EXPERT_GUIDE.md

Expert guide: ÆLYRAITH III

This guide maps the supplied research release to its mathematical contract, executable implementation, and review boundaries. It adds navigation; it does not add a theorem, an external review, a proof-assistant certificate, or a new novelty claim. The authoritative mathematical text is manuscript/aelyraith3.tex, with the typeset manuscript beside it. THEOREM_INDEX.json supplies machine-readable result entries with exact TeX labels.

The contract to review first

States are columns in the full simplex Omega = {x >= 0: e*x = 1}, with e the all-ones row. A finite, known library of strictly positive n x n matrices acts by F_a(x) = T_a*x/(e*T_a*x). A protected readout is (beta*x)/(alpha*x), where every entry of alpha is strictly positive; beta can have either sign. Normalization across outcome branches is an additional obligation for a stochastic instrument, imposed separately for each action.

Equivalence means equality of protected conditional values for every finite supplied branch word, including the empty word. It does not automatically mean equality of branch probabilities, distributions of histories, unconditional rewards, or every planning objective. An application to an actual agent must justify its finite positive model and correspondence to real operations independently.

The exact results are mathematical arguments under these hypotheses. Rational computations check finite instances and certificate identities; floating-point histories check finite numerical cases. Neither alone certifies every written argument or publication priority.

A useful reading order

Review goal	Manuscript location	Supporting files
Understand the object being preserved	Section 1, Definition 1.1 (def:equiv), equations eq:branch, eq:readout	CLAIMS.md
Audit the finite complete compiler	Section 2, Theorem 2.1 (thm:exterior), equations eq:wedge–eq:signature	compile_ratios, compile_exterior, ExteriorCertificate.verify; check_exported.py
Separate signature length from nonlinear dimension	Section 3, Proposition 3.1 (prop:fiber) and Theorem 3.2 (thm:dimension)	fiber_matrix, reconstruct_fiber_float, jacobian_rank
Check the sharp initial-information law	Section 4, Lemma 4.1 (lem:ratio) and Theorem 4.2 (thm:two)	snapshot_space, factor_two_fixture; SnapshotTests
Audit the explicit memory gap and fair control	Section 5, Theorem 5.1 (thm:family), including “A fair conventional control”	separation_family, Family.signature, Family.compact_step, affine_closure; mode-aware test
Audit the rewrite and exact inverse	Section 6, Theorem 6.1 (thm:rewrite), equations eq:perturb, eq:inverse, eq:recover	reconstruct_hidden_from_B_readouts; RewriteTests
Evaluate finite precision rather than exact dimension	Section 7, Theorem 7.1 (thm:approxjump) and eq:quant	results/rewrite_approximation.csv, results/long_horizon.csv
Distinguish exact compatibility from a sufficient span test	Section 8, Theorem 8.1 (thm:admit), eq:QE, eq:suff; Proposition 8.2 (prop:all)	explicit counterexample and test_span_admission_not_necessary
Audit an approximate continuation bound	Section 9, Theorem 9.1 (thm:robust), equations eq:residual, eq:boundC, eq:robust	Written sufficient bound; no generic robust solver is delivered
Evaluate a recursive promotion claim	Section 10, eq:gate, vertex counterexample, bounded protocol	test_vertex_promotion_failure; no generic gain optimizer is delivered
Inspect evidence and novelty boundaries	Sections 11–12; Appendices A–C	results/summary.json, raw certificates/histories, LITERATURE_AUDIT.md

What the finite certificate actually checks

src/continuation.py contains the mathematical reference code. For each supplied readout (beta, alpha), the compiler spans the orbit of beta wedge alpha under the second compound matrices C2(T_a). It selects actual orbit rows and retains their numerator/denominator pairs and word provenance. The global signature comprises their ratios. The exterior basis has at most n*(n-1)/2 rows, but its length is not the intrinsic nonlinear dimension and is not a physical state count.

ExteriorCertificate.verify() checks all of the following: branch positivity and dimensions; full row rank; seed-row coverage; positive stored denominators; agreement between each ratio wedge and its exterior row; genuine word provenance; and every identity R*C2(T_a) = H_a*R. Genuine provenance is needed for the converse implication, while seed coverage and transport establish completeness for all words. The checker permits outcomes from several actions and therefore does not impose one global stochastic normalization. check_exported.py adds normalization for the supplied two-outcome instruments.

Exact entries should be sympy.Rational or integers. Constructing a rational from a binary float can preserve that float's exact binary value rather than an intended decimal. For example, use sp.Rational(1, 100) for delta, rather than an unexamined float conversion. The exported matrix entries are rational strings; reconstruct them as rational data, not by evaluating arbitrary code.

Word and index conventions

Branch indices are zero-based. For the explicit family, 0 means A and 1 means B. A word (a1, ..., aL) is chronological, so its state product is T_aL * ... * T_a1. Corresponding row products therefore multiply matrices in reverse chronological order. Ratio.word retains chronological provenance. Changing this convention can create a plausible-looking but incorrect certificate.

The computed basis need not have the same row scaling as the manuscript's hand-selected basis. Consequently, a transport matrix need not match a printed transport entry-by-entry. Check R*C2(T) = H*R in the actual basis.

The three memory contracts

For the explicit Hadamard family, n is a power of two with n >= 4. The constants are c=1/2, tau=1/4, gamma=kappa=1/8, and lambda=1/(8*n). With feature coordinates z=F*x, its exact global compact memory is (u,v) = (z1, z2/(1+tau*z3)).

Resource	Minimum for the unperturbed family	Meaning
Continuous exact nonlinear recurrent memory	2 real coordinates	Global for this family; the general local rank theorem is only local at regular interior points
Fixed affine initial snapshot	3 real scalars	Initialization need not update these scalars from themselves
Fixed self-updating affine memory	n-1 real scalars	Fixed affine encoding on the entire initial simplex; arbitrary nonlinear decoding/update functions are permitted

A conventional mode-aware controller can acquire three initial affine values, compute the compact statistic, and subsequently use at most one scalar plus a mode flag. The large gap is therefore not a performance advantage over every conventional implementation. Coordinate counts exclude acquisition, model/code storage, temporary workspace, and precision costs.

jacobian_rank(cert, x) returns the rank at the supplied exact state. A single evaluation is not a general algorithm for proving that point regular or for proving a global minimum chart exists. A numerical representative returned by reconstruct_fiber_float is any compatible witness, not recovery of the true hidden state. Its feasibility tolerance is a numerical check, and generic updates can use n-dimensional temporary workspace.

The rewrite result and its engineering interpretation

For 0 < delta < 1/8, replacing B by B_delta = B + delta*f.T*a/n leaves branch masses and individual/aggregate characteristic polynomials unchanged, yet makes complete conditional behavior injective on the entire simplex. This raises the continuous exact minimum to n-1. The inverse reconstructs hidden coordinates from n-3 successive B readouts plus the old initial and next-A values. The inverse becomes ill-conditioned for small delta or deep coordinates; exact recoverability is not a finite-precision stability claim.

The old compact controller, initialized with the true old statistic, still has output error at most 8*delta/3 for every word and time. Quantization of its contraction adds at most Delta/(2*(1-kappa)) under the analyzed rounding assumptions. The finite-bit statement 2*log2(1/epsilon)+O(1) concerns deterministic worst-state accuracy in this explicit family and excludes code, model, acquisition, and branch metadata.

Compatibility and deployment boundaries

The exact rewrite test quantifies over feasible decomposable wedges of equivalent simplex states. A linear transport-span test is sufficient but not necessary. Failure of the span test alone is not an impossibility certificate. A genuine failure witness consists of two feasible states in one old fiber with different new protected behavior.

The general compatibility condition is a finite quadratic implication over rational data, decidable in principle by real quantifier elimination. A scalable general quantifier-elimination implementation is not included. Likewise, Section 10's gain gate requires an actual lower bound on the full feasible uncertainty set. Its vertex example shows why checking only vertices can be unsound for a difference of linear-fractional utilities. The package does not implement a general nonlinear gain minimizer.

For the robust theorem, denominator lower bounds must be valid and positive; represented states must be feasible or a separate off-image update extension must be justified. Lifetime bounds require the stated common contraction and genuine local error charges. Small residuals by themselves do not establish a stable all-horizon controller.

Reproduction and evidence interpretation

Run from the repository root in an isolated environment:

python -m pip install -r requirements.txt
python -m unittest discover -s tests -v
python check_exported.py
python example.py

python run_checks.py regenerates the experiment outputs in results/; copy the repository first if preserving the supplied evidence bytes matters. The recorded original environment and finite counts are in results/summary.json. The release reports 40 test methods, five exact family sizes, 30 generic instruments, 360 extra exact words, six factor-two fixtures, 128 histories of 256 steps, 72 perturbed histories, and hidden recovery at three sizes. These are source-reported observations until independently rerun. Any new run should retain its actual versions, commands, exit status, and log separately from the original record.

The simulations supply labels externally. They do not train an LLM, predict history probabilities, benchmark a frontier agent, measure research productivity, establish universal method dominance, or establish a T9 rating. A targeted literature audit is a scoped comparison, not proof of worldwide publication priority. The creative author attribution and personal dedication are preserved source metadata and are not mathematical evidence.