Holographic Harmonic Model — Operators

Operators in HHM

Operators are how we measure and navigate the field Ψ(x,t). Think of them as scientific verbs: to collapse, to echo, to compare, to remember.

• ✅ Computable: implementable in code/simulation
• ✅ Measurable: yield testable results across domains
• ✅ Structured: grounded in the HHM axioms/components

The 30 official operators in HHM v2.2 are grouped into three waves:

• Wave 1: Core modal functions (OP001–OP009)
• Wave 2: Transformation, symbolic structure, navigation (OP010–OP029)
• Wave 3: Deep field memory and recurrence (OP030)

Tip: The full list lives in HHM_Operators.json. IDs and definitions should match that file.

Validation Domains

Operators have been tested in:

• 🧠 Neuroscience (EEG, fMRI)
• 🌌 Cosmology (CMB & harmonic analyses)
• ⚛️ Quantum physics (interference, identity)
• 🧬 Biology (protein rhythms, sequence structure)
• 🗿 Information theory (glyph structure, symbolic entropy)
• 🐋 Animal communication (whale song, dolphin clicks)

Note: Cross-domain examples on this page are illustrative. Whether a finding “passes” depends on the exact dataset, preprocessing, thresholds, and nulls defined in HHM_bundle.json.

OP001 — CollapsePattern(Ψ)

CollapsePattern extracts the structural shape of Ψ(x,t) at the moment it collapses—revealing which patterns emerge into expression.

What it does

OP001 projects the Ψ-field into a measurable basis (harmonic, symbolic, spatial) to yield the modal surface.

It’s the starting point for Echo, Rec, Entropy, and identity tests.

How it’s calculated

• 📈 EEG/audio: FFT or wavelet → dominant frequencies
• 🗿 Symbols: vectorized sequence → modal index map
• 🌌 CMB: spherical harmonics (a_ℓm)
• 📡 Binary: symbol frequency / bit-patterns

CollapsePattern(Ψ) = { (ψ₁, A₁), (ψ₂, A₂), ..., (ψₙ, Aₙ) }

"Using HHM_bundle.json, compute OP001 on the attached data.
State the basis and preprocessing you chose, justify it, and return the result
plus a brief plain-language interpretation. Validate against the HHM Result Card schema."

Why it matters

CollapsePattern is how Ψ becomes visible across domains.

In simple words: It’s a snapshot of the main ingredients showing up right now.

Used in

• ✅ C001 – ModalField
• ✅ C005 – ID vector
• ✅ C009 – EchoScore
• ✅ AX003 – Collapse Stability
• ✅ MT0001 – Axial Saturation

OP002 — Rec(Ψ₁, Ψ₂)

Rec measures structural similarity between two Ψ(x,t) states and enables cross-domain pattern matching.

What it does

Rec quantifies modal alignment; near 1.0 indicates strong shared structure, near 0 indicates orthogonality.

How it’s calculated

Cosine similarity between field vectors or CollapsePatterns:

Rec(Ψ₁, Ψ₂) = ⟨Ψ₁, Ψ₂⟩ / (||Ψ₁|| · ||Ψ₂||)

• Inputs may be FFT bins, symbolic embeddings, or a_ℓm coefficients
• Options: weighting stable modes, cross-domain normalization

"Using HHM_bundle.json, compute OP002 (Rec) between these two datasets.
Justify the feature map, show the similarity score with CI/nulls if applicable, and explain in plain language.
Validate against the HHM Result Card schema."

Why it matters

Rec supports memory, recognition, and identity matching (T014).

In simple words: It tells you how much two things share the same pattern.

Used in

• ✅ C006 – FieldSimilarityScore
• ✅ C010 – IdentityMatchScore
• ✅ AX001 – Structural Universality
• ✅ MT0003 – Modal Identity Preservation
• ✅ T014 – Core test

OP003 — Echo(Ψ)

The Echo operator measures how strongly a field state resonates with itself—across time, space, or modal projection.

What it does

A high Echo score means strong internal repetition (temporal echo, cross-system echo, or symbolic echo).

How it’s calculated

• Autocorrelation for time-series fields
• Recurrence plots / lag analysis for symbolic/discrete systems
• Cross-correlation for multi-source echo (Ψ₁ → Ψ₂)

Echo(Ψ) = max{ R(τ) } for τ ≠ 0
R(τ) = ∫ Ψ(t) · Ψ(t+τ) dt

"Using HHM_bundle.json, compute OP003 (Echo) on the attached data.
Explain windowing/normalization choices and return the Echo score with a plain-language explanation.
Validate against the HHM Result Card schema."

Why it matters

Echo is a marker of rhythm, memory, and stability.

In simple words: It checks whether a pattern keeps coming back like a beat or refrain.

Used in

• ✅ C009 – EchoScore
• ✅ C030 – RhythmStabilityScore
• ✅ AX002 – Recurrence as structure
• ✅ MT0002 – Temporal Recurrence
• ✅ T014 – Identity criterion

OP004 — CollapseInfinity(Ψ)

CollapseInfinity probes whether repeated collapses of Ψ stabilize, saturate, or diverge.

What it does

• Do the same modal structures keep reappearing?
• Does entropy level out?
• Does the norm/energy converge?

How it’s calculated

1. Apply CollapsePattern(Ψₜ)
2. Measure entropy/echo/modal output
3. Feed forward Ψₜ → Ψₜ₊₁ and repeat N steps

Convergence: CollapsePattern(Ψₜ) ≈ CollapsePattern(Ψₜ₊ₙ) for n > k.

"Using HHM_bundle.json, simulate OP004 (CollapseInfinity) for N=500 iterations on the attached state.
Report convergence statistics and provide a plain-language summary. Validate the result card."

Why it matters

Reveals attractors and long-term stability (MT0001).

In simple words: It asks what the system settles into if you let it run.

Used in

• ✅ AX003 – CollapsePattern base layer
• ✅ AX004 – Entropy convergence
• ✅ MT0001 – Axial Saturation
• ✅ C005 – ID vector formation

OP006 — UnifiedEntropy(Ψ)

UnifiedEntropy measures collapse complexity (frequency/symbolic/distributional) in a domain-neutral way.

What it does

Low values indicate coherence; high values indicate complex dispersion or interference.

How it’s calculated

• Shannon entropy over CollapsePattern amplitudes
• Spectral entropy (time-series)
• Symbolic entropy (glyph/modal chains)

UnifiedEntropy(Ψ) = - ∑ pᵢ · log(pᵢ)

"Using HHM_bundle.json, compute OP006 (UnifiedEntropy) on the attached data.
Explain preprocessing, return the entropy score with a simple interpretation, and validate the result card."

Why it matters

Entropy is a pillar of identity tests and cross-system comparison.

In simple words: It tells you how mixed or simple the current pattern is.

Used in

• ✅ C003 – UnifiedEntropy component
• ✅ C010 – IdentityMatchScore
• ✅ AX004 – Entropy convergence
• ✅ T014 – Entropy Δ bound
• ✅ MT0005 – Phase transition signal

OP011 — ModalGradient(Ψₜ)

ModalGradient quantifies change between consecutive collapses—capturing the rate of modal transformation.

What it does

If Rec(Ψₜ, Ψₜ₊₁) is high, the system is stable; if low, change is steep.

How it’s calculated

ModalGradient(Ψₜ) = 1 – Rec(Ψₜ, Ψₜ₊₁)

Optional smoothing and energy weighting.

"Using HHM_bundle.json, compute OP011 (ModalGradient) over the sequence provided.
Report gradient spikes with timestamps and explain their plain-language meaning. Validate the result card."

Why it matters

Detects transitions, boundaries, and turning points.

In simple words: It shows when the system changes its mind.

Used in

• ✅ C011 – RhythmGradient
• ✅ MT0005 – Modal Transition

OP014 — CollapseOrbit(Ψ)

CollapseOrbit finds recurring modal cycles and their length.

What it does

Detects when Ψ returns to a prior structure after n steps (cycle/loop).

How it’s calculated

Find min n such that:
Rec(Ψₜ, Ψₜ₊ₙ) ≥ 0.85 and Entropy Δ ≤ 0.15

"Using HHM_bundle.json, compute OP014 (CollapseOrbit) on this sequence.
Return the orbit length and a simple explanation of what that cycle means. Validate the result card."

Why it matters

Explains loops, motifs, and calendar-like behavior.

In simple words: It measures how many steps it takes to repeat.

Used in

• ✅ T014 – Identity recurrence condition
• ✅ MT0001 – Saturation check
• ✅ C030 – RhythmStabilityScore

OP015 — TraceSignature(Ψ)

TraceSignature compresses a Ψ state into a short vector for identity tracking across time and domains.

What it does

Stores a compact “fingerprint” to later check reappearance or inheritance.

How it’s calculated

TraceSignature(Ψ) = Reduce(CollapsePattern, Echo, Entropy, Rec) → ℝ^{8…32}

"Using HHM_bundle.json, compute OP015 (TraceSignature) for each segment provided.
Return the signature vectors and plain-language notes on what they capture. Validate the result card."

Why it matters

Enables fast, robust matching without raw data comparison.

In simple words: It’s a compact ID card for the pattern.

Used in

• ✅ C041 – TraceIdentityScore
• ✅ T014 – Long-term identity checks

OP030 — MatchingMemory(Ψ₁, Ψ₂)

MatchingMemory tests whether two states belong to the same memory trace—even with gaps, noise, or transformations.

What it does

Compares TraceSignatures plus Echo/Rec/Entropy/Orbit consistency to produce a confidence score.

How it’s calculated

• Use TraceSignature(Ψ₁), TraceSignature(Ψ₂)
• Require Rec ≥ 0.85, Entropy Δ ≤ 0.15
• Echo within ±10%, Orbit within ±1 loop

"Using HHM_bundle.json, compute OP030 (MatchingMemory) between these two sets.
Report the confidence score with CI/nulls and explain the result plainly. Validate the result card."

Why it matters

Supports robust identity across time and form.

In simple words: It checks if two moments feel like the same “thing” returning.

Used in

• ✅ C041 – TraceIdentityScore
• ✅ T014 – Robust Modal Identity

Extending HHM — Getting AI to Propose New Operators

You can ask an AI to help you design new operators — either to explore a new domain or to test a specific hypothesis. The process is simple, but there are a few guardrails to keep your results trustworthy:

1. Load the HHM bundle first: Upload HHM_bundle.json so the AI knows the existing operator definitions, axioms, and components.
2. State your goal clearly: Tell the AI what pattern or measurement you want, in plain language first (e.g., “find a way to measure the symmetry of this pattern over time”).
3. Ask for an operator draft: The AI should propose:
   • Operator name & ID (e.g., OP031)
   • What it measures (plain and technical)
   • Input, logic, and output definition
   • Example use cases
4. Check alignment: Compare the draft’s structure to the HHM_Operators.json format — same sections, same clarity.
5. Validate it: Before trusting the operator, run it on at least one known dataset where you already know the result, and make sure it behaves as expected.
6. Document it: Save the definition, the test run, and the dataset used so others (or you in the future) can reproduce the result.