Menu: Every claim and its tier

Every claim and its tier

Every claim on the site, section by section and page by page in reading order, each named by its heading and marked with its tier. A claim's page states it in full, with the program that checks it or the proof it rests on. How to read a claim says what each tier promises.

433 claims, by first tier: 67 property, 9 observation, 4 pattern, 50 rule, 42 criterion, 247 theorem, 14 known.

The object: every finite place at once

TOWER — the primorial rungs

The rung property
The first fourteen rungs property
The transparency criterion criterion
The fate rule rule
The birthday formula theorem; observation
The residue code's distance theorem
The Hamming spectrum rule
The unitary spectrum theorem; observation
The escape law theorem; observation
The density theorem theorem; observation
The serial ledger rule
The limit property

FINGERPRINT — one real number reads a residue tuple up to sign

The fingerprint property
The fingerprint theorem theorem; observation

BUDGET — what omitting a place frees at a global constraint

NAMING — which primes divide a set's Euler characteristic

The complex theorem
The naming integer property
The weight law criterion
Every odd prime names a set theorem; rule
The baseline theorem
The slices theorem
No free automorphism of outside prime order theorem

Residue readers: the finite places

SIZE — what a residue window cannot read, and what reading it costs

The locality criterion criterion
The size wall theorem
Infinity at a finite place property
The hiding lemma theorem
The half-coverage law theorem
The conjunctive split theorem
The least comparator known
The least key theorem
A published comparator against the least key rule
The additive labelling law theorem
The F₂[x] control theorem
The erring labelling theorem
Rounding's carry at many labels property
Doubling cuts against the front rule
Growing widths observation; pattern
The rate of the least error theorem; observation
The cell tent bound theorem; property

COORDINATES — which coordinate system reads which operation

The two-readings criterion criterion
The carry wall theorem
Addition and multiplication each decide known
The suffix and floor reads criterion
The Internet checksum theorem
The valuation row theorem
The index coordinates property
The Zech wall theorem
The support log theorem
The exponent wall theorem
The rigidity dichotomy theorem
The bearers theorem

LOGIC — the logic a squarefree ring's own polynomials carry

GATES — what a reader of words decides, and what it pays

BINDING — what multiplication keeps of a code, and what a sum costs

Digit readers: the real place

READING — what a digit-by-digit reader computes

The reading lemma theorem
Non-redundant absolute tiles wall only at the archimedean place theorem
The wall criterion criterion
The Lipschitz criterion criterion
The division criterion criterion
The division trichotomy theorem
Division over F_q[x] criterion
The comb theorem
Floor division on the d-bonacci completion theorem
The scaling certificate rule

RATES — what a reader pays where a digit's scale varies

Walls sit at the preimages of the rationals criterion
The commitment bound theorem
Rate forcing theorem
The unit-index law pattern
The unit criterion criterion
The ladder law theorem

OSTROWSKI — the circle a trailing numeration reads

The circle numeration theorem
The two codings of −α theorem
The containment criterion criterion
The affine maps theorem
Floor division is never MIDDLE here theorem

FLUSH — what a redundant reader of the circle computes

The tear at E = 0 theorem
The overlap reader theorem
Floor division walls at every slack theorem
The lift game theorem
The completion reader is the safety game theorem
The flush floor theorem
The universal reader rule

SHIFTS — a digit shift is a change of numeration

DUAL — the leading end of base b, read against the trailing end

The leading-tile lemma property
The two-ends law criterion
The crown exchange property
The coset rigidity property
The affine class theorem theorem
The least-lookahead law theorem; pattern
The two-ends exchange criterion
The structural split property
The redundant reader theorem
The hedge property
What the purchase spends property
The redundancy leak criterion
The period law theorem
The dual Haar measure property
Each end prices in its own metric property

REDUNDANT — what a signed-digit reader of integers computes

The affine criterion criterion
The two regimes of slope criterion
On one stream and a symmetric set the phase lowers, never raises theorem
Dividing by the radix criterion
The length bound theorem
Addition criterion
The many-stream theorem theorem
The pole units theorem
The selection estimate property; rule
The relative-gradient law property; rule
The piecewise theorem theorem
The segment lemma criterion
The tile-valued maps theorem
The absolute value criterion
The free comparator theorem
The sign count theorem
The full width property
Comparison theorem
The chained delay theorem
The threshold-miss law rule; pattern
The matching law rule

Growth: places priced and compared

GROWTH — a modulus grown by the cheapest move a demand admits

The three fates theorem
The wall is Adams' number property
The wall over F₂[x] theorem
The lock-prime law theorem; rule
The ghosts are the witnesses' records theorem; known
The two-adic prices theorem
The module law theorem; observation
The lock over a number ring theorem; observation
The sprawl over F₂[x] theorem; observation
The sibling shadow theorem
The standing-move dichotomy theorem; observation
The properness criterion criterion
The route is the deleted place's theorem; observation

LIMIT — where one greedy walk goes

The ring is a supply rule
No supplied degree is covered before it opens theorem; rule
The chain theorem
When the clock runs forever theorem
The support theorem
The ideal limit rule
The ceiling theorem
Attainment is the void menu's rule
The strand rule
What stops the widening theorem; rule

CLOCK — what a place charges a growth walk

The ladder is ψ's orbit, save a head at the bend theorem
The head criterion criterion
The width theorem
A ramified quadratic place over 2 reads its width off δ mod 8 known
The two numbers theorem
The walk stops at its high-water price theorem
Under a per-item clock the sup sets the stop theorem
Under the global clock the sup is a floor theorem
A block clock keeps at most one runaway per block theorem
The admission census is the high-water lemma, item by item theorem
Two blocks of a number ring never both run away theorem
A ring's door is one valuation and its clock a count theorem
The staircase column theorem
A norm does not fix a column property
A move's price is an index property
A ring's block keeps one runaway, in counts theorem
A carrier bears on a place down to a ladder member theorem
Which place a ring's block runs theorem
The void tie crosses kinds and the tie-break chooses the place that runs rule
A void walk pays its winner's lone path theorem; criterion
A lidded block locks the walk for good theorem; observation

ELEMENT — the walk that must stay principal

The ring is a supply matrix rule; theorem
The element reading sees only classes criterion
The class gap between principal and all ideals theorem; rule
The coarsest colouring theorem
The minimal rider is unique theorem
The bare door theorem
Greed pays no avoidable rider theorem
The margin step theorem
The eventual chain rule
Where the chain changes hands rule
The orbit law and the rider recursion theorem
The element limit rule
The number rings price in (min, ×) rule
The lock carries its rider rule
The element lid theorem

CASCADE — whether the greedy walk over a number ring must lock

The escape is a conjunction theorem; observation
The budget inequality theorem; rule
The characteristic sweep rule
Dead levels recur at eleven characteristics rule
The residual is inhabited at every bound theorem
The shape of a dead level theorem; rule
The full-price opening at odd p is a dead level over Z criterion; rule

THERMAL — the growth walks at finite temperature

The zeta measure theorem
The hot limit is Ẑ theorem
The closure trichotomy theorem
What each fate needs of the ground theorem
The grading and the radical theorem
The melt over F₂[x] theorem
The two tests criterion
The mortality split theorem
The place spectrum theorem; observation
The 2-adic place is the top theorem
The algebraic clock over F₂[x] theorem; observation
The class split theorem; observation
The F₂ place need not top theorem; observation
The F₂ place tops every quadratic field theorem
The race over cubic fields theorem; rule
The F₂ place loses at about 1 in 200 cubic fields theorem; rule

MEMORY — what a grown world can know of its genesis

The route-weight cancellation theorem
The bounded-evidence law theorem
The unbounded-memory law theorem; observation
A deepening world reads its lock prime theorem; property
The witness gap theorem; observation
The cold blindness theorem
The fossil paradox theorem; observation

BRIDGES — where growth meets open number theory

The seating law theorem; observation
The phoenix theorem; observation
The atom criterion criterion
The least unrealizable odd number rule
Transparent rungs are transparent moves property

HEADROOM — what a probe sees of a grown state

The headroom property
The ledger theorem
The premium in closed form theorem
Silence lands in a blind class theorem; rule
The closure theorem
The resolution criterion criterion
The non-divisor bound theorem; observation
No blindness is permanent theorem; observation
The watcher's guaranteed blindness theorem
The blind-density theorem theorem
The least resolver is unbounded theorem

IMAGE — the set of limits a greedy walk reaches

The colour count theorem
A quadratic ring's ideal image is 2^t rule
The lock's rider erases a choice rule
A curve's image is the continuum rule
The gap rule
The free image has five forms theorem
The free image forgets the ring theorem
The fates are the image's extremes property
Greed's point rule
The c-greedy walk is dammed theorem
Polynomial slack reaches breadth from the void theorem; rule

One place up close

OBSERVATORY — what the digits of a ramified field read

The coefficient triangle theorem
The anchor design of the 2e-th roots of unity theorem; observation
The tame readout theorem
The cofactor readout theorem
The wild readout theorem
The letters of K(ζ₈)/K at e = 4 rule
Pagano's jump set known; observation
The jump set as the class maxima theorem
The jump set reads M·e digits of u theorem
The last point gives the break of the next storey theorem
A radical storey adds e to the first defect and passes the rest down theorem

PRINCIPAL — where a quadratic field's first principal prime sits

The floor theorem
No floor over a real field theorem; observation
Where the omitted prime powers land property
The shortfall is the prime powers pattern
The bias against principal primes known

TRIPLE — what a cubic field's split primes carry

The triple is a homomorphism property
The stable subgroups theorem
The genus index law rule
The conductor's primes are forced rule
The sign of the second line is the regime theorem
Gerth's rank law at conductor 1 known
The partial place is uniform theorem
The classes of the prime powers theorem
The shortfall is a fixed weight at the small primes pattern
The partial fiber has no square term property
The least partial place leans non-square observation

TORSION — which torsion a curve holds, by monomials and by degree

Machines and learning

MACHINE — a state that only multiplies, read as a machine

The depth machine decides its halting theorem
Clearing a place decides and dividing once does not theorem
One form decides and two can run Minsky theorem
Two places at the ratios 1 and α/β run FRACTRAN theorem
The headroom is a bank of Minsky counters theorem
A blind read decides and a sighted read is universal theorem

SUPPLY — the element machine over a growing supply of windows

FORGET — when a state provably no longer holds a datum

The four grades property; observation
Two witnesses buy the robust grade theorem
The quarantine theorem theorem
Recycled worlds need no witness property
Readability descends, spread ascends, flatness neither theorem
The working clause property
Two classes never cure by refinement theorem
The no-majority criterion criterion
Retaining less can leave no cure theorem
A strict cure needs three classes theorem

COLLIDE — when two products of weighted sets are one polynomial

The frame property
Non-uniqueness needs a seed property
A witness-free tie is a regrouping or passes a seed property
Term counts multiply property
Size 2 closes in the elements criterion
Size 3 closes in the exponent vectors criterion
Ljunggren's theorem on trinomials known
Menu size is the grading property
Factorization by number of terms known
The escape at (2,5) property
The descent dimension property
Two factorizations no proper face separates property
The sign scan criterion
The face law criterion
The cycle lemma theorem
The mixing theorem theorem
The family mixes at every e property
The binomial-pair criterion criterion
Yost's indecomposability bound known
The half-size law theorem
The torsion split property
The least free degree rule
Mills's theorem on quadrinomials known
The cofactor theorem theorem
The size-4 closure theorem
The row lemma theorem
The least free size off a line theorem
Mann's bound known
The Filaseta–Solan question known
The size-5 reduction theorem

LEARNER — a reader that learns by exact comparisons of its price

The straddle cover and its readers property; observation
The bottom lemma theorem
Greedy patience dominates every sound reader theorem
The partition valley theorem
The funnel with one blind corner rule
The layers of a holding set theorem
Under an income the rank is common theorem
On the eights row, patience with no bank lags for the whole run rule
Destination universality rule
A density forecast is a length loss theorem
The side loss makes the best reader depend on the stream rule
The stream enters at a reversal theorem
The two poles theorem
Breaking needs no stream and keeping needs no pole rule
The collision grower property
The rung reading theorem
How fine, and what theorem
The encoding bound theorem
The substrate parts by depth or inside a layer theorem
The grower's worst case theorem
Exact readout reads the subdominant criterion
The grown depth is the readable depth theorem

Exact truth

RULER — evaluations that know their optimum

The parity law of the sign bit theorem; observation
One threshold range, one tent: the threshold family theorem; observation
The gauge is the task's: orientation and comparison theorem
The threshold-range law under a tilted prior theorem
The resonance criterion criterion
One live arc: the magnitude parent theorem; observation
The knife: a consistent point inside an inconsistent interval observation
More data, a worse held-out interval observation
One bound at two statistics theorem
The spanning tree is not identified property
The sign law of a stated interval property
Coverage is a function of two ratios observation
The kernel of the inside instruments property
The nested repair reads the term, not the model observation
The audit world and its two roots property
The flip is the sign of one scalar theorem; observation
The audit's band counts one sample of three observation
With σ known, no self-report read off D̂ is unbiased at every distance theorem
The per-example audit states its Gaussian limit property

FLATTEN — the least height flattened to a given depth

Flattening is annihilation property
The least height is a lattice minimum property
At rank 1 the least height is the central binomial property
In the corner the least height is a pure product's rule
On the chart the products fail more often the deeper rule
Through its table the inverse dial is a pure-product degree rule
Read by rank, the failures sit in a band of depths rule
Past the last failure seen the champion attains the least height rule
A failing cell is a break on the dial property
At heights 8, 9 and 11 the dial breaks rule
A traded pure product attains every failing cell of the chart rule
The nine non-cyclotomic factors are reciprocal and need not sit on the circle observation; property
The height-1 column is classical known
Two free bounds property
At rank 2 the least height is the row's largest step theorem
At rank 2 the two minima part by a factor growing as √J rule

SETS — set-valued prediction against its exact optimum

A marginal coverage guarantee cannot rank two set predictors observation
The certificate brackets the optimum property
At equal masses the certificate is the optimum property
Off equal masses the certificate can overstate property
Three tests can prove the certificate exact property
On the geometric walk's tail the anchored gap is one product rule
An exact solver for the marginal optimum property
At equal masses the operative level forbids the wrong forced fates property
Off equal masses the condition fails both ways property
No test on one atom's own numbers decides its fate property
The rest's integer price decides the fate property
Two mass bounds cap an abandoned atom and one wholly below the operative level property
A rescue is an exchange whose masses balance property