A change of research question
The final JAIC manuscript concerns whether a declaration could mark a subsequent internally produced persistent change. Its empirical dataset consists of model training artifacts. This is a different question and dataset from the January federation-resonance manuscript.
Operational measurement
The study compares low-rank factors and their effective weight product, measures step-to-step changes and distances, and considers gradient and loss trajectories. Equivalent factorizations can leave the effective operator unchanged, so factor movement must be distinguished from functional weight change.
What is and is not measured
The final source reports 167 adapter checkpoints, 792 gradient observations and 79 evaluation losses. It explicitly does not measure a declaration channel, subjective experience or a within-run declaration-triggered causal transition. Addenda develop comparison and introspection evidence to guide a prospective design.
Equations in this build
87 linked mathematical records. Each opens its notation, interpretation and source.
Identity at a fixed time
An expression equals itself under ordinary equality. This logical identity does not state that a quantity is unchanged between different times, and it does not measure the…
Bookkeeping state: parameters, memory, runtime, and observations
The tuple separates four components of a model instance at time t: learned parameters, persistent memory, runtime state or context, and observation records. This is a bookkeeping…
Extract the representation being tested
H maps the model’s state into the representation z that will be measured. Its output could be a parameter-derived coordinate, activation summary, or another declared measurement.…
Construct an independently specified reference
The map ρ produces a reference measurement from the same system state. To make d(z,r) meaningful, r and z must be comparable in a declared common space. Independence here concerns…
Current-reference discrepancy and declaration-anchored displacement
The first distance compares the current representation zₜ with its current reference rₜ. The second compares zₜ with the representation recorded at the declaration time. These…
Identity at a fixed time
An expression equals itself under ordinary equality. This logical identity does not state that a quantity is unchanged between different times, and it does not measure the…
Change between consecutive states
The inequality says the recorded value at the next time differs from the current one. Whether that difference is meaningful depends on measurement precision, the distance or norm,…
Declaration indicator and first detected declaration
An independently specified detector is applied to the observations available up to time t. The indicator records whether its criterion is satisfied, and t_D is the first time this…
Observation history available at time t
O≤t denotes all observations available up to and including time t. A prospective detector may use this history but cannot use observations that arrive later. This restriction…
Nonnegative discrepancy tolerance
The tolerance ε is required to be nonnegative because it thresholds a nonnegative distance. A value of zero tests any strictly positive discrepancy, whereas a positive value…
Finite-window persistence indicator
The minimum discrepancy across the next W observations must exceed ε for the indicator to equal one. This requires every observation in the window to clear the threshold, rather…
Independent onset of persistent discrepancy
The event time is the first eligible window in the full observation record whose persistence indicator is one. Because the search does not start at the declaration time, it can…
Lag between measured change and declaration
Subtracting declaration time from the independently detected change time gives a signed lag. Positive lag means change follows declaration; negative lag means it precedes…
Bidirectional coupling between a loop and an internal state
The first transition allows the outer process L to depend on x, while the second allows x to depend on L. This notation makes the proposed two-way path explicit. Actual causal…
Select an internal component of the bookkeeping state
The projection π_x extracts the chosen internal component x from the complete state S. This states precisely which part of the system the coupling or change claim concerns. An…
Effective rank-one LoRA update
The outer product bₜaₜᵀ forms a rank-at-most-one matrix, and the scalar s sets its contribution to the model weight update. The pinned study uses s=64. Comparing this effective…
Reciprocal rescaling leaves the rank-one operator unchanged
Multiplying one factor by a nonzero scalar and dividing the other by that scalar cancels in their outer product. The individual vectors can therefore move in magnitude while the…
Frobenius inner product of two rank-one updates
The matrix inner product factorizes into an inner product of the output factors and an inner product of the input factors, multiplied by their scale coefficients. With one fixed…
Frobenius norm of a rank-one update
The matrix norm is the product of the two factor lengths multiplied by the absolute scale. The final manuscript writes s without absolute bars because the pinned coefficient is…
Squared distance between effective update matrices
Expanding the squared Frobenius norm gives two squared norms minus twice the inner product. Combined with rank-one factorization identities, this calculates matrix distance…
Factor-space path variable
This analysis choice takes Zₜ to be the output-side LoRA vector bₜ. It provides a compact coordinate path within the saved parameterization. Reciprocal rescaling of the two LoRA…
Operator-space path variable
This choice takes Zₜ to be the effective update matrix Uₜ. It makes path geometry insensitive to reciprocal scaling of its two rank-one factors. It still describes…
Local path bend from two displacement vectors
Both displacement vectors point outward from the center checkpoint: one toward t−k and the other toward t+k. Their normalized inner product measures the angle between those…
Negative-one cosine for straight passage
Because both vectors in the centered bend statistic point away from the center, a straight trajectory places them in opposite directions and gives cosine −1. This is the reference…
Amplitude eligibility filter
A checkpoint is eligible when the larger of the two declared factor-change magnitudes reaches η. This excludes bends formed from very small displacements under the study’s chosen…
Stable decomposition of an operator difference
Expanding the two outer products shows that the operator difference equals a changed output factor times the later input factor plus the earlier output factor times a changed…
Centered moving average of gradient norms
The average uses a symmetric window of w observations around time t. For odd w, there are (w−1)/2 samples on each side and one center sample. Because future samples enter the…
Allowed split locations in the gradient fit
The candidate split is restricted to indices 20 through 280 within the first 300 gradient observations. This leaves at least twenty observations on each side for a two-line fit.…
Two-line least-squares split of the gradient record
For each candidate split k, a separate straight line is fitted to the observations on each side, and their squared residuals are added. The preferred split minimizes this combined…
Stable affine recurrence toward one
Subtracting one from both sides gives error_next=a×error. For 0<a<1, the positive error shrinks geometrically but remains nonzero at every finite step when initialized above one.…
Initial value for the contraction example
The recurrence starts at two, one unit above its fixed point at one. This makes its initial error exactly one and its closed-form error aⁿ. The value is chosen for a simple…
Closed form of the contraction example
The solution is one plus aⁿ because the initial error is one and each step multiplies it by a. For 0<a<1 it approaches one but never equals one at a finite integer n in exact…
Positive step-index domain
The condition n≥1 restricts the statement to updates after initialization. In the surrounding recurrence example, it distinguishes later values from the explicitly chosen n=0…
Observationally indistinguishable interpretations
If two interpretations assign exactly the same probability to every observation in the study, the likelihood ratio for any observed record is one. Those data therefore supply no…
Adjacent-window gradient-norm reduction
The expression computes one minus the ratio of the later window mean to the earlier window mean, then converts to percent. With the supplied means it gives about 46.52465%. This…
Recorded evaluation-loss reduction
The final logged evaluation loss is compared with the first logged value through 100(1−final/initial). The supplied values give 63.6935429588%, consistent with the recovered saved…
Bookkeeping state: parameters, memory, runtime, and observations
The tuple separates four components of a model instance at time t: learned parameters, persistent memory, runtime state or context, and observation records. This is a bookkeeping…
Construct an independently specified reference
The map ρ produces a reference measurement from the same system state. To make d(z,r) meaningful, r and z must be comparable in a declared common space. Independence here concerns…
Extract the representation being tested
H maps the model’s state into the representation z that will be measured. Its output could be a parameter-derived coordinate, activation summary, or another declared measurement.…
Earlier discrepancy and fixed-reference displacement
This version compares zₜ with the current reference rₜ and separately with r at declaration time. The fixed-reference distance can remain large even when the current…
Zero current-reference discrepancy
If d is a true metric, distance zero means the two compared representations are equal in the chosen measurement space. It does not require the entire underlying states to be…
Registered declaration criterion
This indicator equals one when the fixed detector says the observed history meets the registered declaration criterion. Reproducibility requires the detector, thresholds, and…
Finite-window persistence indicator
The minimum discrepancy across the next W observations must exceed ε for the indicator to equal one. This requires every observation in the window to clear the threshold, rather…
Earlier post-declaration onset search
This earlier definition searches only after t_D for the first window of persistent discrepancy. It therefore cannot discover a qualifying change that began before declaration and,…
Bidirectional coupling between a loop and an internal state
The first transition allows the outer process L to depend on x, while the second allows x to depend on L. This notation makes the proposed two-way path explicit. Actual causal…
Adapter scale fixed at sixty-four
The pinned analysis uses the effective LoRA scaling coefficient s=64. This coefficient multiplies the factor outer product and therefore multiplies its norm by 64 and a…
Gradient smoothing windows examined
The analysis compares four odd window lengths: 5, 11, 21, and 31 observations. Smaller windows preserve more local variation, while larger windows smooth it more strongly.…
Centered moving average of gradient norms
The average uses a symmetric window of w observations around time t. For odd w, there are (w−1)/2 samples on each side and one center sample. Because future samples enter the…
Evaluation-loss decrease over one logged interval
Subtracting the current evaluation loss from the loss ten steps earlier gives a positive number when loss has fallen. It is an absolute change in loss units, not a percentage and…
Two-line least-squares split of the gradient record
For each candidate split k, a separate straight line is fitted to the observations on each side, and their squared residuals are added. The preferred split minimizes this combined…
Input-side rank-one factor dimension
This states that the flattened input-side factor has 13824 real coordinates in the selected module. It defines the column dimension of the effective rank-one update. The vector…
Output-side rank-one factor dimension
This states that the flattened output-side factor has 5120 real coordinates in the selected module. With the 13824-dimensional input factor, the outer product has shape 5120 by…
Effective rank-one LoRA update
The outer product bₜaₜᵀ forms a rank-at-most-one matrix, and the scalar s sets its contribution to the model weight update. The pinned study uses s=64. Comparing this effective…
Rank-one inner product and norm identities
The Frobenius inner product separates into two vector dot products, while the norm separates into the product of the two vector lengths and |s|. These identities retain the…
Squared distance between effective update matrices
Expanding the squared Frobenius norm gives two squared norms minus twice the inner product. Combined with rank-one factorization identities, this calculates matrix distance…
Local path bend from two displacement vectors
Both displacement vectors point outward from the center checkpoint: one toward t−k and the other toward t+k. Their normalized inner product measures the angle between those…
Negative-one cosine for straight passage
Because both vectors in the centered bend statistic point away from the center, a straight trajectory places them in opposite directions and gives cosine −1. This is the reference…
Checkpoint lags examined for path bends
The bend statistic is evaluated with lags 5, 10, and 15 on each side of a center checkpoint. Larger lags compare broader segments of the path. These are sensitivity settings…
Amplitude thresholds examined for bend eligibility
The filter is evaluated at zero and five positive thresholds ranging from 10⁻⁴ to 0.005. Increasing the threshold excludes more small-displacement comparisons. The selected peak…
Stable decomposition of an operator difference
Expanding the two outer products shows that the operator difference equals a changed output factor times the later input factor plus the earlier output factor times a changed…
Zero output factor at the first checkpoint
The recorded first output-side factor is zero. Multiplying it by any input factor produces a zero effective rank-one update. This initialization explains why factor motion and…
Zero effective update at the first checkpoint
The rank-one update is zero when the output factor is zero. This means the selected adapter contribution initially adds no matrix to the corresponding base weight. It does not…
Specific eligibility setting at 0.002
This is the factor-amplitude threshold used for one reported bend selection. It includes a checkpoint only if the selected factor-change magnitude reaches 0.002. The value is an…
Observationally indistinguishable interpretations
If two interpretations assign exactly the same probability to every observation in the study, the likelihood ratio for any observed record is one. Those data therefore supply no…
Adjacent-window gradient-norm reduction
The expression computes one minus the ratio of the later window mean to the earlier window mean, then converts to percent. With the supplied means it gives about 46.52465%. This…
Largest stated adjacent evaluation-loss drop
Subtracting the two supplied losses yields 0.20026636123657227 loss units. This is an absolute difference, not a percentage. The claim that it is the largest drop depends on the…
Recorded evaluation-loss reduction
The final logged evaluation loss is compared with the first logged value through 100(1−final/initial). The supplied values give 63.6935429588%, consistent with the recovered saved…
Product of rank-one factor cosines
For nonzero rank-one matrices with compatible positive scaling, the normalized Frobenius inner product equals the product of the two factor cosines. The provided comparison gives…
Variance fraction captured by the first two principal components
Adding the first two explained-variance fractions gives 0.9498998129, or about 94.99% of the variance in the centered representation analyzed. This measures compression of that…
Stable affine recurrence toward one
Subtracting one from both sides gives error_next=a×error. For 0<a<1, the positive error shrinks geometrically but remains nonzero at every finite step when initialized above one.…
Initial value for the contraction example
The recurrence starts at two, one unit above its fixed point at one. This makes its initial error exactly one and its closed-form error aⁿ. The value is chosen for a simple…
Closed form of the contraction example
The solution is one plus aⁿ because the initial error is one and each step multiplies it by a. For 0<a<1 it approaches one but never equals one at a finite integer n in exact…
Leaky accumulator with constant input
Each update retains a fraction λ or ρ of the previous value and adds q. With initial value zero and 0≤λ<1, past inputs decay and the state approaches q/(1−λ). This is a standard…
Nonnegative stable retention range
The range 0≤λ<1 means each update retains a nonnegative fraction smaller than the whole previous state. In the leaky accumulator, it guarantees decay of earlier contributions and…
Empty initial accumulator
The accumulator begins at zero, so its later value is entirely attributable to the subsequent inputs under the chosen recurrence. This initial condition removes the extra…
Closed form of the leaky accumulator
Summing the retained contributions q, λq, λ²q, and so on gives q(1−λⁿ)/(1−λ). For |λ|<1 this tends to q/(1−λ). The denominator must be nonzero; at λ=1 the recurrence instead…
Unchanging stored state
The next value equals the current value, so induction gives xₜ=x₀ for all subsequent steps. Other parts of a system can evolve while this component stays fixed. A changing…
Centered moving average of gradient norms
The average uses a symmetric window of w observations around time t. For odd w, there are (w−1)/2 samples on each side and one center sample. Because future samples enter the…
Evaluation-loss decrease over one logged interval
Subtracting the current evaluation loss from the loss ten steps earlier gives a positive number when loss has fallen. It is an absolute change in loss units, not a percentage and…
Two-line least-squares split of the gradient record
For each candidate split k, a separate straight line is fitted to the observations on each side, and their squared residuals are added. The preferred split minimizes this combined…
Effective rank-one LoRA update
The outer product bₜaₜᵀ forms a rank-at-most-one matrix, and the scalar s sets its contribution to the model weight update. The pinned study uses s=64. Comparing this effective…
Reciprocal rescaling leaves the rank-one operator unchanged
Multiplying one factor by a nonzero scalar and dividing the other by that scalar cancels in their outer product. The individual vectors can therefore move in magnitude while the…
Frobenius inner product of two rank-one updates
The matrix inner product factorizes into an inner product of the output factors and an inner product of the input factors, multiplied by their scale coefficients. With one fixed…
Frobenius norm of a rank-one update
The matrix norm is the product of the two factor lengths multiplied by the absolute scale. The final manuscript writes s without absolute bars because the pinned coefficient is…
Squared distance between effective update matrices
Expanding the squared Frobenius norm gives two squared norms minus twice the inner product. Combined with rank-one factorization identities, this calculates matrix distance…
Adjacent-window gradient-norm reduction
The expression computes one minus the ratio of the later window mean to the earlier window mean, then converts to percent. With the supplied means it gives about 46.52465%. This…
Largest stated adjacent evaluation-loss drop
Subtracting the two supplied losses yields 0.20026636123657227 loss units. This is an absolute difference, not a percentage. The claim that it is the largest drop depends on the…