Using the Pythagorean Curvature Correction Theorem to model the Coherence-Pulse Gravity Model

In a previous post, I had used the Pythagorean Curvature Correction Theorem to suggest a geometric argument for gravity.  Since then, I have posted my ideas on the Coherence-Pulse Gravity Model and I want to show that the PCCT doesn't invalidate the CPGM at all. In fact, they augment each other.  

1. The Modified Law of Cosines in Curved Space

In Euclidean geometry the law of cosines is familiar:

$$c^2 = a^2 + b^2 - 2ab\cos\theta.$$

On a curved surface, however, the relationship between the sides of a triangle must be corrected to account for the curvature. One proposal for the modified law of cosines is

$$a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2} = c^2.$$

Here:

• $\mathrm{a,}$ $b$, and $c$ are the side lengths of the triangle,

• $R$ is a measure of the curvature scale (for example, the radius of a sphere or the characteristic curvature for hyperbolic space),

• $h$ is a dimensionless parameter encoding chirality (handedness) and quantum corrections, and

• the $\pm$ sign differentiates between two classes of curvature (the “minus” sign might be used for spherical (positive) curvature and the “plus” sign for hyperbolic (negative) curvature).

A careful expansion of the spherical or hyperbolic law of cosines in the small‑angle limit leads to an extra correction term proportional to $\frac{a^2b^2}{R^2}.$ Here, the chirality factor $h$ appears as a free parameter, which—if determined by underlying quantum geometry—may take on a specific value (or vary between two choices corresponding to “left‑handed” or “right‑handed” configurations).

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2. The Quantum Interpretation and the 32-Dimensional Hilbert Space

The core idea of a quantum geometrical approach is that classical geometric quantities (like distances and angles) are not fundamental. Instead, they emerge from quantum states encoded in a Hilbert space. In one version of this idea, all relevant geometric information—such as the binary choices that determine chirality, sign conventions, and the contributions from curvature corrections—can be encoded in a set of quantum bits (qubits).

Because each qubit has 2 possible states, a system of 5 qubits spans a Hilbert space of dimension $2^5=32$. In other words, there are 32 distinct quantum states available for encoding the information that will eventually give rise to classical geometrical measurements.

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3. Mapping the Correction Term to a 32-Dimensional State Space

Let’s outline how the modified law

$$a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2} = c^2$$

can be understood as having a representation in a 32-dimensional Hilbert space:

3.1 Binary Choices and Quantum Degrees of Freedom

• Chirality Factor:
  The parameter $h$ captures chirality. In a quantum description, chirality is often a binary characteristic (e.g., left-handed or right-handed). Thus, one qubit could encode the “sign” of the chirality correction—assigning, for example, the state $\mathrm{\mid }0⟩$ to the “negative” (spherical) case and $\mathrm{\mid }1⟩$ to the “positive” (hyperbolic) case.

• Curvature Corrections:
  The factor $\frac{a^2b^2}{R^2}$ represents a small correction dependent on both the geometric lengths and the curvature scale. In a quantum picture, additional binary degrees of freedom can encode how strongly curvature affects different parts of the measurement. For example, one might imagine that the overall interaction between the basic lengths $a$ and $b$ has several “channels” (or modes) that can interfere constructively or destructively. Each channel could be associated with a binary variable indicating whether the contribution is “active” or “inactive.” Suppose we have 4 such independent binary choices (each represented by a qubit). Together with the chirality qubit, you then have a total of 5 qubits.

3.2 Constructing the 32-Dimensional Hilbert Space

• Enumeration of States:
  With 5 qubits, the complete basis is

  $$\{\mathrm{\mid }00000⟩,\mathrm{\mid }00001⟩,\mathrm{\mid }00010⟩,\dots ,\mathrm{\mid }11111⟩\},$$

  which is a set of 32 orthonormal states. Each state can encode a particular combination of the binary factors mentioned above (e.g., chirality, curvature influence signs, phase choices in the interference pattern).

• Mapping the Formula:
  Imagine that the correction term $\pm h\,\frac{a^2b^2}{R^2}$ is not a fixed scalar but is the outcome of an operator acting on the Hilbert space. For instance, let’s denote an operator C^ with eigenvalues that depend on the state of our 5-qubit system. Then the classical relation might emerge as an expectation value:

  $$\langle \psi | (a^2 + b^2 + \hat{C}) | \psi \rangle = c^2,$$

  where the operator C^ embodies the quantum corrections and has eigenvalues of the form $\pm h\frac{a^2{b}^2}{R^2}$.
  Because C^ acts on a 32-dimensional space, it can capture 32 distinct “correction patterns.” In many approaches to quantum geometry, these corrections are not arbitrary but are governed by the interference of quantum states. Thus, the 32 possible states represent all the ways in which these corrections can be arranged.

• Interpretation:
  The mapping suggests that the value $c^2$ (a classical measurement) is an emergent outcome from a quantum superposition of states in the 32-dimensional Hilbert space. The conventional Euclidean relation $a^2 + b^2 = c^2$ appears as one special case when the quantum corrections vanish (for example, when the state is such that the eigenvalue of $\widehat{C}$ is zero). When coherence effects are present, one gets the extra term $\pm h\frac{a^2{b}^2}{R^{\mathrm{2 }}}$as a signature of the underlying quantum structure.

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4. Detailed Outline of the Mapping Process

Let’s walk through the mapping process in several steps:

Step 1: Identify the Quantum Variables

• Chirality (1 qubit): Encodes whether the space is “left‑handed” or “right‑handed” (or equivalently, spherical vs. hyperbolic nature).

• Curvature Channels (4 qubits): Represent independent binary decisions (such as whether a particular interference effect contributes positively or negatively) that affect how the lengths $a$ and $b$ combine in the presence of curvature $R$.

Step 2: Construct the Hilbert Space

• Combine the 5 binary degrees of freedom to form the basis states $\mathrm{\mid }{b}_1{b}_2{b}_3{b}_4{b}_5⟩$, with each $b_i$ being 0 or 1. There are exactly 32 basis states.

Step 3: Define the Correction Operator

• Define an operator C^ that acts on the 32-dimensional Hilbert space such that for each basis state $|i\rangle$ (with $i=0,\dots,31$):

  $$\widehat{C}\mathrm{\mid }i⟩={\lambda }_i\mathrm{\mid }i⟩,$$

  where each eigenvalue ${\lambda }_i$ is of the form $\pm h\,\frac{a^2b^2}{R^2}$. (The specific mapping from the 5-bit string to the plus or minus sign and any additional scaling is a model-dependent choice; one simple assignment is to let one of the bits directly determine the sign and combine the others into a binary representation of a coefficient that—when properly normalized—gives $h$.)

Step 4: Emergence of a Classical Measurement

• Imagine that the system of interest is prepared in a superposition state

  $$\mathrm{\mid }\psi ⟩=\sum _{i=0}^{31}{\alpha }_i\mathrm{\mid }i⟩$$

  where the coefficients ${\alpha }_i$ are complex numbers.

• Then the expectation value of the operator $\widehat{C}$ is

  $$⟨\widehat{C}⟩=\sum _{i=0}^{31}\mathrm{\mid }{\alpha }_i{\mathrm{\mid }}^2{\lambda }_i\mathrm{.}$$

• The modified law of cosines then emerges as an effective relation:

  $$a^2+b^2+⟨\widehat{C}⟩=c^2\mathrm{.}$$

  If the state $\mathrm{\mid }\psi ⟩$ is such that the weighted average of the quantum corrections (the contributions from all 32 basis states) yields $\pm h \, \frac{a^2b^2}{R^2}$, then the expression

  $$a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2} = c^2$$

  is recovered as the classical limit of the quantum expectation.

Step 5: Interpretation and Significance

• This mapping demonstrates that the extra term in the modified cosine law is not ad hoc but could be understood as the average effect of many quantum states in a 32-dimensional Hilbert space.

• It bridges the gap between classical geometry (a measurement of distances and angles) and the underlying quantum dynamics (where many binary choices—like chirality and curvature channels—interfere to produce the observed classical values).

• The approach supports the idea that gravity and geometry may be emergent properties resulting from an underlying quantum information structure.

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5. Summary

To summarize, we start with a modified law of cosines:

$$a^2 + b^2 \pm h\,\frac{a^2b^2}{R^2} = c^2.$$

This relation suggests that classical distances (and hence gravitational measurements) include a quantum correction term characterized by a factor $h$ (representing chirality) and a dependence on curvature via $R$. By postulating that the quantum degrees of freedom responsible for these corrections are encoded in 5 independent binary choices, we show that the entire quantum configuration space is 32-dimensional.

We then define a correction operator $\widehat{C}$ acting on this 32-dimensional space such that the eigenvalues correspond to the possible corrections, and demonstrate that the expectation value of $\widehat{C}$ in a general state $\mathrm{\mid }\psi ⟩$ yields the effective classical correction. In this way, the seemingly “ad hoc” term $\pm h\,\frac{a^2b^2}{R^2}$ can be understood as arising from the weighted contributions of 32 quantum states.

This synthesis supports the idea behind Quantum Pulse Gravity and related quantum geometry theories: that our classical understanding of distances—and thus gravity itself—may be the emergent result of underlying quantum processes encoded in a finite-dimensional Hilbert space.

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Final Thoughts

This exploration shows how a modified geometrical relation can be derived by considering corrections to the classical law of cosines and then mapping those corrections into quantum language. The use of a 32-dimensional Hilbert space (via 5 qubits) provides a structured way to account for the quantum degrees of freedom—such as chirality and curvature effects—that could subtly affect classical gravitational measurements.

While this approach is theoretical and remains under active study, it offers a fascinating perspective on how the quantum nature of matter might ultimately give rise to the macroscopic laws of geometry and gravity we observe every day.