Bowl-Mesh Falsifiability Protocol

This file is the falsifiability protocol for the per-node TaoBowl Q-filter that replaces the binary standing-wave mask. It is the prediction ledger, not a results document. Every empirical claim below is marked as a prediction (P*) or a falsification trigger (F*); nothing here is an observation.

Companion documents:

• STANDING_WAVE_STATUS_2026_05_03_AFTER_FALSIFICATION.md (same dir) — the failure being addressed.
• AMPLITUHEDRON_PLAN.md (same dir) — the prefix-cache that composes with this reframe rather than competing with it.
• Gnosis/AmplituhedronFalsifiability.lean (in open-source/gnosis-math/) — the parallel amplituhedron-side falsification module. It established the canonical pattern this protocol follows: name the failure regime as a limit of the correct mechanism, prove a *_runtime_oracle_* pentad to pin the Lean → Rust value contract, exhibit the structural-preservation property the regime violates. The terminology ("high-damping limit", "runtime oracle") is reused here verbatim.
• Gnosis/BowlMeshNode.lean — node types and mask_at_high_damping (the binary mask IS the high-damping limit of a TaoBowl, not the damping=0 limit; see §1).
• Gnosis/BowlMeshQSweep.lean — the sweep theorems (off_resonance_strictly_positive, damping_zero_off_resonance_pass_through, high_damping_monotone_decay, layer_residual_nonzero_under_positive_damping).
• Gnosis/BowlMeshPinkNoiseBound.lean — the spectral-flatness cutoff theorems (flatness_dichotomy, white_flat_argmax_ambiguous, pink_structured_argmax_recovers, bowl_recovers_below_flatness_threshold).
• Gnosis/BowlMeshRuntimeOracle.lean — five *_runtime_oracle_* theorems pinning the exact Nat values the Rust BowlMeshHead must reproduce on canonical witnesses (balancedBowl, highDampingBowl, pejorativeBowl, filledBowl).
• src/bowl_mesh.rs — the Rust per-node Q-filter primitive.
• src/bin/bowl-q-sweep.rs — the measurement harness that emits the curves below.

1. Headline reframe

The binary mask used in the wave-3 / wave-4 PCA-only experiments is the high-damping limit of a TaoBowl, not the damping=0 limit. The distinction matters and is fixed by BowlMeshNode.mask_at_high_damping:

for any bowl b, any off-resonance freq, any amp, amp < b.damping + 1 → route_through_bowl b freq amp = 0.

In words: when damping is large enough that the input amplitude falls under the damping + 1 floor, Nat division (the same truncation that the quantized residual stream sees in practice) crushes the off-resonance output to zero. That is the binary-mask behavior, recovered as a limit of the bowl model — not a different mechanism.

The damping=0 case is a different failure mode, the "Q-amplifier-only degenerate" of BowlMeshQSweep.damping_zero_off_resonance_pass_through: off-resonance is preserved (amp / 1 = amp) but on-resonance loses its gain because qFactor is pinned to 0 by EchoChamberAsTaoBowl.qFactor_eq_zero_of_damping_eq_zero. The damping=0 end of the sweep collapses the amplification side; the high-damping end collapses the off-resonance side. The healthy regime is interior.

The mask falsified on 2026-05-03 lives at the high-damping end; the bowl-lifted route avoids the failure by sitting interior, where both sides of filteredAmplitude carry signal. The residual-stream add-and-norm x_{l+1} = x_l + attn(x_l) + ffn(x_l) then never sees a zero-filled dimension, which is what the binary mask did and what falsified.

2. What the falsification proved

From STANDING_WAVE_STATUS_2026_05_03_AFTER_FALSIFICATION.md §2 and the empirics it cites (docs/speculative-decode-empirical-qwen-0.5b.md and docs/draft-verify-qwen-coder-7b.md):

• K=1 acceptance = 0 / 79 across N ∈ {2,4,8} on Qwen-0.5B PCA-only, pinned exactly to the 1/(2N) worst-case slowdown floor. PCA-only at K=1 is a 4×–16× slowdown, not a speedup.
• Even at coverage 0.95 (only ~5% of residual-stream dimensions zeroed), top-1 argmax preservation was 0 / 9 at the layer-policy chosen by the cliff atlas.
• On Qwen-Coder-7B at K=5, PCA-only, 8 cliff layers: F_eff = 0 / 30 = 0, cosine drift averaged 0.58, KL averaged 6.59 nats, net wire savings = 0.

The cliff structural part (where the cliffs sit, which layers the picker selects) held — see STANDING_WAVE_STATUS_… §3. What failed was the operational fidelity claim on top of binary-mask carving.

3. Why the bowl model predicts this

Gnosis/ThoughtBowlMechanics.lean proves the Daodejing-11 multiplicative form:

bowlUsefulness (bowlOfField waves) = waves.length * dissentingCount waves (field_dao_dejing_usefulness_formula, ibid.)

and Gnosis/EchoChamberAsTaoBowl.lean proves the symmetric collapse:

IsFilledBowl b → bowlUsefulness b = 0 (filled_bowl_dead), IsEmptyBowl b → bowlUsefulness b = 0 (empty_bowl_dead).

The binary mask zero-fills off-resonance dimensions. By BowlMeshNode.mask_at_high_damping, that behavior is reproduced by any bowl whose damping is large enough that the input amplitude falls under the damping + 1 floor. On the zero-filled fraction, the dimension is operating as if its void had been emptied — there is no breathing room left to carry the off-mode signal that the residual stream is about to add into. filled_bowl_dead then forces bowlUsefulness = 0 on that fraction. The wave-3 / wave-4 falsification empirically rediscovered a result the bowl lattice already pins: a high-damping bowl cannot do the work a residual-stream filter is being asked to do.

This is not a post-hoc fit. The bowl model was already in the repository (EchoChamberAsTaoBowl.lean, written before the wave-3 experiments), and the predicate IsFilledBowl was already labelled there as the structural failure mode. The wave-3 mask satisfies IsFilledBowl on the zeroed fraction. The falsification reads as Daodejing-11's "function in the void" expressed in PCA coordinates.

4. The damping sweep

src/bin/bowl-q-sweep.rs sweeps the damping dial of bowl_mesh.rs and emits K=1 argmax accuracy per layer per model. The knob:

damping ∈ {0, 1, 2, 4, 8, 16}

Semantics, mirroring filteredAmplitude in EchoChamberAsTaoBowl.lean and BowlMeshQSweep.lean:

• damping = 0 — Q-amplifier-only degenerate. Off-resonance returns amp / 1 = amp (passthrough); on-resonance returns amp * qFactor = amp * 0 = 0 because qFactor is pinned to 0 at damping = 0. The bowl collapses on the amplification side. This is not the binary mask; it is the opposite failure mode. (BowlMeshQSweep.damping_zero_off_resonance_pass_through.)
• 0 < damping, amp ≥ damping + 1 — healthy interior. Off-mode amplitude is attenuated by damping + 1; on-mode amplitude is amplified by qFactor. The cross-layer add still receives a signal on every dimension. (BowlMeshQSweep.off_resonance_strictly_positive.)
• amp < damping + 1 — high-damping mask regime. Off-mode amplitude is crushed to 0 by Nat division (in f32 it is a vanishing fraction, but in any quantization-aware downstream it is the binary-mask equivalent). This is the regime that falsified on 2026-05-03. (BowlMeshNode.mask_at_high_damping.)

The sweep at damping ∈ {0, 1, 2, 4, 8, 16} spans both endpoints (the amplifier-collapse and the mask-regime crush) on the typical post-RMSNorm amplitude scale of the residual stream. BowlMeshQSweep's monotonicity (high_damping_monotone_decay) guarantees the off-mode amplitude falls monotonically across the sweep, so the K=1 accuracy curve should be unimodal with the peak interior.

The sweep harness records, per (model, layer, damping):

• K=1 argmax accuracy vs the unfiltered baseline.
• Cosine drift of the post-filter residual vs the baseline residual.
• KL divergence of the next-token distribution.
• Wall time (so the measurement is honest about the cost of damping).

5. The pink-noise upper bound

Gnosis/BowlMeshPinkNoiseBound.lean formalizes the cutoff between spectral regimes in which the bowl Q-filter can preserve argmax and those in which it cannot.

• flatness_dichotomy — every nonempty spectrum is either white-flat (listMin = listMax) or pink-structured (listMin < listMax). The dichotomy is the structural carrier of the cutoff.
• white_flat_argmax_ambiguous — when the input spectrum is flat, two off-resonance frequencies carrying equal amplitude route to equal route_through_bowl output. The bowl cannot prefer one over the other; the argmax is structurally tied.
• pink_structured_argmax_recovers — when the spectrum has a dominant bin AT the bowl's fundamental mode and qFactor ≥ 1, the on-resonance route output is at least the dominant amplitude. The carrier survives.
• bowl_recovers_below_flatness_threshold — existence of a flatness threshold (a constructed witness, not a tight bound) below which on-resonance strictly exceeds off-resonance.

The cutoff is the falsifiable predicate: there must exist a measurable spectral-flatness threshold below which the bowl reframe helps and at or above which it cannot. The harness emits per-layer flatness so this threshold can be located empirically. The Lean module fixes the predicate shape; it does not pin the threshold value.

6. Lean → Rust runtime oracle

Gnosis/BowlMeshRuntimeOracle.lean exports five *_runtime_oracle_* theorems, mirroring the AmplituhedronFalsifiability.vandermonde_runtime_oracle_* pentad. These are the value-level contracts the Rust kernel (src/bowl_mesh.rs, BowlMeshHead) MUST reproduce on the same canonical witnesses:

Contract #3 is the only one with an arithmetic gap: Lean Nat division truncates 10 / 100 to 0 (the binary-mask equivalent); Rust f32 division returns 0.1. The qualitative claim (the off-resonance amp vanishes relative to the on-resonance amp under high damping) is preserved across both arithmetics. The exactness gap is the cost of the f32 path and is documented in BowlMeshRuntimeOracle.lean's comment on the regime.

7. Predictions

All five are predictions, not observations. They will be falsified or upheld by bowl-q-sweep.rs running against Qwen-0.5B and at least one of {Qwen-Coder-7B, Llama-1B}.

• P1. At damping large enough that amp < damping + 1 on the layer's typical post-RMSNorm amplitude scale (call this the high-damping end of the sweep — at damping = 16 on residuals normed to ≈1 this regime is already entered), K=1 argmax accuracy on the bowl-mesh equals the K=1 accuracy of the existing binary mask within the measurement noise floor (~ ±1 token across the 79-window protocol). Justification: BowlMeshNode.mask_at_high_damping is a pointwise equality between the bowl-lifted route and the mask's zero-fill in the crushing regime. If the implementation honors the Lean spec the two harnesses must agree.

• P2. For an interior damping (small enough that amp ≥ damping + 1 and large enough that qFactor ≥ 1), K=1 accuracy strictly exceeds both the damping = 0 (Q-amplifier-collapse) end and the high-damping mask end on the same layer / same model. Justification: BowlMeshQSweep.off_resonance_strictly_positive pins the off-resonance branch to ≥ 1 in the interior, and BowlMeshNode.positive_damping_preserves_signal pins the route as a whole to strictly positive amplitude there; both endpoints sit at bowlUsefulness = 0 by filled_bowl_dead / qFactor = 0.

• P3. There exists damping* > 0 strictly interior to the sweep at which K=1 accuracy peaks. Either side of damping* the accuracy is lower: below it, the bowl over-amplifies and on-resonance saturates (the amplifier-collapse approach); above it, the bowl over-attenuates and off-resonance enters the mask regime (high_damping_monotone_decay).

• P4. damping* correlates positively with the residual-stream spectral flatness measured at that layer. Justification: flatter spectra need more aggressive damping to lift a preferred mode above the noise; the pink_structured_argmax_recovers theorem only applies when the bowl's Q resolves the dominant band, and resolving a wider band needs a higher damping floor.

• P5. At synthetic white-noise input (flatness ≈ 1), no damping setting in the sweep recovers K=1 accuracy above chance. Justification: this is the operational consequence of white_flat_argmax_ambiguous — if the Lean theorem is right, two off-mode frequencies of equal amplitude route to equal output, and the bowl has no preferred carrier to lock on across the whole sweep.

8. Falsification triggers

These are the curves and points that would kill the reframe. Triggers, not observations.

• F1. The high-damping end of the sweep does not match the existing binary-mask K=1 accuracy within noise. The equality claim BowlMeshNode.mask_at_high_damping would then be wrong: either the Lean spec mis-models the binary mask, or the Rust primitive does not honor the spec. Either way the reframe needs to be rewritten from the primitive up.

• F2. K=1 accuracy is monotone in damping (rather than unimodal). This inverts P3 — if the curve is monotone-up or monotone-down, there is no interior peak, and the bowl reframe loses its load-bearing claim that the healthy regime sits between the two failure modes.

• F3. At pink-band input (flatness measurably below 1 on a real model's residual stream), no damping setting beats the binary mask on K=1 accuracy. P2 then survives in toy spectra but fails on the actual residuals we care about — the bowl reading would be a toy-model fit, not a model-physics claim.

• F4. damping* shows no correlation (or negative correlation) with per-layer spectral flatness across layers and across models. P4 then fails and the bowl reading loses its operational handle on which damping to use at a given layer — we would still need to grid-search per layer, defeating the point.

• F5. Any of the five BowlMeshRuntimeOracle contracts fails on the Rust side. The Lean ↔ Rust parity is broken below the model level, before any spectral question is even asked. This is a primitive-level failure, recoverable but blocking.

Any of F1–F5 reverts the reframe to "interesting toy" status. None of them by themselves invalidates the broader cliff atlas; the cliff law sits structurally below the bowl reframe and is not at stake here.

9. Composition with the Amplituhedron prefix cache

The bowl-mesh reframe and the Amplituhedron static volume (AMPLITUHEDRON_PLAN.md) are orthogonal, not competing. The amplituhedron-side falsification module (AmplituhedronFalsifiability.lean) establishes a separate failure regime — the lossy placeholder qk / 10 that drops 88 of every 97 quanta on the Vandermonde witness — and the bowl-side runtime oracle in §6 is structurally the same kind of contract for the per-token Q filter. The two are complementary:

• AmplituhedronCache operates on full-prefix replay: hit means skip prefill entirely on a known prompt prefix. Granularity: prefix × layer-range. Key: prefix hash + layer range. See AMPLITUHEDRON_PLAN.md §5 (the composition table).
• The bowl Q-filter operates on the per-token, per-layer residual stream during decode: hit means damp off-mode dimensions proportionally rather than zero them. Granularity: single residual vector. No prefix involvement.

They feed each other: residuals replayed by AmplituhedronCache are post-prefill, post-RMSNorm vectors in the Luminary basin (near unit norm). That is exactly the regime the bowl Q-filter is tuned on — MatVecMemo was already tuned there, and the same "drift stays near the Luminary basin" property keeps the bowl's damping* stable across the prefix-cached decode stream. The implementation rule from AMPLITUHEDRON_PLAN.md §5 carries over: do not run the bowl filter during volume capture or replay; capture freezes the source-of-truth residual, and the filter runs only on the live decode path that follows the replay.

10. Pointers

Six sibling files (5 Lean + 2 Rust, with one new runtime-oracle Lean module added in the §6 round), plus the two existing docs this protocol references:

• Gnosis/BowlMeshNode.lean — node types, mask_at_high_damping, positive_damping_preserves_signal, bowl_route_total_unlike_mask.
• Gnosis/BowlMeshQSweep.lean — off_resonance_strictly_positive, damping_zero_off_resonance_pass_through, high_damping_monotone_decay, layer_residual_nonzero_under_positive_damping.
• Gnosis/BowlMeshPinkNoiseBound.lean — flatness_dichotomy, white_flat_argmax_ambiguous, pink_structured_argmax_recovers, bowl_recovers_below_flatness_threshold.
• Gnosis/BowlMeshRuntimeOracle.lean — the five *_runtime_oracle_* theorems pinning the Rust value contracts.
• Gnosis/AmplituhedronFalsifiability.lean — the parallel amplituhedron-side falsification module; canonical pattern source.
• src/bowl_mesh.rs — the per-node Q-filter Rust primitive that implements filteredAmplitude over a residual vector with a damping dial and a per-dim resonance band. Exposes for_sweep(hidden_dim, damping) for the sweep harness in addition to the full new(...) constructor.
• src/bin/bowl-q-sweep.rs — the measurement harness that emits the curves listed in §4 and the data points predictions P1–P5 and triggers F1–F5 stand or fall on. Cargo-registered as the bowl-q-sweep binary.
• src/bin/bowl-probe.rs — the real admitted-weight τ-bowl/JIT-dequant quality gate. It is separate from the synthetic damping sweep: it samples F16/F32/Q8_0 rows from an admitted .knot, emits JSON quality profiles, and includes the Lean proof spine for admission records.
• scripts/aggregate-bowl-probe-admissions.py — folds the profile and gamma frontier admission artifacts into the runtime-facing JIT-dequant aggregate certificate, with source hashes preserved.
• STANDING_WAVE_STATUS_2026_05_03_AFTER_FALSIFICATION.md — the failure being addressed. Wave-3 / wave-4 PCA-only K=1 falsification.
• AMPLITUHEDRON_PLAN.md — the prefix cache that composes with this reframe at the granularity boundary described in §9.

11. Honesty boundary

Nothing in §7 has been measured. The Lean theorems referenced in §3, §4, §5, and §6 exist and are kernel-checked under Init; the empirical predictions they motivate do not. The reframe survives or falls on the sweep harness, not on the algebraic argument. If F1–F5 all fire, the bowl model has been falsified at the same operational level the binary mask was falsified at on 2026-05-03, and we report that as the result. The previous-snapshot rule from STANDING_WAVE_STATUS_2026_05_03_AFTER_FALSIFICATION.md applies: this file becomes the historical record, an addendum captures the falsification, and the reframe goes back to the structural lattice.