Two 7 April 2025 Lammert fractal models: both ending with an initial 20 November 2025 Crash Nadir ....

There are two operative Lammert fractal models using underlying low valuation trend lines to define fractal groupings. The European STOXX 600 is showing a technical final sharply higher blow-off to a new all-time high. Likewise the Nikkei, which appears to following a 7 April to 7 October 2025 21/53/52 of 53 days :: x/2.5x/2.5x 3-phase fractal blow-off series. The MSCI world index composite is self-assembling, using the underlying trend lines, in a highly ordered Lammert fractal growth pattern since 7 April 2025. There are two models of interest: a 2 phase fractal series and a 4-phase fractal series ... both ending with an initial crash nadir on November 20, 2025 plus or minus a few days.

Lammert Second Fractals: Current Model: Significant SPX Nadir on 14 Oct and Lower Nadir on 20-21 Nov 2025

Everything in the known universe self-assembles ... on the Planck level and macro level, including time- based quark interactions and the more complex molecules of the fused zygote and developing embryo. Combined entities also under decay and dissolution with occasional nonlinear collapse such as the gravitational collapse of a star. This is a natural deterministic process that can be observed and studied. Valuations of assets both debt and nondebt, likewise grow and decay in a time based fractal manner, based on buyer and seller saturation and the work-debt-credit-wage conditions of the underlying economy. The Nonlinear Second Fractal (cycle): From the main web page in the 2005 Economic Fractalist: The ideal growth fractal time sequence is X, 2.5X, 2X and 1.5-1.6X. The first two cycles include a saturation transitional point and decay process in the terminal portion of the cycles. A sudden nonlinear drop in the last 0.5x time period of the 2.5X is the hallmark of a second cycle and characterizes this most recognizable cycle. After the nonlinear gap drop, the third cycle begins. This means that the second cycle can last anywhere in length from 2x to 2.5x. Since 2005, the 1807 36/90/90/54-57 year :: x/2.5/2.5x/1.5-1.6x pattern recognized 4-phase Fractal Series has added 0.5x to the original 2x time length 3rd cycle(fractal). After a 1932 10-11/20-21/-21 year :: x/2x/2x 3 phase series concluded in 1982, an interpolated 13/32 of 33/33/18-20 year 4-phase cycle :: x/2.5x/2.5x/1.5-1.6x can be identified and inferred to complete the larger 1807 4-phase fractal cycle ending in 2074-2076. Composite equity valuation growth of the 1982 to 2026 13/33 year 1st and 2nd fractal series has been propelled in the 2nd fractal terminal 2x-2.5x portion, i.e. years 26-33, by unprecedented % deficit to GDP US spending and mass migration that has created growth the work-debt-credit-wage conditions of the underlying economy. After the Covid March 2020 low, US Equity composites followed a 8/17/11 month fractal : x/2-2.5x/1.5x' 3-phase fractal series. Since the December 2022 8/17/11 month low ending fractal series, equity growth has self assembled in a series of 1st and 2nd fractal groupings: 15/30 weeks :: x/2x ending on 27 October 2023 and 26/52 weeks :: x/2x ending 7 April 2024 . The 2nd fractal nadir low ending valuation has come shortly after a 2x time length and within 2-4% of the 1st fractal base time length x. For example, the 27 Oct 2023 1st and 2nd fractal series were 120/243 days in length. Nonlinearity occurred after 2x or day 240 with nonlinear lower low gaps between day 240 and 241 and day 241 and 242.

The 1st and 2nd fractal series starting 7 April appears to be a 53/107-108 day :: x/2x fractal series ending 20-21 Nov 2024 with an interim low on 14 October. 4 identifiable subfractal series make up the 107-108 day second fractal. The lengths of the 4 subfractal series are shown in diagram. A final 27-28/55-57 day:: x/2x 1st and 2nd fractal series could place a final SPX low on 24-25 March 2026.

A Primer on Quantitative Time-based Self-Assembly Lammert Fractal Growth and Decay of Valuations of The Asset-Debt Macroeconomic System’s Composite Equities Qualitatively …on the longest time-unit cycles (years) (e.g. 1807 36/90/90/54-57 years :: x/2.5x/2.5x/1.5-1.6x and 1982 13/32 of 33/32-33/20 years x/2.5x/2.5x/1.5-1.6x credit expands via governmental, corporate, and citizen debt; assets are produced and over-produced , overvalued and over-consumed; consumers reach maximum debt loads; the population of possible traders/invested are fully invested and composite equity asset valuations reach a singular fractal time-unit (minute, hour, day, week, month) peak valuation and thereafter undergo decay; recessions occur with weakening demand, interest rates fall; excess debt undergoes default and restructuring; individual and corporate bankruptcies occur; and composite equity (and commodity) asset valuations eventually reach a singular fractal time-unit nadir. The cycle thereafter repeats itself. On smaller time-unit fractal cycles (quarters, months, weeks, days, hours, 15-minute/ 5 and 1-minute) : trader/investor population saturation of asset buying occurs ending in a transient peak valuation .. followed by trader/investor population saturation end selling resulting in a transient nadir valuation. These peaks and nadirs occur in one of the two quantitative fractal patterns as above. Empirically composite equity asset valuation growth and decay cycles occur in 2 modes(laws) of mathematical self-organizing, self-assembly time-based fractal series: a 4-phase fractal series: x/2-2.5x/2-2.5x/1.5-1.6x and a 3-phase fractal series: x/2-2.5x/1.5-2.5x In the 4-phase fractal series sequential elements are termed: the 1st, 2nd, 3rd, and 4th fractals and in the 3-phase fractal series: the 1st, 2nd, and 3rd fractals. The 2nd fractal is characterized by terminal gapped nonlinear lower lows between the 2nd fractal’s terminal 2x and 2.5x time period. (These gapped nonlinear lower lows can be observed in weekly valuation units for the DJIA between 1929 and 1932, the terminal portion of the US 90 year 2nd fractal; within the last three day before the terminal 5 August 2024 139 day 2nd fractal low occurring in an interpolated ACWI/SPX 27 October 2023 55/139/135-136 day (SPX-ACWI respectively) :: x/2.5x/2.5x valuation maximum growth series ending on 18/19 February 2025: and can be expected within the last months on a daily and weekly basis of the current 32 of 33 year 2nd fractal of the interpolated 1982 13/32 of 33 year 1st and 2nd fractal series. (The 27 October 2023 dominant fractal series now appears to have a 120 day first fractal base with a 243 day 2nd fractal ending 7 April 2025, whereas the 55/139/136 day 3 phase growth series was a secondary interpolated fractal series). With the exception of the 3rd fractal in the 4-phase series whose fractal grouping is determined by its terminal high or final lower high peak valuation, fractals (fractal groupings) are determined by the nadirs of the first and last time unit in the grouping with all intervening valuations above the connecting nadir trend-line. SPX from 1982 Quarterly:

QQQ from nadir 2008 Quarterly:

QQQ from Dec 2018 nadir Monthly:

QQQ from Dec 2022 nadir Weekly:

QQQ from Oct 2023 nadir weekly:

QQQ from Aril 2025 nadir weekly:

Blow-off Self Ordering Lammert Fractal Peak Valuation 19-22 Sept 2025 and The Crash Low 3-6 October 2025

The Nikkei, The ACWI, The SPX , the QQQ, et. al. - are all undergoing deterministic self-ordering terminal fractal growth to a blow-off peak valuation on 19-22 September 2025 with an expected crash low on 3-6 October 2025. The central banks of the world will rapidly lower federal interest rates to near zero over the ensuing months with final lower lower high of composite equities in about 125-126 trading days after the 3-6 October 2025 crash low. A final crash then will be spectacularly fast (like 1987) and devastating. The fiscal year 2025 2 trillion dollar deficit spending which provided unneeded greater wealth to the wealthiest and bonuses and employment for those gathering illegal aliens made ubiquitous by the bad policies of the preceding administration, will do little to promote job creation for American service workers, now paying considerable import taxes to support that deficit. At the end of the 1982-2026 13//33 year :: x/2.5x 1st and 2nd fractal series, the world will become a much more dangerous place as many citizens are unable to service personal debt load.

Current Fractal Self-Assembly Model: The Global 5 Sept 2025 Equity Peak Valuation and 4/10/10 day :: y/2.5y/2.5y 6 October 2025 Incipient 3-Phase Fractal Series Crash

The recent monthly US job creation and the 4.3 % unemployment rate are consistent with an incipient serious recession made worse by the sudden disruption of globalization, world trade, and international interdependent manufacturing . The US 80% service sector economy is at the edge of a 1982 13/32 of 33 year 1st and 2nd fractal series with expected nonlinear devaluation of asset prices in the terminal portion of the 33 year 2nd fractal. The 2020 Covid dead horse macroeconomy was re-animated by repetitive annual quasi-WW2 US % GDP deficit spending and AI, crypto, real estate, software, and social media exuberant malinvestment. The PE ratio's for palantir stood at 508.84 on 5 September 2025 The final lower high fractal growth series from the 27 Oct 2023 ACWI low appears to be a 120/252/251-252 day :: x/2-2.5x/2-2.5x growth series. The 252 day 3rd fractal appears to be composed two 126 day fractal series, the 1st ending 6 October 2025.

Back to Fractal Grouping Basics: 5 Sept Peak Global Equity Composite Valuation; 29-30 Sept 2025 Incipient Crash Nadir Valuation

Fractal groupings are determined by the underlying trendline from the first time unit to the last time unit of the grouping whereby all interim valuations are above that trendline. From the 7 April 2025 low there are three identifiable fractal groupings consistent with this definition. 1: a 3-phase 10/25/20 day :: x/2.5x/2x growth fractal grouping 2: a 3-phase 6/12/13 day :: x/2x/2-2.5x growth fractal grouping and 3. a potential 4-phase 7/13 of 14/14/9- 10 day :: x/2x/2x/1.5x or 1.5xi growth and crash decay fractal grouping ending 29 or 30 September 2025. A peak global equity composite valuation day on 5 Sept 2025 represents day105 from the 7 April low composed of two 53 day sequences 10/25/20 days and {6/12/13 and 7/14/6 of 14 days} A Primer on Quantitative Time-based Self-Assembly Lammert Fractal Growth and Decay of Valuations of The Asset-Debt Macroeconomic System’s Composite Equities Qualitatively …on the longest time-unit cycles (years) (e.g. 1807 36/90/90/54-57 years :: x/2.5x/2.5x/1.5-1.6x and 1982 13/32 of 33/32-33/20 years x/2.5x/2.5x/1.5-1.6x credit expands via governmental, corporate, and citizen debt; assets are produced and over-produced , overvalued and over-consumed; consumers reach maximum debt loads; the population of possible traders/invested are fully invested and composite equity asset valuations reach a singular fractal time-unit (minute, hour, day, week, month) peak valuation and thereafter undergo decay; recessions occur with weakening demand, interest rates fall; excess debt undergoes default and restructuring; individual and corporate bankruptcies occur; and composite equity (and commodity) asset valuations eventually reach a singular fractal time-unit nadir. The cycle thereafter repeats itself. On smaller time-unit fractal cycles (quarters, months, weeks, days, hours, 15-minute/ 5 and 1-minute) : trader/investor population saturation of asset buying occurs ending in a transient peak valuation .. followed by trader/investor population saturation end selling resulting in a transient nadir valuation. These peaks and nadirs occur in one of the two quantitative fractal patterns as above. Empirically composite equity asset valuation growth and decay cycles occur in 2 modes(laws) of mathematical self-organizing, self-assembly time-based fractal series: a 4-phase fractal series: x/2-2.5x/2-2.5x/1.5-1.6x and a 3-phase fractal series: x/2-2.5x/1.5-2.5x In the 4-phase fractal series sequential elements are termed: the 1st, 2nd, 3rd, and 4th fractals and in the 3-phase fractal series: the 1st, 2nd, and 3rd fractals. The 2nd fractal is characterized by terminal gapped nonlinear lower lows between the 2nd fractal’s terminal 2x and 2.5x time period. (These gapped nonlinear lower lows can be observed in weekly valuation units for the DJIA between 1929 and 1932, the terminal portion of the US 90 year 2nd fractal; within the last three day before the terminal 5 August 2024 139 day 2nd fractal low occurring in an interpolated ACWI/SPX 27 October 2023 55/139/135-136 day (SPX-ACWI respectively) :: x/2.5x/2.5x valuation maximum growth series ending 18/19 February 2025: and can be expected within the last months on a daily and weekly basis of the current 32 of 33 year 2nd fractal of the interpolated 1982 13/32 of 33 year 1st and 2nd fractal series. (The 27 October 2023 dominant fractal series appears to have a 119 first fractal base, whereas the 55/139/136 day 3 phase growth series was a secondary interpolated series). With the exception of the 3rd fractal in the 4-phase series whose fractal grouping is determined by its terminal high or final lower high peak valuation, fractals (fractal groupings) are determined by the nadirs of the first and last time unit in the grouping with all intervening valuations above the connecting nadir trend-line.

Current Global Equity ACWI Fractal Model: 5 Sept 2025 blow-off peak valuation; 25 September2025 Crash low; 3-6 October 2025 lower low or higher low.

The Macroeconomic System is composed of assets with changing daily valuations propelled higher or lower by an expanding or contracting GDP and ongoing debt obligations which represent both assets and liabilities. Money and debt creation are caused by the expansion of private, corporate, and government debt with central banks and private banks acting as the fortuitous intermediaries(and as a primary source with regard to the former element) accommodating 'credit worthy' loan applicants. GDP grows as a result of money and debt creation. In recessions increased government borrowing (and central bank QE) acts as a counter-cyclical measure. Governments backed by nuclear-deliverable and capable military assets offer the greatest reassurance of loan repayment. With an accommodating central bank and ongoing and future social contract promises to the 99% of private citizens, the US government will, per force, create more debt and money expansion than corporations(the billionaire class proxies) and the private sector combined. The end of prominent of major economic cycles and recessions occur for the pleurality of private citizens when the valuation of assets are too high, the personal debt load is collectively too great, the liabilities of held and sought after assets, e.g., property taxes, new home and car loans, and the cost ongoing assets of daily living exceed the ability to pay or to further expand private debt. After lofty overvaluation of asset prices, these devaluations occur periodically and in a nonlinear fractal manner. The major self-ordering US fractal cycle is an 1807 36/90/90/54-57 year :: x/2.5x/2.5x/1.5-1.6x cycle with nadir commodity/equity prices in 1807, 1842-32, 1932, and expected in 2074-2077 with a 90 year 3rd fractal peak in November 2020 extended to higher highs by 7-15% annual US deficit to GDP spending in 2020 -2025. After a 1932 10-11/21-22/21-22 :: x/2x/2x fractal growth cycle ending in 1982, the dominant interpolated fractal cycle from 1982 is 13/32 of 33/33/20 years ending in 2077. A March 2020 9/23/14 month :: x/2.5x/1.5x fractal cycle was followed by a 27 Oct 2023 119/ 243 day cycle ending 7 April 2025. Nonlinearity in the 2nd 243 day 2nd fractal is seen between days 240 and 241 and days 241 and 242 with a final low on day 243 7 April 2025. From the 7 April 2025 nadir an ACWI 105 day 53/53 day peak is expected to occur on 5 September 2025 composed of two sequential fractal growth series of 10/25/20 days and 10/20/25 days. A 5 Spet 2025 15-22 day crash series of 5/11/7 days will take the ACWI to a 11 day 2nd fractal low on 25 September 2020 with a lower low or higher low on day 7/8 of the 3rd fractal on 3-6 October 2025. With the market (and the fed following) creating lower interest rates, a 119 day rebound is expected with a final crash of 15-21/22 days with a low in March/April of 2026.

The ACWI 22 Aug 2025 Peak Valuation and the Initial 25-26 September 2025 Crash Nadir.

In the asset-debt macroeconomic system there is an underlying self-ordering mathematical fractal pattern in composite asset valuation growth to composite asset peak valuation and thereafter decay to composite asset nadir valuation. Governments and central banks can expand credit and the money supply but are constrained by inflation of assets affecting the pyramid base population of the system. Overvaluation of assets represents a limiting barrier to further consumption in the face of constrained base population job availability and the nonlinear limiting factor of ongoing financial stress of debt servicing and further debt expansion. The mathematical self-assembly deterministic fractal process in the tradeable markets represents the optimal growth and decay investment pathway in the global asset debt macroeconomic system - just as does the steady state of the self-assembly process of the growth and decay of galaxies and black holes occur in the infinite universe. In the current system, 40 years of billionaire-led. corporation-led, both US political party led globalization, lowest wage highest corporation profit led, and resulting international production, transport supply chains, and assembly ... are being suddenly rent apart by US tariffs politically intended to bolster American internal product line manufacturing. This populist political anomaly is coincident with the timing of the 1982 credit cycle began by Volcker's 13-15.5% 3 month interest rates which has been fueled by gradually declining interest rates and extraordinary US peacetime percentage GDP-deficit spending. A 1982- 2026 13/33 year x/2.5x fractal credit cycle is an interpolated part of a greater US 1807 36/90/90/54-57 year x/2.5x/2.5x/1.5-1.6x. The 2020 15% US deficit to GDP COVID money expansion economy began with a nadir in the global composite ACWI equity index in March 2020. It is hard to imagine how a president whose name was at the bottom of the treasury checks sent to unemployed individuals resting at home and now making more than their usual salary, lost the Nov 2020 election. The current fractal projection from the 27 Oct 2023 low is: 118-119 //243//119-120 and 124 = 243//147 days :: x/2x+/2x+/1.5x'. An ACWI peak is predicted on 22 Aug 2025 followed by a 5/10-11/10-11 day 3 phase crash decay fractal series ending near 25-26 September 2025. From the April 2025 low a 10/25/20 day :: x/2.5x/2x growth fractal series is followed by a x/2x/2x' growth fractal series of 10/20/16 days with x' = 8 days for the observable 20 day 2nd fractal and the 3rd fractal 2x' or 16 days, reaching a peak on 22 Aug 2025. A growth fractal of 119-122 days is expected with falling US fed funds rates and lowered US interest rates with the US 3 month rate reaching near zero. The following ACWI 119-122 high will not exceed the 22 Aug 2025 high. A nonlinear collapse will the result in a sequential 124 day or total 2x+ 243 day nadir. The 147 day 1.5 x' 4th fractal double low nadir will occur near the 2026 midterms. Tariffs will be perceived as the post hoc ergo propter hoc etiology of an otherwise natural nadir valuation fractal progression.