> Codex v9.10.64 · Track 4 · Codex §Comprehensive Plans
> AI / low-credit: One section/experiment spec per session; on stop log Last worked / Next.
Purpose: Turn Sparse Resonance Hyperlattice Theory (SRHT) from a hyped draft into a credible, peer-reviewable contribution. The plan keeps what is genuinely novel, retires what is unsupported, and specifies experiments a researcher could execute to settle the open questions honestly.
Authoring stance: Every recommendation below is written to make SRHT *survive* scrutiny rather than impress on first read. Where a claim cannot yet be supported, the plan says so plainly and defines the evidence that would be required to make it.
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The defensible core of SRHT is established by the author's own audit (SRHT_CRITIQUE_AND_PROOF.md): when contradiction $\chi$ is bound to constraints that are independent of the cost being optimized, the SRHT selection rule does real, non-trivial work that a same-width greedy beam search cannot replicate. This — not the physics vocabulary — is the contribution worth leading with.
The mechanism, stated cleanly. In a beam/best-first search, the selection score is $C(p) = \rho \cdot R(p)^2 \cdot \frac{1}{2-R(p)} \cdot (1-\chi(p))^2 \cdot \tau$. The audit proves that if $R$ and $\chi$ are both monotone functions of the accumulated cost $d(p)$, then sorting by $C$ is order-equivalent to sorting by $d$, and the entire physics layer is a redundant re-encoding of the cost. The framework becomes interesting precisely when $\chi$ is decoupled from $d$ — for example, $\chi = 0.9$ when a partial TSP tour self-intersects, a structural defect that raw path length does not see. Then $C$ is non-monotone in $d$: the search can rationally prefer a slightly longer-but-valid partial solution over a shorter-but-structurally-doomed one, escaping the local trap that greedily-shortest-segment selection walks into. The $N=20$ self-intersection experiment over 20 seeds (SRHT beating plain beam by up to 22.42 units on the reported seeds) is the seed of real evidence for this.
Reframing. SRHT should be presented as a constraint-aware best-first search framework: a single scalar score that fuses (i) progress toward the objective and (ii) satisfaction of validity constraints that are not reducible to the objective, with a tunable gate that prunes branches whose constraint violation is high regardless of their objective value. This framing is honest, testable, and connects SRHT to a well-understood literature (constrained search, soft/hard constraint penalties, beam search with feasibility scoring) rather than isolating it behind invented physics. The novelty claim narrows but becomes *defensible*: a particular, smooth, multiplicative fusion of progress and independent-constraint scores, plus a pruning gate, that empirically improves beam search on constraint-structured problems.
Action. Restructure the paper so Section 1 is this result. Lead with the proof of when the method is non-redundant, then the controlled experiment that demonstrates it. Everything currently in the abstract about "waves," "Born-rule interference," and "cracking cryptography" moves out of the lead and is either cut or demoted to clearly-labeled speculation (Sections 2 and 4 below).
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Three claims in the current draft are not supported by the evidence presented and will sink the paper in review. Each is handled below.
Section 4.2 of the paper argues that under high constraint density the survival probability drives expected nodes visited to $O((b(1-p))^d)$ and, if $p \to 1 - 1/b$, the search "collapses from exponential to polynomial time $O(d)$." This is a conditional asymptotic claim with an unproven and likely false premise. It assumes a *constant per-node* pruning probability $p$ that is independent across depth and uniformly close to $1-1/b$ — an assumption that has no proof and that, for NP-hard problems, would imply P = NP if it held adversarially. The TSP and SVP timing tables show fast runtimes on *small, specific, randomly generated instances*; they say nothing about worst-case scaling.
Restatement (required). Replace the complexity-collapse claim with: *"On the random instances tested, SRHT pruning reduces the number of expanded nodes relative to unpruned search, yielding empirical speedups at the sizes shown. We make no worst-case complexity claim; the per-node pruning rate is instance-dependent and is not guaranteed to remain bounded away from 1/b as dimension grows."* If the author wishes to retain any asymptotic statement, it must be accompanied by (a) a measured pruning-rate-versus-depth curve across a sweep of sizes, and (b) an explicit statement that the favorable regime is an empirical observation, not a theorem.
Section 4.3's suggestion that an analog SRHT implementation "could crack lattice cryptography in polynomial time" is unsupported and should be cut entirely from any peer-facing version. The SVP results are at dimensions 8, 10, and 12 against a *random-coefficient baseline* (2,000 random combinations) — not against LLL or BKZ, and not at cryptographically relevant dimensions (Kyber/Dilithium operate at module ranks corresponding to lattice dimensions of several hundred to ~1000+). Beating random search in dimension 12 is many orders of magnitude away from beating BKZ in dimension 500+. The claim also conflates finding *a* short vector below the Minkowski bound with solving exact-SVP at scale, which is the actual cryptanalytic requirement.
Action. Remove the cryptography-breaking claim. Replace Section 4.3 with a sober statement: *"Whether SRHT-style constraint-gated search can match or accelerate BKZ's SVP oracle at cryptographically relevant block sizes is an open empirical question. Nothing in our small-dimension experiments bears on the security of deployed lattice schemes."* The BKZ-acceleration idea (Section 4.1) may be retained only as an explicitly-labeled *hypothesis to be tested* (see Section 3 below), never as a result.
The operator $P = C \cdot \cos(\chi \frac{\pi}{2})$ is presented with quantum-mechanical language (Born rule, probability amplitude, phase interference). This is decorative, not physical. There is no Hilbert space, no complex amplitude, no superposition, no measurement; $\chi \in [0,1]$ is a real scalar and $\cos(\chi\pi/2)$ is just a smooth real-valued gate that equals 1 at $\chi=0$ and 0 at $\chi=1$. The quantum framing invites — and will not survive — a referee who knows quantum mechanics.
Restatement (required). Describe $P$ honestly as a classical cosine constraint-gate: a monotone soft penalty that multiplicatively suppresses a candidate's score as its constraint violation $\chi$ rises, reaching exactly zero at full violation. State plainly that the cosine shape is a *design choice* for a smooth, bounded gate with zero gradient at the endpoints, and that alternatives (linear $1-\chi$, $(1-\chi)^2$, sigmoid) should be compared (see Section 4). Remove "Born rule," "quantum," "amplitude," and "interference" from the operator's definition. The same applies to the abstract's "destructive interference" language — call it pruning.
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The current baselines (instant greedy nearest-neighbor for TSP; 2,000 random combinations for SVP) are too weak to support any claim of practical value. A method that beats a deliberately weak baseline has demonstrated nothing about its standing against the methods practitioners actually use. The benchmarking must be rebuilt around strong, standard solvers, larger instances, and statistics over many seeds.
Strong baselines. Compare SRHT against (i) LKH (Lin–Kernighan–Helsgaun, the de facto state-of-the-art heuristic) and (ii) Concorde (exact optimal, where tractable) on standard instances. Include a same-width plain beam search as the *ablation* baseline (to isolate the value of the $\chi$-gate), not as the headline competitor.
Instances and sizes. Use TSPLIB instances and randomly generated Euclidean instances at $N \in \{50, 100, 200, 500\}$ — well beyond the $N \le 20$ currently reported. Report results as mean ± standard deviation of tour length over $\ge 30$ independent seeds/instances per size, plus the optimality gap to Concorde (or best-known) where available.
Success criterion. SRHT is credible on TSP if, *at a matched or lower compute budget* (wall-clock and node-expansions both reported), it achieves tour-length gaps to optimal that are statistically indistinguishable from or better than the same-width beam baseline, with the gap improvement significant at $p < 0.05$ under a paired test (e.g., Wilcoxon signed-rank across seeds). Matching or beating LKH is *not* expected and should not be the bar; the honest, achievable claim is "the constraint-gate improves constraint-structured beam search at equal compute." If SRHT *also* narrows the gap to LKH relative to plain beam, report it — but do not promise it.
Strong baselines. Replace random-coefficient search with LLL and BKZ (e.g., via fpylll / the FPLLL library) as baselines, plus an exact enumeration oracle at small dimension for ground truth. Random search must be dropped as the comparison point.
Instances and sizes. Use standard SVP-challenge-style random lattices at dimensions $D \in \{40, 60, 80, 100\}$ at minimum, reporting the achieved norm as a ratio to the Gaussian-heuristic / Minkowski bound, averaged over $\ge 20$ random bases per dimension with variance reported. Dimensions 8–12 are not informative.
Success criterion. The honest claim to test is whether SRHT, used *as the per-block SVP oracle inside BKZ*, can match BKZ-with-enumeration's output quality (Hermite factor) at lower wall-clock or fewer oracle operations, at a fixed block size $\beta$, with significance over seeds. Any cryptographic relevance requires demonstrating this at $\beta \ge 40$ and showing the advantage does *not* vanish as $\beta$ grows. Until that curve is produced, no statement about lattice cryptography may be made.
Every table reports: instance set and size, seed count, mean and variance, compute budget (wall-clock *and* node-expansions/oracle-calls), and the significance test used. Pin solver versions and random seeds; release the harness so results are reproducible. Speedups quoted without a matched-quality and matched-compute comparison are not admissible.
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The thresholds and step values in SRHT are currently hand-tuned magic numbers: the pruning gate $X > 0.8$, the collapse cutoff $P < 10^{-10}$, the contradiction step function $\chi \in \{1.0, 0.6, 0.1\}$, the resonance scale $\alpha$ in $R = 1 - d/(\alpha\, d_{\text{avg}})$, the commit-readiness exponents ($R^2$, $(1-\chi)^2$), and the $\tau$ scale. None are justified or derived. A reviewer will read each as a degree of freedom quietly fitted to the reported instances.
Sensitivity-analysis protocol (required). For each parameter, run a one-at-a-time sweep across at least one order of magnitude (or the full $[0,1]$ range for thresholds) on a held-out instance set distinct from any used for development, and report the performance curve with variance. A robust method shows broad plateaus, not sharp peaks at the chosen value. Follow the one-at-a-time sweeps with a small randomized joint search (e.g., Latin-hypercube over the parameter box) to check for interactions, and report the best, median, and chosen configurations. Critically: tuning is done on a development split and all headline results in Section 3 are reported on a separate, untouched test split, to rule out the appearance of overfitting parameters to the benchmark.
Derivation where possible. Several parameters can be motivated rather than guessed: the $P < 10^{-10}$ cutoff is effectively a floating-point/budget pruning floor and should be stated as such (and shown to be insensitive across several orders of magnitude). The $X > 0.8$ gate should be re-expressed as "prune when constraint violation exceeds fraction $\theta$ of the maximum," with $\theta$ swept. The $\chi$ step function $\{1.0, 0.6, 0.1\}$ should be replaced by, or compared against, a continuous violation measure, since discrete steps are an obvious tuning surface; if discrete steps are kept, justify the boundaries. The cosine gate shape itself (Section 2.3) is a parameter family — compare cosine against linear and quadratic gates and report which generalizes. The goal is that every constant in the final paper is either *derived*, *shown insensitive*, or *honestly flagged as fitted on the dev split*.
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The cleanest real-world test of the SRHT thesis lives inside KAI itself: constraint-aware decoding for language generation. This is exactly the regime where the contribution holds, because grammatical and logical validity are constraints *independent of* a token's raw probability — the analog of self-intersection being independent of path length. A locally high-probability token can produce a globally invalid sentence, just as a locally short segment can force a later crossing.
The binding. Following the audit's recommendation, $\chi$ must *not* be computed from sequence length or raw token log-probability (that path reproduces the monotonic-redundancy bug, where SRHT collapses to ordinary likelihood ranking). Instead, $\chi$ is bound to explicit grammar/logic rule violations evaluated on the partial generation — e.g., a parse-rule check in KAI's algebra.rs that sets $\chi$ high when a candidate continuation violates the grammatical template (noun-after-noun where the template forbids it), breaks a bracket/quote balance, or contradicts an asserted fact in the working context. The base score plays the role of $R$ (progress/likelihood); the cosine gate $P = C\cos(\chi\pi/2)$ then performs constraint-aware decoding, suppressing high-likelihood-but-invalid continuations during top-k/nucleus sampling.
Concrete test. Construct a held-out set of generation prompts with *machine-checkable* constraints: (a) balanced-delimiter / valid-JSON or valid-expression generation, (b) a controlled grammar where validity is decidable by a parser, and (c) a small factual-consistency task with a checkable knowledge base. For each, compare three decoders at matched compute: (i) standard top-k/nucleus sampling, (ii) the same sampler with a naive length/probability penalty (the redundant control), and (iii) SRHT constraint-gated decoding with $\chi$ bound to the rule checker. Primary metric: rate of constraint-valid outputs. Secondary metrics: fluency/likelihood of the valid outputs (to confirm the gate does not destroy quality) and compute overhead. Success criterion: SRHT decoding produces a statistically significant increase in valid-output rate over both baselines (paired test over prompts, $p < 0.05$) without a meaningful drop in fluency on the valid set. This directly operationalizes the paper's true claim — that constraint-bound $\chi$ does work that cost-bound ranking cannot — in KAI's actual domain, and it is the experiment most worth running first because it is cheap, decisive, and on-mission.
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