Research Archive: What We Learned
Science advances by learning from failures AND by honestly interpreting valid data. This page documents what was overclaimed, what the data actually shows, and what we learned.
Where These Failures Fit
Each failure below is part of the longer research journey. The timeline notes what led to each phase and where it is discussed on this page.
Phase 2 (2013-2015): Single-Frame Hypothesis
Bigeometric Christoffel failure below
Phase 6 (2022): MOO + Cosmology
k(L) numerology, 92% tension claim failures below
Phase 7 (2023): Composite Discovery
Scheme-invariance hypothesis (unproven) below
Phase 9 (2024): Spectral Gap Verification
What finally worked after these failures
What This Framework CAN and CANNOT Do
CAN Do
- +Detect and pre-condition power-law singularities (auto-logarithm)
- +Map where mixed discretization schemes retain or lose a spectral-gap bound
- +Map Pareto trade-offs between competing objectives (MOO)
- +Fit a dark-energy equation of state by curve fitting (historical; no stored artifact)
- +Scan quantum parameter regions for norm drift (the committed scan finds 0 of 1000 unitary)
CANNOT Do
- -Show that it resolves the H0 tension (the stored test was inconclusive: its criterion cannot be met by any single H0)
- -Create super-convergence (it averages gaps, doesn't enhance)
- -Replace established numerical methods (diagnostic tool only)
- -Prove new physics (numerical tool, not physics theory)
- -Predict without fitting (k values are chosen, not derived)
Why Document Failures?
Most research only publishes successes. This creates a biased view of science where dead ends are hidden and the path to discovery looks smoother than it was.
This archive serves three purposes: (1) intellectual honesty, (2) helping others avoid the same dead ends, (3) documenting what was actually tried before claiming anything works.
Failed: Bigeometric Christoffel Symbols
The idea:
What We Tried
Replace classical derivatives with bigeometric derivatives in the Christoffel symbol definition, hoping to get a "bigeometric Riemann tensor" that would have better behavior at singularities.
Why It Failed
- - Bigeometric derivative requires positive-valued functions (g_uv can be negative)
- - The exp() wrapper distorts the tensor transformation properties
- - Resulting "Riemann tensor" doesn't satisfy Bianchi identities
- - No consistent variational principle
Lesson Learned
You can't just swap derivatives in established physics formulas. The structure of GR depends on specific properties of classical derivatives (Leibniz rule, linearity, chain rule) that NNC operators don't preserve.
Clarified: k(L) Pattern Overclaim
The archived constrained fit (retired heuristic):
What IS Valid
- + The archived constrained run can be reproduced
- - Its scale-specific bounds partly encode the reported trend
- - A common-bounds control lowers R^2 from 0.7096 to 0.4370
- - The formula is not a validated physical lookup rule
What We Overclaimed
We originally claimed this was a "universal physical law" explaining why quantum mechanics uses different math than classical physics. That interpretation went too far - the physical mechanism is unknown.
Current Status
The k(L) formula is retained as an archived constrained fit, but the common-bounds control does not support it as a calculus selector. See the full results page for honest assessment.
Failed: "92% Hubble Tension Resolved"
What We Claimed
Meta-Friedmann equations with k=0.077 at early dark energy epoch (z=2263) could reduce the Hubble tension from 5-sigma to sub-sigma. We called this "92% resolved" and displayed it prominently on the homepage.
Why It's Invalid
- - The k value was CHOSEN to fit the tension, not predicted
- - No mechanism for why k=0.077 at that specific redshift
- - BBN/CMB "constraints" were placeholder penalties, not real data
- - The model has as many free parameters as data points to fit
- - Circular: used tension data to find k, then claimed k "resolves" tension
The Honest Assessment
This was curve-fitting dressed up as physics. We found parameter values that matched observed discrepancies, then claimed the discrepancies were "resolved." This is the opposite of the scientific method.
Lesson Learned
A theory must predict BEFORE you look at the data. Post-hoc fitting doesn't count as evidence, no matter how good the fit looks.
Unproven: "Physical = Scheme-Invariant"
The Claim
Observables that remain consistent across multiple calculus schemes are "physical" and trustworthy. High cross-scheme variance indicates representation artifacts.
The Problem
- - This is a reasonable HEURISTIC, not a proven principle
- - Scheme-invariance is necessary but not sufficient for "physical"
- - A wrong answer could be scheme-invariant (all schemes wrong the same way)
- - No experimental validation of this criterion
Current Status
We've downgraded this from "core principle" to "diagnostic heuristic." Cross-scheme consistency is useful for flagging potential artifacts, but it doesn't validate physics. Experimental verification is still required.
What We Actually Learned
About the Math
- + NNC derivatives are well-defined numerical tools
- + Meta-integration fundamental theorems hold
- + Spectral-gap boundary mapped: 79/88 valid cases pass, with 9 counterexamples
- - But physics applications require more than good math
About Scientific Method
- + Always document what you tried and why it failed
- + Distinguish prediction from post-hoc fitting
- + Mechanisms matter more than curve fits
- + Intellectual honesty is more valuable than impressive claims
Simulations: Valid Data, Recontextualized Interpretation
The simulations below produced usable data, with a caveat: the MOO fronts were run without a fixed random seed, so exact Pareto points are not bit-reproducible. The original interpretations were sometimes overclaimed. Here's what the results actually show.
Wheeler-DeWitt Factor Ordering MOO
What the data shows: Pareto front of 20 ordering choices (p = -1.99 to 2.00). Trade-off exists: lowest eigenvalue at p = -1.89 vs lowest sensitivity at p = 1.00. The stored WKB comparisons are ordering-invariant.
Recontextualized: MOO is a useful tool for exploring quantum cosmology quantization ambiguities. The data doesn't pick a "correct" ordering - that requires experiments.
Chiral Anomaly Detection Framework
What the data shows: v1 naive fermions give a false negative (doubling cancels the anomaly). The stored v2 Wilson-fermion calculation did not satisfy its own detection criterion: with the field on it records anomaly_detected = false, Dirac index 0 and phase 1.56e-5 against the continuum value 0.8, and its phase scan is_anomalous = False.
Recontextualized: This is not a demonstration that the framework detects the anomaly. The browser visualizer is schematic: it draws the continuum formula, not this lattice result, so it is not independent detection evidence.
Kramers-Wannier Duality Validation
What the data shows: The stored critical coupling K_c = 0.4406867935097715; the exact value 0.5 asinh(1) = 0.4406867935097715. Duality intact at self-dual point, correctly broken by external field.
Recontextualized: G-scheme language correctly captures Kramers-Wannier duality. Confirms known physics; it does not discover new results.
Dark Energy w(z) Form
What was reported (historical, no stored artifact): a Meta-Friedmann fit reproduced a DESI-like w(z) by curve fitting. The stored cosmology grid test did not resolve the H0 or S8 tensions.
Why "Mimicker": a fitted w(z) is not a prediction. The stored H0 test is inconclusive rather than negative: its pass criterion (within 2 sigma of both SH0ES and Planck) cannot be met by any single H0, and only part of its grid was evaluated. The model itself already varies with redshift, so the result says nothing about uniform versus redshift-dependent corrections.
Open question: Is there a physical Lagrangian connecting meta-calculus to scalar-tensor gravity? Could be curve-fitting, could be deeper.
Wheeler-DeWitt Factor Ordering Trade-off
What the data shows: On the stored front the lowest ground-state eigenvalue is at p = -1.89 and the lowest sensitivity at p = 1.00. Cannot optimize both simultaneously.
Interpretation: This defines a "Numerical Uncertainty Principle" - there is no perfect calculus. You must choose: Precision OR Stability. The solution is a Pareto curve of trade-offs.
Quantum Compatibility MOO
Original failure (historical, no stored artifact): naive componentwise meta-derivatives caused large norm drift.
Superseded: an earlier write-up said constrained regions preserve unitarity (drift < 1e-10). The current committed scan (public/quantum_phase_corrected.json) finds 0 of 1000 configurations unitary (minimum norm drift 2.56e-4), so that claim is withdrawn.
Supported and Conditional Contributions
1. The NNC Numerical Library
The core implementations of Grossman-Katz derivatives and meta-integration are mathematically correct and useful for numerical methods work. This is a tool, not a physics theory.
2. Spectral-Gap Boundary (Conditional)
The baseline Laplacian compositions satisfy gap(P_mix) >= min(individual gaps), but the guarded grid audit found 9 counterexamples among 88 valid cases. This is a scoped empirical observation, not an enhancement or safety guarantee. See verification
3. Log-Space Transform (7.7x Fewer Evaluations, Stored Setup)
In the stored solver configurations, Lane-Emden n=3 drops from 12080 to 1562 right-hand-side function evaluations (7.7x) with a 30x smaller error. These ratios compare the two run setups, not the coordinate alone: The classical x-space runs are capped at max_step = 0.01 by the script (about 2,000 RK45 steps over x up to 20, roughly 12,000 evaluations); the log-space runs are not capped. Much of the reduction comes from that cap, not from the 2/x term alone. The classical xi_1 is the first nonpositive output sample on a fixed output grid; the log-space xi_1 is a linearly interpolated zero crossing. Part of the error difference comes from these two estimators, not from the coordinate. Not universal: Bessel J0 takes 2.9x fewer function evaluations but is about 10x less accurate. See benchmarks
4. k(L) Scheme Selection Heuristic - Retired
The constrained run is reproducible, but the common-bounds control fails the linear-pattern criterion. Do not use length alone to select a calculus. See data
Timeline of Honest Assessment
Initial framework development with ambitious claims
First falsifications documented (bigeometric Christoffel)
Website launched with "92% resolved" claims
Critical advisor review revealed curve-fitting issues
Pivot decision: Reframe as numerical methods tool, archive cosmological claims
The core numerical library and visualizers remain useful. The cosmological applications are archived as exploratory work.