Completing the order grammar: the kit, the clock, the drive, the gate

The order-grammar appendix identified what predicts developmental staging — the depth of each gene in the regulatory cascade, the wiring grammar read on the time axis — and was honest about four things it could not yet close. The pre-registered floor on the absolute strength of that prediction was not met across the whole gene set, only on the wired axial trunk, because the thirty-eight-gene kit had named coverage gaps: limb-bud genes that looked like cascade sources because their inducers lay outside the set, and a Hox axis represented by its terminal group alone. Order was fixed only as a ranking, not yet in days. Edge drive was a uniform placeholder rather than something read from each promoter. And the substrate's wavefront theorem was shown for an OR-gate only. This appendix closes all four, with real data and no retuning. It fills exactly the named gaps with real human promoter stiffnesses and the cited limb and Hox edges, and re-tests the same floor: the floor is now reached, and the lift is shown to require both fixes together, with an honest jitter band that keeps the claim empirical rather than a logical invariant. It attaches the real measured segmentation-clock period and the cited Carnegie stage days to turn the somite cadence into a span of days consistent with the cited window, while leaving the single global zero-point that would pin every gene to absolute time explicitly open. It reads each edge's drive from the promoter as the switch's on-branch amplitude, the square root of the stiffness, and shows the firing order is preserved. And it generalises the wavefront theorem to a quorum gate, where a gene fires once a chosen number of its regulators are on, with the cascade order still intact and the OR-gate recovered as the one-regulator special case. The floor result is graded locked, not verified; the clock is locked for the cadence and open for the zero-point; the drive map is a declared modelling choice whose order-invariance is verified; the quorum gate is verified; and no body is claimed.

The order-grammar appendix carried four open obstacles, and this appendix retires them at the level the firewall allows. The first is gene-set coverage. The earlier appendix pre-registered a floor of zero point seven on the magnitude of the correlation between cascade depth and Carnegie onset rank across all anchored genes, met it only on the axial trunk, and named the dilutors precisely. Here those dilutors are filled with twenty-five new real human driver stiffnesses — seven limb FGF and Wnt inducers, and eighteen Hox genes spanning the A and D clusters rather than the terminal group — and the floor is re-tested under a nested-scope ablation. The inherited kit reproduces the earlier global value of about zero point four eight; adding the limb inducers alone brings it to the floor; adding the full Hox chains brings it to about zero point seven six, the floor met globally; and the Hox chains without the limb inducers fall short, so both fixes are necessary, not either alone. A rank-jitter test perturbing every Carnegie rank by one position, seeded on the substrate's own constant, gives a mean near zero point six nine with a lower edge below the floor — the floor is cleared by the cited ranks but is marginal at the edge of citation uncertainty, so the strength claim is graded locked, an empirical correlation on cited data, and is never upgraded to a verified invariant; the gate fails closed if it is. The second obstacle is absolute timing. The measured segmentation-clock period and the cited somite count predict a somitogenesis span of about eight and three-quarter days, consistent within tolerance with the cited Carnegie window, which converts cadence to days; a single absolute zero-point pinning the whole atlas to days would need a second measured anchor the firewall does not grant, and is left open. The third obstacle is the cis-to-drive map. Each edge's weight is set to the square root of the parent's stiffness, the on-branch fixed-point amplitude of the switch, so a stiffer regulator delivers proportionally more drive; this is a declared modelling choice, but its consequence is an invariant — the firing order under sequence-derived drive matches the uniform-drive order at a Spearman of about zero point nine nine, with no wavefront violations. The fourth obstacle is the gate. The OR-gate wavefront generalises to a quorum gate in which a gene fires once at least a set fraction of its regulators are on; at full conjunction the cascade order still holds with zero violations, convergent nodes never fire earlier than under the OR-gate, and the OR-gate is exactly the one-regulator case. Inheritance is byte-exact: the shape operator and the stiffness operator are unchanged, and all thirty-eight inherited stiffnesses are checked digit-for-digit against the parent appendix's locked database read from disk. The fail-closed gate passes sixteen of sixteen deterministically under a double hash, holding the strength claim at locked and keeping physical completion false.

Why this appendix exists

The order-grammar appendix answered the question the build left open — what predicts the stage at which each developmental program emerges — and answered it cleanly: not the local promoter stiffness, which is the wrong category, but the depth of each gene in the regulatory cascade, the wiring grammar already in the reading, read along the time axis. But it was deliberately honest about four things it had not yet closed, and listed them as named obstacles rather than rounding them away. The strength of the prediction, measured as the magnitude of the correlation between cascade depth and the cited Carnegie order, cleared its pre-registered floor only on the wired axial trunk and fell short across the whole gene set; the reason was named, not hidden, as artificial sources in a sparse kit. The order was fixed only as a ranking, with absolute timing in days left open. The drive carried along each edge of the cascade was a uniform placeholder, not yet read from the promoter the way the rest of the corpus reads everything from sequence. And the theorem that the substrate forces firing to respect the cascade order had been proven for the simplest firing rule, where any single active regulator suffices. This appendix takes those four obstacles as its whole task and closes each one only as far as the firewall permits, refusing to convert a filled gap into a grander claim than the data supports.

Filling the named gaps, and re-testing the same floor

The first and largest obstacle is gene-set coverage, and the test is built so that filling it cannot be mistaken for stacking the deck. The earlier appendix did not merely report that the floor was missed; it named the exact dilutors — limb-bud genes that appeared to be cascade sources only because their true inducers lay outside the thirty-eight-gene set, and a Hox axis represented by its terminal group alone. This appendix adds precisely those missing pieces and nothing tuned: twenty-five new real human promoter stiffnesses, seven of them the limb FGF and Wnt inducers that feed the limb-bud genes, and eighteen of them Hox genes spanning the A and D clusters in their cited three-prime to five-prime temporal order rather than the terminal group alone. Then it re-tests the same pre-registered floor under a nested-scope ablation, so the source of any improvement is visible. With only the inherited kit the correlation reproduces the earlier global value of about zero point four eight, exactly as before. Adding the limb inducers alone brings it to the floor. Adding the full Hox chains brings it to about zero point seven six, the floor met across the whole anchored set. And the Hox chains added without the limb inducers fall short of the floor, which means neither fix alone suffices — both are required, and the lift is therefore localised to the two named gaps rather than smeared across an arbitrary addition of genes. The honest part is the jitter test that follows. Every Carnegie rank is perturbed by one position, repeatedly, seeded on the substrate's own constant, and the distribution of the correlation is reported: the mean sits near zero point six nine and the lower fifth-percentile edge falls below the floor. So the floor is cleared by the cited ranks as published, but it is marginal at the edge of how precisely those ranks are known. That is exactly why the strength claim is graded as locked real-data and not as a verified invariant: there is no theorem in the substrate that forces a particular numerical strength, only an empirical correlation on cited anatomy, and the appendix reports its fragility rather than burying it. The gate enforces this distinction and fails closed if the strength is ever silently upgraded.

The clock, the drive, and the quorum gate

The remaining three obstacles close more briefly and each at its proper grade. For timing, the appendix attaches the real measured period of the segmentation clock and the cited number of somite pairs to predict the span of somitogenesis in days, and that span — about eight and three-quarter days — lands within tolerance of the cited Carnegie window, which is what it means to convert the cadence of the clock into days. It does not claim more: a single absolute zero-point that would pin every gene in the atlas to an absolute day, rather than fixing the somite cadence, would require a second independently measured anchor, and that the firewall does not grant, so the global clock is left explicitly open. For drive, the appendix stops using a uniform placeholder weight on the cascade edges and instead reads each edge's drive from the promoter as the switch's own on-branch amplitude, the square root of the stiffness — the stable on-state value of the switch, so that a stiffer regulator delivers proportionally more drive to its target. This remains a declared modelling choice, because reading the full cis-regulatory code from sequence alone is itself an open problem; but the consequence is an invariant rather than a choice, because the firing order under this sequence-derived drive matches the order under the old uniform drive almost perfectly, at a rank correlation near zero point nine nine and with no wavefront violations. The order does not depend on the particular drive weights, only on the wiring. For the gate, the wavefront theorem is generalised from its simplest form. In the earlier appendix a gene fired once any single regulator was on; here a gene fires once a chosen quorum of its regulators are on, a fraction of its in-degree, and at the extreme of full conjunction, where every regulator must be on, the cascade order still holds with zero violations, convergent nodes that wait for more regulators never fire earlier than they would under the permissive rule, and the original OR-gate is recovered exactly as the one-regulator special case. The firing rule can be made as strict as a quorum demands without breaking the order the substrate forces.