Encyclopedia/All topics/Foundation

Foundation

Articles 1,381–1,440 of 2,979. Alphabetical by title.

• Foundation Pair Kernel Newtonian3 Dirichlet Integral Eq Pi Div Two Sub Error

  A famous improper integral from classical analysis, the Dirichlet integral, evaluates to π/2, and a machine-checked proof pins down exactly how fast it gets there.

• Foundation Pair Kernel Newtonian3 Integrable Inv Norm Profile Comp Norm

  A technical lemma about a sharply cut-off inverse-square profile being integrable, enabling a rigorous Fourier analysis of the Newtonian potential in three dimensions.

• Foundation Pair Kernel Newtonian3 Integrable Inv Sq Profile Comp Norm

  This declaration is a technical lemma, not a standalone claim: it shows a certain function can be integrated, a necessary step toward a larger result about the geometry of three-di

• Foundation Pair Kernel Newtonian3 Integrable On Newtonian Fourier Integrand

  A technical lemma about a Newtonian Fourier integral that turns out to be the load-bearing step in a larger proof.

• Foundation Pair Kernel Newtonian3 Newtonian Fourier Target Of Aligned Axis

  A machine-checked proof shows that Newton's inverse-square law emerges from a three-dimensional Fourier integral, but only after a geometric reduction to a single axis.

• Foundation Pair Kernel Onsite Exclusion Exact Jcost As General Ledger Cost Eval

  A machine-checked theorem shows that the framework's core cost function, when written in ledger form, has no absolute per-site term, only terms comparing sites.

• Foundation Pair Kernel Onsite Exclusion Exact Jcost As General Ledger Cost Onsit

  A machine-checked theorem shows that the framework's fundamental cost function has no absolute per-site term, while a separate hypothesis is needed to rule out non-local inter

• Foundation Pair Kernel Onsite Exclusion Mean Field Ledger Cost Shift Invariant

  A symmetry that forbids one kind of term in a cost function turns out to allow another, and the difference matters for what the framework can claim.

• Foundation Pair Kernel Onsite Exclusion Shift Invariant Iff Onsite Sum

  A machine-checked proof shows that a cost rule's invariance under adding a constant to all values reduces exactly to a condition on its per-site term alone.

• Foundation Pair Kernel Operational Locality S4

  A machine-checked proof that in Recognition Science, the only physically realizable dependencies between ledger states are those that change exactly one bit, and that this locality

• Foundation Pair Kernel Operational Locality S4 Committed Candidate Routes Admit

  A machine-checked theorem shows that several tempting shortcuts to spatial locality all fail, because they still allow a distant event to depend on another.

• Foundation Pair Kernel Operational Locality S4 Conserved Elementary Posting Can

  A single, minimal accounting move can connect distant entries in a ledger, and the framework proves this is not an accident of its definitions.

• Foundation Pair Kernel Operational Locality S4 Global Balanced Ledger Conserves

  A ledger that records no debits and no credits is trivially balanced, and the framework's machine-checked library proves that this trivial balance is conserved through every t

• Foundation Pair Kernel Operational Locality S4 J Minimal Posting Step One Bit Di

  A minimal-cost change in a recognition ledger is always a single coordinate flip, and this fact is what makes the framework's notion of locality operational.

• Foundation Pair Kernel Operational Locality S4 Tiled Jminimal Dependency3 Nonemp

  A machine-checked proof shows that in a discrete ledger, the cheapest possible change is always a single, local move, and that this holds even when the ledger is built from repeati

• Foundation Pair Kernel Operational Locality S4 Tiled Jminimal Dependency3 Operat

  A machine-checked proof shows that the cheapest possible change in a ledger is also the most local one, moving a single bit to a neighboring cell.

• Foundation Pair Kernel Operational Locality S4 Tiled Jminimal Generator Finite R

  In a discrete ledger, the cheapest possible change turns out to be a single, local step, and this theorem proves that any process built from such steps can only affect nearby entri

• Foundation Pair Kernel Operational Locality S4 Tiled Jminimal Separated Dependen

  In a discrete ledger of events, two postings far apart never influence the same later cell: a machine-checked theorem about locality.

• Foundation Pair Kernel Owner Channel Occurrence Bridge

  How a single ledger entry becomes a unique, oriented pair of poles, with the machinery that proves no two entries can collide.

• Foundation Pair Kernel Owner Channel Occurrence Bridge Bridged Occurrence Source

  In the ledger model, a debit and a credit are the same event read from opposite sides, and the framework's source covector proves that symmetry exactly.

• Foundation Pair Kernel Owner Channel Occurrence Bridge Candidate A Fails Occurre

  A machine-checked proof eliminates one proposed way to set the scale of physical sources, leaving a specific open question about which law governs them.

• Foundation Pair Kernel Owner Channel Occurrence Bridge Candidate B Satisfies Occ

  A formal theorem selects one of two candidate scale factors for a physical source term, while leaving the deeper physical law that would justify it open.

• Foundation Pair Kernel Owner Channel Occurrence Bridge Constant Occurrence Contr

  A single account cannot be both a debit and a credit at once, and the framework proves this as a theorem about its own ledger.

• Foundation Pair Kernel Owner Channel Occurrence Bridge Heat Flux Scalar Join Not

  In the framework's ledger, two different events can carry the same numerical boundary value, so a scalar alone never identifies which event happened.

• Foundation Pair Kernel Owner Channel Occurrence Bridge Occurrence Bridged Source

  A machine-checked equivalence ties a proposed normalization rule to a known physical equality, but the rule itself remains unproved.

• Foundation Pair Kernel Pair Cost

  A machine-checked library proves that a pinned two-body interaction energy is non-vacuous and sees locality, a concrete step toward deriving physics from a ledger of recognition ev

• Foundation Pair Kernel Pair Cost Band Dirichlet Eq Adjacent

  A machine-checked theorem gives a simple formula for the interaction energy of two pinned points on a line, and it is careful about what it does not say.

• Foundation Pair Kernel Pair Cost Band Kernel Inverse Distance Decay

  A machine-checked theorem shows that a minimal interaction energy between two pinned points on a line decays no slower than one over their separation, a first discrete step toward

• Foundation Pair Kernel Pair Cost Pair Kernel Discriminates

  A machine-checked theorem shows that the framework's two-body interaction energy can tell a local coupling from a non-local one, a necessary first step toward deriving forces.

• Foundation Pair Kernel Pair Cost Pair Min Band Le Inv Dist

  A machine-checked theorem shows that on a simple chain graph, the minimum interaction energy between two pinned defects falls off at least as fast as one over their separation.

• Foundation Pair Kernel Pair Cost Wpair Band Dist3 Le Third

  A machine-checked theorem places an upper limit on the interaction energy of two pinned defects in a discrete ledger, a small step toward showing how locality shapes cost.

• Foundation Pair Kernel Pair Cost Wpair Mean Field Far Ge One

  A machine-checked proof shows that two pinned points in a fully connected graph always carry at least one unit of interaction energy, a fact that separates a local from a non-local

• Foundation Pair Kernel Periodic3

  A finite, wrap-around grid of points replaces the infinite lattice, and the framework proves the exact Fourier machinery that makes waves on it behave.

• Foundation Pair Kernel Periodic3 Symbol Torus Quantized Wave Ne Zero Of Ne Zero

  On a finite three-dimensional torus, every non-constant pattern of vibration has a non-zero frequency signature, a fact that makes the discrete world's wave mechanics well-def

• Foundation Pair Kernel Periodic3 Torus Fourier Mode Pointwise Orthogonality Fact

  On a finite periodic grid, distinct wave patterns cancel out when added across all points, a fact that lets signals be decomposed into independent components.

• Foundation Pair Kernel Periodic3 Torus Laplacian Torus Fourier Mode Eq Symbol Mu

  On a finite periodic grid, the discrete Laplacian acts on each Fourier mode as a simple multiplication, a fact that turns hard difference equations into algebra.

• Foundation Pair Kernel Periodic3 Torus Laplacian Torus Spectral Green Mode Eq Mo

  On a finite three-dimensional torus, a discrete Laplacian acts on each Fourier mode as a simple multiplication, and the framework's spectral Green's function inverts it e

• Foundation Pair Kernel Periodic3 Torus Laplacian Torus Spectral Response Eq Sour

  On a finite three-dimensional torus, the framework's Laplacian operator can be inverted exactly: applying it to a specially built response field recovers the original source f

• Foundation Pair Kernel Periodic3 Torus Laplacian Torus Spectral Response Normali

  On a finite periodic grid, a certain averaging operator can be undone exactly, which is the discrete analogue of solving a differential equation.

• Foundation Pair Kernel Periodic3 Torus Source Transform Zero Eq Zero Of Neutral

  On a finite periodic three-dimensional grid, a source whose values sum to zero has no zero-frequency component, a fact that underpins the framework's discrete Fourier analysis

• Foundation Pair Kernel Physical Posting Attachment S10

  A machine-checked library shows that if a minimal ledger entry carries the framework's native action quantum, its source magnitude is forced to be the fifth power of the golde

• Foundation Pair Kernel Physical Posting Attachment S10 Ledger Unit Normalization

  A machine-checked proof shows that setting the ledger's unit scale does not, by itself, determine the physical size of an event, a boundary with a concrete counterexample.

• Foundation Pair Kernel Physical Posting Attachment S10 N2 Relation Set Collapses

  On a two-point torus, a single posting pair can count as two directions at once, a collision the framework establishes and then must interpret.

• Foundation Pair Kernel Physical Posting Attachment S10 Native Posting Action Can

  A machine-checked theorem shows that if a posting carries the framework's native action quantum, its source magnitude is forced to be the golden ratio to the fifth power.

• Foundation Pair Kernel Physical Posting Attachment S10 Physical Posting Attachme

  A machine-checked proof shows that if physical magnitudes attach to ledger events in one specific way, the magnitude is forced to be the fifth power of the golden ratio, with nothi

• Foundation Pair Kernel Physical Posting Attachment S10 Physical Posting Magnitud

  A theorem in the Recognition Science library establishes that the physical magnitude assigned to a fundamental posting event does not depend on which event you label first.

• Foundation Pair Kernel Physical Posting Attachment S10 Primitive Posting Pair Ex

  A small formal theorem draws the line between what the framework's mathematics forces and what remains a choice.

• Foundation Pair Kernel Physical Posting Attachment S10 Reciprocal Assignments Sa

  A machine-checked theorem shows that two reciprocal numbers, q and 1/q, satisfy a duality condition in the framework's ledger, but it does not identify which physical quantity

• Foundation Pair Kernel Physical Posting Semantics S11

  A machine-checked result pins down the smallest field-level step that attaches a native action quantum to a realized posting, and proves it forces the golden-ratio scale.

• Foundation Pair Kernel Physical Posting Semantics S11 Field Native Action And Ex

  A machine-checked theorem shows that when a field carries the native action quantum and its source is the reciprocal dual, the source magnitude must be phi to the fifth power.

• Foundation Pair Kernel Physical Posting Semantics S11 Posting Step Ledger Jlog C

  A single machine-checked theorem fixes the cost of a ledger posting at a value that is not the framework's native action quantum, and the proof is a direct numerical compariso

• Foundation Pair Kernel Physical Posting Semantics S11 Realized Posting Field Car

  In the framework's ledger, a realized posting's spatial field is forced to a single, fixed difference: the canonical drop, whose value is not an input but a theorem.

• Foundation Pair Kernel Physical Posting Semantics S11 Reciprocal Torsor Satisfie

  A machine-checked theorem shows that when a posting's action and source magnitude are mutual reciprocals, the source is exactly the dual of the action, and nothing else is for

• Foundation Pair Kernel Physical Posting Semantics S11 Source Magnitude Does Not

  A machine-checked proof shows that the size of a physical source does not by itself determine its electric charge, drawing a precise line in a theory of discrete events.

• Foundation Pair Kernel Physical Posting Semantics S11 Zero Field Same Support Do

  A zero field cannot carry the native action quantum, even when it occupies the same spatial support as a realized posting.

• Foundation Pair Kernel Physical Readout Selection S17

  A module that defines the smallest exact interfaces for turning recognition events into physical measurements, and proves which ones are forced.

• Foundation Pair Kernel Physical Readout Selection S17 Canonical Posting Recognit

  A theorem in the Recognition Science framework shows that a specific, minimal way of pricing physical events is consistent, but it does not prove that nature uses it.

• Foundation Pair Kernel Physical Readout Selection S17 Complete Carrier Channel P

  A machine-checked library shows that when a physical channel fully classifies its events, its price must be the golden ratio raised to the negative fifth power.

• Foundation Pair Kernel Physical Readout Selection S17 Misclassified Five Carrier

  A machine-checked proof shows that a deliberately miscounted physical channel still has dimension five, and the proof's own companion theorems show why that dimension does not

• Foundation Pair Kernel Physical Readout Selection S17 Physical Duration Readout

  A theorem about what a physical clock must be reading, if it is reading anything at all.