The grammar space — one operator, five grammar levels
One operator — the median-centred, MAD-scaled A4 projection — reads five grammar levels from sequence, each as a (γ, A4) pair: G1 material (cell), G2 regulatory (tissue), G3 architecture (organ), G4 dynamical (state), G5 body-plan (body). Structural decoding is complete — every grammatical level is identified, formalized, sequence-readable, orthogonal, and read by the one operator. Functional decoding — that each coordinate predicts a measured phenotype — is open by measurement.
The reading is not a single ladder but a two-axis space: an interpretation axis (G1 material → G2 regulatory → G4 dynamical) and an organization axis (G3 architecture → G5 body-plan), with the regulatory level read across organs. The same A4 = robust_z operator reads every cell, scale- and shift-invariant to machine epsilon. Read on the time axis, G3 is the developmental order grammar — emergence order is cascade depth, the absolute clock pinned by two measured anchors (App. I–L). This appendix states the whole space as one map, the algorithms that reproduce it, and the honest two-axis grade.
The space — one operator, five levels, two axes
Every grammatical level is the same operation on a different signal: take a sequence-derived signal, return its LEVEL (the mean, written γ) and its SHAPE (the median-centred MAD-scaled projection, written A4). The operator does not change up the tower; only the signal it reads changes. The five levels established in Appendix E and Appendix F:
| grammar | level (linguistics) | signal read | what it returns | grade |
|---|---|---|---|---|
| G1 material | cell — phonology | nearest-neighbour stacking ΔG (SantaLucia 1998) | cell stiffness; sets the R19 threshold scale | [V]/[L] |
| G2 regulatory | tissue — syntax | transcription-factor binding syntax | which tissue (cardiac separates from housekeeping ~14×) | [L]/[V] |
| G3 architecture | organ — discourse | spatial layout of the regulatory hotspots | the arrangement map; read on time = the order grammar | [F]/[V] |
| G4 dynamical | state — pragmatics | CpG-island methylation potential (windowed CpG O/E) | one locus → a family of readings by cell state (orthogonal to G1) | [V] |
| G5 body-plan | body — genre | Hox colinearity on the real cluster | position on the DNA = position on the body axis (rank corr 1.000) | [L]+[V] |
The two axes are distinct projections, not the same signal twice: the material level G1 and the regulatory level G2 are orthogonal (corr ≈ −0.04), and a dinucleotide-preserving shuffle preserves the material LEVEL exactly while collapsing the regulatory grammar — so tissue identity is information above the material, not a renaming of it. The SHAPE projection is the identical robust_z operator at every level, verified scale- and shift-invariant to machine epsilon.
The order grammar — G3 read on the time axis
The single-locus stiffness order does not predict developmental staging — that is the measured null of Appendix I, reproduced as a number. What predicts the order is the architecture grammar G3 read along time. A gene cannot fire until its upstream regulators have delivered their drive, so the order key is not a local barrier but a global network property: the cascade depth, the longest chain of regulators that must fire in series upstream of a gene. The order grammar is therefore a composite — depth is primary, the local barrier γ²/4 breaks ties within a tier — and it is closed and absolutely pinned across Appendices I–L: edge concordance with cited staging is exact (zero inversions), depth predicts staging strictly better than stiffness, the floor is jitter-robust, and the absolute clock day(s) = a + b·s is pinned by two independent measured anchors (a segmentation-oscillator slope and a first-heartbeat intercept) with zero free parameters, held-out days reproduced to ≤ 0.56 d in the somite window.
The substrate makes this a theorem, not a fit. Every switch sits on its stable OFF branch s = −√γ with a sub-spinodal baseline, so the only route to ON is an ON parent — firing is forced to respect the cascade partial order. The OR-gate wavefront (a gene fires after its earliest required parent) generalizes to a quorum/AND threshold-k gate (after its k-th-earliest required parent), with the OR gate the k = 1 special case; both are verified at zero violations.
What the emergence reaches, and what it does not — stated precisely
The emergence is read to the exact edge of what a deterministic substrate can entail, and that edge is mapped, not apologized for. What the substrate entails — and what this program therefore demonstrates — is direction, sign, order, and reachability: which gene fires, in what order, after which regulators, on what stage. The cell-switching layer is grounded in chemistry (the γ operator reads real promoter stacking energy), so the switching grammar is sequence-exact. What lies outside the substrate by construction is magnitude: absolute strengths, absolute kPa tissue moduli, and the active continuum mechanics of a tissue (muscle tension, blood pressure, applied stress) are a different physics — a full continuum / finite-element simulation with active stress — that this information-and-topology engine does not claim and is not built to claim. The R19 "stiffness" is the switch barrier γ²/4, an informational stiffness (how hard a switch is to flip), not a mechanical Young's modulus; keeping those two senses apart is exactly why absolute tissue strength is a firewall ceiling. Within the entailable edge the emergence is carried to its maximum: the coupled-R19 wavefront theorems, depth-beats-stiffness, the two-anchor absolute clock, and the jitter-robust order floor are each closed and gated. The limit is analyzed; the result is pushed to that limit.
The algorithms — reproducible without running the code
Each block below is the deterministic specification of one grammar operation: the formula, the procedure, the input/output, and the grade. They reproduce the reading on paper; the locked code and byte-identical expected outputs are in the linked package.