CMS Run-2 Log-Periodic Dimuon Residual

Reproducible open-data analysis of a fixed-frequency residual in the CMS Run-2 opposite-sign dimuon invariant-mass spectrum, including frozen independent-file replication, cross-period replication, a prospective phase-locked holdout, and an independent robustness and identifiability audit.

Paper: Log-Periodic Dimuon Residual in CMS Open Data: Cross-Period Replication and a Prospective Phase-Locked Holdout
Independent audit: docs/CMS_INDEPENDENT_ADVERSARIAL_AUDIT.md
Research: rickyjreyes.github.io

Current status — 2026-09-07: Reproducible residual; significance background-dependent; physical origin unresolved. The fixed-frequency structure reproduces across the historical H2 and G1 tests and on the prospectively phase-locked G2 holdout. The independent audit also reproduces those results. However, the previously quoted extreme significance is not robust to reasonable smooth-background uncertainty: alternative smooth generating means with no explicitly injected sinusoid exceed the observed H2/G1 pair score in 359–442 of 1,000 trials when the original selection, fitting, and scoring pipeline is rerun. The audit therefore assigns Category C: suggestive only / insufficiently robust to the discovery-significance and physical-attribution claim, not to the empirical fact that the residual is reproducibly returned by the declared historical pipeline.

Independent robustness and identifiability audit

The audit first reproduces the historical result and then tests how strongly its interpretation depends on background, selection, binning, frequency, and nuisance assumptions.

Diagnostic	Audit result
Reproduced H2/G1 selected pair score	108.43746
Reproduced G2 locked score	126.27500
Historical selected-background null	0 / 10,000 exceedances
Spline s=1 smooth generator	442 / 1,000 exceedances
Poisson log-polynomial degree 12	359 / 1,000 exceedances
Penalized spline (16 knots, penalty 0.1)	405 / 1,000 exceedances
Permissive degree-12 broad-frequency diagnostic	918 / 1,000 exceedances

The central unresolved issue is background/signal identifiability. Several defensible backgrounds predict held-out spectra better while removing the locked positive component, but flexible backgrounds also absorb genuinely injected waveforms. A low score after increasing continuum flexibility is therefore not, by itself, evidence that the observed residual is an artifact.

The balanced interpretation is:

• Observed recurring structure: reproducible under the declared historical pipeline.
• Prospective evidence: meaningful; G2 was tested with frequency, phase, and sign frozen beforehand.
• Historical conditional significance: strong under the historical generating background.
• Model-robust significance: not established because reasonable alternative backgrounds generate comparable scores frequently.
• Physical origin: unresolved between signal, detector/acceptance structure, Standard Model structure, or another mechanism.
• WCT attribution: open.

The historical analytic p-values and 0/10,000 historical-generator result remain useful conditional diagnostics. They are not validated physical discovery probabilities.

---

What this repository tests

The primary observable is the inclusive opposite-sign dimuon invariant mass

$$ m_{\mu\mu} $$

analyzed in the logarithmic coordinate

$$ x = \ln\left(\frac{m_{\mu\mu}}{1,\mathrm{GeV}}\right). $$

After fitting a smooth continuum background $B(m)$, Pearson-like residuals are defined as

$$ r(m)=\frac{N(m)-B(m)}{\sqrt{B(m)}}. $$

The tested residual model is

$$ r(m)=c+a\cos(\omega x)+b\sin(\omega x) $$

or equivalently

$$ r(m)=c+A\cos(\omega x-\phi). $$

The frozen CMS frequency is

$$ \omega_{\mathrm{CMS}} = 7.025825825825827 $$

in $\ln(m_{\mu\mu}/1,\mathrm{GeV})$.

The important distinction is chronological:

• WCT motivated a pre-existing prediction class of log-periodic collider structure before this CMS analysis;
• the specific numerical CMS frequency $\omega_{\mathrm{CMS}} = 7.025825825825827$ was selected in the first certified Run2016H discovery file;
• that numerical value was then frozen before the subsequent independent-file and cross-period tests.

Do not relabel this CMS frequency as $k \sim 9.7$. That value belongs to a different observable/coordinate in the GWTC analysis.

---

Evidence chain

The analysis progressively removes fitting freedom.

Stage	Dataset	What was free?	Amplitude	Phase (rad)	$\Delta\chi^2$
H1 discovery	Run2016H file 1	frequency + phase	0.7543	-0.1890	75.76
H2 frozen replication	independent Run2016H file 2	phase only at frozen frequency	0.9367121	-0.3059911	118.9148
G1 cross-period replication	preregistered Run2016G file 1	phase only at frozen frequency	0.9348797	-0.1567923	115.8921
G2 phase-locked holdout	previously unused Run2016G file 2	positive amplitude only; frequency + phase frozen	0.9708618	-0.2313917 frozen	126.2832

The H2 and G1 amplitudes differ by only about 0.20% under the historical background convention.

G2 prospective phase-locked holdout

Before inspection of the G2 target file, the file-selection rule, file identity, frequency, phase, positive amplitude sign, event selection, mass range, binning, resonance masks, background model, null sizes, and random seed were frozen.

Observed historical result:

$$ A = 0.9708617746 $$

$$ \Delta\chi^2 = 126.2832399542 $$

$$ p_{\mathrm{analytic}} = 1.3329276765\times 10^{-29} $$

for the one-sided fixed-waveform analytic diagnostic.

Finite Monte Carlo ensembles gave zero exceedances:

residual permutations:                 0 / 1000
end-to-end Poisson background refits: 0 / 500

Therefore the empirical probabilities from those ensembles are limited by their Monte Carlo floors:

$$ p_{\mathrm{perm}} = \frac{1}{1001} \approx 9.9900\times10^{-4} $$

$$ p_{\mathrm{refit}} = \frac{1}{501} \approx 1.9960\times10^{-3}. $$

The extremely small analytic probability is a fixed-waveform diagnostic conditional on the model. The independent audit shows that its physical significance is highly background-model dependent.

---

Background robustness and signal-retention test

The original replication sequence used a degree-7 Chebyshev continuum. A central question is whether background fitting or detrending can manufacture the frozen waveform, or whether increased continuum flexibility can absorb it.

The historical pipeline implements a WCT-blind predictive-background selector over:

Chebyshev degrees 5..12
Bernstein degrees 5, 7, 9, 12
smoothing splines with factors 0.5, 1, 2

Backgrounds are ranked by blocked held-out Poisson deviance without using the WCT frequency, phase, amplitude, or test statistic.

The signal-independent procedure selected:

spline_s2

The conservative H2-G1 pair statistic is

$$ T_{\mathrm{pair}} =\min!\left(\Delta\chi^2_{\mathrm{locked,H2}},\Delta\chi^2_{\mathrm{locked,G1}}\right) =108.4978. $$

Historical end-to-end smooth-null calibration

Each pseudoexperiment reruns:

1. smooth-background generation;
2. background-family selection;
3. continuum refitting;
4. residual construction;
5. the frozen waveform test.

Observed historical calibration:

$$ N_{\mathrm{exceed}} = 0/10{,}000 $$

with add-one Monte Carlo probability

$$ p_{\mathrm{MC}} = \frac{0+1}{10{,}000+1} = 9.9990\times10^{-5}. $$

This shows that the observed score is unusual conditional on that selected generating mean. The independent audit demonstrates that the same conclusion does not hold across other reasonable smooth generating backgrounds: the unchanged analysis pipeline produces exceedance fractions of roughly 0.36–0.44 under three alternatives.

Injection / recovery

The same end-to-end pipeline was tested after injecting the frozen waveform at amplitudes

$$ A_{\mathrm{inj}} \in {0.25,,0.50,,0.75,,1.00}. $$

Across that range, the selected historical flexible-background pipeline retained approximately

$$ R_A \approx 0.76\text{--}0.81 $$

of the injected waveform amplitude and achieved approximately

$$ \mathrm{power} \approx 0.93\text{--}0.98 $$

relative to the historical smooth-null 95th-percentile threshold.

The independent audit additionally confirms that sufficiently flexible backgrounds can absorb injected signal. This is the strongest reason not to interpret a flexible-background disappearance as proof that the residual is false.

The implementation retains its historical filename for reproducibility:

python scripts/run_cms_background_kill.py \
  --null-trials 10000 \
  --injection-trials 1000 \
  --injection-amplitudes 0.25 0.5 0.75 1.0

background_kill.py and run_cms_background_kill.py are legacy filenames. In current documentation this procedure is referred to as the background robustness and signal-retention test.

Run the independent robustness/identifiability suite with:

python -m pip install -r experiments/cms_audit/requirements.txt
python -m pip install -e '.[dev]'
python experiments/cms_audit/fetch_inputs.py
OPENBLAS_NUM_THREADS=1 python experiments/cms_audit/reproduce_initial.py
OPENBLAS_NUM_THREADS=1 python experiments/cms_audit/run_spectrum_attacks.py
OPENBLAS_NUM_THREADS=1 python experiments/cms_audit/run_failure_certificate.py --out results/cms_audit_certificate_clean
OPENBLAS_NUM_THREADS=1 python experiments/cms_audit/run_additional_controls.py
OPENBLAS_NUM_THREADS=1 python experiments/cms_audit/run_model_averaging.py
OPENBLAS_NUM_THREADS=1 python experiments/cms_audit/run_event_controls.py
python -m pytest -q

---

What the current result establishes

The combined historical analysis and independent audit support the following empirical statements:

1. an interior log-frequency selected in one certified Run2016H file reproduced at the frozen frequency in an independent Run2016H file;
2. the same frozen frequency reproduced in a separately preregistered Run2016G cross-period test;
3. a previously unused Run2016G file supported the already-frozen frequency, phase, and positive amplitude sign under the historical background procedure;
4. the fixed-frequency residual is not explained by a single fortunate bin, a simple binning choice, or basic numerical arithmetic;
5. the historical selected-background null produces 0/10,000 exceedances, while alternative defensible smooth generating means produce 359–442/1,000 exceedances through the unchanged pipeline;
6. flexible backgrounds can absorb a genuine injected waveform, so the audit does not establish that the physical residual is an artifact.

The current empirical conclusion is therefore:

A recurring fixed-frequency residual is reproducibly returned by the declared CMS analysis, including a prospective phase-locked holdout. Its discovery significance and physical attribution remain unresolved because the decomposition between smooth background and periodic component is background-model dependent.

---

What it does not establish

The current result does not by itself show that:

• WCT is the unique physical cause;
• the residual is a new particle or resonance;
• CMS detector or reconstruction effects cannot generate it;
• trigger/selection/acceptance structure cannot generate it;
• correlated detector systematics are negligible;
• Standard Model continuum, resonance tails, or interference cannot generate it;
• the analytic local tail is a calibrated physical p-value;
• the historical 0/10,000 result is robust across reasonable background uncertainty;
• the empirical tail probability is $>5\sigma$;
• the CMS result and results in other physical domains are statistically independent evidence for one common mechanism.

There is currently no unique validated global p-value for the CMS claim in this repository.

---

Next discrimination priorities

The goal of the next stage is not to fit the residual away. It is to determine whether the recurring structure belongs to the physical signal or to the detector/background model.

1. Preregister an unseen CMS subset with an independently constrained background — use an efficiency-matched control sample or validated detector-folded Standard Model prediction; freeze frequency, phase, trigger plateau, masks, nuisance/background procedure, and injection-recovery acceptance before opening the target.
2. Independent detector replication — test the frozen observable/signature with ATLAS or another genuinely independent detector chain where compatible data exist.
3. Trigger and reconstruction efficiency controls — test whether known efficiency structure projects onto the frozen waveform.
4. Standard Model and resonance/interference controls — propagate broad continuum, resonance tails, and interference models through the identical residual pipeline.
5. Correlated detector/systematic models — replace independent smooth-Poisson pseudoexperiments with justified correlated uncertainty models.
6. Acceptance and selection tests — stress muon kinematics, IDs, masks, run subdivisions, and detector-era structure.

More trials around the same historical fitted mean do not resolve the background/signal identifiability problem.

---

Frequency conventions

This repository uses

$$ x_{\mathrm{CMS}} = \ln\left(\frac{m_{\mu\mu}}{1,\mathrm{GeV}}\right), \qquad \omega_{\mathrm{CMS}} = 7.025825825825827. $$

The mapped LHCb request-48 work in rickyjreyes/LHC uses $\ln(q^2)$. Since $q^2=m^2$,

$$ k_{\mathrm{LHCb}} = 3.512912912912913, \qquad \omega_{\mathrm{CMS}} = 2k_{\mathrm{LHCb}}. $$

Raw numerical frequencies from different logarithmic coordinates must not be compared without the coordinate conversion.

---

Repository layout

wct-cms/
├── src/cms_wct/
│   ├── analysis.py             end-to-end base pipeline
│   ├── background.py           base smooth background
│   ├── background_families.py  Chebyshev/Bernstein/spline fits
│   ├── background_cv.py        WCT-blind blocked predictive selection
│   ├── background_kill.py      legacy filename: background robustness + injection tests
│   ├── cmsio.py                NanoAOD input + dimuon reconstruction
│   ├── signature.py            fixed-frequency and scanned statistics
│   ├── locked.py               fixed-frequency/fixed-phase directional tests
│   ├── significance.py         Monte Carlo resolution and exact tail bounds
│   ├── plots.py                diagnostic figures
│   ├── models.py               result dataclasses
│   └── cli.py                  command-line interface
├── experiments/cms_audit/      independent robustness/identifiability audit
├── audit/2026-09-06/           compact archived audit evidence and summaries
├── scripts/
├── tests/
├── configs/
├── data/
├── docs/
├── .github/workflows/
├── legacy_single_script.py
├── pyproject.toml
└── requirements.txt

ROOT inputs and regenerable per-chunk Monte Carlo audit checkpoints are intentionally ignored by git. The audit keeps compact manifests, aggregate summaries, regression witnesses, tables, and plots under version control.

---

Install

python -m venv .venv

# Windows Git Bash
source .venv/Scripts/activate

# Linux/macOS
# source .venv/bin/activate

pip install -e .[dev]
pytest -q

---

Input

Create data/files.txt containing one NanoAOD ROOT file or XRootD URL per line:

root://.../file1.root
root://.../file2.root

---

Base frozen-frequency run

cms-wct \
  --input data/files.txt \
  --output-dir results/dimuon_blind \
  --mass-min 2 \
  --mass-max 120 \
  --bins 350 \
  --log-bins \
  --muon-pt-min 4 \
  --muon-eta-max 2.4 \
  --tight-id \
  --fit-degree 7 \
  --omega-min 0.5 \
  --omega-max 80 \
  --omega-steps 3000 \
  --frozen-omega 7.025825825825827 \
  --permutations 2000 \
  --seed 20260827

The unrestricted omega scan is exploratory. The scientific replication statistic is the statistic evaluated at the frequency frozen before the target sample was inspected.

---

Phase-locked prospective test

The sharpest historical holdout script freezes frequency, phase, and positive amplitude sign:

scripts/run_phase_locked_period.py

Canonical G2 result record:

docs/CMS_RUN2016G_FILE2_PHASE_LOCK_RESULT_2026-08-31.json

---

Background robustness test

Quick/default historical diagnostic:

python scripts/run_cms_background_kill.py

Deep run matching the historical selected-background calibration:

python scripts/run_cms_background_kill.py \
  --null-trials 10000 \
  --injection-trials 1000 \
  --injection-amplitudes 0.25 0.5 0.75 1.0

Default outputs are written under the legacy path:

results/cms_background_kill/

including:

selection_freeze.json
cv_scores.csv
spurious_null_trials.csv
spurious_null_summary.json
absorption_matrix.csv
injection_trials.csv
injection_summary.json
summary.json

---

Empirical $>5\sigma$ protocol

For a one-sided Gaussian convention,

$$ 5\sigma \iff p = 2.866515718791946\times10^{-7}. $$

Zero exceedances in 10,000 trials are nowhere near enough to resolve this tail directly. More importantly, the independent audit shows that increasing trial count around the same historical fitted mean would not solve the larger background-model uncertainty.

The repository retains the historical direct-Monte-Carlo planning document in:

docs/EMPIRICAL_5SIGMA_PROTOCOL_2026-08-31.md

For zero exceedances:

criterion	required trials
add-one numerical floor reaches $5\sigma$ p scale	3,488,555
exact one-sided 95% upper bound reaches threshold	10,450,778
exact one-sided 99% upper bound reaches threshold	16,065,391

Those trial counts matter only after the composite background/nuisance model is independently justified. No combined H/G/G2 sigma is reported by multiplying p-values or adding Z values.

---

Interpretation hierarchy

Keep four claims separate:

1. Empirical recurrence: the declared historical pipeline reproducibly returns a fixed-frequency residual across H2, G1, and the phase-locked G2 holdout.
2. Prospective evidence: G2 provides a meaningful frozen frequency/phase/sign test under the historical background convention.
3. Statistical robustness: unresolved; reasonable smooth-background alternatives remove the positive component and generate historical-sized scores routinely in the unchanged pipeline.
4. Physical attribution: whether the recurring structure is signal, detector/acceptance background, Standard Model structure, or another mechanism remains open.

The repository preserves both the positive replication chain and the independent evidence that its extreme significance is background-model dependent. Those findings are complementary rather than contradictory.