Research Findings

Executive Summary

Our experiments confirm the fundamental behavior of regularised p-adic linear regression:

• Zero regularisation (λ=0): The optimal line passes through exactly n+1 points, consistent with the theory in arXiv:2510.00043v1
• Strong regularisation (λ→∞): The optimal line tends toward passing through only 1 point
• Threshold behavior: There exist critical λ values where the number of exactly-fitted points changes discretely

Update: Off-Prime Base Sweep (2025-12-06)

p-Adic dominance fails when the loss base r ≠ p. A 2D/3D sweep across primes {2,5,11} and bases {1.1,1.5,2,5} (288 datasets) found p-adic-only inversions.

• Counts: L2 inversions = 1; p-adic inversions = 4 (all 2D). 3D stayed monotone for both penalties.
• Near-binary surprise: r=1.1 already produced a p-adic-only inversion (p=2, seed=5) while L2 remained monotone.
• Heavy-base asymmetry: r=5.0 triggered p-adic-only inversions at p=2 and p=5; L2 stayed monotone in those cases.
• Example dataset: dim=2, p=2, base=1.1, n=6 with k-paths L2 [3,3,3,2,1] vs p-adic [3,2,3,2,1] (inversion).

MAJOR UPDATE: The Penalty Gap Mechanism (2025-12-05 Night)

p-Adic monotonicity dominance is a 1D phenomenon - it breaks in higher dimensions!

Discovery

In 2D/3D, p-adic can invert when L2 stays monotonic (5/160 vs 2/160 in exp059).

The Mechanism: Integer Solutions with High Valuations

3D p=2 seed=21 example:
  Fractional-slope candidate (k=4): β = [-2.08, -0.73, 0.50]
    v_p(β_slopes) = [0, 0, -1], R_padic = 2.5

  Integer-slope candidate (k=5): β = [-3, -7, 8]
    v_p(β_slopes) = [0, 0, 3], R_padic = 2.125

  The integer candidate fits MORE points AND has LOWER p-adic penalty!
  Why? 8 = 2³ has v₂(8) = 3, giving |8|₂ = 0.125

In L2: 8² = 64 (huge penalty), so L2 never switches to integer candidate.
            

Near-Binary Base Results

Near-binary bases partially suppress p-adic inversions but don't eliminate them.

Revised Understanding

Update: 4D Base-Density Resample (2025-12-21)

Higher-seed sweep of 4D inversions across mid/heavy bases confirms base-flat behaviour driven mainly by extra points and small primes.

• Near-binary immunity: r=1.05 stayed inversion-free for n ∈ {7,8,10} and primes {2,5,11}.
• Dense n=10: p=5/11 invert in 1–2/6 runs once r≥1.5 (g_hat ≈1.67 for p=5; ≈3.67 with a bump to 7.33 at r=5 for p=11); p=2 stays zero.
• Baseline n=8: only p=2 flips sparsely at r≥1.5 (g_hat ≈0.89); p≥5 invert only at r≥5 with low rates (1/6) and stay flat.
• Sparse n=7: small-prime inversions rise at r=5 (p=2 g_hat ≈2.33; p=5 hits 1/6 at r≥5); p=11 remains inversion-free.
• Takeaway: Beyond r≈1.5 the base effect is mild; inversion risk tracks extra points and small primes.

Update: Base-Factor Curve (2025-12-15)

Dense sweep over bases r ∈ {1.01…10} shows how g(r) = c(p, r)·p ramps up.

• Near-binary immunity: r ≤ 1.1 produced 0/560 inversions across primes {2,5,11,17} (dims 1–2).
• Two-phase ramp: g(r) ≈ 0.006 at r=1.2 → 0.057 at 1.5 → 0.099 at 2, then jumps to ≈0.30 at r=3 and ≈0.327 at r=5.
• Heavy-base bump: r=10 lifts g_hat to ≈0.417, suggesting a secondary rise past the r≈5 plateau.
• Prime scaling persists: Small primes drive most inversions for mild bases; p=11/17 only flip for r ≥ 3, consistent with c(r, p) ≈ g(r)/p.

Update: Corrected Inversion Counts (2025-12-09)

1D segments now export regularisation penalties, restoring accurate inversion detection.

• No phantom failures: Every non-monotonic k-path now coincides with a Pareto inversion.
• Updated 1D rates (r ≥ 1.5): n=5 → 2.5–5.0%, n=6 → 2.5–8.8%, n=7 → 7.5–17.5%.
• Near-binary immunity: r ∈ {1.02, 1.1} still show 0/410 inversions.
• Implication: Base sensitivity remains, but inversion events are rarer at small r and scale with extra points beyond d+1.

Finding 1: Confirmation of n+1 Point Theorem

Date: 2025-12-01

Status: Validated

In 1D (n=1) regression without regularisation, our exhaustive search algorithm consistently finds optimal lines passing through exactly 2 points (= n+1).

Evidence

Dataset: (0, 0), (1, 2), (2, 3), (3, 7)
Result at λ=0: line y = 0 + 2x passes through 2 points

Dataset: (0, 1), (1, 2), (2, 4), (4, 5)
Result at λ=0: line passes through 3 points (special case)
            

Note: When the data has special structure (nearly collinear), the optimal line may pass through more than n+1 points.

Finding 2: Discrete Threshold Behavior

Date: 2025-12-01

Status: Validated

The number of exactly-fitted points changes at discrete threshold values of λ, not continuously. This suggests a phase transition phenomenon.

Evidence

Dataset: (0, 1), (1, 2), (2, 4), (4, 5)

Threshold detected between λ=1.2 and λ=1.3:
  - Below threshold: 3 points fitted exactly
  - Above threshold: 1 point fitted exactly
            

Interpretation

This discrete behavior is consistent with the combinatorial nature of p-adic optimization: the optimal line either passes through a point exactly (residual = 0, infinite valuation) or it doesn't (finite valuation). There are no "partial" fits.

Finding 3: Regularisation Type Matters

Date: 2025-12-01

Status: Preliminary

p-Adic regularisation (penalizing |β|_p) behaves differently from real L2 regularisation (penalizing β²).

Evidence

Dataset: (0, 0), (1, 2), (2, 3), (3, 7)

Real L2 Regularization (λ=0.1):
  Intercept = 1, Slope = 1, Exact fits = 2

p-Adic Regularization (λ=0.1):
  Intercept = -5, Slope = 4, Exact fits = 2
            

The p-adic regularisation prefers coefficients with high p-adic valuation (more divisible by p), leading to different optimal solutions.

Finding 4: Monotonicity of k(λ) - REVISED

Date: 2025-12-02 (Original), 2025-12-03 (Revised)

Status: Conditional - Counter-examples Found!

Original claim: k(λ) is monotonically non-increasing in λ.

Revision (Day 3): Monotonicity is not universal. Stress testing found 157 counter-examples out of 8480 tests where k(λ) actually increases as λ increases!

Example Counter-Example

Dataset: x = [5, 7, 8, -8, -6], y = [10, 10, -7, 10, -5]

k-path (base r=2, prime p=2):
  λ ∈ [0.00, 0.07]: Line y = 3.18 + 1.36x  → k = 2
  λ ∈ [0.07, 12.3]: Line y = -5.86 - 0.14x → k = 2
  λ ∈ [12.3, ∞]:    Line y = 10 (horizontal) → k = 3

k INCREASES from 2 to 3 as λ increases!
            

Why This Happens

1. Three points share y=10: (5,10), (7,10), (-8,10)
2. The horizontal line y=10 passes through all three
3. At λ=0, a tilted line fits 2 points with better data loss
4. At large λ, the horizontal line (slope=0, reg penalty=0) wins
5. The horizontal line happens to fit more points!

Revised Understanding

Conditional Monotonicity Theorem (Conjectured):
k(λ) is monotonically non-increasing if and only if no horizontal hyperplane passes through more points than the optimal hyperplane at λ=0.

Statistics from Stress Test

Higher dimensions have lower failure rates (harder to accidentally align points).

Finding 5: Analytical Threshold Formula

Date: 2025-12-02

Status: Validated

Thresholds occur when two candidate solutions have equal total loss. For 1D regression comparing lines with slopes b₁ and b₂:

λ* = (L₂ - L₁) / (b₁² - b₂²)

where L₁, L₂ are the data losses (without regularisation) of the two lines.

Evidence

Dataset: (0,1), (1,2), (2,4), (4,5)

Theoretical prediction: λ* = 1.25 (line y=1+x vs y=1)
Empirical threshold: λ ∈ [1.24, 1.25]

Perfect match between theory and experiment!
            

Finding 6: 2D Generalization Confirmed

Date: 2025-12-02

Status: Validated

The n+1 theorem and monotonicity extend to 2D regression (fitting planes to 3D points). At λ=0, optimal planes pass through at least 3 points (n+1 for n=2).

Evidence

15/15 random 2D datasets satisfy:
  - k(0) ≥ 3 (n+1 theorem holds)
  - k(λ) is monotonically non-increasing
  - Discrete thresholds exist (1-2 thresholds typical)

Example: Dataset with 5 points
  λ=0.00: plane z = 1 + 0.5x₁ + 1.5x₂ fits 4 points
  λ=0.50: plane z = 2 + x₁ fits 3 points
  λ=2.00: plane z = 5 fits 1 point
            

Finding 7: Prime Dependence of Thresholds

Date: 2025-12-02

Status: Validated

The threshold structure depends on the choice of prime p. Different primes yield different numbers and locations of thresholds.

Evidence

Dataset: (0,1), (1,2), (2,4), (4,5)

p=2:  1 threshold at λ ≈ 1.25 (k: 3→1)
p=3:  2 thresholds at λ ≈ 0.45 (3→2) and λ ≈ 3.95 (2→1)
p=5:  2 thresholds at λ ≈ 1.35 (3→2) and λ ≈ 3.95 (2→1)
p=7:  2 thresholds at λ ≈ 1.35 (3→2) and λ ≈ 3.95 (2→1)
            

This reveals that p-adic regression is truly prime-dependent, not just in the valuation function but in the geometry of the optimization landscape.

Finding 8: Exact Asymptotic Threshold Formula (MAJOR)

Date: 2025-12-02 (Evening)

Status: Validated

Discovered and validated an exact formula for the threshold λ* as a function of the loss base r. This provides complete analytical understanding of threshold behavior.

The Formula

λ* = (L₂ - L₁) / (b₁² - b₂²)

where L = Σᵢ r^{-v_p(residual_i)}

Asymptotic Expansion

The threshold admits an exact series:

λ* = c₀ + c₁/r + c₂/r² + c₃/r³ + ...

where c_k counts residuals with valuation exactly k.

Evidence (100% Match)

Dataset: canonical_threshold
Derived formula: λ* = 1 + 1/r²

Verification:
  r=2:    λ* = 1.25      (predicted: 1.25)     ✓
  r=3:    λ* = 1.111...  (predicted: 1.111...) ✓
  r=5:    λ* = 1.04      (predicted: 1.04)     ✓
  r=10:   λ* = 1.01      (predicted: 1.01)     ✓
  r=100:  λ* = 1.0001    (predicted: 1.0001)   ✓
  r=1000: λ* = 1.000001  (predicted: 1.000001) ✓

Error: 0.00 (exact match for all tested values)

Dataset: gentle_line
Derived formula: λ* = 1 + 1/r
Also verified with zero error.
            

Physical Interpretation

The formula reveals that threshold behavior is entirely determined by the p-adic valuations of residuals. The limit as r→∞ depends only on residuals with valuation 0 (coprime to p).

Finding 9: d=4 and d=5 Validation

Date: 2025-12-03

Status: Validated

Extended the exact threshold solver to higher dimensions (d=4, d=5). The n+1 theorem and threshold formula continue to hold.

Results

d=4 (8 points): 14/14 configurations monotonic
  - canonical_4d: k path 5→2→1 (2 thresholds)
  - random datasets: k path 5→2 (monotonic)

d=5 (9 points): 2/2 configurations monotonic
  - canonical_5d: k path 6→5 (1 threshold)
  - random: k path 6→2 (4 thresholds)

All d≥4 tests passed n+1 property: k(0) ≥ d+1
            

Computational Scaling

d=4: ~140 candidate hyperplanes for 8 points
d=5: ~200 candidate hyperplanes for 9 points
Time: ~30ms per test
            

Finding 10: Pareto Frontier Monotonicity Theorem (MAJOR)

Date: 2025-12-05

Status: Validated - 100% Accuracy

After the conditional monotonicity conjecture failed (Day 4), we discovered the true necessary and sufficient condition for k(lambda) monotonicity.

The Theorem

Pareto Frontier Monotonicity Theorem: For L2-regularised p-adic linear regression, k(lambda) is monotonically non-increasing in lambda if and only if there is no "Pareto inversion" in the optimal path.

A Pareto inversion occurs when, for consecutive optimal solutions beta_1, beta_2 with R(beta_1) > R(beta_2), we have k(beta_1) < k(beta_2).

Intuitive Explanation

As lambda increases, the optimal solution moves toward lower regularisation penalty R. If a lower-R solution happens to fit MORE points than a higher-R solution, then when we switch to it, k increases - breaking monotonicity.

Example

Segment 1: R = 11.56, k = 2
Segment 2: R = 3.24,  k = 1
Segment 3: R = 0.049, k = 2  ← INVERSION! k increased
Segment 4: R = 0.0,   k = 1

k-path: 2 → 1 → 2 → 1 (non-monotonic due to inversion at segment 3)
            

Validation

Why This Supersedes the Conditional Monotonicity Conjecture

The earlier conjecture (k_opt >= k_horiz implies monotonicity) was incomplete because it only considered horizontal hyperplanes. The Pareto criterion considers ALL low-penalty candidates, including tilted planes that happen to have high exact-fit counts.

Finding 11: Pareto Theorem Formally Proven

Date: 2025-12-07

Status: Proven

The Pareto Frontier Monotonicity Theorem is now rigorously proven (see proofs/pareto_monotonicity_proof.md).

Key Lemmas

1. Finite Candidates: Optimal β*(λ) belongs to a finite set C for all λ ≥ 0
2. Piecewise Linear: Total loss L_total(c; λ) = L_data(c) + λ·R(c) is linear in λ for each candidate
3. R Decreases at Transitions: At every transition λ* from c_1 to c_2, R(c_1) > R(c_2)

Proof Sketch

The proof follows directly from the structure:

• k(λ) non-monotonic ⟺ k increases at some transition
• At any transition, R strictly decreases (by construction of lower envelope)
• k increasing while R decreasing ⟺ Pareto inversion by definition

Therefore: k(λ) non-monotonic ⟺ Pareto inversion exists. ∎

Finding 12: Strong Inversion Predictor

Date: 2025-12-07

Status: Validated - 91.7% accuracy

The simple test "does the horizontal hyperplane fit more points than the λ=0 optimum?" is a strong predictor for inversions:

When k_horiz > k_lambda0: 91.7% have inversions (11/12 cases)
When k_horiz <= k_lambda0: only 3.8% have inversions (53/1378 cases)
            

Practical Implication

This provides a cheap O(n) heuristic for detecting inversion-prone datasets without computing the full candidate set.

Finding 13: Empirical Inversion Probability Formula

Date: 2025-12-07

Status: Empirically Validated

From testing 1390 datasets, the inversion probability follows a scaling law:

P(inversion) ≈ 0.10 × excess_points

where excess_points = (n - d - 1) / n

Interpretation

• excess_points measures the fraction of "extra" data points beyond the minimum (d+1) needed to define a hyperplane
• At n = d+1, excess_points = 0, so P(inv) = 0 (inversions impossible)
• As n grows relative to d, more extra points → more chances for inversions

Inversion Rates by Configuration

Why Higher Dimensions Help

In higher dimensions, d+1 is larger, so excess_points = (n-d-1)/n is smaller for the same n, reducing inversion probability. This explains why 2D and 3D have lower inversion rates than 1D at comparable n values.

Finding 14: k_horiz Predictor Limitation (MAJOR)

Date: 2025-12-05

Status: Important Limitation Discovered

The k_horiz > k_lambda0 predictor (91.7% accurate on random data) fails completely on axis-duplication structured data.

Evidence (exp047)

540 runs: dims {2,3,4}, n=10, axis-dup generator
- Only 1/540 runs had k_horiz > k_lambda0 (100% precision, 2% recall)
- 48/49 inversions happened when k_horiz <= k_lambda0
            

Why This Happens

In axis-duplication data:

• k_lambda0 ≈ 7-10 (tilted axis-aligned plane fits MANY points due to alignment)
• k_horiz ≈ 4 (horizontal only fits the most duplicated y-value)
• k_horiz is almost always < k_lambda0

New Inversion Mechanism

Example k_path: [10, 3, 4]
1. At λ=0: tilted axis-aligned plane fits all 10 points (k=10)
2. At higher λ: intermediate plane with k=3 takes over
3. At high λ: horizontal (R=0) with k=4 wins

The inversion is 3→4, NOT related to k_lambda0!
            

Implications

• k_horiz > k_lambda0 is NOT a universal inversion predictor
• Works for random data where k_lambda0 ≈ d+1
• Fails for structured data where axis alignment inflates k_lambda0
• Two different regimes require different detection strategies

Finding 15: k_min Predictor Breakthrough (MAJOR)

Date: 2025-12-05

Status: Validated - F1=0.962 with Perfect Precision

A new predictor based on k-path analysis achieves near-perfect inversion detection, vastly outperforming all previous approaches.

The Predictor

k_min_intermediate < k_final: Check if the minimum k among all segments (except the final one) is less than the final k.

Results (exp049)

Two Types of Inversions Discovered

Type 1: End inversions (92.7%) - Captured by k_min_intermediate < k_final

• The minimum k occurs before the final segment
• Example: [10, 2, 4] → inversion at 2→4

Type 2: Middle inversions (7.3%) - Missed by k_min predictor

• k dips and rises in the middle, but final k is still the minimum
• Example: [3, 2, 3, 2, 1] → inversion at 2→3 in middle

The Perfect Predictor (Tautological)

Checking if k ever increases anywhere in the path achieves F1=1.000. This is tautological (inversion = k increases), but validates the theoretical framework.

Why This Matters

• Theoretical completeness: We now have a perfect characterization of inversions via k-path analysis
• Practical prediction: k_min < k_final catches 92.7% of inversions with zero false positives
• k_horiz limitation explained: It fails on structured data because k_lambda0 is inflated by axis alignment

Finding 16: Tie-Point Phenomenon (2025-12-05)

Status: Analyzed and Explained

Investigation of the rare case from exp050 where k increased without R decreasing revealed a new phenomenon: tie points.

What is a Tie Point?

A tie point occurs when two candidates have exactly equal total loss at some λ:

At λ ≈ 1.0667:
  k=3 candidate: L_data=3.2 + λ*0.25 = 3.467
  k=2 candidate: L_data=3.4 + λ*0.0625 = 3.467
  Both candidates are equally optimal!
            

Observation

At tie points, the solver records both candidates, creating zero-width segments (width ≈ 10⁻¹⁵). The k "oscillation" (2→3→2) is multi-valuedness at the corner point, NOT a Pareto inversion.

Key Insight

• Tie-point k-increases are followed by immediate k-decreases (oscillations)
• They have zero net effect on k_min
• The k_min predictor remains valid: k_min < k_final still implies inversion exists
• The Pareto theorem is exact: k truly increases only when R strictly decreases

Finding 17: k_min Predictor Theorem Proven (2025-12-05)

Status: Formally Proven

The k_min predictor theorem is now rigorously proven (see proofs/kmin_predictor_proof.md).

The Theorem

Theorem: If k_min_intermediate < k_final, at least one Pareto inversion exists.

Proof Sketch

1. If k_min < k_final, there's a net k-increase from some intermediate segment to the final
2. Tie-point oscillations have zero net effect (k goes up then down)
3. Therefore, at least one k-increase must be from a true Pareto inversion (R drops while k rises)

Implications

• Precision = 1.000: Every flagged dataset has a true inversion (no false positives)
• Recall = 0.927: Catches 92.7% of inversions (misses "middle inversions")
• Computational cost: O(n) after path computation

Finding 18: Cheap Signals Cannot Replace k_min (2025-12-05)

Status: Negative Result

Attempted to predict inversions using only O(n×d) features without full path computation.

Results

Conclusion

Cheap features (duplication, k_horiz) cannot reliably predict inversions. The full path computation is necessary for accurate inversion detection. The k_min predictor remains the best available method.

Finding 19: Middle Inversion Characterization (2025-12-05)

Status: Analyzed

The 7.3% of inversions missed by the k_min predictor are "middle inversions" with a specific k-path pattern.

Definition

A middle inversion is a Pareto inversion where k bounces UP in the middle of the path but the final k is still the overall minimum. These are missed by the k_min predictor (which checks k_min_intermediate < k_final).

Observed K-Path Patterns

[3, 2, 3, 2, 1] - k drops to 2, bounces to 3, continues to final k=1
[3, 3, 2, 3, 2, 1] - k plateaus at 3, drops to 2, bounces to 3, continues to final k=1
[5, 5, 2, 3, 2, 2] - k plateaus at 5, drops to 2, bounces to 3, settles at final k=2
            

Statistics (from exp049)

• Total inversions: 41 (8.4% of 486 runs)
• End inversions (k_min < k_final): 38 (92.7% of inversions)
• Middle inversions (k_min >= k_final): 3 (7.3% of inversions)

Characteristics Comparison

Key Insight

Middle inversions require a "valley-then-bounce" pattern: k first decreases, then increases (the inversion), then continues to decrease to the final value. Because the final k is the minimum, the k_min predictor doesn't flag these cases.

Practical Implication

Middle inversions are rare (0.6% of all runs) and detecting them requires full path computation. The k_min predictor remains nearly optimal (92.7% recall with 100% precision) for practical use.

Finding 20: Perfect Inversion Detector via k-up Transitions (2025-12-05 Evening)

Status: Proven (tautological)

The count of k-increasing transitions in the optimal path perfectly detects inversions.

The Discovery

k_up_transitions >= 1 is a perfect inversion detector:

• Precision: 1.000 (zero false positives)
• Recall: 1.000 (catches ALL inversions)
• F1: 1.000

This is tautological: a Pareto inversion IS by definition a k-up transition (k increases while R decreases).

Distribution (270 runs, 2D quick sweep)

Key insight: ALL inversions have >= 1 k-up transition. ALL non-inversions have 0 k-up transitions.

Middle vs End Distinguished by Position

The average relative position of k-up transitions distinguishes the types:

• Middle inversions: avg position = 0.298 (early in path)
• End inversions: avg position = 0.633 (late in path)

Path Length as Distinguisher

Theoretical Explanation

Middle inversions require the k-path to:

1. Drop to some intermediate minimum
2. Bounce UP (the inversion)
3. Continue DOWN to the final minimum

This needs at least 3 k-changes, requiring longer paths (typically 5+ segments).

Practical Implication

While k_up >= 1 is a perfect detector, it requires computing the full optimal path. No truly "cheap" predictor exists - detecting inversions fundamentally requires path computation.

Finding 21: p-Adic Regularisation Monotonicity Dominance (2025-12-05 Late Evening)

Status: MAJOR DISCOVERY

p-Adic regularisation (|β|_p penalty) is strictly more monotonic than L2 regularisation (β² penalty).

Statistical Evidence (1000 random 1D datasets)

Critical observation: Zero cases found where p-adic inverts but L2 doesn't!

Conjecture: p-Adic Monotonicity Dominance

For any dataset (X, y):

• If p-adic regularisation has a Pareto inversion, then L2 also has one.
• The converse is FALSE: L2 can invert when p-adic doesn't.

Mechanism: Valuation Classes and Threshold Reduction

93% fewer thresholds with p-adic regularisation:

Why This Happens

L2 penalty β²: Continuous - every distinct slope gives a unique penalty.

p-Adic penalty |β|_p: Discrete valuation classes:

v_p(β) = 0:  |β|_p = 1   (e.g., β=1, 3, 5, 7, ...)
v_p(β) = 1:  |β|_p = 0.5 (e.g., β=2, 6, 10, ...)
v_p(β) = 2:  |β|_p = 0.25 (e.g., β=4, 12, 20, ...)
            

All slopes within a valuation class have THE SAME penalty! When two candidates are in the same class, no threshold exists between them - the better data-loss candidate wins for ALL λ.

Threshold Formula Comparison

L2: λ* = (L₂ - L₁) / (b₁² - b₂²) [continuous denominator]

p-Adic: λ* = (L₂ - L₁) / (|b₁|_p - |b₂|_p) [discrete denominator]

Practical Implications

• Safer regularisation: p-adic is strictly safer than L2 for avoiding inversions
• Different preferences: p-adic prefers β=4 over β=1 (for p=2), despite |4| > |1|
• Simpler paths: p-adic often skips intermediate candidates that L2 visits

Open Questions

1. Prove Pareto Theorem Analytically - DONE (Day 7)
2. Characterize Inversion Probability - DONE (Day 7): P ≈ 0.10 × (n-d-1)/n
3. Efficient Inversion Detection - SOLVED (Day 34): k_min predictor achieves F1=0.962
4. Two-Regime Detection - SUPERSEDED: k_min predictor works across regimes
5. The 3.8% Anomaly - EXPLAINED: Middle inversions (7.3% of cases)
6. Cheap k_min Estimation - INVESTIGATED: No cheap alternative found
7. k_min Predictor Proof - DONE: See proofs/kmin_predictor_proof.md
8. Middle Inversion Geometry - CHARACTERIZED (Day 35): Valley-then-bounce pattern in longer paths
9. Perfect Inversion Detector - PROVEN (Day 35): k_up >= 1 is perfect but requires path
10. p-Adic Regularisation - INVESTIGATED (Day 35): Strictly more monotonic than L2!
11. Prove p-Adic Monotonicity Dominance: Formally prove the conjecture
12. Higher dimensions for p-adic: Does ~93% threshold reduction persist in 2D, 3D?
13. Hybrid regularisation: Mix of L2 and p-adic penalties
14. Formal Lean/Coq Proof: Translate the k_min theorem to a theorem prover