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Foundation

Articles 1,921–1,980 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Grow Integer Divisibility Dvd Z Add

  A formal theorem about divisibility on signed orbits shows that if one number divides two others, it also divides their sum, mirroring a basic fact of ordinary arithmetic.

• Foundation Primitive Recognition Calculus Grow Integer Divisibility Dvd Z Refl

  A machine-checked proof that every signed orbit divides itself, the first rung of an integer divisibility ladder built from recognition events.

• Foundation Primitive Recognition Calculus Grow Integer Divisibility Dvd Z Trans

  A machine-checked proof that divisibility flows through chains of integers, a small but load-bearing step in the framework's arithmetic foundation.

• Foundation Primitive Recognition Calculus Grow Integer Divisibility One Dvd Z

  In integer arithmetic, one divides every number. Recognition Science's formal library proves the same fact for its own signed orbits, and nothing more.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Dense Mediant

  Between any two ratios on a recognition orbit, a third ratio always lies between them, and the machine-checked proof shows why the orbit never gaps.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Dense Mediant Lt Q If

  A machine-checked theorem translates the framework's ordering of ratios into plain arithmetic on whole numbers, and nothing more.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Dense Mediant Lt Q Me

  A simple theorem about fractions shows how a discrete counting process can pass through every rational ratio without ever skipping one.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Dense Mediant Mediant

  The mediant is a simple way to slide one fraction between two others, and the framework's library proves it always lands strictly between them.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Neg

  A small formal lemma about flipping ratios reveals the symmetry that keeps the framework's growth calculus consistent.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Neg Le Q Neg Neg I

  In the framework's discrete growth order, negating two ratio orbits reverses their comparison, a structural symmetry with a precise scope.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Refl Total

  A small formal module proves that the rational numbers, viewed as signed orbits, always admit a total order, a fact that anchors the framework's growth dynamics.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Refl Total Le Q

  A rational number can be ordered by comparing cross-products; leQ is the machine-checked proof that this order is reflexive and total.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Refl Total Le Q Re

  A machine-checked proof that every rational number is less than or equal to itself, and why that small fact matters for a larger framework.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Trans Antisymm

  A ratio orbit is a discrete path of ratios generated by repeated growth; the module proves the order along that path is transitive and antisymmetric, the two properties that make i

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Le Trans Antisymm Le

  A formal theorem about ordering growth ratios shows that if two ratios are mutually no larger than each other, they are the same ratio, with no exceptions.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy

  A trichotomy law guarantees that every two growth orbits in the primitive recognition calculus can be compared in exactly one of three ways.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Cross E

  A machine-checked proof that any two ratio orbits can be compared, and what that comparison does not settle.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Lt Q

  A formal definition of "less than" for growth ratios, proved to behave like ordinary number ordering.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Lt Q Ir

  A strict ordering relation never relates an object to itself; the theorem ltQ_irrefl proves this for ratio orbits, the framework's discrete growth states.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Lt Trichotomy Lt Q Tr

  Within the framework's growth model, every pair of growth ratios is strictly ordered, equal, or reversed, a trichotomy that makes the ledger's comparisons total.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Mul Pos

  A ratio orbit is a pair of counts that tracks a growing ledger; the module proves that multiplying two such orbits preserves the ledger's direction of growth.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Mul Pos Lt Q Mul Pos

  A small formal lemma about ordered ratios, and the exact boundary of what it proves.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Mul Pos Mul Strictpos

  When two positive ratios are multiplied, the product stays positive; the framework's machine-checked library proves it in a few lines.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Mul Pos Zero Lt Q Iff

  A machine-checked theorem gives a simple arithmetic test for whether one growth ratio is larger than another, and the proof rests on counting, not on any physical assumption.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono

  A small formal module shows that the order on ratio orbits respects addition, a step in building the framework's arithmetic from recognition events.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono Le Iff

  A small lemma about comparing signed orbits shows how a discrete ledger orders its entries without invoking choice, and what that bridge does not say.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono Le Q A

  A formal theorem shows that in a discrete recognition ledger, adding the same step to two ordered states preserves their order, a property that anchors the framework's growth

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono To Int

  A small lemma converts a signed orbit into an ordinary integer difference, and it does so without invoking any choice principle.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Mul Nonneg

  A small formal lemma about ordered ratios shows that multiplying by a nonnegative ratio preserves order, a step toward building the recognition framework's arithmetic.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Mul Nonneg Le Q

  A formal theorem about ordered ratios shows when multiplying by a nonnegative value preserves comparison, a small but load-bearing step in the framework's growth calculus.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Zero Lt One

  A tiny formal proof that the ratio orbit's zero sits below its one, and why that ordering underpins the framework's growth dynamics.

• Foundation Primitive Recognition Calculus Grow Ratio Orbit Zero Lt One Zero Lt Q

  A machine-checked theorem pins down the first step of a discrete growth sequence, and the proof method shows what it does not say about that sequence.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Left Of Bal

  In the framework's discrete ledger, two histories that have consumed the same number of steps are interchangeable on the left of every ordering comparison.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Left Of Balanced Choice Free

  A theorem about ordered orbits in the framework's recognition calculus, proved by reducing to natural numbers.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Of Balanced

  A theorem about ordered lists in a formal recognition calculus: if two entries are balanced against each other, then comparing them is stable.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Of Balanced Choice Free

  When two paths through a recognition ledger are balanced, they order their successors identically, a fact that lets the framework compare choices without picking favorites.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Right Of Ba

  In the framework's primitive calculus, a balanced pair of signed orbits is indistinguishable from the right for the order relation.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Right Of Balanced Choice Free

  A formal theorem about ordered structures shows when two objects can be swapped without changing any comparison.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Mul Right Iff Of

  In the framework's discrete arithmetic of recognition events, multiplying both sides of an ordering inequality by a positive, unbalanced element preserves the comparison, and

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Mul Right Iff Of Nonneg Flag Of Not Balanced Zero Choice Free

  A signed orbit is a list of +1 and -1 steps; the lemma says multiplying by a nonnegative, unbalanced orbit preserves order.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Of Product Right

  When two growth records agree in their balance, multiplying either one by the same factor preserves their ordering, a machine-checked fact about how recognition costs accumulate.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Of Product Right Factor Iff Of Balanced Choice Free

  A small formal lemma about signed orbits shows that multiplying on the right preserves order exactly when the two factors are balanced, a stepping stone in the framework's gro

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Product Right Fac

  A theorem about signed orbits shows when two entries in a recognition ledger can be swapped without changing what comes next, and it stays silent on every other kind of comparison.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Le Product Right Factor Iff Of Balanced Choice Free

  A signed orbit pairs a positive and negative count; this lemma shows that swapping one factor for a balanced partner never changes which products sit below a given bound.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Mul Balanced Zero Of

  A small formal lemma about signed orbits: if one factor is balanced, then multiplying by it preserves balance.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Mul Balanced Zero Of Balanced Zero Right Choice Free

  A tiny formal proof shows that multiplying two balanced signed orbits keeps the ledger balanced, a closure property that underpins the framework's arithmetic.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Nonneg Flag Mul Of O

  A machine-checked theorem shows that multiplying a signed orbit by any nonzero distinction leaves its sign unchanged, a structural fact with a narrow scope.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Nonneg Flag Mul Of Orbit Right Of Ne Zero Choice Free

  In the framework's primitive recognition calculus, a sign flag on an orbit remains unchanged when the orbit is multiplied by any nonzero distinction.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Order Choice Free

  A signed orbit is a pair of natural-number counts; the module shows how to compare them without invoking the axiom of choice.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Order Choice Free Le

  A signed-orbit order is a way to compare two discrete records; this declaration shows the comparison can be made without invoking a choice principle.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Order Choice Free No

  A machine-checked proof shows that a signed number's sign can be read directly from its parts, with no hidden logical assumptions.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Zero Le Iff Nonneg F

  A signed orbit is nonnegative exactly when its flag is set, a small theorem that anchors how the framework recognizes order.

• Foundation Primitive Recognition Calculus Grow Signed Orbit Zero Le Iff Nonneg Flag Choice Free

  A signed orbit is a pair of counts, one for each direction, and the framework's library proves a simple test for when such an orbit is nonnegative.

• Foundation Primitive Recognition Calculus Hard Problem Certificate Audits

  A machine-checked library shows how four famous open problems can be reduced to finite bookkeeping, without claiming any of them is solved.

• Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Domain

  A machine-checked library sets up finite certificate inventories for several famous open problems, without claiming to solve any of them.

• Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Hard P

  The framework's machine-checked library declares that four famous open problems admit finite certificate audits, without claiming any of the problems is solved.

• Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Navier

  A machine-checked proof shows the Navier-Stokes energy problem can be reduced to a finite bookkeeping task, not that the equations are solved.

• Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Prime

  A machine-checked theorem shows that certifying the Riemann hypothesis reduces to checking a finite list of certificates, without proving the hypothesis itself.

• Foundation Primitive Recognition Calculus Hard Problem Certificate Audits Yang M

  A machine-checked library proves that its Yang-Mills certificate audit never mistakes a legitimate display for a pathological one, without claiming the mass gap itself.

• Foundation Primitive Recognition Calculus Hilbert Display Completion

  The module shows that a finite quantum-like state space is exactly a display of the framework's native amplitudes, with no information lost.