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Bilateral stimulation as hierarchical multi-modal pattern — 2026-05-11

Role: Concertmaster (dispatched by conductor) Round date: 2026-05-11 Method: MPM-discipline closed-form linear algebra (numpy / scipy eigh); deterministic seed 20260511; reproducible 24-record NDJSON output; 33s runtime. Artifacts: - Script: bilateral-stimulation-hierarchical-pattern-script.py - Per-level records: bilateral-stimulation-hierarchical-pattern-per-level-2026-05-11.ndjson

Provisional verdict

Math-doesn't-lie verdict on the user's load-bearing question: the srmech framework genuinely covers hierarchical multi-modal bilateral stimulation patterns, end-to-end, with no new math required. All three load-bearing claims hold; the only framework discipline needed is choosing the segment-transition graph weighting (uniform vs inverse-duration) appropriately for variable-duration patterns. The user's "active-system" stance is documented but does not extend the framework.

Five math identities verified at machine precision (max deviation ~1e-14):

Identity Layer Max deviation
K_2 eigenvalues exactly {0, 2} §3.5 row 5 (degenerate case) 0.0
13-segment uniform cycle = 2(1−cos(2πk/13)) §3.5 row 5 (general graph) 2.66e-15
Fiber-bundle Cartesian sum = λ_base + λ_fiber §3.5.4 (operator orthogonality) 9.19e-15
Envelope × pattern Cartesian product Laplacian §3.5.3(F) Imrich-Klavžar 5.30e-15
Envelope × pattern strong-product adjacency: μ_env + μ_seg + μ_env·μ_seg §3.5.3(F) Imrich-Klavžar 8.88e-15
Triple Cartesian product (envelope × pattern × modality) §3.5.3(F) + §3.5.4 composition 1.07e-14

Three-level math summary

Level 1 — Trivial bilateral alternation (DEGENERATE)

Two motors A, B; K_2 adjacency; L = [[1,−1],[−1,1]]; eigenvalues {0, 2}; anti-phase eigenvector (1,−1)/√2. CTQW lift U(t) = exp(−i L t) propagates anti-phase mode at rate 2 on T².

Framework-already-covers verdict. EMDR Phase 2 NTP-style time sync (±100μs first-pair handshake; ±30μs converged drift over 90 min stress test) IS the T² phase-coherence preservation primitive (srmech §3.5.1 layer b). Phase 6r drift continuation under disconnect IS the magnitude-shadow fallback (exp(−L t)) when fresh measurements pause but extrapolation continues. EMDRIA standard 0.5–2 Hz @ 125 ms ON / 500 ms cycle is one operating point on this 2-node graph's T² ambient.

Honest verdict: not a "complex object" in the project's sense. One phase relationship, one frequency. The user's framing is correct — this is the degenerate case.

Level 2 — Multi-segment pattern (THE EMERGENCE POINT)

Test pattern (user's "emergency lights" example): [ON-300ms × 3 with 200ms gaps] [OFF-1500ms] [ON-100ms × 5 with 50ms gaps] [OFF-3000ms] — 13 segments, 6500 ms total cycle, segment durations vary 50ms–3000ms, intensities {0, 1}.

Three layers stack cleanly per srmech §3.5:

  1. Per-segment Euclidean-time-axis (§3.5 row 1). Step-function segments have only the constant (λ=0) mode within each segment; sub-mode structure is trivial. Recorded as a degeneracy-check finding, not a load-bearing computation.

  2. Segment-transition graph-Laplacian (§3.5 row 5). Cycle graph of 13 segment-nodes. Uniform-weight version matches the closed-form 2(1−cos(2πk/n)) cycle spectrum at machine precision (max deviation 2.66e-15); inverse-duration-weighted version (edge weight 1000/d_ms in Hz) gives a deformed spectrum where short segments contribute more to high-frequency modes. Fiedler value is the natural pattern-period proxy. Higher eigenvalues correspond to within-pattern substructure (the 3-flash vs 5-flash grouping).

  3. Modality fiber bundle (§3.5.4). Base = 13 segments; fiber = rank-2 (vibe, light); total space = 13 × 2 = 26-dim. Constructed via Cartesian sum L_total = L_base ⊗ I_2 + I_n ⊗ L_fiber (the §3.5.4 operator-orthogonality identity). Empirical eigendecomposition matches the closed-form sum-of-eigenvalues prediction λ_total = λ_base + λ_fiber to machine precision (9.19e-15).

CTQW lift on the bundle Laplacian produces eigenphase variance on T²⁶; documented at t=1.0 graph units for record-keeping.

Framework-handles-cleanly verdict. All three layers are explicit instantiations of existing srmech rows. The framework was already there.

Level 3 — Hierarchical pattern-of-patterns (THE RECURSION POINT)

Level-2 envelope (2-node graph, K_2) × level-1 pattern (segment-transition graph). Tested for two level-1 patterns:

  • Envelope × emergency-light-13: Cartesian sum match 5.30e-15; strong-product adjacency match 8.88e-15.
  • Envelope × simple-bilateral-2: Cartesian sum match 2.66e-15; strong-product adjacency match 1.78e-15.

Triple Cartesian product (envelope × pattern × modality, dimensions 2 × 2 × 2 = 8): match 1.07e-14.

Framework-handles-cleanly verdict. The Imrich-Klavžar product-graph universality (srmech §3.5.3(F)) gives closed-form composition at any depth. The sum-of-eigenvalues formula λ_{ij} = λ_i(env) + λ_j(pattern) for Cartesian product and the μ_{ij} = μ_i + μ_j + μ_i·μ_j formula for strong-product adjacency both verified empirically.

Anomaly findings

Anomaly 1: uniform-weight segment-transition Laplacian loses duration information. Any two patterns with the same segment count have identical eigenspectrum under uniform weighting (the graph topology is identical). This is bounded — inverse-duration weighting or per-segment Euclidean-time-axis lifting (§3.5 row 1 with segment duration as lattice extent) recovers the information. The framework already supports both responses; the engineering discipline is choosing per use-case.

Anomaly 2: Cartesian vs strong product produce distinguishable bundles for modality fusion. Cartesian sum (separable channels: vibration alone or light alone gives the same eigenphase structure on different bands) vs strong product (fused channels: vibration AND light produces a NEW phase relationship). The framework supports both math-identity-cleanly; which one therapy benefits from is a clinical EMDRIA question downstream and out of scope for this spike. Documented for future clinical hypothesis-testing.

Anomaly 3: step-function per-segment time-axis is degenerate. All within-segment energy lives at the constant mode (λ=0); sub-mode structure trivial for piecewise-constant intensity. This is consistent with the pattern domain (intensity is binary per segment) and is not a framework gap — it's the math saying "this segment carries no sub-structure to decompose."

No catastrophic gaps surfaced. The framework's (Transform, λ_k, g) decomposition + product-graph universality + fiber-bundle composition fully express the hierarchical multi-modal bilateral pattern problem.

Cross-domain analog instances

The hierarchical-multi-modal-active-stimulation structure has clean analogs across already-absorbed srmech domains:

Domain Analog Provenance
Music Rhythm = beat × measure × phrase × form (Tonnetz cyclic-group hierarchy) srmech §5.2 audio absorption
EEG / neural oscillations Theta-gamma cross-frequency coupling = T^n eigenphase lift between scales srmech §3.5.1 layer (b)
Cardiac PVC patterns / arrhythmias Multi-segment nested rhythmic structure (HRV multi-scale) New analog (cross-pollination candidate)
Morse code Discrete-time analog: dot × dash × character × word × sentence hierarchy tested in script as P_morse_V
Emergency vehicle sirens wail / yelp / phaser / hi-lo each have multi-segment patterns; change-of-mode = level-2 envelope the user's load-bearing example
OFDM cyclic prefix subcarrier telecom signal as T^N periodic-CP basis (srmech §3.5 row 3 + §5.4 telecom round identity) srmech §5.4

Most-direct cross-pollination: EEG theta-gamma coupling is the cleanest published-spectral-literature analog. It is literally T^n eigenphase coupling between two scales of brain activity, IS the §3.5.1 layer (b) structure, and has decades of clinical EEG-coherence literature (Buzsáki, Canolty et al). A future srmech absorption round on neural oscillations would extend the cross-manifold §3.5 row 1+5 coverage with biomedical-electrophysiology evidence.

Active-vs-passive stance (the "active system" observation)

User: "what's odd here is it's an active system whose outputs can be vibration or light or both."

Documented as a stance record, not a framework extension. The (Transform, λ_k, g) decomposition (srmech §3.0) applies identically to active and passive systems. The difference is direction:

  • Passive (audio analysis, NMA, EEG): input is external; system computes the eigendecomposition; output is a structural fingerprint.
  • Active (EMDR device, emergency-light controller, metronome, pacemaker): pattern is user-specified; system generates the field-domain excitation matching the eigenphase trajectory on T^N ambient.

Per MFO §VII.1.1 two-level ontology: active stimulation IS a field-domain excitation generator; the multi-modal output coupling (vibration AND light) IS the modality fiber bundle product structure. No new math needed.

Ship-mode mapping — EMDR Phase 7 P7.3 PWA Pattern Designer

The EMDR firmware project's Phase 7 P7.3 (next milestone per CLAUDE.md) ships a PWA Pattern Designer. This spike maps directly:

Pattern descriptor schema. List of (duration_ms, vibe_intensity, light_intensity[, audio_intensity]) tuples. TOML-shaped per project memory feedback_ndjson_over_bloated_json.md (descriptor-shaped data → TOML; results-style data → NDJSON).

Spectral fingerprint. Top-k eigenvalues from inverse-duration-weighted segment-transition graph-Laplacian. Pattern similarity via cosine-of-fingerprint-vectors. Pattern library indexable as a Path-D-style spectral index over the descriptor heavy-store (srmech §2). Pattern interpolation via eigenmode-weighted blending (future work).

Framework layers, in order of complexity supported:

  1. §3.5 row 1 (Euclidean grid, per-segment time axis — degenerate for step functions)
  2. §3.5 row 5 (general graph, segment-transition Laplacian — the load-bearing layer)
  3. §3.5.4 (fiber bundle, modality × segment Cartesian sum — multi-modal layer)
  4. §3.5.3(F) (product-graph, envelope × pattern composition — hierarchical layer)
  5. §3.5.1 layer (b) (eigenphase T^N via CTQW — phase-coherent dynamics)

Phase 6r drift-continuation (CLAUDE.md) already implements §3.5.1 layer (b) magnitude-shadow fallback for the 2-node case; the multi-segment generalization follows the same drift extrapolation pattern at the segment level.

Conductor decision points (fermata records)

The conductor should integrate these findings to decide:

  1. §1.5 cross-domain pollination map gains a 7th absorption round — "Bilateral stimulation patterns (with EMDRIA as canonical citation; project-internal direct-mission fit)." This round is unusual because the domain is the project's actual mission (not a stretch test) — it deserves notebook treatment commensurate with the audio round (§5.2). My recommendation: add §5.6 as the absorption-round section, with this notes file as the detailed scoping reference.

  2. §3.5.3 gains a new motif (G) — "Hierarchical multi-modal active stimulation as Cartesian-product-of-graph-Laplacians × fiber-bundle modality structure." The motif is empirically anchored by the six machine-precision math-identity matches in this spike; provenance pattern matches the existing motif (C) chess + MFO Phase B + finance instances.

  3. §3.5.4 fiber-bundle table gains a new row — "EMDR bilateral pattern: segment-transition graph × (vibe, light, audio) modality fiber rank 2-3."

  4. EMDR Phase 7 P7.3 specification gains a srmech-compatible descriptor — TOML schema + spectral-fingerprint computation + Path-D-style pattern library index. Recommended as concrete deliverable for P7.3 integration with the spectral framework.

  5. Future-work bullet for clinical EMDRIA hypothesis: Cartesian-product (separable modalities) vs strong-product (fused modalities) bilateral multi-modal stimulation produces math-identity-different eigenspectra; downstream clinical question is which is therapeutically superior, but the framework expresses both. This is a research-direction record only; the project does not make clinical-efficacy claims.

What's NOT in scope here (honest framing)

  • No clinical efficacy claims. The framework expresses Cartesian-vs-strong modality fusion as math identity; whether one therapeutic configuration outperforms another is downstream of this spike.
  • No HRV biofeedback adaptive control. That's a substrate primitive (state-dependent), not a closed-form g(λ) entry, per srmech §4.2.
  • No phase-extraction-from-observation methods. This spike is forward-direction (generate pattern → eigenphase trajectory); inverse problems (extract user-physiological-phase from sensor data) are different math.
  • No microtonal / non-12-EDO audio extension. Z₁₂ audio cyclic-group framing per srmech §5.2 carries through; microtonal needs different modular arithmetic and is out of scope for this spike.

References

  • srmech §3.5 cross-manifold table (six rows)
  • srmech §3.5.1 layer (b) eigenphase torus T^n (CTQW lift)
  • srmech §3.5.3(F) product-graph universality (Imrich-Klavžar Product Graphs)
  • srmech §3.5.4 fiber-bundle structure (chess 2D 64×10 + 4D 4096×11 as load-bearing instances)
  • srmech §5.2 audio absorption round — bilateral audio as peer modality
  • CLAUDE.md Phase 2 NTP-style time sync (±30 μs drift over 90 min)
  • CLAUDE.md Phase 6r drift continuation (magnitude-shadow fallback at 2-node level)
  • CLAUDE.md Phase 7 P7.1 Scheduled Pattern Playback + P7.3 PWA Pattern Designer
  • MFO §VII.1.1 two-level ontology (active stimulation = field-domain excitation generator)
  • Memory user_explanation_discipline.md — Feynman compression: trust the user's compressed framing