Harmonic Field Theory begins with a single mirror rule: \(R^{2}=I\). Every measurable thing must have a lawful reflection; only those pairs that close: <R-even, dimensionless, and boundary-complete>, count as reality. The mirror rule \(R^{2}=I\) is not metaphoric—it formalizes why conservation, symmetry, and measurability coincide. Whenever a physical equation closes under reflection, it expresses both an energetic balance and an informational consistency; this is what makes a law measurable in the first place. 

In practice, HFT functions as a constraint-driven framework and working compiler: it tests any equation for mirror closure, enforcing tiers, units, and inflow. Equations that close survive as invariants; those that don’t are flagged as unclosed energy propagation of quantum information.

Please explore this toolset hands-on: run the Python tools, feed it your own math.

    One goal of this work is to enforce this meta-constraint framework on other scientific systems to predict their dimensional and geometric structure, behaviors, and attributes under reflection mechanics and to test whether this meta-hypothesis holds as a substrate independent frame work. Here "observation" is defined by lawful closure, not by the physical medium. Another goal is to use the toolset to probe paradoxes and edge cases, to see how shifting to a reflection-based framework can resolve long-standing ontological tensions in current theories.

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Basics of HFT
• Single law, hard filter: \(R^{2}=I\) forces even, positive, unitless readouts; non-closing forms are rejected.
• Typed tooling, not talk: linters enforce tiers, units, and boundary inflow; the compiler runs Maxwell, Dirac\(\to\)KG, and GR with boundary.
• Built-in limits: the \(1-2-4-8\) composition ceiling and closure rules constrain extensions (e.g., gauge structure) before numerics.

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Mirror Law — General Axiom
For every action, there exists a twin, equal and opposite in angle: one of informational inference, one of geometric result. Reflection is their closure; only closures are physical.

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R² = I
 

Admissibility (what the pipeline is doing)

We take a claim \(F\) (fields + units), build a candidate scalar that is even under the mirror \(R\) and unitless, and then extract a boundary readout. Only candidates that are \(R\)-even and unitless pass. Anything that doesn’t close is recorded (typed \(\varepsilon\)), not discarded.
 

Steps (explicit):
1. Input claim: \(F(\text{fields},\ \text{units})\).
2. Form candidate: \(S=\langle F,RF\rangle/S_{0}\)  (even by construction; \(S_{0}\) makes \(S\) dimensionless).
3. Extract at •⊸: check \(RS=S\) and \([S]=1\).  Pass \(\to\) admissible readout.  Fail \(\to\) \(\varepsilon\)-ledger with type \((\Omega,\eta,\mathrm{op})\).
 

Quick legend (non-standard symbols):
• \(R\): mirror/involution, \(R^{2}=I\). “\(R\)-even” means \(RX=X\).
• •⊸: boundary/measurement extraction functional (readout).
• \(\varepsilon^{(\Omega,\eta,\mathrm{op})}\): typed remainder (topology \(\Omega\), parity \(\eta\), operator class “op”); kept for bookkeeping, not thrown away.

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Worked example:
\[
S=\frac{E^{2}-(pc)^{2}}{(mc^{2})^{2}}\ \xrightarrow{\ \text{•⊸}\ }\ 1
\quad\text{(even, unitless, passes).}
\]

Tiles referenced by the diagram:
• LA: \(R^{2}=I,\ \mathrm{spec}\{+1,-1\},\ P_{\pm}=(I\pm R)/2\).
• Geo: double reflection \(\Rightarrow\) identity/isometry.
• Phys: energy–momentum norm; example above \(\to 1\) at •⊸.
• Defects: odd/unitful pieces \(\to \varepsilon^{(\Omega,\eta,\mathrm{op})}\) (recorded).

Rule (global): only even, unitless, \(R\)-even scalars extract at •⊸. All odd or dimensional remainders are \(\varepsilon\) (bookkept).

 

🔗 Begin here → HFT PRIMER - Basic Walk Through of Tiered Physics