Technical Overview & Roadmap

Harmonic Field Theory (HFT) is an axiomatic program built on a single principle, the Mirror Law: Apply reflection twice and you return to identity (\(R^{2}=I\) ). Every structure is treated as a two-faced unit (“thing” and “record”). From this axiom HFT defines:

When an equality is lawful (two-pass closure to an even, dimensionless invariant)
How defects are recorded and balanced
A tier cascade that organizes when adding new generators doubles state counts (1 → 2 → 4 → 8).

Rule and Promise
Rule: Apply the reflection twice (R² = I). Only even, unitless, boundary readouts count as facts; everything else is a typed remainder (ε) that must pair to close.

Promise: What the checker does. Enforces that rule across substrates and places relations in the 1→2→4→8 stack.
Try to break it. See Falsifiers ↓ for clean counterexample tests.

🔗 Begin here → HFT PRIMER - Basic Walk Through of Tiered Physics using Only Mirror Law

This page gives researchers the essentials before diving into the papers.

Try It Yourself – Quick Run of Mirror Notation v7.x

If you want to see the Mirror Law checker in action before reading the full papers, you can run the v7.x reference compiler directly. It’s a short, readable Python file that enforces the bullet rule, checks for dimensionless closure, and reports any ε-defects.

1. Install and run

Save the script from Paper 0 – Mirror Notation v7.x (Chapter 3: Compiler Spec) as mn.py, then in a terminal:

python
from mn74 import analyze_pixel
print(analyze_pixel("{ E ↔ m } ← (c^2) •⊸"))

You’ll get a PixelReport like:

closure_ok: True
bullet_scalar_unitless: True
eps: []
tier: [2,4]

2. What this means

The ratio E / (m * c^2) is unitless and passes the mirror test, so the bullet “publishes.”
If a relation still carries phase, parity, or metric debt, the report shows ε^(Ω), ε^(op), or ε^(η) instead of closing.
The same tool can analyze any equation written in Mirror Notation and will tag its tier, realm (quantum or classical), face, algebra, and closure status, revealing whether the relation fully closes (lawful equality) or leaves a typed remainder ε that quantifies the unclosed energy or information channel and points to the missing bridge or partner needed for balance.

Why this matters for science

Science advances by finding where current models fail in a structured way. A typed ε converts “hand-wavy mismatch” into a concrete, reproducible diagnostic.

Conservation and publication criteria are universal across domains; a substrate-agnostic ε means anomalies become comparable across quantum, classical, geometric, and statistical settings.

If ε is typed (phase/coherence, parity/topology, metric/boundary) and localized at the bullet, the tool can algorithmically show which kind of energy or information remains unclosed and what bridge, boundary term, or dual partner is needed to restore balance.

Because every check is dimensionless and R-even, it separates “bad units or parity” from “genuinely new physics,” reducing false leads.

A stable diagnostic that outputs the same ε for the same structural fault creates shareable test cases, enabling replication and cumulative correction.

Beyond that, the realm and tier detection hint at something deeper: the tool isn’t classifying by syntax but by algebraic parity.

Odd tiers (1,3,5,7) are prime generator states—anti-mirror moves that cannot close without a frame—so they appear as quantum, demanding a partner to publish.

Even tiers (2,4,6,8) are mirror-even frames—points on the Cayley–Dickson ladder (complex, quaternionic, octonionic) where multiplicative norms exist and closure is lawful—so they appear classical.

A note about Ontology
In HFT, the tiers are literal reflection stages—each one a do/undo operation under the Mirror Law (R² = I). Their parity determines the physical realm.

Odd tiers (1, 3, 5, 7) are generative or “quantum” stages: open reflections that introduce new degrees of freedom but cannot self-close.

Even tiers (2, 4, 6, 8) are framing or “classical” stages: they supply the mirror closure that makes an equality lawful and measurable.

The Mirror Law itself expresses reducibility—a relation is real only when its mirrored parts cancel to a unitless, boundary-safe invariant.
Its opposite, irreducibility, is what we experience as quantum openness: the half-step that seeks a frame. Together they form a natural pixel pair—<mirror / anti-mirror>—that drives the harmonic cascade.

This pattern continues until Tier 8, where the outward reflections lock into a stable octonionic (8-D) unit and reflection turns inward. Beyond this point, the system writes its own ledger: every unhealed remainder (ε) is recorded as a memory tilt.

A non-zero tilt that resists all lawful factorizations behaves as a prime residual—it cannot be canceled and therefore persists as structure—while factorizable tilts resolve as composites.

The even tiers line up with the Cayley–Dickson sequence of normed algebras:

Tier 2 – real (1)
Tier 4 – complex (2)
Tier 6 – quaternion (4)
Tier 8 – octonion (8)

These are the only algebras that naturally support a self-consistent quadratic norm, exactly what the Mirror Law requires for closure.
Thus, the tier ladder and the algebra ladder are not identical but resonate: each even tier supplies both a classical frame and a normed seat, while each odd tier opens a new quantum channel that the next even tier stabilizes.

After seven reflections, the system has doubled its information capacity seven times—reaching a 7 bit 128-state depth—forming a harmonic cascade where geometry, algebra, and information growth are three aspects of one reflective process.

What this unlocks

• Targeted anomaly triage: Distinguishes setup errors from real effects.

• Hypothesis generation on rails: Typed ε narrows the hypothesis space and points directly to the missing bridge or partner.

• Cross-domain unification test: The same closure grammar audits QM, GR, gauge theories, actions, and inference models; mismatches reveal where unification still leaks.

• Experimental leverage: ε-signatures predict how controlled changes (like flipping handedness or boundary conditions) will behave, producing crisp falsifiers.

• Boundary honesty: The ε(η) flag enforces explicit boundary accounting, improving reproducibility.

• Discovery filter: A clean 0-ε pass confirms lawful equality; a persistent ε with no lawful partner marks unexplained structure worth investigation.

• Comparable ledgers: Typed ε provides a common diagnostic language for physics, statistics, and neuroscience models.

• Automation and scaling: Because the diagnostic is algorithmic, large sets of models can be batch-scanned to locate genuine leaks and the minimal fixes that close them.

3. Where to go next

Open Paper 0 – Mirror Notation v7.x to explore the full CLI examples, tier detection rules, and the complete ε-ledger. The code there is the same engine used for all tests in the HFT papers.

Defining R

Formal Core

Objects we manipulate

States: preparations, written x: I -> X.
Processes: arrows (maps) between objects.
Scalars: endomorphisms of the unit object I (the numbers you can publish).
Togetherness (“tensoring”): A ⊗ B means “A and B considered jointly.”

What R is
R is a single reflection with the two-pass property: applying it twice returns identity (R ∘ R = Id). Operationally, R swaps the two faces of a pixel (thing ↔ record; image ↔ weight) and enforces “do → undo → check.”

The bullet (how a fact is published)
Given a state x: I -> X, its bullet is the scalar extracted at the publication point. A helpful picture:

I --x--> X --(id ⊗ mirror-of x)--> X ⊗ R X --swap--> R X ⊗ X --ev_X--> I

(Prepare x, bring in its mirror/adjoint, swap order, evaluate to a number.) This is the shared pattern behind ⟨ψ|ψ⟩, quadratic invariants like E^2 - (p c)^2, and normalized likelihoods.

Bullet rule (when equality may publish)
A written equality may publish at the bullet • only if the extracted scalar is:

Unitless (after explicit normalization by physical constants),
R-even (unchanged under a two-pass mirror), and
Boundary-closed (no missing surface/envelope term).

If all three pass, write the pixel as:

{ A ↔ B } ← ( C ) •

and require A / (B · C) = 1. Otherwise the checker refuses and logs a typed remainder ε.

Normalization (what “unitless” means in practice)
The checker verifies that units cancel. For auditability, show the normalization exponents you used, e.g.:

Normalization (example): c^-2 · ħ^+1 · G^0

(H0 and Λ can appear in worked examples but are not assumed built-ins.)

Typed residues (ε) and what they mean

ε(Ω) — phase/evolution debt (e.g., evolution without a lawful readout closure).
ε(op) — parity/topology debt (odd/handed density without a lawful carrier/inflow).
ε(η) — metric/boundary debt (missing surface/envelope; evaluation factors through a non-trivial boundary).

Faces (what is being mirrored)
Every pixel is tracked by face (image vs weight). Some lawful pixels live on a single face; others (e.g., gravitational couplings) link faces. The checker records face bookkeeping explicitly. (“3+1 :: 3+1” is a useful physical picture, not a required axiom.)

Tiers (why odd vs even matters)

Odd tiers 1,3,5,7 generate new degrees of freedom and cannot publish alone.
Even tiers 2,4,6,8 provide frames where quadratic closure is lawful. This echoes the normed-algebra ladder 1,2,4,8 (reals, complex, quaternions, octonions).

Concrete pixels (check by hand)

de Broglie closure (T1 ↔ T2):

{ p ↔ k } ← ( ħ ) • Bullet ratio: p / (ħ · k) → must equal 1 (unitless) → publishes. If ħ omitted → unit defect; handed density without lawful envelope → ε(op).

Relativistic mass–energy (T3 ↔ T4 frame):

{ E ↔ m } ← ( c^2 ) • Bullet ratio: E / (m · c^2) = 1 → publishes when boundary-closed.

Continuity equation (T5 ↔ T6 frame):

{ ∂t ρ + ∇·j ↔ 0 } ← ( 1 ) • Default failure tag if it doesn’t close: ε(Ω). Topology/envelope issues can add ε(op) or ε(η) depending on context.

Practical corollary (reversals)

Single-pass “reversal” is not admissible. Physical reversals are paired protocols that pass the bullet, or they log ε. Treat any one-shot rewind claim as ε until a lawful partner is shown.

R by Substrate — how the one reflection shows up in practice

Purpose: R is a single, substrate-independent involution (apply twice → same report). These boxes don’t redefine R; they show how it is instantiated on the substrates you actually use so a reader can verify what “R-even at the boundary” means in each context.

Cross-domain use of R assumes that involutional closure preserves dimensional homomorphism — i.e., the form of ‘do/undo’ remains invariant though substrates differ.

Two-face geometry (Image ↔ Weight)

Two-face geometry (Image ↔ Weight)
Role: The global scaffold that every R operates within — all substrates instantiate the same two-face symmetry.
Object type: Paired reports — Image (3+1 “now”/instrumental) and Weight (3 curvature + 1 memory) [illustrative physical picture].
Map R: Face-swap via the Presenter (involution at the H-pixel). Presenter = the minimal readout gate that produces a boundary-safe scalar.
Boundary extraction: Presenter’s single “now” readout from the accumulated record.
R-even scalar: The Presenter’s dimensionless boundary invariant; this defines “R-even at the boundary” for all substrates below.
Mirror Law ↔ Prime Obstruction (Closure ↔ Irreducibility)
Object type: lawful relation vs. its minimal unpairable residue.
Map R: closure attempt → irreducible limit.
Boundary extraction: failure of closure tagged as ε (prime defect).
R-even scalar: 1 (identity) when closure succeeds; ε (prime) when not.

HFT Interpretation:
Structure arises where perfect reflection (unitarity) and perfect invariance (conservation) co-exist. Each new tier is a lawful recursion of closure: the universe repeating its own symmetry through a new orthogonal channel. Complexity is not symmetry breaking — it is symmetry continuing in higher order.

Mirror Law ↔ Prime Obstruction (Closure ↔ Irreducibility)

Quantum (amplitudes)

Object type: state amplitudes / wavefunctions
Map R: complex conjugation / bra↔ket pairing (involution)
Boundary extraction: amplitude → outcome report channel
R-even scalar: modulus-squared probability (dimensionless, non-negative)

Classical fields / EM

Object type: fields/forms used for energy/flux statements
Map R: the field mirror you adopt (specific dual/parity consistent with closure)
Boundary extraction: field → energy/flux report, then normalize to a unitless scalar.
R-even scalar: a dimensionless quadratic invariant (e.g., energy density normalized by the appropriate constant/scale you accept).

Relativistic kinematics / actions

Object type: 4-vectors / action relations for closure checks
Map R: time/space reflection pairing appropriate to the identity in use
Boundary extraction: form the unit-normalized check so units cancel
R-even scalar: invariant “mass-style” check (dimensionless, non-negative)

Wave ↔ Field (ψ ↔ H) — The Core Reflective Pair

Object type: quantum amplitude ↔ classical condensate
Map R: phase collapse ↔ field stabilization
Boundary extraction: measurement resolves amplitude into a fixed field configuration (lawful readout).
R-even scalar: a closed amplitude–field invariant (dimensionless; e.g., an action/energy functional normalized to 1 when the pair publishes).

Statistics / inference (FEP/Bayes)

Object type: model / observation distributions
Map R: forward model ↔ normalized readout pairing
Boundary extraction: normalization to a publishable distribution / scalar
R-even scalar: normalized likelihood / probability (unitless, sums to 1)

Qualia ↔ brain (co-report) [exploratory]

Object type: experiential report vs. neural field correlate
Map R: report ↔ field mirror you endorse (self-mirror/predictive closure)
Boundary extraction: normalized behavioral/phenomenological report paired with neural measure
R-even scalar: unitless co-report index invariant under the swap

Interaction / gauge organization (Tier 6)

Object type: gauge potentials / currents (lawful wiring)
Map R: gauge-compatible mirror pairing that preserves closure
Boundary extraction: conserved-quantity check in dimensionless form
R-even scalar: the conserved scalar you accept for “law-like and local” outcomes

Octonionic cap (Tier 7)

Object type: the 1+7 non-commutative unit at the cap
Map R: cap mirror that locks faces as a single 8-direction object (still involutive at report)
Boundary extraction: cap-closure test (no new outward directions)
R-even scalar: cap-closure indicator (dimensionless)

Time / ledger (Tier 8)

Object type: record vs. present (memory ledger vs. presented “now”)
Map R: presenter ↔ record swap
Boundary extraction: Presenter’s publication of the “now” from accumulated record
R-even scalar: ledger-safe invariant (dimensionless, non-negative); reflections beyond 8 write the ledger

Complementary roles
(example: predator ↔ prey)

Object type: co-defined roles in a taxonomy
Map R: complementary role swap (predator ↔ prey)
Boundary extraction: pair-defined status in the same system (boolean or normalized co-occurrence)
R-even scalar: unitless pair index; absence of the complement → log as ε (missing mirror)

The H-Pixel and the Law That Holds the Scales

(Reflection Mechanics: a closure calculus for physics)

1. Every theory leaks

Einstein’s field equations need the Gibbons–Hawking–York boundary term to balance.
Quantum evolution needs the Born norm to produce observables.
Gauge fields need anomaly inflow to keep charge conserved.

Each theory stands until a residue appears at its edge — a missing partner, a term that refuses to cancel.
Different domains, same leak.

2. The pattern behind the leaks

From Newton’s third law to Noether’s theorem to Wheeler’s participatory universe, one principle repeats:

Every measurable thing is a pair hunting its twin.

Newton: action ↔ reaction
Noether: symmetry ↔ conservation
Wheeler: observer ↔ observed

Reflection Mechanics formalizes this recurring structure as the basic unit of reportable reality — the H-pixel — and the rule that governs when a relation is complete enough to publish as truth.

3. The mirror law as a scale invariant

Anything that flips sign under R is called R-odd (a generator).
Anything that survives two reflections unchanged is R-even (a frame).

When a generator meets its proper frame and the two cancel their dimensions and boundaries, one H-pixel is formed.
That pixel is the smallest complete report the universe can make about itself. Since it is scale invariant the whole universe is also considered a holographic pixel, so it's also the largest complete report the universe can make about itself.

4. The bullet rule (the law of publication)

A relation may publish equality only at the bullet — •⊸ — when it passes three tests:

Unit test: The ratio of quantities is dimensionless.
Reflection test: The total result is R-even.
Boundary test: No unhealed residue remains.

The relation is written as:

{A ↔ B} ← (C) •⊸

Read it as “A and B are mirrors through C, publishing at the bullet.”

If the ratio A / (B·C) equals 1 and all reflections close, the bullet passes.
If not, the system logs a residue ε, tagged by its failure type:

ε(Ω) — phase or evolution debt (incomplete conjugation)
ε(op) — parity or topological debt (odd density without lawful envelope)
ε(η) — metric or boundary debt (unclosed surface or frame)

Each ε marks a place where the theory leaks — where an open generator still waits for its frame.

Historical equivalents

The rule expressed here isn’t new; it’s a generalization of structures physics already trusted.
What changes is that Reflection Mechanics puts them into one operational grammar.

Newton’s third law introduced the first closure form: action ↔ reaction.
In mirror terms, that’s generator ↔ frame — two complementary operations that only make sense together.
The third law is the first appearance of the bullet condition in disguise.

Noether’s theorem formalized the same idea in algebraic language.
A continuous symmetry (R-odd generator) produces a conserved quantity (R-even frame).
Every invariance under R produces a reportable scalar — exactly what the bullet rule demands.

Wheeler’s participatory principle extended the pattern to epistemology.
Measurement is incomplete until the act of observation closes it.
The observer serves as the final ledger frame, completing the physical report.

Lawful equality in any equation corresponds to the bullet itself — the moment where the mirrored terms have reconciled and can publish.

When the reconciliation fails, the theory records a residue ε — a quantifiable debt, classed by its defect (phase, parity, or metric).

Across these steps — Newton, Noether, Wheeler, and the present formalism — the same logic persists:
each law of nature is a reflection that becomes true only when its generator and frame close without remainder.

The closure sequence — the ladder of equal and opposite reflection tiers

Every closure happens between adjacent levels.
Each odd tier creates a new generator (a new degree of freedom).
Each even tier provides the frame that reads it and produces a unitless invariant at the bullet.

T0 → T1 : seed to phase

T0: empty capacity, no distinguished direction.
T1: the first oscillation, the birth of phase.
Phase alone is R-odd — it cannot publish.
The world gets its first mirror, but no ledger.
Without its frame: ε(Ω) — a phase that cannot report.
With a frame: phase becomes conjugate; the complex plane opens.

T1 ↔ T2 : phase meets kinematics

T1: hidden wave.
T2: visible track — the first frame of motion.
Closure pixel: {p ↔ k} ← (ħ) •⊸
Momentum to wavenumber with the Planck bridge.
The ratio [p]/([k][ħ]) → 1, so the pixel passes.
This is de Broglie’s law written in closure form.
Action divided by ħ gives the first dimensionless invariant.

T2 ↔ T3 : straight path meets twist

T2: straight propagation.
T3: intrinsic twist — spin, handedness.
A pure topological density is parity-odd, so it fails closure.
{θ ↔ Tr(F⋆F)} ← (1) •⊸ ⇒ ε(op)
The twist needs a lawful carrier to publish.
Anchor math: Dirac spinors and their bilinears — local curls that cannot report without an envelope.

T3 ↔ T4 : twist meets propagation

T3: local curl wanting transport.
T4: the causal envelope — the light cone that carries what T3 built.
Closure pixels that pass:
- {E ↔ m} ← (c²) •⊸ → [E]/([m][c]²) = 1
- {p²/(2m) ↔ E} ← (1) •⊸ → passes in Newtonian regime
Anchor math: Lorentz invariants, intervals, and the d’Alembertian.
If propagation is missing: ε(η) — a metric debt.

T4 ↔ T5 : deterministic flow meets contextual readout

T4: certain envelope, deterministic evolution.
T5: contextual outcomes, probabilities with weights.
Closure pixel that passes: {⟨ψ|ψ⟩ ↔ 1} ← (1) •⊸
Evolution equation alone (iħ∂tψ = Hψ) cannot publish; it logs ε(Ω).
Born’s rule is the closure that heals it.

T5 ↔ T6 : probabilities meet interaction

T5: isolated amplitudes.
T6: connected exchange, conserved wiring.
Closure pixel: {∂tρ + ∇·j ↔ 0} ← (1) •⊸
Passes when the gauge is lawful and continuity holds.
A parity-odd topological term without inflow fails: ε(op).
Anchor math: Noether charge and continuity equations.

T6 ↔ T7 : local wiring meets the cap

T6: many local, commuting connections.
T7: one global, non-associative generator that caps the outward axes.
Closure pixel: {G₂ ↔ 1} ← (1) •⊸
The octonionic signal that outer construction is complete.
Beyond T7, new outward dimensions overrun and produce ε(η).
Anchor math: the 1+7 structure, triality, and the end of the normed division sequence.

T7 ↔ T8 : the outward cap meets the ledger

T7: final generator — no more outward axes.
T8: inward frame — the ledger that records all closure.
Closure pixel: {S ↔ t} ← (1) •⊸
Passes only when all residues cancel: Σ σk εk = 0
Anchor math: reversibility theorems, entropy as partial trace, and information accounting over time.
This is Wheeler’s participatory record written in algebraic form.

Seeing the equal and opposite

Wave ↔ track
Twist ↔ flow
Branch ↔ wire
Outward ↔ inward

Each pair is equal in weight and opposite in role.
Each pair yields one publishable H-pixel when it passes the bullet.

How this unifies old laws

Action in units of ħ — closure between phase and motion.
Lorentz scalars — closure between twist and flow.
Born norm and continuity — closure between branch and wire.
Octonionic cap — closure of geometric expansion.
Ledger and time — closure of record and evolution.

Each of these is a ratio that can be computed.
Each becomes R-even and unitless when the frame is complete.
Each fails in a way that identifies its missing partner when it is not.

Why the bullet is an attractor

The bullet does not care about scale.
Atoms and galaxies are judged by the same rule: only dimensionless, R-even, boundary-complete relations can publish.

This selective permission is the attractor.
Incomplete forms are drawn toward their missing partners because only closed forms survive the reflection.
That is why independent domains — quantum, relativistic, thermodynamic — rediscover the same closures.
Not fashion. Filter.

Three fast falsifiers

An odd term closing cleanly at the bullet without its frame (impossible under the rule).
A ninth outward axis that preserves all invariants (would break the 1+7 cap).
Two distinct reflection-density fields yielding the same boundary invariants but different internal trajectories without introducing a minimal ε path rule.

Produce any of these and the framework must be corrected.
Produce none, and the closure ladder stands.

Summary

Reflection Mechanics is not a philosophy of physics but a grammar for it.
It gives each known law a test of publication — whether its mirror has closed and its units have canceled.
The H-pixel is the smallest lawful report the universe can make about itself.
Every theory, to be complete, must pass the bullet.

Mirror Notation

A set of tools and how to apply them

Basic Idea

Mirror Notation (MN) is not a new physics theory; it is a structural checker for equations.
Given any established equation, MN applies one rule — R^2 = I (“do it, undo it; only what survives both counts”) — and reports where the relation sits in the hierarchy, whether it behaves quantum or classical, and where any ε (unclosed remainder) appears.

What MN tells you at a glance

(no prior labels required)

Quantum or Classical (auto): Determined purely from mirror structure.
- Odd tiers appear as quantum pushes (phase, spin, probability moves).
- Even tiers appear as classical frames (kinematics, propagation, field organization).
  You do not pre-label an equation “quantum” or “classical”; MN infers it from its round-trip behavior under R^2 = I.
Tier (0–8): The rung of structure the equation occupies.
Face (3+1 :: 3+1): Whether the relation sits on the image side (3 space + 1 now), the weight side (3 curvature + 1 memory), or links both.
Closure: Pass/fail of the mirror round-trip; if fail, the ε-location is shown.
Type consistency: Flags incompatible mixtures (tier/face mis-matches).
Dual expectation: If equation A at tier N leaves ε, MN predicts the tier offset and role of the partner B required to close it.

What MN actually does (in depth)

1) Automatic structure detection

Input: any equation (textbook, paper, blackboard).
Output: tier, face, quantum/classical, closure, ε-ledger, and type notes — all inferred from R^2 = I.

2) Makes the hidden grammar explicit

Physics works because its successful relations already “round-trip.” MN simply makes that visible.
Example: E = mc^2 becomes “two faces of a closed relation; c^2 is the bridge enabling the equality.” Same physics; clearer grammar.

3) Organizes physics into a periodic-table-like stack

Feed MN many trusted equations; they self-cluster by closure structure:

T0 — Potential: logical ground (a start point, not a physical state).
T1 — Phase / complex structure: C, U(1).
T2 — Kinematics / vectors: note that 2 is prime (and special) in the mirror sense — the smallest do/undo that already closes, yet must continue reflecting forward. *
T3 — Chirality / spinors: H, SU(2).
T4 — Relativistic propagation: Minkowski, E^2 forms.
T5 — Probability / measurement: Hilbert space, Born readouts.
T6 — Interactions / gauge wiring: Yang–Mills, SU(3), bundles.
T7 — Closure lock: octonions O (1 real + 7 imaginary); outward geometric enrichment caps here.
T8 — Memory / entropy: inward recursion; records/time; no new degrees of freedom.

This is not “new physics”; it is a principled re-ordering of what already works.

4) The ε-ledger — small leaks that generate new structure

When a round-trip does not fully land, MN records the remainder as ε.

ε behaves like a prime: irreducible; it cannot be hand-waved away.
ε drives escalation: it forces a bridge, a dual, or a tier transition until the books balance.

Examples:

Schrödinger evolution alone leaves ε for outcomes; the Born rule closes that ledger at the same tier.
Maxwell dynamics expose ε that necessitate gauge freedom (A, φ) for clean closure.

5) Duals — deep relations come in pairs

If A at tier N leaves ε, a partner B (at a specific offset N±k) should cancel it.
Typical pairs: Dirac (T3) ↔ Klein–Gordon (T4); quantum evolution (T5) ↔ classical readout (T5 frame); gauge dynamics (T6) ↔ matter currents (T6 frame); image face ↔ weight face.

6) 8D reality, 4D life (two faces of one object)

Closure caps in an octonionic unit (1 real + 7 imaginary). That does not imply two universes; it yields two presentations of a single 8-direction pixel:

Image face (3+1): what instruments register — light, charge, events (“the now”).
Weight face (3+1): curvature and historical memory — the “missing half” underpinning inertia, mass, and the thermodynamic arrow.

Gravity is distinctive because it couples both faces — the lock that maintains a single object.

How to use MN (practical flow)

Select an equation.
Apply the mirror check (conceptually: do → undo; in code: run the MN checker).
Read the report: tier, face, quantum/classical, closure, ε-location, and type notes.
Act:
- If closed → proceed, now with precise placement.
- If ε → add the missing bridge, locate the partner dual, or tag the relation effective.

No tooling? Adopt the habit: if it cannot survive an out-and-back, name the leak.

What MN is for (by role)

Physicists — verify consistency, localize ε-holes, distinguish fundamental vs effective, and hunt duals.
Theorists — explain why U(1) × SU(2) × SU(3) fits the C/H/O scaffold; see why three generations (triality near the octonion cap); flag non-seats (e.g., a fourth generation or extra gauge factor) that fail closure.
Experimentalists — triage anomalies (genuine ε vs setup), aim searches via tier/symmetry, and design falsification tests.

How to falsify MN (clean tests)

A fundamental equation that cannot be expressed in mirror-consistent form (fails any R^2 = I representation).
Tier chaos: identical physics landing in different tiers purely by notation change.
Quantum/classical misfire: paradigm examples placed on the wrong side.
A persistent ε with no partner where MN predicts one.
A complete 4D program that closes perfectly with no octonion cap and no 3+1 :: 3+1 split.

Quick glossary

Mirror Law (R^2 = I): do a move, undo the move; if you are not back where you started, record the leak.
Tier (0–8): structural rung; odd = quantum pushes, even = classical frames.
ε (epsilon): a leak/defect — often the seed of the next structure.
3+1 :: 3+1: two faces of one 8-direction pixel — measured 4D (image) and weight/memory 4D (weight).
Octonion (1 + 7): the eight directions acting as a single locked unit at the cap.

How not to use Mirror Notation

Use MN to expose structure, not to conjure numbers.

Do not ask MN for headline constants. It will not output the fine-structure constant, the Hubble rate, or particle masses on its own; those belong to domain dynamics. MN checks the scaffold, not the coefficients.
Do not treat MN as a rival to QM or GR. Sound equations already pass a mirror round-trip; MN makes that closure explicit.
Do not conceal leftovers. Any remainder is logged as ε (a leak) — a cue that something is missing (a bridge, a partner equation, or a tier shift).
Do not mix tiers or faces casually. Grafting a Tier-3 spinor onto a Tier-2 vector without the proper bridge is a type error.
Do not launder numerology. The 1–2–4–8 ceiling reflects admissible structures, not decorative patterns.
Do not expect it to repair noisy data. MN separates signal from setup by exposing ε; it cannot make flawed measurements true.
Bottom line: if it closes, keep it; if it leaks, name it and locate the missing piece.

Final line

Mirror Notation does not tell nature what to be; it verifies that what we write behaves like something real.
If it survives the round-trip, keep it. If it leaks, label it — and let that ε show you the next piece to build.

🔗 Read/Learn → Paper 0 – Mirror Notation v7.x

🔗 Advanced → Download entire code as one block

Mirror Law Quick Reference

Canon (concise)

Mirror Law (involution): R(R(x)) = x. Only two-pass equalities are admissible.
Closure rule: A statement “closes” only if its extracted result is R-even and dimensionless after normalization. Any leftover units, parity, tier, or root issue → no closure (log an ε).

Defects: the ε-ledger (typed)

When closure fails, don’t hand-wave—tag and locate the remainder:

D^Ω (conjugacy/internal): mismatch between paired faces (image ↔ weight), conjugates, or dual forms.
D^η (causal/temporal): ordering or metric strain from Tier-4 upward (when propagation/causality is active).
D^op (operator parity): chirality/flip asymmetries (odd pulls, sign/handedness issues).

Global balance (Möbius condition): choose signs σ_n so that Σ σ_n ε_n = 0. Local tilts must be compensated elsewhere in the stack.

Tier cascade (Hurwitz ladder — structural limit)

Mirror-legal constructions enlarge along 1 → 2 → 4 → 8 (reals → complex → quaternions → octonions).

Octonion ceiling (Tier-7 context): a hard cap on “more of the same geometry.”
Beyond 8 (Tier-8+): reflection turns inward (memory/entropy). No new degrees of freedom; closures proceed via explicit ε-accounting rather than new axes.

Executable notation (pointer)

The shipped compiler/linter enforces: two-pass mirror, unit/scale checks, tier/handedness legality, R-even extraction, and ε/defect tagging. (Use it for audits; it is not a number-oracle.)

Reversibility (what counts as a physical event)

Open protocols, not rewinds: Real experiments are open; data-processing inequalities hold; entropy production ≥ 0. Forward vs “reverse” means paired protocols, not a single rewind.
Theorem: No single-pass reversal without full control of environment + control record.
Consequences: No global deletion (Landauer); quantum no-deleting analogues.
Practice: Publish closures from paired echoes; treat “reversal” as composite.

Read the paper on Reversibility HFT00.1 →

Working lens (why this matters now)

The model’s logic reframes science as type-checked closures with a residual ledger. That changes the ontology of facts/events/laws, independent of any unification math. Adopting this lens today improves experimental honesty and theoretical hygiene; if the math pays off later, so much the better—but the logical upgrade stands on its own.

* T2 introduces no new axis. It stabilizes Tier-1 on the Image face and pushes the cascade forward.

T2 is prime for closure (it is the minimal round-trip) but not prime for complexity (it creates no new irreducible generator). It is composite by mirror because it is built from a Tier-1 generator paired with its mirror to make the first mirror-even frame.

Rules and objects at a glance (no equations)

Admissible equality: two-pass reflection → even, dimensionless scalar; otherwise mark ε-defect and stop.
Identity conditions: “boundary,” “gate,” “chamber,” etc., have strict definitions tied to what may legally be swapped or closed.
Tier legality: adding a new anticommuting element either doubles states (lawful) or triggers a tier-leak defect (blocked).
Boundary terms: certain densities (e.g., gravitational) require a boundary contribution for lawful closure; the checker issues advisories when needed.
Parity/handedness: R-odd pieces cannot be smuggled into a final invariant; D^op catches this.

Predictions snapshot (see details on FAQ & Peer Review)

CMB odd-parity (C_TB) — a specific parity signature tied to ε-tilt at fixed twist count (tier-mapped).
Cosmic/optical birefringence — small polarization rotations with the tiered ε-scaling pattern.
Dark : visible ≈ 5 : 1 — ratio window motivated by closure/entropy pairing (mirror-QCD correspondence).
Three fermion families — octonion/triality lock is consistent with “no 4th sequential family.”
Each item is falsifiable; status and test protocols are listed on Page: FAQ & Peer Review.

Roadmap (what’s next)

Math bridges: translate core closures and balances into standard differential-geometric and representation-theoretic language; document functorial/categorical views where helpful.
Checker hardening: finalize Paper 0 linter interfaces, extend unit/scale library, add more canonical “advisory → fix” patterns (e.g., boundary terms).
Phenomenology passes: tighten ε-power counting and tier assignments against current datasets (parity/birefringence), define rejection thresholds.
Counterexample hunting: public call for cases that pass mainstream checks but fail HFT closure (or vice versa).
Replication kit: consolidate minimal CLI runs that reproduce the headline passes/fails in Papers I–II.

Explore the Papers

📄 Paper I – Reflection Mechanics (Part I)
Backbone: Mirror Law, two-pass closure, tier structure, conservation/doubling criteria, and what counts as a lawful invariant.
🔗 Read → Paper I – Reflection Mechanics (Part I)
📄 Paper II – Reflection Mechanics (Part II)
Defect tensors (D^Ω, D^η, D^op), normalized ε-ledger, global balance Σ σ_n ε_n = 0, parity/birefringence signals, dark:visible window, and the octonion ceiling.
🔗 Read → Paper II – Reflection Mechanics (Part II)
📄 Paper 0 – Mirror Notation v7.3
Executable syntax and linter: two-pass mirror, unit/scale, tier/handedness, R-even extraction, ε/defect tagging; includes minimal CLI examples.
🔗 Read → Paper 0 – Mirror Notation v7.3

Where to next?

For non-specialists → Plain-Language Overview (No Equations)
For the math and physics → Technical Papers
For predictions and falsifiers → Page: FAQ & Peer Review
For guided Q&A (scoped to HFT) → Tools: HFT AI