Two Cascades, One Mechanism
Multi-Asset E_q Scanner · APP & PLTR · Feb 2026
FVLT v1.5.5 · §8.2 Sub-Shell Extension · §8.3 Primorial Classification · Remark 8.5e Sliding Scanner
Data through Feb 11, 2026 · N=60 · P₃=30 · W₀=120
0
APP Alerts
1
PLTR Alerts
3
PLTU Alerts
0
NDQ Alerts
5
Assets Scanned
E_q z-score > 2.0 && volume > 2.0× trailing 20d → ALERT | Shell: N=60, d(60)=12, P₃ boundary at q=30
1
The Finding
APP and PLTR Show Identical Spectral Signatures
Two unrelated companies — ad-tech and defense software — beat earnings
estimates and dropped 19% and 17% respectively within days of each other.
The E_q spectrum tells us why. Both show the same energy migration:
normal state concentrates in low divisors (q ≤ 30), pre-cascade floods
the high-order modes (q > 30), then the structure fragments.
This isn't company-specific news. It's the same rebalancing mechanism operating on two different stocks with identical positioning characteristics: high retail ownership, leveraged ETF products, crowded momentum trades.
This isn't company-specific news. It's the same rebalancing mechanism operating on two different stocks with identical positioning characteristics: high retail ownership, leveraged ETF products, crowded momentum trades.
APP — E_q Spectrum at Three Phases
PLTR — E_q Spectrum at Three Phases
Normal State (Sep 2025)
Both stocks: ~23–27% energy above P₃.
The remaining ~73–77% sits in low divisors — the lattice is relaxed.
Price movements reflect company-specific information,
not structural coupling.
Pre-Cascade Build
Energy above P₃ doubles to ~50–51%,
concentrating overwhelmingly at q=60 —
the shell boundary itself. The lattice is locking up.
All divisibility resonance collapses into the highest-order mode
before the structure fragments.
Cascade Resolution
After the drop: energy partially redistributes back to lower modes,
but remains elevated at 36–42% above P₃.
The cascade has released some tension but the lattice hasn't fully relaxed —
the structural stress isn't resolved yet.
2
APP Scanner
The E_q Build That Waited Three Months
APP's E_q z-score crossed 2.0 five times in early November 2025 — peaking at
z = +2.73 on Nov 7 — but volume never confirmed.
The stock was at $630. The scanner correctly withheld the alert: structural stress
without the kinetic trigger. When earnings provided the catalyst on Feb 11, the
stored lattice tension released in a single session. APP dropped 19% the next day.
FVLT Cascade Detection — APP · Price + Volume + Frac > P₃ + Z-Score
Why Zero Alerts
The dual-gate design works. Five z > 2.0 readings in November all failed
the volume gate (max 1.97×, below the 2.0× threshold).
By the time earnings hit Feb 11, the z-score had mean-reverted.
This means the E_q build preceded the event by ~3 months —
structural stress was measurable long before the catalyst arrived.
What This Shows
The framework distinguishes between potential energy (E_q build)
and kinetic energy (volume-confirmed cascade).
APP had the potential since November. The earnings miss converted it
to kinetic. The scanner measures what IS — structural
stress — not what will happen.
3
PLTR Scanner
One Alert in 328 Scanned Days
With 507 days of data (Feb 2024–Feb 2026) and a 120-day calibration window,
the scanner had 328 days to evaluate. It produced exactly one alert:
Feb 4, 2026 — the day PLTR began its 17.7% two-day drop.
Zero false positives across 16 months of scanning.
FVLT Cascade Detection — PLTR · 328 Scanned Days · 1 Alert
The Feb 2 Signal
The highest z-score was actually z = +3.13 on Feb 2,
but volume was only 1.81× (below 2.0× threshold). Two days later,
volume surged to 2.37× and the z-score was still
elevated at +2.23 — dual confirmation, alert triggered.
This is what "build then break" looks like in the lattice.
Frac > P₃ Trajectory
The purple area plot (panel 3) tells the story: Frac > P₃ climbed steadily
from late 2025, reaching ~50% at peak —
more than double the ~22% normal baseline. That's the lattice locking:
energy migrating from low-order to high-order Ramanujan subspaces,
conditional correlations building, everyone on the same side.
PLTR Feb 4 Alert: z = +2.23 | frac = 0.443 | vol = 2.37× |
Next 2 days: $157.88 → $130.01 (−17.7%)
4
Path-Deepening
Leveraged ETFs as Extraction Amplifiers
The 2× leveraged ETFs (PLTU, APPX) don't simply double the underlying return —
they amplify the path, creating rebalancing flows that deepen the Stern-Brocot
lattice traversal. This is measurable: the cumulative difference between
the leveraged ETF and 2× the underlying return is pure path-dependent decay —
value transferred to the market makers who rebalance the instruments daily.
Leveraged ETF Path-Deepening — Cumulative Decay = Value Extracted
PLTU vs 2×PLTR
−114.8% cumulative path-deepening over 12 months.
PLTR rose significantly over this period, yet the 2× ETF destroyed more
than the total return it was supposed to amplify. This is not volatility drag
in the textbook sense — it's the rebalancing mechanism systematically
transferring value at each daily reset.
APPX vs 2×APP
−61.1% cumulative decay in 10 months.
Shorter history (APPX launched Apr 2025) but the same downward drift.
Note how the decay accelerates during cascade periods —
the staircase drops in the lower panels. Each high-volatility episode
ratchets the path-deepening down further.
The Mechanism
Daily rebalancing creates forced buying at highs and forced selling at lows.
The market maker sitting at the Vernier point knows the rebalancing flow
that's coming — it's deterministic from the closing price and the fund's
stated leverage ratio. This is the information asymmetry
the FVLT framework identifies: not insider knowledge, but structural
knowledge about forced flows that's visible to anyone who reads the
prospectus, yet systematically exploited by those with execution infrastructure.
5
Signal Amplification
PLTU Fires Three Alerts Where PLTR Fires One
The leveraged ETF acts as a signal amplifier in the E_q scanner. PLTU triggered
alerts on Feb 2, 4, and 5 — confirming the cascade was in progress across three
consecutive trading days. The Nasdaq Composite shows zero alerts.
This is stock-specific structural breakdown, not a market event.
E_q Z-Score — Five Assets Compared
Reading the Five Panels
APP: z-scores elevated early (Nov 2025),
then subsided — energy build without volume confirmation.
PLTR: 328 days of mostly sub-threshold readings,
then the sharp spike into Feb 2026.
PLTU: The leveraged ETF shows persistently
positive z-scores from launch, reflecting the constant rebalancing
stress — then three confirmed alerts.
APPX: Only 22 scanned days (short history)
but trending upward into Feb.
NDQ: Flat. Peak z of only +1.50.
The broad market is fine.
Why the ETF Is Louder
The leveraged ETF amplifies both the E_q energy migration and the volume.
Where PLTR needed one day for the volume gate to confirm, PLTU's
daily rebalancing creates self-reinforcing volume spikes. The z-score
in PLTU is structurally higher because the instrument is the cascade
mechanism — it doesn't just suffer the cascade, it participates in creating it.
6
Correlation Structure
Conditional Correlation — The Giant Component
APP and PLTR have no business reason to correlate — one sells mobile advertising
technology, the other sells government intelligence software. Yet their 60-day
rolling correlation has been ~0.5–0.7 for most of the past year.
This is the "giant connected component" forming — the same phenomenon Yen
& Cheong (2021) found in their TDA analysis of the Taiwan Stock Exchange
before the March 2020 crash. Leverage builds conditional correlations
between otherwise unrelated names.
Rolling Correlation — APP ↔ PLTR and both vs Nasdaq Composite
APP ↔ PLTR
60-day correlation averaged ~0.55 over the past year.
It collapsed briefly in Oct 2025 (the APP E_q build period), then
reconverged. When two unrelated stocks move together, the explanation
isn't fundamentals — it's common ownership structure.
Same retail base, same momentum factor loading, same leveraged ETF
rebalancing flows.
vs Nasdaq
Both stocks maintain 0.5–0.8 correlation with the Nasdaq, but
the Nasdaq itself shows zero cascade signals.
The structural stress is concentrated in the specific names where
positioning is crowded. The broad index dilutes the signal —
which is why the FVLT scanner works at the individual stock level,
not the index level.
7
Traditional Metrics
What a Quant Desk Already Sees
None of the following requires the FVLT framework. A standard factor model
flags these two stocks as occupying the same risk bucket. The question isn't
whether they correlate — any Bloomberg terminal shows that. The question is
why two fundamentally unrelated companies move together, and what the
mechanism is that converts shared positioning into synchronized drawdowns
absent any macro catalyst.
1.96
APP β to NDQ
1.78
PLTR β to NDQ
0.534
APP ↔ PLTR ρ
72.3%
Same-Direction Days
35
Shared Big Down Days
Factor Profile
Both are high-beta momentum names: APP β = 1.96,
PLTR β = 1.78 to the Nasdaq.
Annualized volatility: APP 84.6%,
PLTR 67.0%,
NDQ 23.6%.
They move in the same direction on 72.3% of trading days —
and share 35 "big down" days (>2% loss) out of the 57
big down days PLTR had in the period.
The Drawdown Test
Both peaked in late 2025 — APP at $734 (Dec 22),
PLTR at $207 (Nov 3) —
and both are down roughly a third:
APP −37.7%,
PLTR −34.5%.
YTD 2026: APP −26.1%, PLTR −19.2%.
Nearly parallel descent curves for an ad-tech company
and a defense intelligence platform.
| Date | APP Return | PLTR Return | Note |
|---|---|---|---|
| Jan 2 | −8.6% | −5.7% | First trading day of 2026 |
| Jan 16 | −6.5% | −3.5% | |
| Jan 30 | −18.5% | −3.5% | APP cascade begins |
| Feb 4 | −17.6% | −12.3% | PLTR alert day · Shared big-down |
| Feb 5 | −3.2% | −7.1% | PLTR cascade continues |
| Feb 11 | −3.5% | −2.8% | APP Q4 earnings day |
What Traditional Analysis Shows
High-beta momentum unwind. Factor crowding. Correlated drawdown
in names with similar retail ownership and ETF exposure.
A quant desk would say: "same factor, same result." And they'd be
right — about the what.
What Traditional Analysis Doesn't Show
The spectral migration preceding the event
by weeks or months. The leveraged ETF rebalancing
as a measurable extraction channel. The difference between
potential energy (E_q build without volume)
and kinetic energy (confirmed cascade).
Beta tells you they'll move together. The FVLT tells you
the lattice is locking up before they move.
The April 4 Distinction
Both stocks hit max drawdown on the same day:
April 4, 2025 — "Liberation Day" tariff shock.
That shared drawdown was a macro event, not a structural cascade.
Everything sold off. The Jan–Feb 2026 cascade is different: no exogenous
shock, both companies beat earnings, yet synchronized decline.
Traditional correlation can't distinguish the two.
The E_q spectrum can — April 4 wouldn't trigger the scanner because
the energy migration pattern is absent when the cause is external.
Scanner Results
Complete Multi-Asset Scan
| Asset | Type | Days | Scanned | Alerts | Peak z (2026) | Peak Frac > P₃ | Decay vs 2× Base |
|---|---|---|---|---|---|---|---|
| APP | Underlying | 257 | 78 | 0 | +2.73 (Nov 7) | 51.3% | — |
| APPX | 2× Leveraged | 201 | 22 | 0 | +1.08 (Feb 6) | 40.6% | −61.1% |
| PLTR | Underlying | 507 | 328 | 1 | +3.13 (Feb 2) | 50.4% | — |
| PLTU | 2× Leveraged | 257 | 78 | 3 | +2.86 (Feb 2) | 46.6% | −114.8% |
| NDQ | Composite Index | 257 | 78 | 0 | +1.50 (Feb 3) | 37.1% | — |
Cascade Timeline
Sequence of Events
SEP 2025
Normal state.
APP Frac > P₃ = 26.5%. PLTR Frac > P₃ = 22.9%.
Both stocks rising on fundamentals.
NOV 4–10, 2025
APP E_q build.
Z-scores cross 2.0 five times (peak +2.73 Nov 7). Frac > P₃ reaches 51.3%.
Volume fails to confirm. APP at $630.
DEC 2025
APP E_q z-scores subside. Price begins declining from highs.
PLTR Frac > P₃ begins climbing.
FEB 2, 2026
PLTR z = +3.13 — highest reading in 16 months.
Frac > P₃ = 50.4%. Volume 1.81× — just below threshold.
PLTU alert #1 fires (z = +2.86, vol = 2.47×).
FEB 4, 2026
PLTR cascade alert fires. z = +2.23, vol = 2.37×.
PLTR closes at $139.54 — drops to $130.01 next day (−17.7% in 2 days).
PLTU alert #2 (z = +2.29, vol = 2.75×).
FEB 5, 2026
PLTU alert #3 fires (z = +2.36, vol = 2.10×).
Leveraged ETF confirming sustained cascade. PLTU has lost half its value in a week.
FEB 11, 2026
APP Q4 earnings. Beat on all metrics.
Stock drops 19% next session. The E_q build from November finally resolves.
Methodology
The Algorithm
The scanner implements Remark 8.5e of FVLT v1.5.5. At each trading day, it takes
a 60-day trailing window of log returns, computes the DFT, then decomposes the
spectral energy into Ramanujan subspaces indexed by divisors of 60. It measures what
fraction of total energy lies above the P₃ boundary (q > 30), z-scores this against
a 120-day rolling baseline, and flags when both the z-score exceeds 2.0 and volume
exceeds 2.0× the trailing 20-day mean.
for each day t: frac(t) = Σ{q > 30} E_q / Σ_all E_q |
z(t) = (frac(t) − μ₁₂₀) / σ₁₂₀ |
alert ← z > 2.0 ∧ vol > 2.0×
What E_q Measures
E_q is the Ramanujan subspace energy at period q — the amount of signal power
that resonates with the divisibility structure at that period. When energy migrates
from low q (simple patterns) to high q (complex patterns), the market's
correlation structure is complexifying. This is Yen et al.'s "cavitation"
seen from the spectral side.
The Dual Gate
The volume gate prevents false positives from quiet structural shifts
(APP November). The z-score gate prevents triggering on high-volume
days without structural stress (earnings beats in normal regime).
Both conditions must fire simultaneously.
Result: 1 alert, 0 false positives on PLTR
across 328 scanned days.
Perception, Not Prediction
The scanner does not predict price direction. It measures structural stress —
what the lattice IS doing, not what it will do. An alert means:
"the divisibility resonance structure has departed significantly from
baseline while volume confirms active participation." What happens next
depends on the catalyst, not the measurement.