To Kill Three Stones with Six Birds: A Common Grammar for the SM, QM, and GR

Can one small mathematical vocabulary organize two very different open problems in fundamental physics? We test this with the Six Birds emergence calculus, whose working core is simple: a coarse description is a quotient of a finer one, a quantity either passes through the quotient or leaves a checkable obstruction, and two descriptions may share a common refinement. We apply this vocabulary to two problems: why the Standard Model (SM) has its gauge structure, and how quantum (QM) and gravitational (GR) descriptions relate. Every result lives on an explicitly declared finite model (a finite enumeration, a fixed finite graph, or matrices of fixed size), holds only under the stated caps and conventions, and ships with a public validator. On the SM side, a census of 1,066 chiral gauge structures shows that, within the declared windows, only the SU(2)×SU(3) family admits a “clean” breaking branch, and every clean branch breaks the SU(2) factor. Selection is therefore a property of a structure-plus-branch pair, not of a bare gauge group. A separate toy theorem rules out single-factor SU(N) candidates. An exact construction of the regular SU(5) embedding forces the hypercharge direction and reproduces sin2θW = 3/8 and kY = 5/3. These are standard values. They are recovered under named conventions and do not single out SU(5). On the QM–GR side, an exact theorem shows that the declared QM and GR readouts of an abstract carrier are incomparable: neither is a coarse-graining of the other, although a common refinement carries both. On finite graphs, an exact max-flow/min-cut certificate realizes Area(min cut) = ∑e ce ye with per-edge shadow prices ye, and sampled tensor-network entropies stay below the cut. In a toy holographic carrier, a general linear-algebra theorem shows that the full raw boundary readout loses only internal gauge data, while an exact witness shows that a half-boundary readout loses physical information. Three questions are developed as construction programs. For the first, nineteen exact equal-cut graph pairs cannot be joined by forward reduction moves, while six state-level pairs turn out to be gauge-equivalent. The other two concern gravitational record formation and the persistence of records; their finite precursors are, respectively, conditional on an assumed channel and sensitive to how records are defined. Negative results, such as the exact commutation of the natural “quantize” and “curve” completions, are reported as results. No claim about nature is made beyond these finite carriers. Version 3 follows a mathematics and mechanization review of all 42 claims-registry records: the information-loss result now rests on a general GL(d) factorization theorem with exact rank certificates and an exact partial-readout witness; the tensor gauge-collapse lemma is certified on every connected carrier; several validators were strengthened; and imported statements are scoped precisely. The text was rewritten for readability with new figures; Appendix E of the paper records the version history. Code, artifacts, validators, and the claims registry: github.com/ioannist/six-birds-sm-qm-gr.