Previously we validated Informational Energetics (IE) against fundamental physics, deriving the fine-structure constant from first principles with zero free parameters and applied it to the brains cleanup system to give us an equation for Alzheimer’s. This post asks whether the same architecture appears in pure mathematics.

Riemann Hypothesis and the Mystery of the Primes

In analytic number theory, the distribution of prime numbers is governed by a series of beautiful, frustrating heuristics. We know the primes thin out at a rate of x/ln(x) (The Prime Number Theorem). We know the “error” in this distribution behaves remarkably like a random coin flip, bounded by sqrt(x)\ln(x) (Cramér’s model). And we strongly suspect that the zeros of the Riemann Zeta function all sit perfectly on a single critical line (The Riemann Hypothesis). When we look at these properties through the lens of IE, they are the exact thermodynamic signatures of a persistent information network.

The IE Rosetta Stone

Any system that persists against informational entropy must balance a specific six-pillar architectural ledger. It must have a Capacity (a substrate to hold states), a Map (irreducible, compressed data), a Protocol (the generative rules connecting them), a Governor (a stabilizing boundary), a Toll (the irreducible cost of encoding), and a Margin (a resolution floor of unresolvable noise).

If we treat the integers as an informational substrate, the architecture of analytic number theory align flawlessly:

• Capacity: The total system Capacity (the integer range x)

• Map: The primes

• Protocol: The Euler Product formula provides a standardized translation layer that multiplies the isolated primes back into the global integer substrate.

• Governor: The Riemann Hypothesis (zeros on the Re(s)=1/2 critical line). If the integers are a lossless information processor, their transfer matrix must be perfectly unitary (eigenvalues on the unit circle). The functional equation of ζ(s) enforces a reflection symmetry across Re(s)=1/2. For the system to be lossless (unitary), its spectral data, the zeros, must respect this symmetry exactly, confining them to the critical line. This is the Governor: the boundary constraint that prevents the system from exponentially diverging.

• Toll: The logarithmic encoding cost, ~ln x

• Margin: Cramér’s random-walk error bound (sqrt(x) ln x). The primes fluctuate precisely at the limit of irreducible statistical white noise.

The Prime Number Theorem emerges as the Shannon Channel Capacity and the balance equation: Capacity divided by Toll yields Map (x / ln x ≈ π(x)).

The Meta-Claim

IE does not prove the Riemann Hypothesis, nor does it replace the deep analytic machinery of number theory. But it makes a structural requirement: any persistent information system requires a Governor. Because the primes persist as a coherent informational structure, a stabilizing boundary constraint must exist. The Riemann Hypothesis is simply the Governor that number theory produced.

This reveals something arguably more profound than anything around the Riemann Hypothesis. The integers naturally surface the exact same objects that physical thermodynamics surfaces: channel capacities, Landauer encoding costs, standardized protocols, white-noise margins, and unitary boundaries.

The Birch and Swinnerton-Dyer (BSD)

BSD is another unsolved Millennium Prize problem and is structurally identical to IE’s Entropic Balance Sheet, perfectly equating the continuous thermodynamic drag of an elliptic curve to its discrete hardware capacity and causal waste.

The BSD Entropic Balance Sheet:

• Capacity: The torsion subgroup E(Q_tors): the finite “hardware” of rational points available at the base.

• Map: The algebraic rank r: the compressed dimensionality of the curve’s solution space, measuring how many independent generators exist.

• Protocol: The real period Omega: the transduction interface converting between the continuous analytic side (integration) and the discrete algebraic side (arithmetic).

• Governor: The Tamagawa numbers c_p: local constraints at bad primes that prevent the arithmetic from diverging (the boundary conditions).

• Toll: The central value L(E,1): the irreducible encoding cost of the curve’s arithmetic information at the critical point.

• Margin: The Shafarevich-Tate group Sha(E/Q): the unresolvable cohomological noise, information lost to the ambiguity of principal homogeneous spaces.

Just as the Prime Number Theorem balances x/ln(x) against the zeta zeros, BSD balances the product of these discrete invariants against the analytic drag of the L-function. It is the same ledger, written in the language of schemes rather than channels.

What IE Is and Is Not

IE is built from information theory, control theory, and thermodynamics. These are disciplines typically thought of as grounded in physical reality. Yet here it applies in a platonic realm completely devoid of mass, time, or energy.

IE does not tell you what the Governor is for elliptic curves, only that a Governor must exist if the system persists. It does not derive the specific Toll cost for Alzheimer’s, only that a Toll must exist and can be quantified. And if my particular BSD-to-pillar assignment is incorrect, the framework itself should make the correct one visible.

The fact that all systems that persist have these six pillars unlocks something new when IE is used as a Rosetta Stone.

• Diagnosis: For complex persistent systems that fail (such as Alzheimer’s), rather than reinventing understanding from scratch, identify the pillars and import solutions from domains where failure is cheap to test and well-documented.

• Design: When building new systems, predict the requirements and failure points before they emerge.

• Analysis: Given any existing system, perform a rapid structural audit in a common language. Comparing two companies, two organisms, or two theories becomes structurally identical and easy to compare.

• Translation: When a new field appears (such as AI), instead of reinventing basic concepts, instantly import knowledge from domains that have already solved the same architectural problems.

IE is a Rosetta Stone enables finding solutions to problems in any domain when the same solution has not yet been found in your local domain.

For these two Millennium Prize problems, the remaining work belongs to specialists. What IE contributes is a radically narrower target: RH becomes the problem of identifying the specific operator whose spectrum realizes the Governor (a question Hilbert and Pólya posed a century ago, and which IE suggests is not merely a useful analogy but a structural requirement.¹). BSD becomes the problem of bounding Sha: showing that the Margin is finite and accounting for it explicitly. Neither reformulation is a proof. Both are now actually tractable and solvable unlike the originals.

Why does the fine-structure constant in physics have this approximate value?

To figure out what it is and its exact value we utilize Informational Energetics (IE) to cascade through three systems to the fine-structure constant.

System 0 : The Requirements

Mapping the six IE pillars to the vacuum we determine the three requirements:

• Finiteness: (Capacity, Margin) To prevent energetic and informational divergence.

• Unitarity: (Map, Protocol) To ensure the lossless conservation of information.

• Causality: (Governor, Toll) To enforce a well-defined temporal evolution.

System 1 : The Substrate

System 0’s requirements uniquely constrain a single solution giving us a substrate and the required six pillars:

Substrate: An E8 Manifold split into two D4 dimensions, one of which contains spacetime.

• Capacity: The vacuum has a finite clock: 43 fundamental ticks per cycle, with 32 channels of information per tick. This is the universe's total processing budget, fixed by the geometry.

• Map: To exist as a distinct thing rather than dissolving into the vacuum, a particle needs 5 pieces of information: 3 to say where it is, and 2 to describe the shape of its boundary. Nothing more, nothing less.

• Protocol: Information on this substrate requires exactly 16 bits to propagate without loss, the minimum needed to encode both phase (what a signal is) and handedness (which way it flows). This is why the universe can tell left from right.

• Governor: The boundary of any persistent particle must be a sphere. Only a sphere perfectly separates inside from outside with no ambiguity, and this topological fact is what makes electric charge come in discrete units rather than continuous values.

• Toll: Time itself is the cost of processing and the geometry allocates 11 of the 43 ticks per cycle as overhead, the irreducible slack required for state transitions to complete without destroying information.

• Margin: There is a noise floor below which no signal can persist. The electron sits exactly at this floor: it is the lightest thing that can exist.

System 2 : The Fine-Structure Constant

The substrate presents a natural impedance to sustaining topological structure. This geometric impedance is also a persistent system and contains all six pillars:

where

and results in a value that is only 0.58σ away from the Morel (2020) experimental value:

On this April 1st confirm for yourself by following this link to WolframAlpha and checkout the full derivation in the preprint paper on complex systems that persist: Informational Energetics: A Universal Architecture of Persistence.

Alzheimer’s occurs when the Pathogenic Entropy (Tau/Amyloid) scales faster than the Governor(GSK3β) prevents it and the Protocol (Estrogen/Choline) can regulate it (via the glymphatic system), exacerbated by an exponential rise in Substrate Friction (Fibrosis) that reduces/prevents waste export.

Variable Breakdown

1. The Numerator: Pathogenic Entropy [Kτ​⊗KAβ​]

The Complexity Load: Represents the exponential “coupling” of Tau Tangles and Amyloid Plaques. Structural Noise that prevents the “Model” (ΔI) from communicating with the “Vessel” (ΔE).

INLRP3​ The Accelerant - Neuroinflammation:

• The Concept: Plaques and Tangles are inert trash until the immune system reacts. The NLRP3 Inflammasome is the driver that turns “trash” into “war.” Note this can often be driven by intestinal permeability.

• The Mechanic: When Microglia sense Amyloid, they activate NLRP3, releasing Cytokines (IL-1β, TNF-α). This turns the brain into a scorched-earth war zone, actively killing neurons that might otherwise have survived the plaque load.

• The Multiplier: If Entropy is high but Inflammation is zero, the numerator remains manageable (Asymptomatic AD). If Inflammation spikes,ΨAD goes vertical

2. The Denominator Part A: Governor Control G(Li⋅GSK3β−1)

The Lithium Brake: This is the Governor (G). Lithium inhibits GSK3β, which is the enzyme that “writes” the Tau error. Without Lithium or Estrogen, the Governor fails, and the numerator grows unchecked.

3. The Denominator Part B: Protocol Fidelity MI(ERα⋅PEMT)

The Choline Currency: This is the Signal Fidelity (MI). ERα (Estrogen) activates the PEMT gene to create Phosphatidylcholine. This process is rate-limited by Methylation (B-Vitamins/Homocysteine). This provides the “Bandwidth” for Acetylcholine. If this term drops (as in menopause or APOE4), the system loses its ability to process information.

4. The Denominator Part C: ΩATP​ (Mitochondrial Efficiency):

• The Constraint: Repairing DNA, synthesizing Choline, and exporting waste via the Glymphatic system are all active transport mechanisms. They cost ATP.

• The Failure State: In Insulin Resistance (Type 3 Diabetes), neurons cannot uptake glucose. ΩATP​ approaches zero. As the denominator shrinks, the total equation value (AD Risk) skyrockets.

5. The Exponential Term: Glymphatic Stagnation e(…)

The Friction Coefficient: This represents the Temporal Cost (T).

• Γfibrosis​: The perivascular “clogging” that stops waste removal.

• EPA & Natto: These act as the divisors of friction, keeping the exponential decay in check by ensuring the “pipes” remain open for the sleep-based cleanup cycle via reduction of PAI-1. (Note: the clinical data on Nattokinase penetrating the blood-brain barrier and significantly clearing perivascular fibrosis in humans is still emerging)

  ---

Every Alzheimer’s prevention strategy comes down to balancing this equation.

Crucially, this model defines the velocity of decay. It is highly accurate for Prevention (Stage 0–2). However, for Reversal (Stage 3+), simply balancing the equation is necessary but insufficient. Halting the entropic slide stops the progression, but replacing dead neurons requires a separate “Regeneration Equation” (driven by BDNF and Neurogenesis), which is not modeled here.

While individual biology varies, the core system requirements to stabilize the equation are:

• Regulate Entropy : Inhibit the error-writing enzyme GSK3β (Lithium Orotate) to stop the hyper-phosphorylation of Tau.

• Modulate Input Gain (The Stabilizer): Dampen Hippocampal Hyperexcitability (Levetiracetam/Keppra) to prevent activity-dependent Amyloid generation. High neural “noise” accelerates the production of the numerator.

• Reduce Inflammation: Downregulate the NLRP3 Inflammasome & TNF-α (Curcumin, Fasting/Ketosis) to stop the immune system from attacking the structure.

• Restore Energy: Maximize Metabolic Capacity ΩATP​ (Insulin Sensitivity, MCT Oil/Ketones, Mitochondrial Support) to power the repair systems.

• Signal Fidelity (The Protocol):
  Support the PEMT pathway and membrane integrity (Phosphatidylcholine, Citicoline, improved Estrogen signaling/HRT and ensure adequate Methylation (B-Complex/Folate) to synthesize Choline) to prevent neuronal cannibalization.

• Waste Export (The Flow):
  Reduce Substrate Friction/Fibrosis to maintain Glymphatic clearance (Nattokinase, EPA, Deep Sleep).

If we look at Alzheimer's treatments through the lens of this equation, we see why they keep missing. They are mathematically incomplete. Brand name (Aduhelm / Leqembi, Aricept, Namenda) only solve part of the equation. Current hope (Namenda, GLP-1, Rapamycin, Viagra) also only solve part of the equation. B-Complex (B6/B9/B12) can improve Choline synthesis and HRT can provide estrogen to produce Choline (as long as metabolism is good to allow for estrogen signaling), but they only change Choline. You cannot solve a multi-variable differential equation by changing just one number.

Appendix: APOE gene e4 variant breakdown

This equation shows that APOE4 carriers aren't just "unlucky"; they are Systemically Solvent only when Estrogen is high. Once ERα drops, the Φ term collapses, and because η is already low, the system enters Insolvency (Alzheimer's) almost immediately.

Welcome, and thanks for signing up!

You’ll start receiving new posts right here in your inbox. You can also log into the website to read the full archives or access a clean, ad-free reading experience in the Substack app.

Some housekeeping…

If you can’t find the newsletter, check your Spam folder or Promotions tab, and move this email to your primary inbox.

Get more in the Substack app

We hope you'll take a few minutes to answer our New Reader Survey and help us make Meyer better!

Take Survey

Thanks again, and please tell a few friends if you feel like it.