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T-Router: Learning Thalamic Routing for Reasoning with Parameter-Efficient Reinforcement Learning

Liuxian Ma, Jiale Dai, Jiaqi Li, Lu Mi

arXiv:2609.39109v1cs.LGcs.AIcs.NE

TL;DR

Reasoning reinforcement learning needs a compact way to improve a pretrained model without training its full parameter set. T-Router learns to route completed computations through a compressed source bank and recurrent controller around a frozen backbone. It reports higher mathematical-reasoning MathAvg than full-parameter GRPO and matched-retry LoRA, using 0.466% of the backbone’s parameters.

  • Problem

    The paper asks how to place trainable capacity so reward feedback can improve reasoning while preserving most pretrained parameters.

    Source: §1 · p. 1

  • Method

    T-Router uses compressed, addressable block changes and a depth-recurrent controller to select reusable contributions and regulate their writeback around a frozen backbone.

    Source: §1 · p. 2

  • Results

    T-Router reaches 83.64 ± 1.16 MathAvg versus 73.79 ± 1.83 for full-parameter GRPO and 77.28 ± 1.95 for matched-retry LoRA, while improving all three mathematical families over LoRA.

    Source: §1 · p. 2, §4.2 · p. 6

  • Takeaways & Limitations

    The results support controlled computation reuse as a compact route to reinforcement-learning adaptation while preserving pretrained transformations and layer order.

    Source: §5 · p. 9

Abstract

from arXiv · show

Parameter-efficient reinforcement learning aims to improve reasoning with a compact trainable interface to a pretrained model. We introduce the Thalamic Router (T-Router), which concentrates adaptation on the reuse of completed computations. A compressed, addressable bank preserves block changes; a depth-recurrent controller conditions their selection and relative-scale writeback. This coupling gives thalamic context-dependent routing a concrete computational form: learn which earlier contributions a receiving layer uses, and with what influence. Correctness rewards train the interface while preserving backbone parameters and layer order. On an 8.95B-parameter backbone, T-Router allocates 41.73M parameters (0.466% of the backbone) and achieves 83.64 +/- 1.16 MathAvg after GSM8K RL, compared with 73.79 +/- 1.83 for full-parameter GRPO across three evaluation rounds. At a comparable parameter budget and with matched retries, it exceeds LoRA's 77.28 +/- 1.95 MathAvg, improving all three task families and raising mean AIME accuracy from 48.33 to 60.56. Capacity-controlled comparisons favor addressable block changes and recurrent context; separate search training extends the interface to tool-mediated reasoning. These results establish controlled computation reuse as an effective route to parameter-efficient reasoning reinforcement learning.

1 Introduction

The paper proposes T-Router, a compact interface that learns how later layers reuse completed computations while keeping the pretrained backbone frozen. On mathematical reasoning, it reports higher MathAvg than full-parameter GRPO and matched-retry LoRA with a small fraction of the backbone’s parameters.

1 Introduction

  • Learning which completed changes later computations reuse adds an adaptation target beyond changing weights or local transformations.LoRA and adapters place trainable capacity in weight updates or local transformations; separately addressable contributions expose reuse and influence as another learning target.Source: §1 · p. 1
  • Context-dependent routing motivates the design: T-Router controls which completed block changes a later layer reuses and at what strength.The paper draws the routing principle from thalamic regulation of cortical communication according to task demands.Source: §1 · pp. 1–2
  • T-Router stores compressed, addressable block changes and uses a depth-recurrent controller to select their mixture and regulate signed writeback before frozen layers execute in order.Correctness rewards train this communication interface end to end while preserving pretrained transformations.Source: §1 · p. 2
  • With 41.73M parameters, T-Router reaches 83.64 ± 1.16 MathAvg versus 73.79 ± 1.83 for full-parameter GRPO and 77.28 ± 1.95 for matched-retry LoRA.The reported comparison follows GSM8K reinforcement learning on an 8.95B-parameter backbone.Source: §1 · p. 2
  • The contributions include capacity-controlled comparisons and a separate application of the compact interface to agentic search.The paper presents controlled computation reuse as a route to efficient reinforcement-learning adaptation.Source: §1 · pp. 2–3

2 Related work

Related work locates T-Router at the intersection of parameter-efficient adaptation and learned communication across depth. The paper distinguishes its coupled content-and-control interface from prior depth-connectivity approaches.

2 Related work

  • Prior parameter-efficient methods reduce adaptation footprints through adapters and LoRA, while PE-RLHF applies parameter-efficient learning to reward modeling and reinforcement learning (Houlsby et al., 2019; Hu et al., 2022; Sidahmed et al., 2024).The section also contrasts GRPO, reward-shaped LoRA mixtures, sampled-token training, and bounded retries (Shao et al., 2024; Lee & Tong, 2025; Yu et al., 2025).Source: §2 · p. 3
  • Depth-connectivity methods range from dense feature reuse to dynamic or attention-based combinations of earlier computation (Huang et al., 2017; Pagliardini et al., 2024; Xiao et al., 2025; Kimi Team et al., 2026).Delta Attention Residuals provide a close precedent by adding attention-weighted sublayer or block changes to the residual (Luo et al., 2026a).Source: §2 · p. 3
  • Frozen-backbone approaches also learn read/write routing over multiple residual streams, while hyper-connection methods expand depth connectivity (Oldenburg et al., 2026; Xie et al., 2025; Zhang et al., 2026).These works establish depth connectivity as an adaptation axis alongside local weight updates.Source: §2 · p. 3
  • T-Router couples origin-indexed compressed changes, a separate depth-recurrent controller, and calibrated signed writeback while preserving pretrained transformations and layer order.The controller context conditions both source selection and writeback strength.Source: §2 · p. 3
  • Thalamic findings motivate context-dependent control over distributed computation, which T-Router implements through source selection and writeback strength.The related-work discussion separately notes hippocampal indexing theory as inspiration for HippoRAG’s external knowledge retrieval (Gutiérrez et al., 2024).Source: §2 · p. 3

3 T-Router

T-Router wraps a frozen decoder in a trainable communication pathway that records completed block changes, maintains recurrent depth context, routes selected sources, and calibrates writeback. Correctness-based reinforcement learning trains this auxiliary interface while pretrained layers continue to execute in order.

3 T-Router

  • T-Router stores compressed block changes in a source bank and uses a recurrent controller and query-dependent readout to condition source attention and writeback strength.The resulting intervention is added before the receiving frozen layer, and pretrained layers execute in order.Source: §3 · p. 3, Figure 2 · p. 4

3.1 Recording changes across depth

  • Completed blocks are compressed into records of their changes, and only blocks completed before the receiving layer are available for reuse.The bank stores changes at the same token position without adding attention over token positions; with 32 layers and block size 4, later layers can read up to seven completed blocks.Source: §3.1 · p. 4

3.2 From depth state to controller context

  • A recurrent controller updates per-token slots from the current residual, completed-record mean, and receiving-layer embedding, then attention reads context P from the updated slots.The slots and bank reset at each forward pass and maintain separate depth trajectories for each token.Source: §3.2 · pp. 4–5

3.3 Routing completed computations

  • Source routing combines the current state, controller context, and layer identity to mix visible completed sources into content for the receiving layer.The hidden state and controller context determine the query, while source and layer embeddings identify origin and destination.Source: §3.3 · p. 5

3.4 Calibrating the intervention

  • The projected source mixture is calibrated relative to the receiving state before a signed gate controls its writeback magnitude.The gate can reinforce or oppose the retrieved direction; subsequent nonlinear computation can still amplify or attenuate its effect.Source: §3.4 · p. 5

3.5 Training the auxiliary module

  • A sequence-normalized group-relative policy gradient trains source transport and control end to end, alongside KL and routing regularization terms.The loss combines policy, KL, and routing-regularization terms; the module allocation is 41,730,332 parameters, or 0.466% of the backbone.Source: §3.5 · p. 6

4 Experiments

Across mathematical reasoning and separately trained search, T-Router achieves strong results with compact capacity. Controlled comparisons favor addressable block changes and recurrent context, while further ablations examine source identity, writeback, and the retry recipe.

4.1 Evaluation setting

  • Evaluation uses GSM8K (7,473 training prompts; 1,319 test questions), MATH-500 (500 questions), and 30 questions each from AIME 2024 and 2025, with means and SDs across three evaluation rounds.The Qwen3.5-9B-Base backbone has 8,953,803,264 parameters; T-Router uses 41.730M trainable parameters, 0.466% of the backbone.Source: §4.1 · p. 6

4.2 Parameter-efficient mathematical reasoning

  • 83.64 ± 1.16 MathAvg with 41.73M parameters (0.466%) exceeds full-parameter GRPO (73.79 ± 1.83) and matched-retry LoRA (77.28 ± 1.95).T-Router also leads all three mathematical families, including 92.73 on MATH-500 and 60.56 mean AIME accuracy.Source: §4.2 · p. 6, Table 1 · p. 7

4.3 Agentic search

  • 36.98 ± 1.16 on BrowseComp Plus and 73.98 ± 0.74 on ASearch Test accompany paired gains over full-parameter GRPO of 3.75 ± 0.91 F1 and 4.04 ± 1.27 points.Search models are trained separately using ASearcher data and multi-turn retrieval-and-summary tool trajectories.Source: Table 2 · p. 7, §4.3 · p. 7

4.4 Addressable content and recurrent control

  • At equal 41,730,332-parameter capacity, block-change memory scores 83.64 ± 1.16 MathAvg versus 67.46 ± 2.19 for hidden-state memory; recurrent control reaches 83.64 ± 1.16 versus 72.80 ± 4.10 for a matched-capacity MLP.Removing cross-layer retrieval yields 69.48 ± 2.36 MathAvg.Source: §4.4 · pp. 7–8, Table 3 · p. 8

4.5 Source identity and relative-scale writeback

  • Removing layer and block identities yields 58.10 MathAvg; removing token-conditioned gating yields 63.78, and omitting RMS alignment yields 56.39.RMS alignment defines writeback influence relative to the receiving residual.Source: §4.5 · p. 8

4.6 Learning and executing the reuse policy

  • 121.23 GPU-hours and 83.64 ± 1.16 MathAvg with informative retries contrast with 37.92 GPU-hours and 62.53 ± 1.20 without retries.The no-retry setting also changes the generation budget; matched-retry LoRA reaches 77.28 ± 1.95 MathAvg.Source: §4.6 · p. 9, Figure 4 · p. 9

5 Conclusion

T-Router learns how completed computations are reused, reaching higher reported MathAvg than full-parameter GRPO and matched-retry LoRA with 0.466% of the backbone’s parameter count. The appendices provide computational specifications, analysis, and configuration-level results supporting the method and study.

5 Conclusion

  • With 0.466% of the backbone’s parameter count, T-Router reaches 83.64 MathAvg versus 73.79 for full-parameter GRPO and 77.28 for matched-retry LoRA.The conclusion reports these mathematics results and says separate search training extends the interface to tool use.Source: §5 · p. 9
  • T-Router’s compressed source bank, depth-recurrent controller, and relative-scale writeback learn how earlier contributions serve subsequent reasoning while preserving pretrained transformations.The conclusion also reports that capacity-controlled comparisons favor addressable changes and recurrent context.Source: §5 · p. 9

AI use statement

  • The authors report using AI assistants for manuscript preparation and coding, say they verified assisted output, and take responsibility for the paper.They state that final decisions on the method, experimental design, and implementation were theirs.Source: AI use statement · p. 10

Guide to the appendices

A Implementation and evaluation details

  • The implementation records completed block changes after execution and makes them available to later receivers; the canonical configuration uses 32 layers and source blocks of size four.The bank is chronological in depth, and each record retains its compressor and source identity.Source: A Implementation and evaluation details · p. 14, §A.1 · p. 14

A.3 Layer schedule and source availability

  • Before each frozen layer, the controller reads existing records and applies an intervention; block-ending layers create a source for later receivers.A stored displacement follows the block’s executed path, including writebacks made within that block.Source: §A.3 · pp. 16–17

A.4 State lifetime, precision, and differentiation

  • During a forward call, each token position has its own depth-local bank and controller, while the backbone cache carries context across calls.The training procedure retains the first mixed-correctness group or the final retry attempt.Source: §A.4 · p. 17, §A.5 · p. 19

A.6 Evaluation tasks, repeated rounds, and scoring

  • MathAvg averages three task-family scores within each round, and reported table variability is the sample standard deviation across evaluation rounds.The mathematics suite covers grade-school arithmetic, broader problem solving, and competition mathematics.Source: §A.6 · p. 19, Table 12 · p. 19

A.7 Agentic search data and evaluation

This appendix specifies search-data and evaluation roles, answer-scoring procedures, auxiliary metrics, resource costs, and how architectural dimensions allocate T-Router capacity. It distinguishes validation from benchmark evaluation and reports execution settings and analytical resource counts rather than treating them as measured equal costs.

A.7 Agentic search data and evaluation

  • The ASearcher column is held-out validation, not an independent test partition; the reconstructed split contains 8 exact question–answer matches among 279 held-out records.BrowseComp Plus uses its 830-question source file for validation-only execution, while prompt filtering determines the effective evaluation count.Source: §A.7 · p. 20
  • The default search score is penalized token-F1 ×100, not the percentage of questions answered correctly.The scorer extracts the last complete answer span, applies repetition, length, and answer-tag penalties, and uses string-based scoring rather than a language-model judge.Source: §A.7 · pp. 20–21
  • Search trajectories cap prompts at 4,096 tokens, trajectories at 35,000 positions, total length at 40,000, and tool feedback at 2,048 characters.Training samples eight trajectories per question; validation samples one, and these ceilings do not establish equal token or tool-use costs across methods.Source: §A.7 · p. 21, Table 14 · p. 21

A.8 Prompt construction and final-answer extraction

  • Mathematical prompts request boxed final answers, and evaluation checks the selected final answer rather than an intermediate occurrence in the reasoning trace.Truncated responses without an explicit final answer are scored incorrect; an emitted final-answer region can still be evaluated.Source: §A.8 · pp. 21–22

A.9 Auxiliary metric definitions

  • IFEval strict@50 weights prompts equally, while its instruction-level average weights individual constraints; WikiText-2 perplexity remains on a lower-is-better scale.The perplexity evaluator scores newly exposed suffixes across overlapping windows.Source: §A.9 · p. 22

A.10 Analytical storage and projection work

  • The named auxiliary live state totals 7,936 fp32 scalars, or 31 KiB, per batch/token position before final receiving layers.This excludes temporary projection operands, backbone state, and differentiation history; controller history adds a leading O(BTLKp) scalar term during reverse-mode differentiation.Source: §A.10 · p. 23, Table 15 · p. 23

A.11 How architectural dimensions allocate capacity

  • At canonical dimensions, T-Router allocates 41,730,332 parameters; smaller blocks increase transport allocation and source–receiver pairs while reducing each record’s history interval.Figure 5 shows allocation by source rank and block size, with controller width fixed; larger blocks reduce addressable-history resolution.Source: §A.11 · pp. 23–24, Figure 5 · p. 24

B Formal analysis of cross-layer communication

This section formalizes T-Router’s causal cross-layer communication, describing its records, recurrent controller state, and interventions through frozen layers, along with the associated gradient flow, group-relative update, routing regularization, bounded retry procedure, and reporting conventions.

B.1 Notation and completed-block event ordering

  • T-Router’s routing decisions use only completed blocks, preserving token causality; the canonical configuration has 112 source–receiver pairs.The terminal record has no consumer, so deleting its append leaves the final hidden state and training loss unchanged.Source: §B.1 · p. 25

B.2 What a block change contains

  • Each compressed block record represents the change realized by the intervened block, excluding the first injection’s explicit additive term but retaining its downstream effects.The record is differentiable through both its output and anchor endpoints, whose adjoints flow through shared earlier computation.Source: §B.2 · p. 26

B.3 Source transport and the receiving subspace

  • At each receiving layer, interventions lie in a shared subspace of dimension at most r, even though the input-dependent routing interface can be nonlinear.Adaptive attention changes coefficients, not the receiving column space; subsequent frozen layers can transform the intervention.Source: §B.3 · pp. 26–27

B.4 Controller history, rank, and state bounds

  • Controller slots encode finite mixtures of past proposals whose contributions depend on both earlier slot assignments and current slot reads.The decay factor alone is not the full controller’s Jacobian norm; a contraction bound requires fixed zℓ and ∥a∥2∥ν∥2 < 2.Source: §B.4 · p. 28

B.5 RMS geometry, gradients, and threshold behavior

  • On the active calibration branch, positive rescaling of a raw projection leaves its calibrated direction unchanged, while the signed gate sets intervention magnitude.Calibration’s local derivative acts tangentially to the raw vector; the gate supplies a separate radial control.Source: §B.5 · pp. 28–29

B.6 Stop-gradient and scale-invariant optimization

  • Detaching the reference RMS removes its gradient path, while gradients still propagate through residuals, records, controllers, and frozen-layer input operations.Positive rescaling of the receiving projection leaves calibrated interventions unchanged on the active branch, but AdamW need not produce a monotone projection-norm trajectory.Source: §B.6 · pp. 30–31

B.7 Conditional propagation through the frozen decoder

  • Under Lipschitz assumptions on the frozen layers, intervention effects propagate conditionally along the actual routed trajectory, without requiring each intervention to depend only on the current hidden state.With active calibration and bounded gates, the analysis also bounds intervention size relative to the intervened hidden state.Source: §B.7 · p. 31
  • Local intervention effects depend on subsequent frozen maps: with Λ = 1 they grow as nδ, with Λ < 1 they are bounded, and with Λ > 1 they grow geometrically.The analysis distinguishes local relative scale from final-output change.Source: §B.7 · p. 31

B.8 The differentiated group-relative objective

  • The differentiated group-relative surrogate has the same gradient as sequence-normalized REINFORCE for retained observations and advantages, but not the same scalar loss value.Sequence normalization weights valid responses equally after averaging their token scores, while the KL term weights scored tokens equally across retained completions.Source: §B.8 · p. 32

B.9 Routing regularization and its reduction

  • Routing-deviation regularization acts on selection coefficients, while source-mixture regularization acts before calibration; neither directly penalizes the final calibrated writeback norm.With one available source, the routing-deviation term and its logit gradient are zero.Source: §B.9 · p. 33

B.10 Bounded informative resampling and generation budgets

  • Bounded retries increase the mass of mixed-correctness groups while preserving failed-group mass, but pre-filter mixedness need not yield valid groups with nonzero advantages.Retry acceptance occurs before preference-validity filtering.Source: §B.10 · p. 34
  • In the independent Bernoulli illustration, retry probability is symmetric between nearly always-correct and nearly always-incorrect prompts, selecting group correctness diversity rather than accuracy alone.Source: §B.10 · p. 34
  • 29,892–119,568 completions and at most 119,568M tokens are budgeted for 7,473 prompts with G = A = 4, while the number of differentiated retained groups remains 7,473.Source: §B.10 · p. 34

C Complete empirical results and design analysis

  • Generative scores are means ± sample standard deviations across seeds 42, 43, and 44; bold and underlining mark within-column best and second-best means, with lower resource costs and perplexity preferred.Source: C Complete empirical results and… · p. 35

C.1 Method comparisons across mathematics and search

T-Router outperforms full-parameter GRPO on MathAvg with a small share of the backbone’s parameters, exceeds matched-retry LoRA, and records competitive agentic-search results.

Mathematics and search comparison

  • 83.64 ± 1.16 MathAvg with 0.466% of backbone parameters compares with 73.79 ± 1.83 for full-parameter GRPO, a 9.85 ± 2.93-point improvement.MathAvg aggregates GSM8K, MATH-500, and AIME mean; differences pair evaluation-seed indices.Source: §C.1 · p. 35
  • At a similar adapter budget, matched-retry LoRA records MathAvg 77.28±1.95 versus T-Router’s 83.64 ± 1.16, a 6.36-point difference in means.Source: §C.1 · p. 35
  • On agentic search, T-Router records means of 36.98 versus 36.90 on BrowseComp Plus and 73.98 versus 72.80 on ASearch.LoRA-MoE records 37.00 BrowseComp F1 with 173.112M trainable parameters, compared with T-Router’s 36.98 and 41.730M.Source: §C.1 · p. 35, Table 19 · p. 35

C.2 Complete component ablations

Component comparisons favor addressable completed block changes and a recurrent, query-dependent controller over hidden-state memory and static routing.

Computation content and addressed retrieval

  • 83.64 ± 1.16 MathAvg for full T-Router compares with 69.48 ± 2.36 without cross-layer retrieval and 67.46 ± 2.19 with hidden-state memory.The corresponding MATH-500 means are 92.73, 79.40, and 77.80; removing source/layer identity yields a paired MathAvg difference of −25.53 ± 0.56 points.Source: §C.2 · p. 36, Table 20 · p. 36

History, selection, and relative influence

  • 72.80 ± 4.10 MathAvg for the capacity-matched MLP and 64.98 ± 1.71 for static routing support recurrent depth history and query-dependent selection.Removing RMS alignment gives a paired difference of −27.25 ± 1.24 points, while removing the token-conditioned gate gives −19.86 ± 1.47.Source: §C.2 · p. 36
  • The full interface combines calibrated relative intervention scale with a signed gate that adjusts influence to the current token.The component comparisons associate these design choices with lower MathAvg when RMS alignment or the token-conditioned gate is removed.Source: §C.2 · p. 36

C.3 Parameter and training-cost profiles of the ablations

The ablation profiles compare parameter allocation and training cost, with the full configuration using more GPU-hours than the no-retry variant.

Resource-profile scope

  • Table 21 reports exact parameter allocations and original GPU-hour records, while its optimizer-step counts remain estimates.The study separates update count from response generation and computation within an update.Source: §C.3 · p. 36, Table 21 · p. 36

Parameter allocation

  • The capacity-matched MLP uses 41,731,978 parameters, only 1,646 more than full T-Router’s 41,730,332, yet its paired MathAvg difference is −10.84 ± 3.09 points.Hidden-state memory and the no-RMS variant retain exactly the full allocation; removing the token-conditioned gate changes it by 4,352 parameters.Source: §C.3 · p. 36, Table 21 · p. 36

Training cost and retries

  • 121.23 GPU-hours and 83.64 ± 1.16 MathAvg for full T-Router compare with 37.92 GPU-hours and 62.53 ± 1.20 without retries.Their paired MathAvg difference is 21.11 ± 2.26 points; retry generation changes available completions while the estimated optimizer-step count remains the same.Source: §C.3 · pp. 36–37, Table 21 · p. 36

C.4 Configuration landscape and annual task profiles

Full T-Router has the largest mean across the four annual-profile tasks in this panel, while annual means and gaps vary across configurations.

C.4 Configuration landscape and annual task profiles

  • Full T-Router has the largest mean on all four tasks; its annual means are 63.33 and 57.78, versus Top2’s 58.89 and 41.11.The approximately 5.69-point MathAvg difference from Top2 comprises 0.33 from GSM8K, 1.84 from MATH-500, and 3.52 from pooled AIME.Source: §C.4 · p. 37, Table 22 · p. 37
  • Zero initialization has the largest annual mean gap in this panel, at 21.11 points.These annual gaps compare two task sets; seed-paired SDs are not reported for them.Source: §C.4 · p. 37
  • Zero and orthogonal initialization alter the standard-gate branch, while Local16 and Local64 change token-local branch rank and Wide512 changes controller dimensions.Top2 is a composite sparse-routing configuration specified in Appendix C.8.Source: §C.4 · p. 37

C.5 Benchmark contributions to aggregate differences

The section decomposes aggregate mean differences into family-level contributions and reports that both full T-Router and Top2 gain across all three mathematical families relative to the standard gate.

C.5 Benchmark contributions to aggregate differences

  • Full T-Router’s approximately 15.42-point difference from the standard gate comprises 3.35 points from GSM8K, 2.62 from MATH-500, and 9.45 from pooled AIME.Top2’s approximately 9.73-point difference comprises 3.02, 0.78, and 5.93 points, respectively; both configurations gain across all three families.Source: §C.5 · p. 37, Figure 8 · p. 38
  • Zero and orthogonal initialization produce net mean differences of approximately +2.93 and +1.55, respectively.Local16 contributes approximately (+1.06, −0.96, +0.74) points; Local64 contributes (+0.66, −4.60, −7.96), and the no-KL branch contributes (+0.49, −4.40, −7.78).Source: §C.5 · p. 37
  • Figure 8 decomposes each family’s signed difference from the standard gate into one-third contributions, with the net column summing those mean contrasts.The decompositions use supplied rounded means, so small rounding differences from separately reported aggregates can occur.Source: §C.5 · p. 37, Figure 8 · p. 38

C.6 Writeback configurations and task response

The writeback study relates relative intervention scale to task scores and compares learned and fixed RMS configurations. Endpoint diagnostics show that final scale alone does not order accuracy.

C.6 Writeback configurations and task response

  • Full T-Router records 83.64 ± 1.16 MathAvg at 4.0211% final writeback, with 92.73 MATH-500 and 60.56 pooled AIME means.Learned RMS initialized at 0.5% and 1% records MathAvg means of 78.89 and 78.96, with pooled AIME means of 55.00 for both.Source: §C.6 · p. 39, Table 23 · p. 39

C.7 Writeback coefficients and endpoint diagnostics

  • Fixed RMS uses constant coefficients while retaining input-dependent source routing; learned RMS uses a learned gate with a 5% bound.The standard gate has a 20% bound and 5% initial coefficient; fixed RMS configurations use 0.5%, 1%, or 2% coefficients.Source: §C.6 · p. 38, §C.7 · p. 39
  • For learned RMS initializations of 0.5%, 1%, and 2%, tail medians exceed final diagnostics by 0.2609, 0.2227, and 0.1854 points.Final scales lie closer together than the initial coefficients; fixed RMS 0.5% has a larger final diagnostic than the standard gate, yet MathAvg means of 66.41 versus 68.22.Source: §C.7 · p. 39, Figure 10 · p. 40

C.8 Structure–writeback combinations

Across structure–writeback tests and design interpretation, the paper examines how configuration changes relate to task response and frames T-Router as distinct roles for stored content, routing context, and writeback influence.

C.8 Structure–writeback combinations

  • Under fixed RMS writeback relative to original writeback, dense routing and Wide512 change by +9.05 ± 2.26 and +12.60 ± 3.41 percentage points, while Top2 changes by −13.96 ± 1.95 percentage points.Source: §C.8 · p. 40, Table 24 · p. 40
  • Wide512 with fixed RMS records 78.69 ± 0.66 MathAvg with 45.546M allocated parameters, versus 83.64 ± 1.16 with 41.730M for the selected learned-RMS design.The 3.816M allocation difference accompanies a 4.95-point difference in mean MathAvg; allocation is distinct from active memory and execution arithmetic.Source: §C.8 · p. 41

C.9 Auxiliary task profiles

  • Full T-Router records the highest IFEval mean, 38.67 ± 4.16; the compared RMS settings also report separate MuSR, GPQA-D, and WikiText-2 measures.The RMS settings share a MuSR point reference of 61.00, GPQA-D references range from 38.00 to 40.00, and WikiText-2 perplexities are 7.0890, 7.0758, and 7.1185 for fixed 2%, learned 1%, and full T-Router, respectively.Source: §C.9 · p. 42, Table 25 · p. 42

C.10 Sensitivity to the choice of family weights

  • Full T-Router’s and the 1% learned-RMS profile define the upper envelope, with full T-Router leading for equal family weighting and along the specified slice until GSM8K weight reaches about 0.9713.The 1% initialization has the largest GSM8K mean, while full T-Router has the largest MATH-500 and AIME means.Source: §C.10 · p. 43, Figure 13 · p. 43

C.11 Question resolution and alternative aggregation

  • One extra correct answer changes per-seed MathAvg by approximately 0.0253, 0.0667, or 0.5556 points for GSM8K, MATH-500, or pooled AIME.Equal-question reaggregation uses weights approximately (0.7020, 0.2661, 0.0319); full T-Router’s reaggregated mean is approximately 95.14 versus its equal-family MathAvg of 83.64, and it remains first under every nonnegative family weighting summing to one.Source: §C.11 · pp. 43–44, Figure 14 · p. 44

C.12 Evaluation-seed dispersion and complete variance record

  • The reported sample variances use three evaluation seeds, and aggregates and paired differences are formed within each seed before variance is computed.Averaging component variances would not produce the same statistic; reported SDs describe evaluation variability for the configurations.Source: §C.12 · p. 44, Table 26 · p. 45, Table 27 · p. 45

D.1 Computation and coordination as separate roles

  • T-Router’s design assigns computation to frozen decoder blocks and coordination to the auxiliary pathway, which regulates communication while preserving each receiving block’s full transformation.Source weights determine which completed changes contribute to a mixture, and the gate controls their signed influence on the receiving block.Source: §D.1 · p. 47, Table 29 · p. 47, D Design interpretation and functional… · p. 47

D.2 Why the source bank and controller carry different information

  • The source bank stores completed block content while controller slots summarize depth history; a current-state query reads that context to condition routing and gating.Source selection, receiver-specific projection, calibration, and signed gating have distinct roles; T-Router’s low-rank transport acts on completed block changes selected through an input-dependent depth route.Source: §D.2 · pp. 47–48, §D.3 · p. 48, §D.4 · p. 49

D.5 Connecting the empirical comparisons to the interface

The comparisons connect T-Router’s compact-budget reasoning performance to addressable block changes, recurrent control, and separate decisions about selection and influence.

D.5 Connecting the empirical comparisons to the interface

  • At the same allocation, block-change memory scores 83.64 MathAvg versus 67.46 for hidden-state memory; removing retrieval gives 69.48, while a capacity-matched MLP scores 72.80.The current hidden state queries accumulated depth history to obtain context for reuse.Source: §D.5 · p. 49
  • Learned static routing scores 64.98 and removing the token-conditioned gate scores 63.78; all backbone layers still execute in order.These variants address distinct receiving-layer decisions: which earlier changes contribute and how strongly their mixture enters the residual.Source: §D.5 · p. 49
  • 83.64 MathAvg with 41.73M parameters compares with 73.79 for full-parameter GRPO and 77.28 for matched-retry LoRA.The paper links this performance to composing existing intermediate contributions through banked content, depth context, and writeback.Source: §D.5 · p. 49
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