Quadratic Contraction Lemma — “No Linear Terms Survive”

A referee-facing repo with proofs, stress tests, and cross-fit validation.

This folder contains:

• A rigorous, self-contained derivation of why the linear term vanishes (the order-1 Ursell/cumulant is zero after local extraction/centering).
• A proof of the Quadratic Contraction Lemma: for the KP norm of post-RG activities, $$ \eta_{k+1} \le A,\eta_k^2 \quad\text{with }A\text{ geometry-only}. $$
• Reproducible numerical evidence:
  1. Pure quadratic dynamics vs. a linear leak.
  2. A cumulant test (centered vs. uncentered).
  3. A cross-fit test with z-scores and a Hessian≈Covariance check.

The results are packaged as figures/CSVs and compiled into a LaTeX report.

---

Quick Start

1. Python Environment Setup

python3 -m venv env
source env/bin/activate
pip install --upgrade pip
pip install numpy pandas matplotlib

2. Run the Quadratic Recursion Stress Tests

This script tests the stability of the η_k recurrence.

How to Run:

python3 contraction_stability_analyzer_v2.py run --file scenarios.json

(Requires a scenarios.json file, see section 5 for an example)

Outputs (in contraction_analysis_results/):

• eta_k_decay.png: Shows that η_k decays double-exponentially under a pure quadratic flow.
• loss_product.png: Shows that the cumulative loss product $\prod (1 - C,\eta_j)$ remains strictly positive for ideal scenarios but vanishes if a linear leak is present.
• results_summary.csv: A table with detailed metrics for all scenarios.

Interpretation:

• PASS: Ideal runs show is_summable: True. This means the loss product is greater than zero, providing numerical support for the survival of the string tension.
• FAIL (by design): Scenarios with a linear leak (eps > 0) show is_summable: False, demonstrating the fragility of the proof to such a term.

3. Run the Cross-Fit “No Linear Term” Validation

This script provides referee-grade numerical evidence that the centering procedure eliminates linear terms.

How to Run:

python3 no_linear_terms_survive_crosssplit.py

Outputs (in cv_outputs/):

• zscore_per_coordinate.png: A plot showing the z-scores (|gradient / std_error|) for each coordinate. All should be below the significance threshold (e.g., 3.0).
• summary.json: A one-line verdict with the max z-score and the Hessian vs. Covariance error.

Interpretation:

• PASS: A PASS in summary.json (e.g., z_max_direct <= 3.0) means the residual gradient is statistically indistinguishable from zero. This provides strong numerical support that the linear term vanishes.

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Theory in a Nutshell

• Extraction/Centering: An operator is split into its expectation and a centered remainder, $begin:math:text$K = \mathbb{E}[K] + \widetilde K$end:math:text$, where by construction $begin:math:text$\mathbb{E}[\widetilde K]=0$end:math:text$.
• Connected Expansion (BKAR/Ursell): The post-RG activity is a sum over connected clusters of centered fine-scale operators: $$ K_{k+1}(\Gamma') ;=; \sum_{n\ge 1}\frac{1}{n!}!!\sum_{\Gamma_1,\dots,\Gamma_n} \Phi_T(\Gamma_1,\dots,\Gamma_n),\prod_i \widetilde K(\Gamma_i). $$ The $begin:math:text$n=1$end:math:text$ term is the first cumulant, which is $begin:math:text$\mathbb{E}[\widetilde K]=0$end:math:text$. Therefore, the expansion has no linear term and starts at the quadratic ($begin:math:text$n=2$end:math:text$) level.
• Quadratic Bound: The tree-graph inequality for the $begin:math:text$n\ge 2$end:math:text$ terms, combined with combinatorial counting and the finite-range geometry of the RG, yields the quadratic contraction $begin:math:text$\eta_{k+1}\le A\,\eta_k^2$end:math:text$, where $begin:math:text$A$end:math:text$ depends only on the geometry.

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Configuration File (scenarios.json)

To run batch experiments, create a scenarios.json file. It should contain a list of scenario objects.

Example scenarios.json:

[
  {
    "name": "Ideal (A=50, Strong Seed)",
    "A": 50.0,
    "eta0": 0.01
  },
  {
    "name": "Noisy A (rho=0.25)",
    "A": 50.0,
    "eta0": 0.01,
    "rho": 0.25
  },
  {
    "name": "Fatal Leak (eps=0.05)",
    "A": 50.0,
    "eta0": 0.01,
    "eps": 0.05
  }
]

Parameters:

• name (str): A descriptive name for the scenario.
• A (float): The ideal quadratic contraction constant.
• eta0 (float): The initial value of the norm at the seed scale, η₀.
• steps (int, optional): The number of RG steps to simulate.
• eps (float, optional): The “linear leak” coefficient. A small positive value (> 0) simulates a failure of pointwise centering.
• rho (float, optional): The fractional noise on A, simulating variability in the RG map.
• C_loss_factor (float, optional): The geometric constant C in the multiplicative loss factor (1 - C η_k).

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One-line Summary for Referees

The RG step uses centered cumulants, so the order-1 Ursell coefficient is exactly zero (no linear term). BKAR tree bounds, combined with KP smallness and finite-range geometry, imply the quadratic contraction $begin:math:text$\eta_{k+1}\le A\,\eta_k^2$end:math:text$ with a scale-independent constant $begin:math:text$A$end:math:text$. The included numerics illustrate both facts: centered gradients are statistically zero (cross-fit z-scores ≤ 3), and any injected linear component destroys the contraction required for summability.