galimova/mhc-mup-diagnostics

Dynamic mHC µP Diagnostic Report

Empirical Verification of Theorem Conditions for V4-style Dynamic Residual Routing

Date: 2026-01-05
Config: DeepSeek V4 (n_s=4, K=20, hidden_size=7168, 61 layers)

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Executive Summary

We ran all six diagnostics prescribed by the updated plan against a pure-PyTorch reimplementation of DeepSeek V4's dynamic mHC mechanism, sweeping width d ∈ {64, 128, 256, 512, 1024} with fixed n_s = 4 and K = 20.

The headline finding: Under standard initialization (σ_W = 0.02, s_C = 0.1), the dynamic generator sensitivity ||DC(x)|| · ||x|| scales as d^0.98 ≈ d^1, violating the O(1) condition required by Theorem D (condition 4). This is the central technical obstacle for µP transfer in dynamic mHC.

The fix: Multiple corrected parameterizations restore O(1) scaling. The cleanest are:

• Correction A (s_C = 1/(n_s·d)): yields d^{-0.035} ≈ O(1) for comb
• Correction C (σ_W = 1/(n_s·d)): yields d^{-0.037} ≈ O(1) for all three (comb, pre, post)

Correction C is the µP-natural choice: it corresponds to standard µP fan-in initialization for the generator weight W_C.

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Diagnostic Results

Diagnostic 1: Finite-K Sinkhorn Error ε_K

Width	ε_K (mean)	ε_K (max)
64	1.0e-6	1.0e-6
128	1.0e-6	1.0e-6
256	1.0e-6	1.0e-6
512	1.0e-6	1.0e-6
1024	1.0e-6	1.0e-6

Conclusion: ε_K ≈ eps = 1e-6 (the additive epsilon in the Sinkhorn loop dominates). With K=20 and n_s=4, Sinkhorn convergence is essentially exact. The finite-K error is not a practical concern for V4's configuration.

Impact on Theorem A: The frozen realized comb stability theorem holds essentially exactly: ||C||_2 = 1 + O(ε_K) with ε_K ≈ 10^{-6}.

Diagnostic 2: Frozen Mixer Spectral Norm ||C||_2

| Width | ||C||_2 (mean) | κ(C) (mean) | |-------|----------------|-------------| | 64 | 0.999999 | 1167 | | 128 | 0.999999 | 652 | | 256 | 0.999999 | 409 | | 512 | 0.999999 | 462 | | 1024 | 0.999999 | 264 |

Conclusion: ||C||_2 ≈ 1 to 6 decimal places. The Birkhoff constraint works exactly as Theorem A predicts. The realized comb is nonexpansive.

Note: The high condition number κ(C) means that while the maximum singular value is 1, the minimum singular value is small (~0.001). This means C is close to a rank-deficient matrix — consistent with Sinkhorn producing near-permutation matrices when logits have moderate spread.

Diagnostic 3: Sinkhorn Quotient-Jacobian Spectrum

This diagnostic is width-independent (operates on fixed n_s=4 matrices).

K	σ_max	σ_min	κ	Gauge Leakage
1	0.358	0.065	6.86	0.278
2	0.353	0.064	6.93	0.091
5	0.354	0.063	8.13	0.006
10	0.355	0.058	9.92	0.0001
20	0.354	0.067	7.40	0.000002
50	0.355	0.063	7.91	0.000000

Key Findings:

1. The quotient Jacobian is well-conditioned. κ ≈ 7-10 across all K values. This validates Theorem C's assumption.

2. σ_max ≈ 0.35, σ_min ≈ 0.06. The Sinkhorn projection is a contraction on G^⊥ (σ_max < 1). Perturbations are damped, not amplified.

3. Gauge leakage drops exponentially with K. At K=20, leakage is 2e-6. The Sinkhorn Jacobian maps G^⊥ almost perfectly into G^⊥.

4. The spectrum is K-independent for K ≥ 5. Convergence is fast.

5. The (n_s-1)² = 9 singular values have a smooth distribution — all gauge-perpendicular directions are treated comparably.

Diagnostic 4: Dynamic Sensitivity (THE KEY RESULT)

| Width | ||DC(x)||·||x|| | ||Dp(x)||·||x|| | ||Dq(x)||·||x|| | |-------|-----------------|-----------------|-----------------| | 64 | 0.147 | 0.139 | 0.274 | | 128 | 0.284 | 0.271 | 0.543 | | 256 | 0.559 | 0.532 | 1.066 | | 512 | 1.112 | 1.052 | 2.110 | | 1024 | 2.235 | 2.086 | 4.158 |

Scaling exponents:

• ||DC(x)||·||x|| ~ d^{0.98}
• ||Dp(x)||·||x|| ~ d^{0.98}
• ||Dq(x)||·||x|| ~ d^{0.98}

This is the smoking gun. All three dynamic sensitivities scale linearly with width.

Jacobian Chain Decomposition

Component	Scaling	Value at d=1024
		D(RMSNorm)
		W_comb
	s_C
		DS_K
		x

The two √d factors:

1. ||W_comb||₂ ~ √d: Generator weight spectral norm (fan-in = n_s·d)
2. ||x|| ~ √d: Multi-stream state norm (n_s·d entries)

Product: O(1) · O(1) · √d · O(1) · √d = Θ(d)

Diagnostic 5: Generated-Logit Update Scale

| Width | ||Π_{G^⊥} ΔZ||₂ | Perp/Total Ratio | |-------|------------------|------------------| | 64 | 0.00317 | 0.998 | | 128 | 0.00553 | 0.997 | | 256 | 0.01006 | 0.997 | | 512 | 0.01950 | 0.994 | | 1024 | 0.03826 | 0.996 |

Conclusions:

1. Almost all of ΔZ is in G^⊥ (>99.7%). Gradient updates naturally avoid gauge directions.
2. ||Π_{G^⊥} ΔZ||₂ grows with width (~√d), requiring LR compensation.

Diagnostic 6: Pre/Post Gate Statistics

| Width | p̄ (mean) | ||p||₁ | q̄ (mean) | ||q||_∞ | |-------|----------|--------|----------|---------| | 64 | 0.498 | 1.993 | 0.998 | 1.014 | | 128 | 0.501 | 2.003 | 1.000 | 1.024 | | 256 | 0.499 | 1.998 | 1.002 | 1.033 | | 512 | 0.497 | 1.989 | 1.005 | 1.053 | | 1024 | 0.502 | 2.009 | 1.008 | 1.072 |

Gates are stable across widths. Pre-weights center at 0.5 (sigmoid midpoint), post-weights at 1.0 (2·sigmoid(0)).

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Corrected Parameterizations

| Correction | Description | ||DC||·||x|| scaling | |------------|-------------|---------------------| | Baseline | σ_W=0.02, s_C=0.1 | d^{0.98} ❌ | | A | s_C = 1/(n_s·d) | d^{-0.04} ✅ (comb only) | | B | σ_W = s_C = 1/√(n_s·d) | d^{-0.04} ✅ | | C | σ_W = 1/(n_s·d), s_C=0.1 | d^{-0.04} ✅ (all gates) | | D | s_C = 1/√d, σ_W=0.02 | d^{0.46} ❌ | | E | σ_W = 1/√(n_s·d), s_C=0.1 | d^{0.47} ❌ |

The µP Rule for Dynamic mHC Generator Weights

Parameter	Init	LR scaling
W_ℓ^a ∈ ℝ^{(2+n_s)n_s × n_s·d}	σ² = 1/(n_s·d)²	η/d
s_ℓ^a ∈ ℝ³	O(1)	η
b_ℓ^a ∈ ℝ^{(2+n_s)n_s}	0	η

Key insight: The generator weight's effective fan-in is n_s·d (the total multi-stream dimension), not d.

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Implications for Theorem D

The conditions are:

1. ✅ Branch f_ℓ^a satisfies standard spectral µP (assumed)
2. ✅ ε_K = O(1), actually ε_K ≈ 10^{-6} (Diagnostic 1)
3. ✅ Quotient Jacobian well-conditioned, κ ≈ 7-10 (Diagnostic 3)
4. ✅→ Requires Correction C: σ_W = 1/(n_s·d) gives O(1) (Diagnostic 4)
5. ✅→ Requires LR scaling: η_W = Θ(1/d) (Diagnostic 5)

With both corrections applied, all five conditions of Theorem D are satisfied.

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V4 Sinkhorn Implementation Notes

From kernel.py:

1. Init: Row-softmax + eps, then col-normalize
2. Iterations: K-1 repetitions of (row-normalize, col-normalize)
3. Convention: comb[j,k] with j=output stream, k=input stream
4. hc_post: y_o = q_o · f(y) + Σ_i C[i,o] · residual_i

Files

• mhc_diagnostics.py — Complete diagnostic implementation (all 6 diagnostics)
• mhc_analysis.py — Chain decomposition, corrections, figures