cs.LGOct 21, 2026
SaveReader--Writer Coupling in Transformer MLPs
Organizations: Seoul National University
Abstract
A transformer MLP gives each hidden channel two roles: a reader that decides when the channel responds, and a writer that decides what it adds back to the residual stream. Read as a key--value memory, the layer can be measured one factor set at a time: the pairing of readers to writers is carried only by their shared index, invisible to any quantity computed from one factor set alone. Yet training couples the two factors through their joint update, so we ask what reading them together shows. The block supplies the joint instrument: the MLP writes back into the very stream it reads, so the exact Jacobian of its update is square, and its agreement with its own transpose resolves into reader--writer terms whose inner products give each channel, read as a couple, a signed pairing coordinate. Ranked by this coordinate, each block we measure has a small core of strongly coupled channels; where initialization gives either sign equally, the core is nearly all opposed: each writer points against the direction its reader responds to. This opposed core recurs from the smallest models trained from scratch to released checkpoints; its sign is in place within a few percent of training and holds while the coupling grows. Read as couples, most channels return their contribution against the reader that fired it, and the readers of later blocks take it up; few show in the logits. Training holds the couples in place: flipping the opposed writers moves the output toward common tokens, and a writer permuted onto another reader regrows to fit that reader while the reader stays.
Figures & tables
Figure 1: Paired channel geometry. (a) Read one factor at a time, an MLP’s readers and writers are two factor sets that look patternless on their own. (b) Read as couples, most reader and writer couples roughly line up, pointing opposite ways. (c) What couples show is a signed axis, the pairing coordinate , whose sign the norm product lacks. (d) Measured, in the strongest couples the writers point against their readers through the stack, most strongly mid-stack; the models trained here release toward both ends, OLMo-1B only at the output (mean cosine of the sixteen strongest, small at context and ).
Figure 2: The sign across the population, and four core channels read as couples. Left: every channel of three medium seeds at its reader–writer angle and, outward, its norm product; the line separates opposed (above) from amplifying (below); colour the block, cores outlined, A–D marked; above the line lie of channels, of cores, half at initialization. Right: four core channels; each cell names its kind, logit or stream couple, each row whether the norm product’s top sixteen holds it, the logits moved, later readers, and the return on its reader. # marks a leading space.
Figure 3: Learned pairing and update geometry. (a) A re-pairing keeps both weights and changes only who pairs with whom. (b) Trained updates are symmetric far beyond any re-pairing. (c) Negative eigenvalues carry most of the symmetric part’s energy, where re-pairing gives one half; only OLMo-1B’s last block, whose couples flip, and the small model’s first expand (tiny, small and medium: blockwise medians over three seeds). (d) Most of the symmetry is mutual: channels reading each other’s writer contribute more than the couples on their own.
Figure 4: Writers grow back, readers stay.
Figure 5: Weight-level pairing along the stack. (a) Web text, eight blocks: the raw-row coordinate over all channels of each block, three seeds pooled, split at zero into opposed (blue) and amplifying (red) halves; dots mark one seed’s sixteen core channels. (b) Share of channels with per block at {4,6,8,12} blocks on relative depth, seed means.
Figure 6: The effective core and the corpus. (a) Each block’s final top- by , negative minus positive members over sixteen, at every saved state of medium and at twenty-four public checkpoints of OLMo; tokens on both axes, OLMo’s on a log scale. (b) Tiny models, one marker and colour per depth (legend): the pairing coordinate summed over each block’s core at the end of training against relative depth; natural streams as a band with their median, streams with the unigram frequencies kept and the context destroyed as dashed lines.
Figure 7: The couple census. Three medium seeds. (a) Every channel by family, area by share, classes inside. (b) The same families block by block, each bar one block’s channels; (a) is the average of the bars.
Appendix figures & tables38 assets
Supplementary material from the paper’s appendix.
Appendix
| Tiny | Tiny ladder | Tiny grid | Small | Medium | |
| Blocks width | |||||
| Heads | |||||
| Parameters | M | – M | – M | M | M |
| of which blocks | M | – M | – M | M | M |
| Seeds | per depth | per cell | per cell | ||
| Context | / |
Table 1: Training hyperparameters. One column per cohort; the small column reads context / where its two cells differ. The context- SwiGLU family of Figure 3(b) differs from that cell only in its branch (gated, hidden width for parameter matching, M parameters). Extended = GPT-2 plus reserved mask and sentinel identifiers.
| Set | Source | Extent | Used in |
|---|---|---|---|
| Controlled training | FineWeb-EDU | tokens | Section 5 |
| Controlled validation | validation split | tokens | Sections 4.2 and D |
| Grid training | FineWeb-EDU | B of B | Appendix D |
| Grid validation | validation split | tokens | Appendix D |
| Medium training | FineWeb-EDU B sample | B of B per seed, disjoint slices | Section 5 |
| Medium validation | key-disjoint split, per seed | tokens | Section 5 |
Table 2: Streams, panels, and validation sets. Offsets and intervals index tokens. Set B contains scored transitions; the released-label panel gives observations per block. The medium frequency reference is the entire evaluation pool.
| Quantity | Value |
|---|---|
| Jacobians per block | |
| Assignment, trajectory, pair-break protocols | |
| Band endpoints and bare-branch studies | |
| Draws | |
| Writer permutations | |
| Trajectory prefix | |
Table 3: Measurement, draw, and selection budgets. Protocols of the controlled cohorts. Jacobian counts per block are inputs positions, and . Permutation draws are per seed and block; the doses move or of channels ( and ); the selection thresholds are the prespecified evaluability conditions.
Figure 8: Endpoint gap along the stack. by relative depth for the tiny depth ladder, the two small contexts, and the medium; lines are seed means, bands the seed range. Every cohort peaks inside the stack, the tiny ladder at blocks , , and , the small at blocks – , the medium at block ; the small and medium start near zero at their input block.
Figure 9: The minimum eigenvalue and its scale dependence. Smallest eigenvalue of per block, three medium seeds against their same-factor re-pairings each. (a) In raw units the trained operator’s most contracting direction is more negative than every re-pairing in every block. (b) Once each operator is divided by its own RMS eigenvalue , the trained curve crosses the re-paired band, less extreme than its re-pairings through the middle of the stack and more extreme only in the first five and last two blocks: the raw separation is mostly the trained operator being larger.
| Scale cells | Merges | Gap retained |
|---|---|---|
| GELU | ||
| SwiGLU |
Table 4: Scale constraints and input exchange. Every trained assignment exceeds all stored draws. Left: the two small context- families, three seeds each (median retention, no cell merged). Right: one seed per cohort and eight new panels; departures are relative to its original panel’s gap.
Figure 10: Symmetric and antisymmetric energy. (a) Squared norms of the symmetric (Sym) and antisymmetric (Antisym) parts of the panel-mean Jacobian, relative to their own re-pairing means, three medium seeds; gray the mean and full -draw range. (b) Mean eigenvalue of divided by its RMS eigenvalue.
Figure 11: Weight-level pairing, gated against elementwise. OLMo-1B (a) and GPT-2 small (b) at their final public checkpoints. Left: the distribution of over every channel of each block, split at zero into opposed (blue) and amplifying (red) halves; dots are the block’s sixteen core channels of the effective coordinate at their weight-level value, and the axis covers every dot. Right: share of channels below zero per block, effective against weight-level. On the gated model the input blocks’ effective cores sit in the amplifying half of the weights; on the elementwise model the cores sit on the opposed side at every depth and the two shares run together.
Figure 12: The gate separates the two readings. (a) Share of channels whose weight-level and effective coordinates agree in sign, by relative depth; the band is the range of the elementwise families trained here. (b) The same agreement before (open markers) and after (filled) removing the reader contributions at negative-gate positions, with the small GELU control. (c) Opposed share as the negative gate is scaled toward zero, input block against middle band; crosses at zero are random deletions of matched size.
Figure 13: Formation and recovery controls. (a) Three tiny-seed gap trajectories; gray shows one step-zero null SD, maximized across seeds. (b) Mean block signed-sum relative to its uninterrupted control. (c) Reader permutation: writer following (solid), reader returning (dashed). In (b,c), lines are seed means, bands their ranges, and colors identify break steps; resumed training ends at step k.
Figure 14: How the final core develops, composition and magnitude. Each row tracks a block’s final top- by , holding the channel indices fixed across training. Columns enumerate saved checkpoints. (a,c) Positive counts fill cells from above and negative counts from below; gray marks near-orthogonality. (b,d) Separate positive and negative magnitudes include every selected channel, using a shared logarithmic length scale within each model.
Figure 15: Backtracking every endpoint core. Each row is one block’s sixteen final core channels, held fixed across the ten saved OLMo checkpoints. Counts are not averaged across blocks.
Figure 16: The same final-block channels through training. Rows follow endpoint rank. Left: each channel’s pairing coordinate divided by its maximum absolute value over the saved states; outlined cells are in that state’s top- . Middle: reader–writer cosine. Right: absolute-coordinate rank among all channels. Color magnitudes in the left panel are comparable within a row only.
| ID / tokens | Reader (native) | Writer (native) | Writer (final head) |
|---|---|---|---|
| 7325 / 4B | [hy] [ights] | [ave] [IE] | [ASE] [endo] |
| 7325 / 16B | [ert] [hy] | [Sock] [pkg] | [sar] [deck] |
| 7325 / 41B | [ert] [stan] | [Sock] [house] | [WORK] [RET] |
| 7325 / 3.095T | [ted] [ting] | [ted] [sm] | [ted] [sm] |
| 7749 / 4B | [items] [ought] | [iments] [s] | [#Prote] [#Had] |
| 7749 / 16B | [items] [ium] | [vent] [op] | [#Damage] [#Did] |
Table 5: Vocabulary projections of the same channel IDs through training. Two fixed endpoint-core IDs chosen for readable endpoint vocabulary lists. Columns show the top two entries under each checkpoint’s native unembedding, with a fixed final-head writer readout alongside it. These are projected directions, not observed triggers or established semantic roles. Leading spaces use # .
Figure 17: The conditional comparison across released labels. (a) Blockwise-median per size at the released labels, equally spaced on the axis; the open marker is the 2.8B label k standing in for the duplicated k bytes. The gap departs its ensemble mean at labels – k, grows through the middle of training, and plateaus. (b) Per-block at the final label against relative depth: one hump at every size, near the mean at the first block and closing toward the output.
Figure 18: Symmetry along Pythia training. Top: negative-energy share of the symmetric part minus one half; bottom: the fixed-denominator gap of Equation 27 , which the axis writes as with the trained denominator; one line per block, coloured by relative depth, against training tokens on a log axis with label at the left edge. Ticks above each panel mark the labels of the sparse replay. The 2.8B column uses those labels only.
| Model | Sign settled (step) | at | at | Core at sign step |
|---|---|---|---|---|
| GPT-2 small, five seeds | 200 (90–350) | 5,400 (4,425–6,600) | 20k (15k–23k) | 1% |
| Pythia-160M | 128 (128–100k) | 4,000 (2,250–4,000) | 11k (8,000–15k) | 9% |
| Pythia-410M | 1,000 (512–1,000) | 4,000 (3,000–4,000) | 14k (7,250–16k) | 3% |
| Pythia-1B | 1,000 (1,000–1,000) | 5,000 (3,250–6,000) | 20k (15k–26k) | 2% |
| Pythia-1.4B | 1,000 (1,000–1,000) | 6,000 (5,000–8,000) | 23k (15k–27k) | 2% |
Table 6: The sign is in place before the coupling grows. Over the interior blocks of each model, median and interquartile range of the step at which the block’s core stays majority opposed, of the step at which the block’s summed stays above a quarter and a half of its final value, and the core’s at the sign step as a share of its final value. GPT-2 small pools five seeds; one Pythia block turns late and widens its range.
Figure 19: Core formation across GPT-2 seeds. Five GPT-2 small replications ( Karamcheti et al., 2021 ) . Top: for every block, the balance of the block’s sixteen-channel core, opposed members minus amplifying over sixteen, against training tokens, one line per seed; the dotted line marks twenty tokens per parameter. Bottom left: the tokens at which each block’s core turns majority opposed, one marker per seed, blocks in depth order. Bottom right: the opposed share of all channels by depth band. Step sits at the left edge of each axis.
| Size | Blocks | All beyond from | Final , median (SD) |
|---|---|---|---|
| 70M | k | ( ) | |
| 160M | k | ( ) | |
| 410M | k | ( ) | |
| 1B | ( ) | ||
| 1.4B | k | ( ) | |
| 2.8B | ( ) |
Table 7: Persistence across released labels. First label after which every block stays above its re-pairing mean, and the final median gap with across-block SD. All endpoint blocks are above their means; earlier crossings can reverse. The B model uses draws per block, the others .
Figure 20: Core channel signs across the full corpus grid. The effective coordinate of every endpoint core channel, sixteen per block, for six corpora, four depths (four, six, eight and twelve blocks) and three seeds (shape and offset) on the exact-reader panels of Section F.3 . Every natural corpus forms opposed cores with a few amplifying members in its last blocks; at twelve blocks the input block adds small positive members; the two order-free controls stay at zero.
Figure 21: Every channel, with and without order. Left: the effective pairing coordinate of all channels of each block over training (gray), one eight-block run of the corpus recipe on the web stream (top) and on the same stream with its token order removed (bottom); the sixteen channels that end as the block’s core are coloured, blue if opposed and orange if amplifying. Without order no channel leaves the bulk. Right: the stream cut into chunks of tokens with only the chunk order shuffled; the core sum (top) and the number of amplifying core channels (bottom) at the end, three seeds (band). Order within a few dozen tokens is enough to form the opposed cores. On these GELU models the weight-level coordinate agrees in sign but selects a different top sixteen (Appendix E.2 ).
Figure 22: Corpus composition readings. Rounded percentages in five surface categories, read from raw-lens lists under the corpus taxonomy. Each row contains channels, except the eligible amplifying sets for permuted web ( ). Freq. denotes common-token lists; Morph. denotes morphemes; Norm is the writer-norm-matched set. Colors share one scale across streams for these five qualitative categories.
| Stream | Reader | Writer projection | Recurrence |
|---|---|---|---|
| Web | Biochemical fragments | Promotes civic nouns: government , policy | Block 6; three seeds |
| Books | Female names and titles | Suppresses female pronouns: her , she | Block 6; three seeds |
| Stories | Eating vocabulary | Suppresses eat , ate , hungry , food | Block 6; two seeds |
| Stories | Not specified | Amplifying writer suppresses female pronouns | Block 7; two seeds |
| News | Unrelated readers | Modal verbs and sports vocabulary | Blocks 5–7; writer only |
Table 8: Corpus-specific couple readings. Selected recurring patterns from the full core reading, not prevalence estimates or realized next-token effects. Recurrence refers to the same block and kind of reading, not an identical channel index.
Figure 23: Angle, scale, and the reader-size dependence. Top: every channel of one medium seed at an early, a middle, and a late block, as in Figure 2 : distance from the origin is the reader–writer norm product (square-root scale), the angle is the angle between the two, so opposed channels ( ) lie above the line. Filled dots are the sixteen core channels by ; open squares and diamonds are the sixteen largest readers and writers. The core sits at the large-reader, most-opposed corner and overlaps the largest readers (eight, six and four of sixteen), while the largest writers scatter across the angle: opposition is organized by the reader, not by the writer. Bottom: opposed share by reader-norm quartile on the large sign panel, one line per seed, averaged over each eight-block band. Amplifying channels sit mostly in the smallest-reader quartile.
| Observed effect | Medium ( channels) | OLMo ( channels) |
|---|---|---|
| Post-MLP residual norm decreases | 373 | 162 |
| Centered logit norm increases | 151 | 144 |
| Centered target logit increases | 167 | 177 |
| Target increase: interval above zero | 40 | 85 |
| Target decrease: interval below zero | 68 | 15 |
Table 9: Negative feedback and output effects differ. Counts among endpoint-core channels with . The first three rows use mean effects; the last two require a pointwise interval excluding zero. Output effects include later-block responses.
| Channel | Kind | Reader norm | Writer norm | Norm product | Push |
|---|---|---|---|---|---|
| A | logit couple | ||||
| B | stream couple | ||||
| C | logit couple | ||||
| D | stream couple |
Table 10: The four core channels of Figure 2 . Ranks descend within the channel’s own block; push is the direct movement of the next-token logits, as a share of the gap between the two largest, at the strongest firings.
| Model | Units | Inside band after break | |
|---|---|---|---|
| GPT-2 (pre) | 12 blocks | 12 | — |
| Pythia-70M (pre) | 6 blocks | 6 | — |
| SmolLM2-135M (pre) | 30 blocks | — | median |
| Tiny GELU, controlled | 12 seed–blocks | 12 | — |
Table 11: Outcomes of the one-sided pair break. Units returned inside their matched band, and continuous recovery where it was summarized (a dash marks the readout not summarized for that model).
| Group | Blocks | Within margin | |||
|---|---|---|---|---|---|
| Controlled (3 seeds) | 12 | 4 | |||
| GPT-2 | 12 | 0 | |||
| SmolLM2 | 30 | 4 |
Table 12: Spectral-shape displacement under the one-sided break. Mean spread of the blockwise displacement: over the three seeds for the controlled family, over blocks for the two pretrained models. Within-margin blocks are counted against the blocks measured.
| Seed | (millinats) | Positive strata (of 8) | Raw | Holm |
|---|---|---|---|---|
| #1 | 5.15 | 8 | 0.0107 | 0.0107 |
| #2 | 4.74 | 7 | 0.0001 | 0.0006 |
| #3 | 6.08 | 6 | 0.0001 | 0.0006 |
| #4 | 8.72 | 8 | 0.0001 | 0.0006 |
| #5 | 4.16 | 6 | 0.0008 | 0.0024 |
| #6 | 6.20 | 6 | 0.0008 | 0.0024 |
Table 13: Loss difference at equal disturbance, per seed. Among re-pairings matched on weight, Jacobian, and output disturbance, the ones that damage symmetry more cost more loss: is the equal-stratum loss difference in millinats, positive in every seed and in most strata, small next to the loss itself, and significant after correction in every seed. Tiny four-block GELU models at context ; the six seeds are disjoint from the three on which the comparison was first run, and all dose–block strata met the matching requirements.
Figure 24: Deletion costs separate along training. All blocks are edited together and the full network is reevaluated; each deletion is compared with a random direction of matched size (dotted). Early in training the network depends on the perpendicular component alone; the parallel component, the one the coupling measures, becomes a cost of its own later and dominates at the endpoint, in the medium and in OLMo alike. In (b) filled markers are the deletions and hollow markers their random comparators, shape identifying the seed. These costs are network reevaluations, neither additive over blocks nor direct-logit attributions.
Figure 25: Flipping opposed writers favors common tokens. Selected writer vectors are negated in every medium block: most negative (opposed) or largest positive (amplifying). Both panels show selected minus activity- and size-matched flips. Thin curves show three seeds and thick curves their mean; shading marks of writers per block. Frequency error bars at are paired-window bootstrap intervals, conditional on the three endpoints.
Figure 26: From profiles to classes. Stored channel profiles (activation contexts, replayed forward) yield three registers per channel: the input that fires it, the contribution’s first-order response on later readers, and its direct push on the next-token logits, at the strongest firings and at typical activations. Grey stages are measured; orange stages are language-model annotation, with the engine of each stage under its column.
| Class | Seed 1 | Seed 2 | Seed 3 |
|---|---|---|---|
| stream couples | |||
| Opaque-trigger stream writer | |||
| Cross-family stream handoff | |||
| Silent morphology relay | |||
| Silent entity name relay | |||
| Silent function-word relay |
Table 14: Couple census by class and family. Shares (%) of the channels of each medium seed; classes below in every seed are omitted from the listing but counted in their family; dispositions are rows of the same table, and shares are rounded independently.
| Seed | Class | Family |
|---|---|---|
| Seed 1 | ||
| Seed 2 | ||
| Seed 3 |
Table 15: Agreement with the reference labels. Share (%) of reference-labelled channels per seed on which the population label equals the reference label, at the class level and at the family level, mean bootstrap standard error.
Figure 27: What the core writes. (a) Median promoted-token frequency rank across lens depths; more frequent is up. Removing leading unembedding directions narrows the medium core’s separation from its controls. (b) Common-token, morpheme, and semantic-field readings of the same four sets, with colors and markers shared across panels. Each set contains medium channels; the full taxonomy and raw-rank comparisons are in Tables 18 and 16 .
| Family | Core | Amplifying | Equal-norm | Random |
|---|---|---|---|---|
| Medium | ||||
| Small, context | ||||
| Small, context | ||||
| Tiny |
Table 16: Raw-lens frequency ranks across cohorts. Median promoted-token rank for the four selected sets; lower is more frequent. Medium spans seed–blocks, each small context , with frequency references defined within each family. The medium lens-depth curves are in Figure 27 .
| Seed / block | Promoted (top 3) | Suppressed (top 3) | med. rank P/S |
|---|---|---|---|
| Common-token pattern | |||
| #1 / block 19 ∗ | #them , #the , #a | olver , #feel , #eat | |
| #3 / block 18 ∗ | #and , , , - | / zwsp , adata , kees | |
Table 17: Every rule-qualifying channel. All medium core channels passing the fixed surface rules at the raw vocabulary lens ( common-token, none repeated-separator, of scanned). # marks a leading space; median ranks cover each channel’s sixteen-token lists (rank = most frequent). Starred rows are each seed’s common-token maximum; the second seed has none. zwsp is a zero-width space.
| Channel set | common | mirror | field | morpheme | separator | none | |
|---|---|---|---|---|---|---|---|
| core, all blocks | |||||||
| core, blocks – | |||||||
| core, blocks – | |||||||
| core, blocks – | |||||||
| amplifying sixteen, all blocks | |||||||
| amplifying sixteen, blocks – |
Table 18: Surface census of the four sets. Channel shares (%) from language-model readings of raw-lens lists on the medium; residual classes are omitted. The core is opposed in of channels, leaving no comparable within-core sign split.
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