The equation library
Every recovered mathematical record has an individual page. Browse original equations, code expressions, indexed components, parameter definitions, and explicitly unresolved source slots.
Counts refer to mathematical records, including implemented expressions and generated instances. They are not counts of independently validated theories. Exact occurrences and unresolved slots remain traceable. Read the complete 1–516 source index ↗ · Inspect the scan decisions ↗
Coherence-weighted state magnitude term
Multiplies the coherence driver dx by the Hermitian squared-magnitude expression ψ†ψ. For a complex vector, ψ†ψ is real and nonnegative, while the sign of the whole contribution follows dx.
Complex loop-phase representation
Places the real loop phase on the unit circle using a complex exponential. The magnitude is one, and the complex angle tracks the phase modulo 2π. This is a representation of a phase, not an independent measure of consciousness.
Compressed binary-state placeholder
Applies a function named compress to a binary representation. The source must specify an encoder, decoder and exactness or distortion criterion; naming the function does not establish an achievable compression ratio.
Compression size ratio
Divides input size by output size. A value above one indicates a smaller output. The ratio describes size only; recoverability and distortion require separate checks.
Confidence-controlled plasticity
Adds a confidence-dependent gain to the base learning rate, then clips it between η_min and η_max. Ordered finite bounds and a well-defined confidence value are required.
Cost-based neuron coupling
Forms a coupling weight from a weighted combination of the two neuron costs; some historical versions also add a coherence-driver contribution. These variants are retained separately because the extra term changes the coupling rule.
Cost-based neuron coupling
Forms a coupling weight from a weighted combination of the two neuron costs; some historical versions also add a coherence-driver contribution. These variants are retained separately because the extra term changes the coupling rule.
Cost-based neuron coupling
Forms a coupling weight from a weighted combination of the two neuron costs; some historical versions also add a coherence-driver contribution. These variants are retained separately because the extra term changes the coupling rule.
Cost-based neuron coupling
Forms a coupling weight from a weighted combination of the two neuron costs; some historical versions also add a coherence-driver contribution. These variants are retained separately because the extra term changes the coupling rule.
Cross-layer or cross-device message
Constructs a message from differences between an optimized local state/design and the receiving state/design, together with a change signal. The coupling function or weight determines how those differences influence the receiver.
Cross-layer or cross-device message
Constructs a message from differences between an optimized local state/design and the receiving state/design, together with a change signal. The coupling function or weight determines how those differences influence the receiver.
Cross-layer or cross-device message
Constructs a message from differences between an optimized local state/design and the receiving state/design, together with a change signal. The coupling function or weight determines how those differences influence the receiver.
Cross-layer or cross-device message
Constructs a message from differences between an optimized local state/design and the receiving state/design, together with a change signal. The coupling function or weight determines how those differences influence the receiver.
Cross-layer or cross-device message
Constructs a message from differences between an optimized local state/design and the receiving state/design, together with a change signal. The coupling function or weight determines how those differences influence the receiver.
Data-link cost
Combines frame-loss fraction and bandwidth utilization with weights a₂ and b₂. Whether high utilization should be penalized depends on the intended operational objective.
E. Error Propagation Operator · \epsilon
This relation uses a gradient: the vector of local changes in a scalar objective with respect to its variables. It supplies a local direction or stationarity condition; an implementation still needs the differentiation variables, domain and update convention given in the surrounding model.
Environment feedback transition
Updates an aspect-level controller or environment state using the previous state and messages from layers. The transition function is named but not fully supplied in the shorthand.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Environment-aspect loss aggregation
Adds the layer’s per-aspect losses with λ weights. Every aspect can contribute to the same layer objective, avoiding an implicit assumption that each layer observes only one isolated part of the environment.
Fairness accounting variable
Defines the service count or target-share term used in the deficit calculation. Consistent accounting is necessary for the largest-deficit rule to represent the intended allocation.
Fairness service deficit
Computes target share times total service events minus actual services for family f. A positive deficit means the family has received less service than its target share.
Forward and reverse operator shorthand
Uses a derivative-like operator for optimization and an inverse-derivative notation for reverse engineering. These symbols do not uniquely define an inverse mapping: differentiation generally discards constants and reverse problems need constraints and boundary information.
Frame-loss fraction
Divides the number of lost frames by the number sent. The metric requires aligned counts over the same period and a nonzero sent-frame count.
Grounded verification confidence κ real
κ real is the source’s verification-confidence factor in the effective substrate-allocation rule. Its intended purpose is to suppress updates from unverified content. The claimed protection depends on how verification confidence is computed and whether zero confidence actually gates every update…
Hardware feasibility constraints
Lists limits on device resources such as power, temperature, radio use or memory. These conditions define an allowed operating region; they are constraints on optimization rather than terms proving the system has reached that region.
Hardware feasibility constraints
Lists limits on device resources such as power, temperature, radio use or memory. These conditions define an allowed operating region; they are constraints on optimization rather than terms proving the system has reached that region.
Hardware feasibility constraints
Lists limits on device resources such as power, temperature, radio use or memory. These conditions define an allowed operating region; they are constraints on optimization rather than terms proving the system has reached that region.
Ignore weaker shared evidence
States the intended policy that a pool whose confidence does not exceed the local confidence contributes zero. It is a boundary case of the proposed bounded pool-sharing rule.
Illustrative activation regime
The notes describe an expected activation level for this qualitative pressure/confidence regime. The displayed approximation is illustrative because numeric inputs for “high” and “low” are not supplied.
Illustrative activation regime
The notes describe an expected activation level for this qualitative pressure/confidence regime. The displayed approximation is illustrative because numeric inputs for “high” and “low” are not supplied.
Illustrative activation regime
The notes describe an expected activation level for this qualitative pressure/confidence regime. The displayed approximation is illustrative because numeric inputs for “high” and “low” are not supplied.
Illustrative activation regime
The notes describe an expected activation level for this qualitative pressure/confidence regime. The displayed approximation is illustrative because numeric inputs for “high” and “low” are not supplied.
Implementation of saturating substrate learning
Computes an increment proportional to remaining coherence capacity and learning drive, then stores the incremented value with an upper clip at one. Nonnegative increments require corresponding nonnegative input factors.
Indexed nine-operator composition
C is the composition of nine operators indexed 1 through 9. C₁ acts first and C₉ acts last. This compact form preserves pipeline order while suppressing the descriptive filter names; it still depends on each operator being well-defined on the preceding output.
Intelligence and recursive coherence: proposed equivalence
This statement proposes an equivalence between intelligence and recursively evaluated coherence approaching one. An equivalence would require operational definitions and evidence in both directions. The recovered source supplies the proposed criterion but does not prove it.
Inverse discrepancy coherence proposal
Takes the reciprocal of the absolute difference between a transformed state and its expected value. The expression becomes singular when the discrepancy is zero, so a numerical implementation needs a convention or regularizer. It is distinct from the bounded 1/(1+error) score elsewhere in the notes.
Joint behavior–design objective
Adds a norm measuring mismatch between desired design D′ and observed behavior B, weighted operating cost, state regularization R and design regularization R_D. It supplies a scalar optimization target only after the terms, units and weights are defined.
LEARNING FUNCTION · π²_s
The expression uses a saturating nonlinear transform to turn its input combination into an activation or bounded score. The coefficients control sensitivity and the offset controls the transition region; a bounded score by itself does not identify a scientific consciousness measure.
LEARNING FUNCTION · π³_s
This relation contains a squared-state or squared-error term. Squaring makes that contribution nonnegative and increases the penalty for larger magnitudes; the surrounding definition specifies whether it is a cost, normalization or summary statistic.
LEARNING FUNCTION · π¹_s
The expression uses a saturating nonlinear transform to turn its input combination into an activation or bounded score. The coefficients control sensitivity and the offset controls the transition region; a bounded score by itself does not identify a scientific consciousness measure.
LEARNING FUNCTION · π⁰
This relation declares the set in which the stated variable is allowed to take values. It is a domain constraint or type declaration, not a state-update algorithm.