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Knowledge distillation works by sparsifying complex word interactions in student LLMs while keeping teacher-aligned simple ones.

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2026-07-13 08:02 UTC pith:A2NUGNC7

load-bearing objection Clean empirical story that multiple KD methods sparsify complex interactions (with teacher-aligned simple ones retained), plus a cheap CIP regularizer that reliably lifts six baselines; the v(x) choice is a soft spot but not fatal. the 3 major comments →

arxiv 2607.08776 v1 pith:A2NUGNC7 submitted 2026-05-05 cs.LG cs.AIcs.CLcs.GT

A Unified Approach to Interpreting Knowledge Distillation for Large Language Models via Interactions

Qingzhuo Wang , Ruiyang Qin , Zhenxin Qin , Wen Shen , Zhihua Wei This is my paper
classification cs.LG cs.AIcs.CLcs.GT
keywords knowledge distillationlarge language modelsgame-theoretic interactionsAND interactionsinteraction sparsitycomplex interactionsComplex Interaction Penaltymodel compression
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Knowledge distillation is widely used to compress large language models into smaller students, yet why so many different distillation recipes all improve performance has been unclear. This paper treats an LLM's output score as the sum of many AND-style interactions among input words, each interaction encoding a nonlinear relationship among a set of words. Across several families of models and six standard distillation methods, the common change is that the student becomes sparse in these interactions: most effects are driven near zero, while the remaining salient interactions are those the teacher itself uses, especially the simpler ones. Performance differences among methods track how well they sparsify complex interactions (those involving many words) while still retaining the teacher's salient complex ones. A lightweight penalty that explicitly shrinks complex-interaction magnitudes therefore improves existing distillers on both in-domain and out-of-distribution instruction-following benchmarks.

Core claim

The shared mechanism of diverse knowledge-distillation methods for LLMs is sparsification of interactions: after distillation a student relies on fewer high-effect interactions and drives the rest near zero, preferentially retaining teacher-aligned simple interactions and discarding complex ones as noise. A method yields higher performance precisely when it produces higher sparsity of complex interactions together with higher student–teacher overlap among the remaining complex interactions.

What carries the argument

AND (Harsanyi) interactions: the model output score on any masked input equals the sum of interaction effects I_S, each I_S quantifying the nonlinear contribution of a particular subset of input words when all of them are present. Complexity of an interaction is simply its order |S|; Gini coefficient and entropy of the |I_S| measure sparsity, and top-k overlap with the teacher measures alignment.

Load-bearing premise

The argument rests on treating a particular scalar—log-odds of the length-normalized geometric mean of ground-truth token probabilities—together with its complete AND-interaction decomposition as a faithful record of the inference logic that distillation actually transfers.

What would settle it

If student models whose complex-interaction Gini and teacher-overlap rise under CIP (or under a stronger distiller) nevertheless fail to improve ROUGE-L or GPT-judge scores on held-out instruction sets, or if the same sparsity/alignment pattern appears under ordinary supervised fine-tuning without any distillation, the claimed mechanism is refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims that diverse knowledge-distillation (KD) methods for LLMs share a common mechanism of sparsifying interactions (especially complex ones of high order |S|) extracted from a scalar output v(x) = log(¯p/(1-¯p)), where ¯p is the length-normalized geometric-mean probability of the ground-truth sequence, while preferentially retaining teacher-aligned salient simple interactions. Performance differences among KD methods are attributed to how effectively they sparsify and align complex interactions (positive Pearson correlations ~0.81). Motivated by this, the authors introduce a plug-and-play Complex Interaction Penalty (CIP) that samples and penalizes the magnitude of complex interactions (Eqs. 5–7, with m≤3 partition approximation), and show that adding CIP improves ROUGE-L and GPT-5 scores for six KD baselines across GPT-2/OPT/LLaMA families on in-domain Dolly and OOD SelfInst/Vicuna/SNI benchmarks.

Significance. If the interaction-based account is correct, the work supplies a unified, game-theoretic explanation for why KD works (sparsification akin to Occam’s razor) that goes beyond prior theoretical or representation analyses, and yields a practical, low-overhead regularizer that consistently lifts multiple existing KD methods. Strengths include the rigorous invocation of the universal-matching property (Theorem 3.1, proven in App. D.1), clean definitions of Gini/entropy/Overlap@k metrics on simple/complex partitions, extensive multi-family multi-size experiments with SFT/Base controls, λ-sensitivity and CIP-sparsity ablations (Fig. 9, Table 6), and robustness checks for β (App. G.3). The CIP intervention itself is falsifiable and empirically useful even if the full mechanistic story is incomplete.

major comments (3)
  1. [Section 3 and Appendix C] Section 3 + Appendix C: the entire mechanistic diagnosis (sparsity rise in Figs. 2–3, simple-vs-complex ΔGini in Fig. 4, Overlap@k gaps in Fig. 5, and the ~0.81 vs ~0.34 Pearson correlations with ROUGE-L in Figs. 6–7/15–18) is computed exclusively on the scalar v(x) = log(¯p/(1-¯p)) with ¯p the geometric mean of ground-truth token probabilities. Theorem 3.1 guarantees an exact AND decomposition for any scalar, so the reported patterns are properties of this particular functional. No ablation is provided with alternative scalars that more directly reflect the soft next-token distributions or teacher–student KL that KD actually optimizes (e.g., average log-prob of the full vocabulary, token-level logits, or teacher soft labels). Without such checks the claim that sparsification of complex interactions is the common mechanism of KD (rather than a description of this v) remains unvalidated a
  2. [Section 4.1 and Appendix E] Section 4.1 and Appendix E: interaction extraction is restricted to a fixed n=13 meaningful words drawn only from the “Instruction” segment (system prompt and remaining tokens held as unmasked background). While computationally necessary, this choice means the reported sparsification and teacher-overlap statistics capture only a narrow slice of the input; it is unclear whether the same simple/complex patterns hold when variables are taken from the full prompt or from student-generated outputs used by ImitKD/MiniLLM/GKD. A sensitivity check with variable n or full-sequence masking would be required to support the generality of the “common mechanism.”
  3. [Section 5, Eqs. (5)–(7)] Section 5, Eqs. (5)–(7) and Appendix D.2: CIP is approximated by randomly partitioning into m≤3 subsets and averaging absolute interaction effects over non-empty unions. The paper shows that CIP raises complex-interaction Gini and improves most (but not all) ROUGE-L/GPT-5 entries in Table 5/Fig. 8. However, there is no direct verification that the sampled L'_CIP correlates with the true (intractable) sum over Ω_complex, nor an ablation that replaces the complex-only penalty with a simple-only or all-interaction penalty. Without these controls it remains possible that any magnitude penalty on higher-order terms would produce similar gains, weakening the claim that the benefit specifically tracks the complex-interaction sparsity diagnosed earlier.
minor comments (5)
  1. [Figure 1] Figure 1 caption and surrounding text: the visualization of “fading color of the bars” is described but the actual figure (as rendered) does not make the pre-/post-distillation contrast or the red-box teacher alignment quantitatively readable; adding numerical effect values or a side-by-side density plot would help.
  2. [Table 1] Table 1 and Table 4: Overlap@k is reported only for k∈{0.05,0.1,0.15}; a continuous curve or AUC-style summary would make the alignment claim easier to compare across methods and model sizes.
  3. [Appendix C] Appendix C: the clipping ϵ=10^{-7} and the log-odds transform are stated, but the numerical stability of the Möbius inversion for I_S when many masked v(x_T) approach the extremes is not discussed; a short note on conditioning or regularization would be useful.
  4. Throughout: the terms “salient interactions,” “dross,” and “essential” are used rhetorically; defining them strictly via the top-k absolute-effect sets already introduced would improve precision.
  5. [Section 2] References: several recent KD works on f-divergences and speculative sampling are cited, but the interaction literature (e.g., earlier Harsanyi-value papers) could be more completely linked to the LLM-specific citations already present.

Circularity Check

0 steps flagged

No definitional or predictive circularity; post-hoc interaction statistics on frozen models motivate an independent regularizer whose gains are measured on external benchmarks.

full rationale

The derivation chain is observational then interventional, not self-referential. Theorem 3.1 (universal matching via Möbius inversion of any scalar v) is a mathematical identity proven in Appendix D.1 and independent of KD; interactions are extracted from already-trained models by the fixed formula I_S = sum (-1)^{|S|-|S'|} v(x_{S'}). Sparsity (Gini/entropy), simple/complex partition, and teacher overlap are measured after the fact on Base/SFT/KD students (Figs. 2–7, Tables 1,4). The correlation with ROUGE-L is empirical, not forced. CIP (Eq. 5–7) is a new sampling-based penalty on |I_S| for complex S; it is added to existing L_KD, and the paper reports that the same sparsity metrics rise while ROUGE-L/GPT-5 scores improve on held-out sets (Fig. 8, Tables 5–6). Self-citations to the authors’ prior interaction papers supply only the analytical toolkit and the interpretive claim that simple interactions are more generalizable; they do not define the KD mechanism or the CIP objective by construction. No parameter is fitted to a subset and then ‘predicted’ on a related quantity, and no uniqueness theorem is invoked to forbid alternatives. The choice of scalar v(x) (log-odds of geometric-mean GT probability) is an unvalidated modeling assumption, but that is a validity concern, not circularity. Score 1 reflects only the minor, non-load-bearing self-citation of the interaction framework.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 2 invented entities

The central claims rest on the Harsanyi/AND interaction calculus imported from prior work, a particular scalarization of sequence likelihood, an arbitrary but robustness-checked complexity threshold β, a sampling approximation for CIP, and a handful of free hyper-parameters (λ, n=13, m≤3). No new physical entities are postulated; CIP is an engineered loss, not an ontological invention.

free parameters (4)
  • λ (CIP weight)
    Trade-off coefficient between original KD loss and CIP; chosen by validation (default 0.001) and shown to have an optimal range beyond which performance degrades.
  • β (simple/complex threshold)
    Set to ⌈n/3⌉ (with robustness checks at n/5 and n/2); defines which interactions are penalized by CIP and which are analyzed as “complex.”
  • n (number of input variables)
    Fixed at 13 words extracted from the Instruction segment for computational tractability of 2^n interactions.
  • m (partition size for CIP sampling)
    Constrained to m≤3 so that every non-empty union is complex; reduces 2^n cost to 2^m.
axioms (4)
  • domain assumption Universal matching property: any LLM scalar output v(x_T) equals the sum of AND-interaction effects I_S for S⊆T (Theorem 3.1 / Li & Zhang 2023).
    Imported from prior interaction literature; the entire analysis treats interactions as the complete internal logic of the model.
  • ad hoc to paper v(x) = log(¯p/(1-¯p)) where ¯p is the geometric mean of ground-truth token probabilities (Appendix C).
    A design choice that maps sequence likelihood into an unbounded additive space required by the interaction calculus; alternative scalarizations could alter measured interactions.
  • domain assumption Simple interactions encode generalizable knowledge while most complex interactions are non-generalizable noise (Zhou et al. 2024).
    Used to interpret why sparsifying complex interactions improves performance; not re-proven here.
  • domain assumption Masking a word with a dedicated [MASK] token cleanly removes its contribution without introducing out-of-distribution artifacts.
    Standard but non-trivial assumption of the attribution literature; different baselines can change interaction values.
invented entities (2)
  • Complex Interaction Penalty (CIP) no independent evidence
    purpose: Explicit training loss that approximates the expected absolute effect of complex interactions and is added to any KD objective.
    Engineered regularizer derived from the empirical findings; not claimed to be a natural law.
  • Interaction sparsity / alignment metrics (Gini, entropy, Overlap@k on simple vs complex partitions) no independent evidence
    purpose: Quantify the hypothesized sparsification and teacher-alignment mechanisms.
    Standard statistical tools applied to the interaction spectrum; the simple/complex split is paper-specific.

pith-pipeline@v1.1.0-grok45 · 33451 in / 3250 out tokens · 42211 ms · 2026-07-13T08:02:07.811342+00:00 · methodology

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read the original abstract

Despite the success of knowledge distillation (KD) in Large Language Models (LLMs), the underlying mechanism behind its efficacy remains unclear. In this paper, we propose a unified approach to explore the common mechanism of various KD methods using interactions. Specifically, we decompose the output score of the LLM into the sum of numerous interactions. Each interaction represents a nonlinear relationship involving a set of input variables (e.g., words). Based on the decomposed interactions, we discover that the common mechanism underlying various KD methods is the sparsification of interactions, i.e., student models retain fewer interactions for inference while suppressing other interactions to zero effects. Furthermore, we discover that the performance variance across different KD methods arises from their capabilities in handling complex interactions. A KD method typically yields better performance if it enables the student model to achieve higher sparsity of complex interactions. Motivated by these insights, we propose a plug-and-play loss function called Complex Interaction Penalty (CIP) to explicitly enforce the sparsity of complex interactions during the distillation process. Extensive experiments demonstrate that integrating CIP consistently improves the performance of diverse KD methods on both in-domain and out-of-distribution benchmarks.

Figures

Figures reproduced from arXiv: 2607.08776 by Qingzhuo Wang, Ruiyang Qin, Wen Shen, Zhenxin Qin, Zhihua Wei.

Figure 1
Figure 1. Figure 1: Comparing the interaction patterns encoded by the student model before and after distillation. The distillation process drives the sparsification of interactions by suppressing most of the interactions to near-zero effects, which is visualized by the fading color of the bars. Meanwhile, the student model improves the alignment of interactions with the teacher model by learning more salient interactions fro… view at source ↗
Figure 2
Figure 2. Figure 2: plots the probability density of absolute values of the normalized interaction effects for a representative input sample x. Specifically, the normalized interaction effect is calculated as IeS = IS Max , where Max is the maximum absolute values of all 2 n − 1 interaction effects of x. Results show that the non-distilled model exhibits a relatively flat and uniform distribution of interactions. In contrast,… view at source ↗
Figure 3
Figure 3. Figure 3: The distribution of the Gini coefficient and entropy across all samples. The results demonstrate that, compared to non-distilled models (Base models and SFT models), distilled models consistently exhibit higher Gini coefficients and lower entropy. It indicates that the interactions encoded by the LLM become more sparse after distillation [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the changes in Gini coefficients between distilled and base models for simple versus complex interactions. GPT-2-0.1B OPT-1.3B Complex Interactions (𝜏 = complex) Simple Interactions (𝜏 = simple) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Correlation between the sparsity of complex interac￾tions and model performance. Results show that increased sparsity (higher Gini coefficients and lower entropy) of complex interac￾tions is positively associated with higher ROUGE-L scores. How￾ever, there is no distinct correlation between the sparsity of simple interactions and model performance, as shown in [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Correlation between the student-teacher overlap rate of complex interactions and model performance. Results show that increased student-teacher overlap rate of complex interactions is positively associated with higher ROUGE-L scores. shown in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Exploring the effects of the weight λ for on the sparsity of complex interactions (a and b), the performance of the student model (c), and the student-teacher overlap rate of complex interactions (d). Here the base model is GPT-2-0.3B. counterparts without CIP. This confirms that the CIP loss can truly increase the sparsity of complex interactions, thus effectively improving the model performance. Impact o… view at source ↗
Figure 10
Figure 10. Figure 10: The prompt template for training and evaluation of instruction-following task experiments. [System] Please act as an impartial judge and evaluate the quality of the response provided by an AI assistant to the user question displayed below. Your evaluation should consider factors such as the helpfulness, relevance, accuracy, depth, creativity, and level of detail of the response. Begin your evaluation by p… view at source ↗
Figure 11
Figure 11. Figure 11: The prompt template for single-answer grading of GPT-5 feedback from Zheng et al. (2023). 18 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: illustrates the distributions of Gini coefficient G(x) and Shannon entropy H(x) across all samples in the test set. Results show that distilled student models demonstrate an obvious distributional shift towards higher Gini values and lower entropy compared to corresponding base models and SFT models. GPT-2-0.1B OPT-1.3B LLaMA-7B GPT-2-0.3B OPT-2.7B GPT-2-0.7B OPT-6.7B [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 13
Figure 13. Figure 13: plots the change of Gini coefficients between distilled and base models for simple and complex interactions. Results show that the sparsity of complex interactions exhibits an obvious increase after distillation. GPT-2-0.1B OPT-1.3B GPT-2-0.3B OPT-2.7B GPT-2-0.7B OPT-6.7B LLaMA-7B GPT-2-0.1B OPT-1.3B GPT-2-0.3B OPT-2.7B GPT-2-0.7B OPT-6.7B LLaMA-7B Gini Coefficient Entropy Complex Interactions (𝜏 = comple… view at source ↗
Figure 14
Figure 14. Figure 14: shows the overlap rate of salient simple interactions consistently surpasses that of complex interactions. This indicates that the student model learn more simple salient interactions than complex salient interactions from the teacher model. Top5% GPT-2-0.1B Top10% Top15% GPT-2-0.3B GPT-2-0.7B OPT-1.3B OPT-2.7B OPT-6.7B LLaMA-7B Complex Interactions (𝜏 = complex) Simple Interactions (𝜏 = simple) [PITH_FU… view at source ↗
Figure 15
Figure 15. Figure 15: shows that the sparsity of complex interactions is positively correlated with model performance. Student models that exhibit higher Gini coefficients and lower entropy (indicating higher sparsity) in their complex interactions generally achieve superior ROUGE-L scores. GPT-2-0.1B GPT-2-0.3B OPT-1.3B OPT-2.7B GPT-2-0.7B LLaMA-7B OPT-6.7B [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: shows that there is no distinct relationship between the sparsity of simple interactions and the model performance. GPT-2-0.1B GPT-2-0.3B OPT-1.3B OPT-2.7B GPT-2-0.7B LLaMA-7B OPT-6.7B [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Correlation between the student-teacher overlap rate of complex interactions and model performance. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Correlation between the student-teacher overlap rate of simple interactions and model performance. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: plots the change of Gini coefficients between distilled and base models for simple and complex interactions. Results show that the sparsity of complex interactions exhibits an obvious increase after distillation. GPT-2-0.7B OPT-6.7B LLaMA-7B GPT-2-0.7B OPT-6.7B LLaMA-7B Gini Coefficient Entropy Complex Interactions (𝜏 = complex) Simple Interactions (𝜏 = simple) Complex Interactions (𝜏 = complex) Simple In… view at source ↗
Figure 20
Figure 20. Figure 20: shows the overlap rate of salient simple interactions consistently surpasses that of complex interactions. This indicates that the student model learn more simple salient interactions than complex salient interactions from the teacher model. Top10% GPT-2-0.7B OPT-6.7B LLaMA-7B Complex Interactions (𝜏 = complex) Simple Interactions (𝜏 = simple) [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: shows that the sparsity of complex interactions is positively correlated with model performance. Student models that exhibit higher Gini coefficients and lower entropy (indicating higher sparsity) in their complex interactions generally achieve superior ROUGE-L scores. GPT-2-0.7B LLaMA-7B OPT-6.7B [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Correlation between the student-teacher overlap rate of complex interactions and model performance. Here β = ⌈n/5⌉. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: plots the change of Gini coefficients between distilled and base models for simple and complex interactions. Results show that the sparsity of complex interactions exhibits an obvious increase after distillation. GPT-2-0.7B OPT-6.7B LLaMA-7B GPT-2-0.7B OPT-6.7B LLaMA-7B Gini Coefficient Entropy Complex Interactions (𝜏 = complex) Simple Interactions (𝜏 = simple) Complex Interactions (𝜏 = complex) Simple In… view at source ↗
Figure 24
Figure 24. Figure 24: shows the overlap rate of salient simple interactions consistently surpasses that of complex interactions. This indicates that the student model learn more simple salient interactions than complex salient interactions from the teacher model. Top10% GPT-2-0.7B OPT-6.7B LLaMA-7B Complex Interactions (𝜏 = complex) Simple Interactions (𝜏 = simple) [PITH_FULL_IMAGE:figures/full_fig_p033_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: shows that the sparsity of complex interactions is positively correlated with model performance. Student models that exhibit higher Gini coefficients and lower entropy (indicating higher sparsity) in their complex interactions generally achieve superior ROUGE-L scores. GPT-2-0.7B LLaMA-7B OPT-6.7B [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Correlation between the student-teacher overlap rate of complex interactions and model performance. Here β = ⌈n/2⌉. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_26.png] view at source ↗

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    Considering all the cases, the complete derivation of the sum of AND interactions is as follows. X S⊆T,S̸=∅ IAND S = X S⊆T,S̸=∅ X L⊆S (−1)|S|−|L|vand(xL) = X L⊆T X S:L⊆S⊆T (−1)|S|−|L|vand(xL)−v and(x∅) =v and(xT )| {z } L=T + X L⊆T,L̸=T vand(xL)· X|T|−|L| m=0 C m |T|−|L| (−1)m | {z } =0 −vand(x∅) =vand(xT )−v(x ∅) (10) Therefore, we have proved that ∀∅ ̸=...

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    We find that the KD methods with CIP generate more detailed and accurate responses compared with the baselines. Table 7.Instruction-following cases from thedatabricks-dolly-15kdataset. Model Type Content Case #1 InstructionName some famous rock bands from the 1960s InputNone Ground-truth The 1960s had a number of famous rock bands including The Beatles, T...

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    w/ CIP Claughaun won the 57th edition of the Limerick Senior hurling championship after defeating St

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    MiniLLM w/o CIP Claughaun won the 63rd staging an went on to win the next four senior hurling championships

    w/ CIP Claughaun won the championship. MiniLLM w/o CIP Claughaun won the 63rd staging an went on to win the next four senior hurling championships. w/ CIP Claughaun won the Limerick Senior Hurting Championship on 29 September 1957, after a 7-07-3 victory over St. Patrick’s. GKD w/o CIP Claughaun w/ CIP Claughaun won the championship on 29 September 1957 a...

    1957