at the 99 cells of the table and 15 cells past it, each lattice verdict resting on the enumeration's slack",
     "scripts": [
      "flatten.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Read by rank, the failures sit in a band of depths",
     "anchor": "read-by-rank-the-failures-sit-in-a-band-of-depths",
     "tier": "rule",
     "shown": "rule",
     "range": "ranks 1…22: ranks 1…4 to depth 140, ranks 5…10 to eight depths past the last failure at each of which the champion attains h, ranks 11…22 to the first",
     "scripts": [
      "band.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Past the last failure seen the champion attains the least height",
     "anchor": "past-the-last-failure-seen-the-champion-attains-the-least-height",
     "tier": "rule",
     "shown": "rule",
     "range": "ranks 2…10 over the scanned depths; rank 2 is the theorem below",
     "scripts": [
      "band.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "A failing cell is a break on the dial",
     "anchor": "a-failing-cell-is-a-break-on-the-dial",
     "tier": "property",
     "shown": "property",
     "range": "proved; the fence a rule on the dial's table, and the heights read at the 83 failing cells",
     "scripts": [
      "flatten.py",
      "members.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "At heights 8, 9 and 11 the dial breaks",
     "anchor": "at-heights-8-9-and-11-the-dial-breaks",
     "tier": "rule",
     "shown": "rule",
     "range": "exhaustive at each cell: the witness exhibited, the lattice empty one degree below, its emptiness resting on the enumeration's slack",
     "scripts": [
      "flatten.py",
      "members.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "A traded pure product attains every failing cell of the chart",
     "anchor": "a-traded-pure-product-attains-every-failing-cell-of-the-chart",
     "tier": "rule",
     "shown": "rule",
     "range": "exhaustive over the 83 failing cells of the chart",
     "scripts": [
      "members.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The nine non-cyclotomic factors are reciprocal and need not sit on the circle",
     "anchor": "the-nine-non-cyclotomic-factors-are-reciprocal-and-need-not-sit-on-the-circle",
     "tier": "observation",
     "shown": "observation; property",
     "range": "nine factors",
     "scripts": [
      "members.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The height-1 column is classical",
     "anchor": "the-height-1-column-is-classical",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "P. Borwein and M. J. Mossinghoff, \"Polynomials with height 1 and prescribed vanishing at 1\", Experiment. Math. 9 (2000) 425–433."
    },
    {
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     "heading": "At rank 2 the two minima part by a factor growing as √J",
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   "url": "/flush/",
   "title": "FLUSH — what a redundant reader of the circle computes",
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     "heading": "The lift game",
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     "heading": "The completion reader is the safety game",
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     "heading": "The flush floor",
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     "heading": "The universal reader",
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    {
     "heading": "The quarantine theorem",
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    {
     "heading": "Recycled worlds need no witness",
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     "range": null,
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      "quarantine.py"
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    },
    {
     "heading": "Readability descends, spread ascends, flatness neither",
     "anchor": "readability-descends-spread-ascends-flatness-neither",
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     "proof": true,
     "source": null
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    {
     "heading": "The working clause",
     "anchor": "the-working-clause",
     "tier": "property",
     "shown": "property",
     "range": null,
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      "forget.py"
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     "proof": true,
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    {
     "heading": "Two classes never cure by refinement",
     "anchor": "two-classes-never-cure-by-refinement",
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      "forget.py"
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     "proof": true,
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    {
     "heading": "The no-majority criterion",
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     "tier": "criterion",
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      "forget.py"
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     "proof": true,
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    {
     "heading": "Retaining less can leave no cure",
     "anchor": "retaining-less-can-leave-no-cure",
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      "forget.py"
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    {
     "heading": "A strict cure needs three classes",
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   "url": "/gates/",
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      "gates.py"
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    {
     "heading": "The crystallographic skeleton",
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    {
     "heading": "The alphabet ladder",
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    {
     "heading": "The meadow closure",
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      "gates.py"
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    {
     "heading": "The orbit-cost theorem",
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      "gates.py"
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    {
     "heading": "The height floor",
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   "url": "/growth/",
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   "kind": "doc",
   "parent": "/walks/",
   "source": "/growth.md",
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     "heading": "The three fates",
     "anchor": "the-three-fates",
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      "growth.py"
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    {
     "heading": "The wall is Adams' number",
     "anchor": "the-wall-is-adams-number",
     "tier": "property",
     "shown": "property",
     "range": "the closed form is Adams' theorem; checked at every L ≤ 200",
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      "growth.py"
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     "source": null
    },
    {
     "heading": "The wall over F₂[x]",
     "anchor": "the-wall-over-f₂x",
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      "growth.py"
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    },
    {
     "heading": "The lock-prime law",
     "anchor": "the-lock-prime-law",
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      "bridges.py",
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      "standing.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The ghosts are the witnesses' records",
     "anchor": "the-ghosts-are-the-witnesses-records",
     "tier": "theorem",
     "shown": "theorem; known",
     "range": null,
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      "ghosts.py"
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     "proof": true,
     "source": "M. Goldfeld, On the number of primes p for which p + a has a large prime factor, Mathematika 16 (1969), 23–27, Theorem 1."
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    {
     "heading": "The two-adic prices",
     "anchor": "the-two-adic-prices",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
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      "growth.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The module law",
     "anchor": "the-module-law",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
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      "module_law.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The lock over a number ring",
     "anchor": "the-lock-over-a-number-ring",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "the ideal world's lock iff its costs are bounded infinitely often, at p^(ef",
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      "lid.py",
      "module_law.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The sprawl over F₂[x]",
     "anchor": "the-sprawl-over-f₂x",
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      "sprawl.py"
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    {
     "heading": "The sibling shadow",
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      "sprawl.py"
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     "source": null
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    {
     "heading": "The standing-move dichotomy",
     "anchor": "the-standing-move-dichotomy",
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     "shown": "theorem; observation",
     "range": "the dichotomy, the depth reading and the jump",
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      "standing.py"
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     "proof": true,
     "source": null
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    {
     "heading": "The properness criterion",
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     "tier": "criterion",
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     "range": "proved both ways",
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      "sizeless.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The route is the deleted place's",
     "anchor": "the-route-is-the-deleted-places",
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     "range": "the route, and the arrival rule's shape",
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      "sizeless.py"
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   "url": "/headroom/",
   "title": "HEADROOM — what a probe sees of a grown state",
   "kind": "doc",
   "parent": "/walks/",
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     "heading": "The headroom",
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      "headroom.py"
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    },
    {
     "heading": "The ledger",
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      "headroom.py"
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     "source": null
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    {
     "heading": "The premium in closed form",
     "anchor": "the-premium-in-closed-form",
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     "range": null,
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      "headroom.py"
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    {
     "heading": "Silence lands in a blind class",
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     "range": null,
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      "headroom.py"
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    {
     "heading": "The closure",
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      "headroom.py"
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     "proof": true,
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    },
    {
     "heading": "The resolution criterion",
     "anchor": "the-resolution-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": "proved both ways",
     "scripts": [],
     "proof": true,
     "source": null
    },
    {
     "heading": "The non-divisor bound",
     "anchor": "the-non-divisor-bound",
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     "range": null,
     "scripts": [
      "headroom.py"
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     "proof": true,
     "source": null
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    {
     "heading": "No blindness is permanent",
     "anchor": "no-blindness-is-permanent",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "headroom.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The watcher's guaranteed blindness",
     "anchor": "the-watchers-guaranteed-blindness",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
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      "headroom.py"
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     "source": null
    },
    {
     "heading": "The blind-density theorem",
     "anchor": "the-blind-density-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
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      "headroom.py"
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     "proof": true,
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    {
     "heading": "The least resolver is unbounded",
     "anchor": "the-least-resolver-is-unbounded",
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      "headroom.py"
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   "url": "/image/",
   "title": "IMAGE — the set of limits a greedy walk reaches",
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   "parent": "/walks/",
   "source": "/image.md",
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     "heading": "The colour count",
     "anchor": "the-colour-count",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "image.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "A quadratic ring's ideal image is 2^t",
     "anchor": "a-quadratic-rings-ideal-image-is-2t",
     "tier": "rule",
     "shown": "rule",
     "range": "the tie shape and the conjugate never seated proved; the count verified at Z[√−5] and Q(√−23",
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      "image.py",
      "race.py"
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    {
     "heading": "The lock's rider erases a choice",
     "anchor": "the-locks-rider-erases-a-choice",
     "tier": "rule",
     "shown": "rule",
     "range": "verified at the void and every principal ideal of norm ≤ 40: 26 seeds at Z[√−5] and 27 at Q(√−23",
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      "element_ring.py",
      "image.py"
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    {
     "heading": "A curve's image is the continuum",
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    {
     "heading": "The gap",
     "anchor": "the-gap",
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     "shown": "rule",
     "range": "the point proved, the continuum as the curve section's rule, the finite count verified at the two quadratic rings on the seeds run",
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      "growth.py",
      "image.py"
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     "proof": false,
     "source": null
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    {
     "heading": "The free image has five forms",
     "anchor": "the-free-image-has-five-forms",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "free_image.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The free image forgets the ring",
     "anchor": "the-free-image-forgets-the-ring",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "free_image.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The fates are the image's extremes",
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     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "free_image.py"
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     "proof": true,
     "source": null
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    {
     "heading": "Greed's point",
     "anchor": "greeds-point",
     "tier": "rule",
     "shown": "rule",
     "range": "resting on GROWTH.md's lock rules, this page's curve rule and the five forms; the places at ∞ read at the 27 principal seeds of norm ≤ 40 of Q(√−23",
     "scripts": [
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      "free_image.py",
      "image.py"
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    {
     "heading": "The c-greedy walk is dammed",
     "anchor": "the-c-greedy-walk-is-dammed",
     "tier": "theorem",
     "shown": "theorem",
     "range": "the dam, the striking rule for the tide ceiling, τ(P",
     "scripts": [
      "cgreedy.py",
      "dam.py"
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     "proof": true,
     "source": null
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    {
     "heading": "Polynomial slack reaches breadth from the void",
     "anchor": "polynomial-slack-reaches-breadth-from-the-void",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the floor τ(P",
     "scripts": [
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      "track.py"
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     "proof": true,
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   "url": "/learner/",
   "title": "LEARNER — a reader that learns by exact comparisons of its price",
   "kind": "doc",
   "parent": "/machines/",
   "source": "/learner.md",
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     "heading": "The straddle cover and its readers",
     "anchor": "the-straddle-cover-and-its-readers",
     "tier": "property",
     "shown": "property; observation",
     "range": null,
     "scripts": [
      "forecast.py",
      "learner.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The bottom lemma",
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     "tier": "theorem",
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     "range": null,
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      "learner.py"
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     "proof": true,
     "source": null
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    {
     "heading": "Greedy patience dominates every sound reader",
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      "learner.py"
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     "proof": true,
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    {
     "heading": "The partition valley",
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      "learner.py"
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     "proof": true,
     "source": null
    },
    {
     "heading": "The funnel with one blind corner",
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      "learner.py"
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    {
     "heading": "The layers of a holding set",
     "anchor": "the-layers-of-a-holding-set",
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      "rationed.py"
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    },
    {
     "heading": "Under an income the rank is common",
     "anchor": "under-an-income-the-rank-is-common",
     "tier": "theorem",
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      "rationed.py"
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    {
     "heading": "On the eights row, patience with no bank lags for the whole run",
     "anchor": "on-the-eights-row-patience-with-no-bank-lags-for-the-whole-run",
     "tier": "rule",
     "shown": "rule",
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    {
     "heading": "Destination universality",
     "anchor": "destination-universality",
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     "range": "verified at 16 settings on nine rows, two orders",
     "scripts": [
      "rationed.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "A density forecast is a length loss",
     "anchor": "a-density-forecast-is-a-length-loss",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [],
     "proof": true,
     "source": null
    },
    {
     "heading": "The side loss makes the best reader depend on the stream",
     "anchor": "the-side-loss-makes-the-best-reader-depend-on-the-stream",
     "tier": "rule",
     "shown": "rule",
     "range": "verified on the nine rows over 120 steps, steps 8 to 119 counted",
     "scripts": [
      "forecast.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The stream enters at a reversal",
     "anchor": "the-stream-enters-at-a-reversal",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "forecast.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The two poles",
     "anchor": "the-two-poles",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [],
     "proof": true,
     "source": null
    },
    {
     "heading": "Breaking needs no stream and keeping needs no pole",
     "anchor": "breaking-needs-no-stream-and-keeping-needs-no-pole",
     "tier": "rule",
     "shown": "rule",
     "range": "verified over all bands of the 117 positions on the nine rows",
     "scripts": [
      "forecast.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The collision grower",
     "anchor": "the-collision-grower",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "collision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The rung reading",
     "anchor": "the-rung-reading",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "collision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "How fine, and what",
     "anchor": "how-fine-and-what",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "collision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The encoding bound",
     "anchor": "the-encoding-bound",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "collision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The substrate parts by depth or inside a layer",
     "anchor": "the-substrate-parts-by-depth-or-inside-a-layer",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "collision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The grower's worst case",
     "anchor": "the-growers-worst-case",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "collision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Exact readout reads the subdominant",
     "anchor": "exact-readout-reads-the-subdominant",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "readout.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The grown depth is the readable depth",
     "anchor": "the-grown-depth-is-the-readable-depth",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "readout.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/limit/",
   "title": "LIMIT — where one greedy walk goes",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/limit.md",
   "claims": [
    {
     "heading": "The ring is a supply",
     "anchor": "the-ring-is-a-supply",
     "tier": "rule",
     "shown": "rule",
     "range": "the door proved; the walk verified at 720 states over six ring supplies",
     "scripts": [
      "limit.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "No supplied degree is covered before it opens",
     "anchor": "no-supplied-degree-is-covered-before-it-opens",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the lcm reading, from Bang's theorem",
     "scripts": [
      "limit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The chain",
     "anchor": "the-chain",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "limit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "When the clock runs forever",
     "anchor": "when-the-clock-runs-forever",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "limit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The support",
     "anchor": "the-support",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "limit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The ideal limit",
     "anchor": "the-ideal-limit",
     "tier": "rule",
     "shown": "rule",
     "range": "proved at the corner; verified over six ring supplies, 10 moves branched and 290 more on every branch",
     "scripts": [
      "limit.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The ceiling",
     "anchor": "the-ceiling",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "limit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Attainment is the void menu's",
     "anchor": "attainment-is-the-void-menus",
     "tier": "rule",
     "shown": "rule",
     "range": "verified at 54 rows: 14 schedules over three ring supplies and 12 over a designed one, every tie branched over 10 moves",
     "scripts": [
      "limit.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The strand",
     "anchor": "the-strand",
     "tier": "rule",
     "shown": "rule",
     "range": "verified at the 54 rows",
     "scripts": [
      "limit.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "What stops the widening",
     "anchor": "what-stops-the-widening",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the opened-set half",
     "scripts": [
      "limit.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/logic/",
   "title": "LOGIC — the logic a squarefree ring's own polynomials carry",
   "kind": "doc",
   "parent": "/residues/",
   "source": "/logic.md",
   "claims": [
    {
     "heading": "The truth values",
     "anchor": "the-truth-values",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The meadow",
     "anchor": "the-meadow",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The defect",
     "anchor": "the-defect",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The Kleene reading",
     "anchor": "the-kleene-reading",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The pair",
     "anchor": "the-pair",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The envelope",
     "anchor": "the-envelope",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The envelope's sharpness",
     "anchor": "the-envelopes-sharpness",
     "tier": "theorem",
     "shown": "theorem",
     "range": "read-once formulas, every p ≥ 5",
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The ladder",
     "anchor": "the-ladder",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The order-6 shadow",
     "anchor": "the-order-6-shadow",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The ladder's reach",
     "anchor": "the-ladders-reach",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "logic.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/machine/",
   "title": "MACHINE — a state that only multiplies, read as a machine",
   "kind": "doc",
   "parent": "/machines/",
   "source": "/machine.md",
   "claims": [
    {
     "heading": "The depth machine decides its halting",
     "anchor": "the-depth-machine-decides-its-halting",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "machine.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Clearing a place decides and dividing once does not",
     "anchor": "clearing-a-place-decides-and-dividing-once-does-not",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "borrow.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "One form decides and two can run Minsky",
     "anchor": "one-form-decides-and-two-can-run-minsky",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "comparison.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Two places at the ratios 1 and α/β run FRACTRAN",
     "anchor": "two-places-at-the-ratios-1-and-αβ-run-fractran",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "comparison.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The headroom is a bank of Minsky counters",
     "anchor": "the-headroom-is-a-bank-of-minsky-counters",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "machine.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A blind read decides and a sighted read is universal",
     "anchor": "a-blind-read-decides-and-a-sighted-read-is-universal",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "borrow.py",
      "machine.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/memory/",
   "title": "MEMORY — what a grown world can know of its genesis",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/memory.md",
   "claims": [
    {
     "heading": "The route-weight cancellation",
     "anchor": "the-route-weight-cancellation",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The bounded-evidence law",
     "anchor": "the-bounded-evidence-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The unbounded-memory law",
     "anchor": "the-unbounded-memory-law",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A deepening world reads its lock prime",
     "anchor": "a-deepening-world-reads-its-lock-prime",
     "tier": "theorem",
     "shown": "theorem; property",
     "range": null,
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The witness gap",
     "anchor": "the-witness-gap",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "the split",
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The cold blindness",
     "anchor": "the-cold-blindness",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The fossil paradox",
     "anchor": "the-fossil-paradox",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "the one-window world",
     "scripts": [
      "memory.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/naming/",
   "title": "NAMING — which primes divide a set's Euler characteristic",
   "kind": "doc",
   "parent": "/object/",
   "source": "/naming.md",
   "claims": [
    {
     "heading": "The complex",
     "anchor": "the-complex",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The naming integer",
     "anchor": "the-naming-integer",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The weight law",
     "anchor": "the-weight-law",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Every odd prime names a set",
     "anchor": "every-odd-prime-names-a-set",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The baseline",
     "anchor": "the-baseline",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The slices",
     "anchor": "the-slices",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "No free automorphism of outside prime order",
     "anchor": "no-free-automorphism-of-outside-prime-order",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "naming.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/observatory/",
   "title": "OBSERVATORY — what the digits of a ramified field read",
   "kind": "doc",
   "parent": "/local/",
   "source": "/observatory.md",
   "claims": [
    {
     "heading": "The coefficient triangle",
     "anchor": "the-coefficient-triangle",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "triangle.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The anchor design of the 2e-th roots of unity",
     "anchor": "the-anchor-design-of-the-2e-th-roots-of-unity",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "triangle.py",
      "wild.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The tame readout",
     "anchor": "the-tame-readout",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "tame.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The cofactor readout",
     "anchor": "the-cofactor-readout",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "cofactor.py",
      "quarter.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The wild readout",
     "anchor": "the-wild-readout",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "wild.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The letters of K(ζ₈)/K at e = 4",
     "anchor": "the-letters-of-kζ₈k-at-e--4",
     "tier": "rule",
     "shown": "rule",
     "range": "the letters are functions of the cell by the mod π⁹ argument below, checked at each of the 256 cells of the Eisenstein quartics against two random lifts of its coefficient digits at levels 9 to 16; 5 is not a square at each cell's polynomial with no coefficient digit past level 8; the letters by w are read off the table printed over those cells",
     "scripts": [
      "wild.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Pagano's jump set",
     "anchor": "paganos-jump-set",
     "tier": "known",
     "shown": "known; observation",
     "range": null,
     "scripts": [
      "relative.py"
     ],
     "proof": false,
     "source": "C. Pagano, Jump sets in local fields, arXiv:1810.09975 (2018), Theorems 1.4, 1.5, 1.7, 1.8, 1.11 and 10.1, Propositions 3.35, 3.39 and 5.5, and property (P.1) of the reduction maps in the proof of Theorem 9.1."
    },
    {
     "heading": "The jump set as the class maxima",
     "anchor": "the-jump-set-as-the-class-maxima",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "cofactor.py",
      "rung.py",
      "staircase.py",
      "weld.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The jump set reads M·e digits of u",
     "anchor": "the-jump-set-reads-me-digits-of-u",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "precision.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The last point gives the break of the next storey",
     "anchor": "the-last-point-gives-the-break-of-the-next-storey",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "storey.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A radical storey adds e to the first defect and passes the rest down",
     "anchor": "a-radical-storey-adds-e-to-the-first-defect-and-passes-the-rest-down",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "storey.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/ostrowski/",
   "title": "OSTROWSKI — the circle a trailing numeration reads",
   "kind": "doc",
   "parent": "/digits/",
   "source": "/ostrowski.md",
   "claims": [
    {
     "heading": "The circle numeration",
     "anchor": "the-circle-numeration",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "ostrowski.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The two codings of −α",
     "anchor": "the-two-codings-of-α",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "ostrowski.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The containment criterion",
     "anchor": "the-containment-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "ostrowski.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The affine maps",
     "anchor": "the-affine-maps",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "ostrowski.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Floor division is never MIDDLE here",
     "anchor": "floor-division-is-never-middle-here",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "ostrowski.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/principal/",
   "title": "PRINCIPAL — where a quadratic field's first principal prime sits",
   "kind": "doc",
   "parent": "/local/",
   "source": "/principal.md",
   "claims": [
    {
     "heading": "The floor",
     "anchor": "the-floor",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "principal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "No floor over a real field",
     "anchor": "no-floor-over-a-real-field",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "principal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Where the omitted prime powers land",
     "anchor": "where-the-omitted-prime-powers-land",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "principal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The shortfall is the prime powers",
     "anchor": "the-shortfall-is-the-prime-powers",
     "tier": "pattern",
     "shown": "pattern",
     "range": "imaginary fields |D| ≤ 4000 with h ≥ 2, the (class number, cell",
     "scripts": [
      "principal.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The bias against principal primes",
     "anchor": "the-bias-against-principal-primes",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "M. Aoki and S. Koyama, Chebyshev's bias against splitting and principal primes in global fields, J. Number Theory 245 (2023), 233–262, Corollary 3.5 and Example 3.7."
    }
   ],
   "withheld": 0
  },
  {
   "url": "/rates/",
   "title": "RATES — what a reader pays where a digit's scale varies",
   "kind": "doc",
   "parent": "/digits/",
   "source": "/rates.md",
   "claims": [
    {
     "heading": "Walls sit at the preimages of the rationals",
     "anchor": "walls-sit-at-the-preimages-of-the-rationals",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "rates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The commitment bound",
     "anchor": "the-commitment-bound",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "rates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Rate forcing",
     "anchor": "rate-forcing",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "rates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The unit-index law",
     "anchor": "the-unit-index-law",
     "tier": "pattern",
     "shown": "pattern",
     "range": "η = ε^i exactly at 130 quadratic points over 15 real quadratic fields, conductors up to 20",
     "scripts": [
      "rates.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The unit criterion",
     "anchor": "the-unit-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "rates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The ladder law",
     "anchor": "the-ladder-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "rates.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/reading/",
   "title": "READING — what a digit-by-digit reader computes",
   "kind": "doc",
   "parent": "/digits/",
   "source": "/reading.md",
   "claims": [
    {
     "heading": "The reading lemma",
     "anchor": "the-reading-lemma",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Non-redundant absolute tiles wall only at the archimedean place",
     "anchor": "non-redundant-absolute-tiles-wall-only-at-the-archimedean-place",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The wall criterion",
     "anchor": "the-wall-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The Lipschitz criterion",
     "anchor": "the-lipschitz-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The division criterion",
     "anchor": "the-division-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The division trichotomy",
     "anchor": "the-division-trichotomy",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Division over F_q[x]",
     "anchor": "division-over-f_qx",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The comb",
     "anchor": "the-comb",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "reading.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Floor division on the d-bonacci completion",
     "anchor": "floor-division-on-the-d-bonacci-completion",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "bonacci.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The scaling certificate",
     "anchor": "the-scaling-certificate",
     "tier": "rule",
     "shown": "rule",
     "range": "proved per pair at 47 pairs, d ≤ 10, m ≤ 12",
     "scripts": [
      "bonacci.py"
     ],
     "proof": false,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/redundant/",
   "title": "REDUNDANT — what a signed-digit reader of integers computes",
   "kind": "doc",
   "parent": "/digits/",
   "source": "/redundant.md",
   "claims": [
    {
     "heading": "The affine criterion",
     "anchor": "the-affine-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "redundant.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The two regimes of slope",
     "anchor": "the-two-regimes-of-slope",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "redundant.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "On one stream and a symmetric set the phase lowers, never raises",
     "anchor": "on-one-stream-and-a-symmetric-set-the-phase-lowers-never-raises",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "redundant.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Dividing by the radix",
     "anchor": "dividing-by-the-radix",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "redundant.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The length bound",
     "anchor": "the-length-bound",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "redundant.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Addition",
     "anchor": "addition",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "redundant.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The many-stream theorem",
     "anchor": "the-many-stream-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "manystream.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The pole units",
     "anchor": "the-pole-units",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "poles.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The selection estimate",
     "anchor": "the-selection-estimate",
     "tier": "property",
     "shown": "property; rule",
     "range": null,
     "scripts": [
      "poles.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The relative-gradient law",
     "anchor": "the-relative-gradient-law",
     "tier": "property",
     "shown": "property; rule",
     "range": null,
     "scripts": [
      "chains.py",
      "poles.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The piecewise theorem",
     "anchor": "the-piecewise-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "piecewise.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The segment lemma",
     "anchor": "the-segment-lemma",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "piecewise.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The tile-valued maps",
     "anchor": "the-tile-valued-maps",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "comparators.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The absolute value",
     "anchor": "the-absolute-value",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "comparators.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The free comparator",
     "anchor": "the-free-comparator",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "comparators.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The sign count",
     "anchor": "the-sign-count",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "sign.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The full width",
     "anchor": "the-full-width",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "sign.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Comparison",
     "anchor": "comparison",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "sign.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The chained delay",
     "anchor": "the-chained-delay",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "chains.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The threshold-miss law",
     "anchor": "the-threshold-miss-law",
     "tier": "rule",
     "shown": "rule; pattern",
     "range": "the gain proved, by the budget bounded by the producer's room and past it by a tree of hulls; the budget's converse false; the silence at slack 2 or more, a± ≤ b − 1, open",
     "scripts": [
      "chains.py",
      "floor.py",
      "hulls.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The matching law",
     "anchor": "the-matching-law",
     "tier": "rule",
     "shown": "rule",
     "range": "the least total an upper bound on the floor, and at slack 1 reached greedily and the naive sum less ν outside the adjacent range, proved; naive − ν checked to radix 200 behind a product, greedy least to 40, both heads; not the floor at the two-unit chain at (20, 8, 12",
     "scripts": [
      "chains.py"
     ],
     "proof": false,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/ruler/",
   "title": "RULER — evaluations that know their optimum",
   "kind": "doc",
   "parent": "/instruments/",
   "source": "/ruler.md",
   "claims": [
    {
     "heading": "The parity law of the sign bit",
     "anchor": "the-parity-law-of-the-sign-bit",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "squarefree N",
     "scripts": [
      "ceiling.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "One threshold range, one tent: the threshold family",
     "anchor": "one-threshold-range-one-tent-the-threshold-family",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "ceiling.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The gauge is the task's: orientation and comparison",
     "anchor": "the-gauge-is-the-tasks-orientation-and-comparison",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "ceiling.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The threshold-range law under a tilted prior",
     "anchor": "the-threshold-range-law-under-a-tilted-prior",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "ceiling.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The resonance criterion",
     "anchor": "the-resonance-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "ceiling.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "One live arc: the magnitude parent",
     "anchor": "one-live-arc-the-magnitude-parent",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "every N, squarefree or not",
     "scripts": [
      "ceiling.py",
      "sets.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The knife: a consistent point inside an inconsistent interval",
     "anchor": "the-knife-a-consistent-point-inside-an-inconsistent-interval",
     "tier": "observation",
     "shown": "observation",
     "range": "the ratio derived asymptotically; 40 trials at n = 500 to 32,000, on the ring and on bare Bernoulli bins",
     "scripts": [
      "estimators.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "More data, a worse held-out interval",
     "anchor": "more-data-a-worse-held-out-interval",
     "tier": "observation",
     "shown": "observation",
     "range": "40 trials per point; the direction on the ring and on bare bins, the recovery on bare bins to n = 512,000",
     "scripts": [
      "estimators.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "One bound at two statistics",
     "anchor": "one-bound-at-two-statistics",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "estimators.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The spanning tree is not identified",
     "anchor": "the-spanning-tree-is-not-identified",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "estimators.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The sign law of a stated interval",
     "anchor": "the-sign-law-of-a-stated-interval",
     "tier": "property",
     "shown": "property",
     "range": "proved, up to the binomial bar's normal approximation; measured at 8 settings, two worlds at four sample sizes, 400 trials each, and under non-identification in the audit's Gaussian worlds at mixture weights 1/5 and 4/5 with equal class variances, and in one where the second root has left the box, 200 trials each",
     "scripts": [
      "audit.py",
      "statements.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Coverage is a function of two ratios",
     "anchor": "coverage-is-a-function-of-two-ratios",
     "tier": "observation",
     "shown": "observation",
     "range": "44 settings over five worlds, four sample sizes and five kinds of statement; the formula follows from construction under normality, and what is measured is that the approximation holds",
     "scripts": [
      "statements.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The kernel of the inside instruments",
     "anchor": "the-kernel-of-the-inside-instruments",
     "tier": "property",
     "shown": "property",
     "range": "the kernel and the shrinkage identity, proved; the knife factor derived asymptotically; the detection rates measured at 400 trials",
     "scripts": [
      "statements.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The nested repair reads the term, not the model",
     "anchor": "the-nested-repair-reads-the-term-not-the-model",
     "tier": "observation",
     "shown": "observation",
     "range": "three pooled worlds at four sample sizes, 400 trials; Δ̂ ≥ 0 and its limit, proved",
     "scripts": [
      "statements.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The audit world and its two roots",
     "anchor": "the-audit-world-and-its-two-roots",
     "tier": "property",
     "shown": "property",
     "range": "proved; the depth-cell roots and the tangency identity checked in exact rational arithmetic, the Gaussian second root 1 − κ + γ/4 in floating point at κ = 1/5 and the four γ in {0, 1/4, 1/2, 4/5} where it lies in the box, and found outside the box at γ = 1 and 2",
     "scripts": [
      "audit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The flip is the sign of one scalar",
     "anchor": "the-flip-is-the-sign-of-one-scalar",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "off the box",
     "scripts": [
      "audit.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The audit's band counts one sample of three",
     "anchor": "the-audits-band-counts-one-sample-of-three",
     "tier": "observation",
     "shown": "observation",
     "range": "two depth cells and three Gaussian worlds, κ = 1/5 and 4/5 at γ = 0 and κ = 1/5 at γ = 2, three sizes, 200 replicates of 200 resamples",
     "scripts": [
      "audit.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "With σ known, no self-report read off D̂ is unbiased at every distance",
     "anchor": "with-σ-known-no-self-report-read-off-d-is-unbiased-at-every-distance",
     "tier": "theorem",
     "shown": "theorem",
     "range": "in the normal model with σ known; the laws measured at two depth cells, three sizes, 1,000 replicate pairs each",
     "scripts": [
      "selfreport.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The per-example audit states its Gaussian limit",
     "anchor": "the-per-example-audit-states-its-gaussian-limit",
     "tier": "property",
     "shown": "property",
     "range": "proved; the convergence measured at (12, 3",
     "scripts": [
      "lira.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/sets/",
   "title": "SETS — set-valued prediction against its exact optimum",
   "kind": "doc",
   "parent": "/instruments/",
   "source": "/sets.md",
   "claims": [
    {
     "heading": "A marginal coverage guarantee cannot rank two set predictors",
     "anchor": "a-marginal-coverage-guarantee-cannot-rank-two-set-predictors",
     "tier": "observation",
     "shown": "observation",
     "range": "the four ring cells, samples of 2,000, 8,000 and 32,000, 20 trials each, every coverage exact",
     "scripts": [
      "sets.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The certificate brackets the optimum",
     "anchor": "the-certificate-brackets-the-optimum",
     "tier": "property",
     "shown": "property",
     "range": "proved; read at 3,404 cells, the sweep's and the four ring cells, the certificate's sizes covering T + D with D ≥ 0",
     "scripts": [
      "sets.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "At equal masses the certificate is the optimum",
     "anchor": "at-equal-masses-the-certificate-is-the-optimum",
     "tier": "property",
     "shown": "property",
     "range": "proved; exact at 402 equal-mass cells, two of them ring cells, the form paying at 117",
     "scripts": [
      "sets.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Off equal masses the certificate can overstate",
     "anchor": "off-equal-masses-the-certificate-can-overstate",
     "tier": "property",
     "shown": "property",
     "range": "proved by two exhibited cells; how often, an observation on a sampled sweep",
     "scripts": [
      "fates.py",
      "sets.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Three tests can prove the certificate exact",
     "anchor": "three-tests-can-prove-the-certificate-exact",
     "tier": "property",
     "shown": "property",
     "range": "proved; the three tests read at the 3,400 sweep cells and the four ring cells, the anchored gap at every ring cell but TILT-4-WIDE, whose 105 atoms are not searched",
     "scripts": [
      "sets.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "On the geometric walk's tail the anchored gap is one product",
     "anchor": "on-the-geometric-walks-tail-the-anchored-gap-is-one-product",
     "tier": "rule",
     "shown": "rule",
     "range": "proved for every real n ≥ 1451",
     "scripts": [
      "gaplaw.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "An exact solver for the marginal optimum",
     "anchor": "an-exact-solver-for-the-marginal-optimum",
     "tier": "property",
     "shown": "property",
     "range": "proved; with and without the fixing equal to enumeration at 3,400 sweep cells, and at TILT-3 the unfixed search equal to the fixed, both in sets.py::section_controls",
     "scripts": [
      "sets.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "At equal masses the operative level forbids the wrong forced fates",
     "anchor": "at-equal-masses-the-operative-level-forbids-the-wrong-forced-fates",
     "tier": "property",
     "shown": "property",
     "range": "proved; the condition exact at 400 equal-mass cells, the eight-atom optimum by enumeration, its conformal reading 20 trials at each sample size",
     "scripts": [
      "fates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Off equal masses the condition fails both ways",
     "anchor": "off-equal-masses-the-condition-fails-both-ways",
     "tier": "property",
     "shown": "property",
     "range": "proved by two exhibited cells, each with one optimum",
     "scripts": [
      "fates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "No test on one atom's own numbers decides its fate",
     "anchor": "no-test-on-one-atoms-own-numbers-decides-its-fate",
     "tier": "property",
     "shown": "property",
     "range": "proved by an exhibited pair",
     "scripts": [
      "fates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The rest's integer price decides the fate",
     "anchor": "the-rests-integer-price-decides-the-fate",
     "tier": "property",
     "shown": "property",
     "range": "proved; the fractional key's collisions counted on the sweep",
     "scripts": [
      "fates.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Two mass bounds cap an abandoned atom and one wholly below the operative level",
     "anchor": "two-mass-bounds-cap-an-abandoned-atom-and-one-wholly-below-the-operative-level",
     "tier": "property",
     "shown": "property",
     "range": "proved; both attained exactly on the sweep",
     "scripts": [
      "swaps.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A rescue is an exchange whose masses balance",
     "anchor": "a-rescue-is-an-exchange-whose-masses-balance",
     "tier": "property",
     "shown": "property",
     "range": "proved; on the sweep the swap fires 991 times and the full partner 123, every rule they build an optimum, and an atom rescued only by gain is printed",
     "scripts": [
      "swaps.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/shifts/",
   "title": "SHIFTS — a digit shift is a change of numeration",
   "kind": "doc",
   "parent": "/digits/",
   "source": "/shifts.md",
   "claims": [
    {
     "heading": "The change of numeration",
     "anchor": "the-change-of-numeration",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The landing criterion",
     "anchor": "the-landing-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The copy reader",
     "anchor": "the-copy-reader",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The landing walk",
     "anchor": "the-landing-walk",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The uniform lookahead",
     "anchor": "the-uniform-lookahead",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The combs",
     "anchor": "the-combs",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The block code",
     "anchor": "the-block-code",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The borrow code",
     "anchor": "the-borrow-code",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The even residues",
     "anchor": "the-even-residues",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The parity law",
     "anchor": "the-parity-law",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The drop",
     "anchor": "the-drop",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": null,
     "scripts": [
      "shifts.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/size/",
   "title": "SIZE — what a residue window cannot read, and what reading it costs",
   "kind": "doc",
   "parent": "/residues/",
   "source": "/size.md",
   "claims": [
    {
     "heading": "The locality criterion",
     "anchor": "the-locality-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The size wall",
     "anchor": "the-size-wall",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Infinity at a finite place",
     "anchor": "infinity-at-a-finite-place",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [],
     "proof": true,
     "source": null
    },
    {
     "heading": "The hiding lemma",
     "anchor": "the-hiding-lemma",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The half-coverage law",
     "anchor": "the-half-coverage-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The conjunctive split",
     "anchor": "the-conjunctive-split",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The least comparator",
     "anchor": "the-least-comparator",
     "tier": "known",
     "shown": "known",
     "range": null,
     "scripts": [],
     "proof": false,
     "source": "Babenko, Piestrak, Chervyakov, Deryabin, Electronics 10 (2021) 1041; Dimauro, Impedovo, Pirlo, IEEE Trans. Comput. 42 (1993) 608."
    },
    {
     "heading": "The least key",
     "anchor": "the-least-key",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A published comparator against the least key",
     "anchor": "a-published-comparator-against-the-least-key",
     "tier": "rule",
     "shown": "rule",
     "range": "proved from the pseudocode; no mismatch at 118,485 pairs",
     "scripts": [
      "rns_compare.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The additive labelling law",
     "anchor": "the-additive-labelling-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The F₂[x] control",
     "anchor": "the-f₂x-control",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "size.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The erring labelling",
     "anchor": "the-erring-labelling",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "erring_label.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Rounding's carry at many labels",
     "anchor": "roundings-carry-at-many-labels",
     "tier": "property",
     "shown": "property",
     "range": null,
     "scripts": [
      "erring_label.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Doubling cuts against the front",
     "anchor": "doubling-cuts-against-the-front",
     "tier": "rule",
     "shown": "rule",
     "range": "verified s = 3, N = 60, t = 1..29; s = 4, N = 40, t = 1..9",
     "scripts": [
      "erring_label.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "Growing widths",
     "anchor": "growing-widths",
     "tier": "observation",
     "shown": "observation; pattern",
     "range": "the limits computed at m = 162, 400, 406, 1000 and 1100",
     "scripts": [
      "erring_lower.py",
      "erring_widths.py",
      "erring_zones.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The rate of the least error",
     "anchor": "the-rate-of-the-least-error",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "erring_lower.py",
      "erring_zones.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The cell tent bound",
     "anchor": "the-cell-tent-bound",
     "tier": "theorem",
     "shown": "theorem; property",
     "range": "over nondecreasing widths; the error and G₂ on the half-shifted grid; E ≥ G₂(sort P",
     "scripts": [
      "cell_exact.py",
      "erring_cells.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/supply/",
   "title": "SUPPLY — the element machine over a growing supply of windows",
   "kind": "doc",
   "parent": "/machines/",
   "source": "/supply.md",
   "claims": [
    {
     "heading": "A window holds the free evaluation of its own suffix",
     "anchor": "a-window-holds-the-free-evaluation-of-its-own-suffix",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "supply.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A bounded supply is a finite automaton on its word",
     "anchor": "a-bounded-supply-is-a-finite-automaton-on-its-word",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "supply.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "An unbounded supply is universal",
     "anchor": "an-unbounded-supply-is-universal",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "supply.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "One zero-test per register needs no saved window",
     "anchor": "one-zero-test-per-register-needs-no-saved-window",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clocks.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "One zero-test over every register is an automaton on the trip word",
     "anchor": "one-zero-test-over-every-register-is-an-automaton-on-the-trip-word",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clocks.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "One trip reads whether the modulus it grows is coprime to the earlier ones",
     "anchor": "one-trip-reads-whether-the-modulus-it-grows-is-coprime-to-the-earlier-ones",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clocks.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "Base extension crosses no class line but hands over the opening word",
     "anchor": "base-extension-crosses-no-class-line-but-hands-over-the-opening-word",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "supply.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A clock trip that reads coprimality waits at least of the order of the lcm",
     "anchor": "a-clock-trip-that-reads-coprimality-waits-at-least-of-the-order-of-the-lcm",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "clocks.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "A rider on a fixed schedule lands where the supply falls behind it",
     "anchor": "a-rider-on-a-fixed-schedule-lands-where-the-supply-falls-behind-it",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "clocks.py"
     ],
     "proof": true,
     "source": null
    }
   ],
   "withheld": 0
  },
  {
   "url": "/thermal/",
   "title": "THERMAL — the growth walks at finite temperature",
   "kind": "doc",
   "parent": "/walks/",
   "source": "/thermal.md",
   "claims": [
    {
     "heading": "The zeta measure",
     "anchor": "the-zeta-measure",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The hot limit is Ẑ",
     "anchor": "the-hot-limit-is-ẑ",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The closure trichotomy",
     "anchor": "the-closure-trichotomy",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "What each fate needs of the ground",
     "anchor": "what-each-fate-needs-of-the-ground",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "floors.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The grading and the radical",
     "anchor": "the-grading-and-the-radical",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The melt over F₂[x]",
     "anchor": "the-melt-over-f₂x",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The two tests",
     "anchor": "the-two-tests",
     "tier": "criterion",
     "shown": "criterion",
     "range": "proved; checked on 20 states against every monic of degree ≤ 9, the fresh law on every Δ ⊆ {1, …, 12} at d ≤ 14",
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The mortality split",
     "anchor": "the-mortality-split",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "thermal.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The place spectrum",
     "anchor": "the-place-spectrum",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "spectrum.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The 2-adic place is the top",
     "anchor": "the-2-adic-place-is-the-top",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "spectrum.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The algebraic clock over F₂[x]",
     "anchor": "the-algebraic-clock-over-f₂x",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": null,
     "scripts": [
      "spectrum.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The class split",
     "anchor": "the-class-split",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "the headroom reads the splitting law, and the ratio of the normalizers tends to the class number h",
     "scripts": [
      "spectrum.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The F₂ place need not top",
     "anchor": "the-f₂-place-need-not-top",
     "tier": "theorem",
     "shown": "theorem; observation",
     "range": "the comparison and the wild chain",
     "scripts": [
      "cubtop.py",
      "fieldtop.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The F₂ place tops every quadratic field",
     "anchor": "the-f₂-place-tops-every-quadratic-field",
     "tier": "theorem",
     "shown": "theorem",
     "range": "a finite reduction; its one computation brackets every configuration over the whole range where the F₂ point can lie",
     "scripts": [
      "quadtop.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The race over cubic fields",
     "anchor": "the-race-over-cubic-fields",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "the bound decides every column but the column of norm 3 at 675 configurations; the F₂ place tops every cyclic cubic field with one",
     "scripts": [
      "cubrace.py",
      "cubtop.py",
      "realrace.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The F₂ place loses at about 1 in 200 cubic fields",
     "anchor": "the-f₂-place-loses-at-about-1-in-200-cubic-fields",
     "tier": "theorem",
     "shown": "theorem; rule",
     "range": "a finite reduction, the bound's sweep fixing the 675 configurations that race: ordered by |d_K|, the cubic fields of a signature where the F₂ place loses have a density, the same in both signatures, and the density where it tops is positive",
     "scripts": [
      "lossdensity.py",
      "realrace.py"
     ],
     "proof": true,
     "source": "M. Bhargava, A. Shankar and J. Tsimerman, On the Davenport–Heilbronn theorems and second order terms, Invent. Math. 193 (2013), 439–499, Lemma 18 and Theorems 7 and 8 (arXiv:1005.0672); P. Lévy, Sur les séries dont les termes sont des variables éventuelles indépendantes, Studia Math. 3 (1931), 119–153, as stated in N. Bouleau, The lent particle method, application to multiple Poisson integrals, arXiv:1004.2819, Theorem 12(a)."
    }
   ],
   "withheld": 0
  },
  {
   "url": "/torsion/",
   "title": "TORSION — which torsion a curve holds, by monomials and by degree",
   "kind": "doc",
   "parent": "/local/",
   "source": "/torsion.md",
   "claims": [
    {
     "heading": "The fiber criterion",
     "anchor": "the-fiber-criterion",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The menu law",
     "anchor": "the-menu-law",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The pentagon conic",
     "anchor": "the-pentagon-conic",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The exponent reduction",
     "anchor": "the-exponent-reduction",
     "tier": "criterion",
     "shown": "criterion",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The triangle cover number",
     "anchor": "the-triangle-cover-number",
     "tier": "rule",
     "shown": "rule",
     "range": "verified q ≤ 149; the bracket proved",
     "scripts": [
      "torsion.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The class capacity",
     "anchor": "the-class-capacity",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The squared-prime exclusion",
     "anchor": "the-squared-prime-exclusion",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The coverage cap",
     "anchor": "the-coverage-cap",
     "tier": "rule",
     "shown": "rule",
     "range": "proved: analytic for ℓ ≥ 213, exhaustive at every prime below",
     "scripts": [
      "coverage_cap.py"
     ],
     "proof": false,
     "source": null
    },
    {
     "heading": "The two-prime majority",
     "anchor": "the-two-prime-majority",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The gon theorem",
     "anchor": "the-gon-theorem",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The coset collapse",
     "anchor": "the-coset-collapse",
     "tier": "theorem",
     "shown": "theorem",
     "range": null,
     "scripts": [
      "torsion.py"
     ],
     "proof": true,
     "source": null
    },
    {
     "heading": "The realizability law",
     "anchor": "the-realizability-law",
     "tier": "rule",
     "shown": "rule",
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