Scale — root dossier (source of truth v1.7, current 2026-07-12)

This is the long-form root dossier for Scale — the deep root that answers the blunt question why does the universe have the sizes and masses it does, and why are they so wildly separated? The gate’s fixed grade is MEASURED-ANCHOR / RESOLVED +0. It closes not by discovering a mechanism that shrinks a large number into a small one, but by a disciplined sequence of separations — cutting each tangled magnitude question into two clean ones until every clean piece lands on a theorem, an arithmetic identity, or an openly-declared measurement. What follows renders the full substantive content: the typed definition, the frozen thirteen-dimensional arena at full precision, the complete construction and derivation chain (every leg), the anchor ledger, the per-gate interface, the residual register, the closure argument, and the negative controls.

1. Executive summary & honest status

Headline. The sizes and masses of the world — from the Planck mass down through the electroweak scale to the QCD scale and the dark-energy density — rest on a small, openly-charged set of measured rulers, not on a hidden derivation dressed up as a prediction. The celebrated “hierarchy” between the largest and smallest of these scales is not a third mystery requiring new physics to explain but the arithmetic ratio of two rulers already on the books, and it comes out for free the moment both rulers are honestly charged. The genuine output of the frozen thirteen-dimensional construction is not any absolute magnitude at all — it is the dimensionless content (mass ratios, mixing angles, the running of couplings) that rides on top of those rulers and can be checked against data. This gate exists to police that distinction with zero tolerance for slippage, and it closes as a measured-anchor floor: honest, load-bearing, and permanently non-zero.

The precise claim, in four load-bearing statements — and only four:

1. Existence is forced. Unit-gauge invariance — the physical content of Buckingham-\(\pi\) dimensional analysis — proves, as a theorem and not an assumption, that any theory with dimensionful mass content must carry at least one absolute dimensionful ruler. This is a consequence of the statement “physics does not care what units you measure it in.” A theory can never legitimately claim zero anchors. This existence half is genuinely derived: it is the Invariance root (R1) doing real, checkable work.
2. The value of the primary ruler is measured, full stop. The ordinary (non-reduced) Planck mass is charged as \(M_{\rm Pl} = (\hbar c/G_N)^{1/2} = 1.220900000000000\times10^{19}\ \text{GeV}\) (four-significant-figure source precision). Its reduced form is the pure arithmetic rescaling \(\bar M_{\rm Pl} = M_{\rm Pl}/\sqrt{8\pi} \approx 2.4353\times10^{18}\) GeV — not a second independent measurement, just \(M_{\rm Pl}\) viewed through a \(\sqrt{8\pi}\) convention factor. Nothing in the thirteen-dimensional geometry produces the number \(1.2209\times10^{19}\) GeV; it is read off gravity’s observed strength, exactly as in ordinary General Relativity, and entered into the ledger as input, never as output.
3. A second, independent ruler is likewise measured and independently certified irreducible. The electroweak scale \(v_{\rm EW}\approx246.02\) GeV — extracted the ordinary way from \(M_Z\) and the Fermi constant — is realized inside the frozen geometry as a Wilson-line (Hosotani) vacuum expectation value, \(v_{\rm EW} = \theta_H^\star/(2\pi R_\gamma)\), where \(\theta_H^\star \approx 2.46\times10^{-14}\) is the location of the minimum of the one-loop Hosotani potential. Critically, \(\theta_H^\star\) is read off that minimum, not derived from a first-principles exponent. This second ruler survives a dedicated certification that closes both routes by which one might hope to demote it to a derived consequence of \(M_{\rm Pl}\) alone; it stands as a genuine, irreducible second anchor, +1 on the floor.
4. The hierarchy between them is a ratio, not a mystery. Once both rulers are honestly on the books, the hierarchy \(H \equiv v_{\rm EW}/M_{\rm Pl} \approx 2\times10^{-17}\) (equivalently \(v_{\rm EW}/\bar M_{\rm Pl}\sim10^{-16}\) in reduced-Planck convention) is arithmetic — the quotient of two numbers already charged to the ledger. It is emphatically not a third quantity owed a derivation of its own. Whether the electroweak scale’s own microphysical origin, and the magnitude of the hierarchy at the mechanistic level, deserve a deeper story is a separate, still-open question, addressed honestly in the residuals below — but it does not reopen the arithmetic fact that \(H\) is a ratio of two anchors.

Three further certified findings round out the picture. The existence of a cosmological-constant term is forced by a complete classification theorem (Lovelock, 1971) once diffeomorphism invariance, four spacetime dimensions, and second-order field equations are imposed — so “why is there a \(\Lambda\) slot at all” is closed, even while “why does \(\Lambda\) have the value it has” is not. The value \(\Lambda = (2.3\ \text{meV})^4 \approx 5\times10^{-10}\ \text{J/m}^3 \approx 10^{-122}M_{\rm Pl}^4\) is charged as the framework’s fifth measured invariant. And this gate’s own completion-run deliverable — a from-scratch, two-independent-algorithm computation of \(\Lambda_{\rm QCD}\) from the frozen geometry’s \(b_3=-7\) beta coefficient and the measured \(\alpha_3(M_Z)\) — demonstrates by explicit removal-and-recompute that the thirteen-dimensional shape’s distinctive geometric content (the Kaluza–Klein threshold vector) has zero channel into the QCD scale, closing off the one place a skeptical reader might suspect a hidden hierarchy bridge.

1.1 The explicit non-claims — read this list as carefully as the claims

This gate is exactly as valuable for what it refuses to say as for what it asserts, and each refusal is a permanent, standing prohibition, not a stylistic hedge:

• It is forbidden, permanently, to say “\(M_{\rm Pl}\) is derived from the geometry.” The geometry’s derived scale \(M_* = 7.467050992135091\times10^{16}\) GeV is fixed by \(M_{\rm Pl}\) together with the compactification volume (\(M_*^{11}=M_{\rm Pl}^2/\mathrm{Vol}(X_{\rm active})\)); the arrow of dependency runs from the measured \(M_{\rm Pl}\) into \(M_*\), never the reverse.
• It is forbidden to say “the electroweak hierarchy \(v_{\rm EW}/M_{\rm Pl}\) is derived.” It is the ratio of two measured rulers. Deriving it would smuggle in a third ruler where the honest floor is exactly two.
• It is forbidden to say “this exact ratio proves the absolute magnitude” of anything. Ratios are dimensionless and unit-gauge invariant by construction; they can never, by themselves, discharge a debt stated in absolute (dimensionful) terms.
• It is forbidden to say “the value is stable, therefore derived,” in reference to the cosmological constant. The radiative-stability result (the trace-free identity, magnitude-blind because the vacuum energy \(V\) cancels identically) dissolves the tuning sub-problem but says nothing about why \(\Lambda\) has its particular measured value.
• It is forbidden to say “\(10^{-122}\) in Planck units is the derivation of \(\Lambda\).” Reproducing a smallness ratio after the fact is not the same act as forcing a value from first principles.
• It is forbidden to treat a coupling scale such as \(\Lambda_{\rm QCD}\) as an existence proof of the Yang–Mills mass gap. A running coupling crossing a Landau-pole-like scale is a statement about perturbation theory’s breakdown radius, not a nonperturbative existence theorem.
• It is forbidden to describe “measurement” itself as a bridge connecting two scales. Measurement supplies exactly one anchor point; it does not carry magnitude information across sixteen or seventeen orders of magnitude. A number without a window is a number.
• It is forbidden to conflate “dissolved” with “solved.” A dissolution removes a false debt; it does not retroactively pay a real, still-open debt.
• Finally, it is forbidden, in either direction, to move this gate’s grade. The roll-up is ANCHORED / TERMINAL + RESIDUALS-SHOWN — a distinct, permanent status, not a synonym for “solved,” “derived,” or “trivial,” and equally not to be talked back down into “OPEN” because residuals remain.

Single-sentence endpoint preview. The sizes and masses of the physical world are anchored — not derived, not dissolved to zero, and not left ungrounded — on two openly-measured rulers whose ratio is the hierarchy, whose dimensionless consequences are the program’s genuine and falsifiable predictions, and whose remaining magnitude-bridge questions are carried forward as named, bounded, testable bets rather than hidden gaps.

2. The community gap & state of the art

2.1 Two logically separate questions

Strip away the machinery and the question is the one every graduate student meets: the electroweak scale sits at \(v_{\rm EW}\approx246\) GeV; the Planck scale sits at \(M_{\rm Pl}=1.2209\times10^{19}\) GeV; the ratio is \(H = v_{\rm EW}/M_{\rm Pl}\approx2\times10^{-17}\). Sixteen to seventeen orders of magnitude of desert separate the weak force from gravity’s own quantum description. A second, far more extreme separation sits alongside it: the observed dark-energy density expressed in natural units is \(\Lambda/M_{\rm Pl}^4\approx10^{-122}\) — the number famously called “the worst prediction in the history of physics.” These are the two canonical fine-tuning cruxes of the field.

The central methodological move of this gate is to separate two questions the community routinely runs together:

1. The existence question: must a theory with mass content possess at least one absolute, unit-dependent numerical input (a “ruler”), or can all of physics in principle be stated with zero dimensionful anchors? This has a clean yes/no theorem-grade answer.
2. The value question: given that at least one ruler is required, why does it take the particular value it does, and why is the second ruler so much smaller?

Decades of technicolor, supersymmetry, large-extra-dimension, and landscape model-building have almost universally aimed at the second question while implicitly assuming an answer to the first that is rarely stated as a theorem. This gate does the reverse: it proves the first rigorously and treats the second question’s value half as forever measured, refusing to dress a fit as a derivation. That refusal is its distinctive, deliberately unglamorous contribution.

2.2 History of the hierarchy problem, and why prior attempts fall short

The electroweak hierarchy problem in its modern form dates to the naturalness literature of 1979–1980 (Wilson, Susskind, ’t Hooft): once the Higgs is taken as elementary, its mass-squared parameter receives quantum corrections quadratically sensitive to the cutoff, so keeping \(m_H^2\) small up to the Planck scale requires an enormous, unexplained cancellation. The community’s responses fall into a small number of families, each declined here and each worth naming:

• Supersymmetry cancels the quadratic divergences between fermion and boson loops, replacing “why is the hierarchy stable” with “why is SUSY broken near the TeV scale.” LHC null results through Run 2–3 have pushed superpartner masses well above the natural scale, eroding the naturalness motivation.
• Technicolor / composite Higgs generates the Higgs mass dynamically at a compositeness scale; precision electroweak and flavor constraints make simple versions difficult, and the descendants introduce their own new scales as inputs.
• Large extra dimensions / warped geometry (Arkani-Hamed–Dimopoulos–Dvali; Randall–Sundrum) reformulate the hierarchy as a geometric separation, trading “why is a mass ratio small” for “why is a length or warp exponent what it is” — important prior art, but the warp exponent itself is tuned to reproduce the observed hierarchy, relocating the problem.
• Anthropic / landscape reasoning explains why an observed value could be typical of a habitable slice of a landscape without predicting the number — a selection argument, not a derivation.
• Relaxion mechanisms dynamically scan and stop at a small Higgs mass during inflation, trading a static fine-tuning problem for a dynamical, still-parameter-laden one.

None has produced an experimentally confirmed resolution. The honest state of the art is that even the weak (“why is the ratio small at all”) form has no first-principles derivation that does not relocate the smallness into another input.

2.3 The cosmological-constant problem, dimensional transmutation, and Lovelock’s theorem

The cosmological-constant problem shows an even starker mismatch: effective-field-theory estimates of the vacuum energy sit tens of orders of magnitude above the observed value. Weinberg’s 1989 framing splits it into the “old” problem (why the net value is not of order the largest contributing scale) and, after 1998, the coincidence puzzle (why the value is small and positive rather than zero). The standard benchmark burdens quoted in the literature illustrate the unprotected mismatch: a bare QFT zero-point estimate of order \(3\times10^{11}\) J/m\(^3\); a QCD condensate-scale estimate of order \(3\times10^{34}\) J/m\(^3\) (roughly \(10^{44}\times\) observed); and an electroweak-condensate estimate of order \(v^4\sim10^{45}\) J/m\(^3\) (roughly \(10^{55}\times\) observed). No accepted mechanism removes this residual burden at the value level; it stays open community-wide.

Two older, genuinely solved pieces of prior art enter centrally. Dimensional transmutation (Coleman–Weinberg 1973; asymptotic freedom, Gross–Wilczek–Politzer 1973) is the mechanism by which a classically scale-invariant theory generates a dimensionful scale purely through quantum running; the QCD scale \(\Lambda_{\rm QCD}\) has been defined this way for fifty years — measure \(\alpha_3(M_Z)\), run down, read off the scale where the one-loop inverse coupling extrapolates to zero. \(\Lambda_{\rm QCD}\) has never been an independent free parameter; it is always a function of the measured coupling and the calculable beta coefficients. Lovelock’s 1971 classification theorem establishes that in \(D=4\) the only symmetric, divergence-free, at-most-second-order metric tensor is a linear combination of the Einstein tensor \(G_{\mu\nu}\) and the metric \(g_{\mu\nu}\); the coefficient of \(g_{\mu\nu}\) is the cosmological constant. So the existence of a \(\Lambda\) slot is forced — while the number that fills it is not, a conflation this gate refuses.

2.4 The gap as inherited by this gate

The community has, after four decades, no accepted first-principles derivation of either \(v_{\rm EW}/M_{\rm Pl}\approx2\times10^{-17}\) or \(\Lambda/M_{\rm Pl}^4\approx10^{-122}\) that does not relocate the mystery into an equally unexplained new parameter or rely on statistical selection over an unverified ensemble. The best-established prior art this gate honestly stands on is: Buckingham’s \(\pi\)-theorem (existence of \(\geq1\) absolute scale is forced), Lovelock’s theorem (existence of a \(\Lambda\) slot is forced), and dimensional transmutation (fixes \(\Lambda_{\rm QCD}\) from a measured coupling). This gate’s own contribution is not a fourth mechanism competing with SUSY or the landscape — it is the explicit, target-blind bookkeeping that shows the hierarchy is the ratio of two honestly-charged rulers, that the QCD scale computed on the frozen geometry carries no hidden bridge (demonstrated, not assumed), and that the cosmological-constant tuning burden dissolves via a magnitude-blind identity whose limits are stated as strictly as its power.

3. The frozen 13D arena at full precision

The Scale root is not a free-floating claim that “\(M_{\rm Pl}\) is measured”; it is a claim about which numbers a completely pinned thirteen-dimensional arena is and is not permitted to manufacture on its own. Showing the arena at full precision is the proof that nowhere in the geometry, the rulebook, or the operator content is a second hidden ruler being smuggled in to manufacture a derived hierarchy.

3.1 The complete layered object

The branch this gate is anchored to is a three-layer composite:

with \(K_6 = SU(3)/T^2\), the complete flag manifold of the \(A_2\) root system, and \(S^1_Y/\mathbb{Z}_2\) the active orbifold boundary domain carrying hypercharge. Only the \(\times\)-layer carries metric dimension:

The \(\oplus\) (rulebook) and \(\otimes\) (actors) layers are non-metric — zero-dimensional in the ordinary sense — but they are part of the frozen branch and are never dropped when a gate reasons about what is or is not forced. A claim like “the compactification volume is a free size-bridge that could be tuned to produce the hierarchy” would be a claim about the \(\times\)-layer alone, smuggling past the \(\oplus\)-layer rule that fixes the chamber center and the \(\otimes\)-layer operator content that fixes what propagates.

3.2 The × Stage — four metric factors

Binding rule: gauge forces are isometries of the internal metric factors, and the sizes of those factors are geometric quantities, derived once the compactification scale is fixed — never a second free size-bridge that could be dialed to close the \(v_{\rm EW}/M_{\rm Pl}\) hierarchy. Dimensionless ratios (radii-to-radii, volume-to-volume) come out of \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) once and for all; they are not levers available to the hierarchy computation.

Radii, at full precision (chamber center \(\vec u=(1,1,1)\); compactification scale tied to the unification scale via \(R_0\equiv(2\pi M_U)^{-1}\), \(M_U=1.0\times10^{16}\) GeV, closure residual \(9.6\times10^{-11}\)):

The squashing chamber is \(\vec u\in[1/2,3/2]^3\), Weyl-rigid; the witness value used everywhere downstream is the symmetric center \(u_1=u_2=u_3=1\). Off-center points fail Weyl-rigid admissibility and are eliminated by the \(\oplus\)-layer selector — the rulebook, not a per-gate choice. The chamber center is selected by the admissibility rulebook, not tuned by whichever gate is being computed.

Volumes, at full precision:

The \(S^1_Y\) volumes are exact because the \(2\pi\) in the volume formula cancels the \(2\pi\) inside \(R_0=1/(2\pi M_U)\), leaving the clean rationals \(1/M_U\) and \(1/(2M_U)\). This exactness is load-bearing: the orbifold volume carries no hidden irrational fudge factor that could later be read as a second scale bridge.

3.3 Where \(M_{\rm Pl}\) and \(M_*\) meet the geometry — the Planck normalization

This is the single equation at which the Scale root’s measured ruler and the Shape root’s derived geometry touch, and it is the exact point where the “measured, not derived” line is drawn:

Reading this equation correctly is the whole content of the “\(M_{\rm Pl}\) is not derived” discipline: \(M_*\) is fixed by geometry and \(M_{\rm Pl}\) together — not a second independent ruler, but a consequence of already having charged \(M_{\rm Pl}\) as measured input. Nothing in this equation lets the theory produce \(M_{\rm Pl}\)’s numerical value from the 13 dimensions alone; the equation only redistributes an already-measured number across an extra 9 dimensions. Reading it the other direction (deriving \(M_{\rm Pl}\) from \(M_*\)) would require \(M_*\) to be an independent input, which it is not.

3.4 \(K_6\) curvature invariants at full precision

\(K_6=SU(3)/T^2\) carries the \(A_2\) root system: simple roots \(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\); positive roots \(\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}\); half-sum \(\rho=(1,0,-1)\) with \(\|\rho\|^2=2\); Weyl group \(S_3\), order 6. Two metric normalizations are used consistently: (A) frozen physical (\(R_6\)) normalization, curvature in GeV\(^2\); (B) Killing-form normal metric, curvature dimensionless. Both are quoted because a dimensionful curvature number is meaningless without stating which is in force — the Scale root’s discipline applied one level down.

The genuine dimensionless outputs are the metric-scale-invariant ratios, identical in both normalizations — exactly the kind of number the Scale discipline says is allowed to be a physical prediction:

Anti-drift certification carried verbatim: \(\|\mathrm{Riem}\|^2/\mathrm{Scal}^2=23/75\) — it is never \(31/147\), and \(\|\mathrm{Riem}\|^2\) is never \(=60\) (that value belongs to the round unit \(S^6\), a different space used only as a calibration control). \(\|\nabla\mathrm{Riem}\|^2=1/4\neq0\), so \(K_6\) is homogeneous but not locally symmetric — unrelated to whether \(K_6\)’s volume could serve as a second scale ruler (it cannot: it is fully fixed once \(R_0\) and the chamber center are fixed). The Euler characteristic is exact and topological: \(\chi(K_6)=6\).

3.5 Representation content and the KK spectrum

Quadratic Casimir and dimension for \(SU(3)/T^2\), Dynkin labels \((p,q)\): \(C_2(p,q)=\tfrac{p^2+q^2+pq+3p+3q}{3}\), \(\dim(p,q)=\tfrac{(p+1)(q+1)(p+q+2)}{2}\). Notable exact values: \((1,0)=\mathbf3\), \(C_2=4/3\) (quark color triplet); \((1,1)=\mathbf8\), \(C_2=3\) (gluons); \((3,0)=\mathbf{10}\), \(C_2=6\). KK mass spectra scale as \(m^2_{(p,q)}=(C_2(p,q)+\Delta)/R_6^2\): every KK mass this arena produces is a dimensionless Casimir number divided by the single radius-squared \(R_6^2\) — exactly one scale-carrying denominator in the whole tower, reinforcing that Shape supplies ratios/spectra, not an independent absolute scale.

3.6 The threshold vector \((\delta_1,\delta_2,\delta_3)\) — the one number this gate must show has no slot in the hierarchy

The full Shape KK-threshold correction vector, summed packet by packet from the heat-kernel ledger (GUT-normalized \(\alpha_1=\tfrac53\alpha_Y\); two-loop SM RG, \(\overline{\rm MS}\), \(M_Z=91.1876\) GeV):

The associated one-loop beta coefficients (GUT-normalized) are exact fractions fixed by Standard Model field content: \(b_1=41/10=4.1\), \(b_2=-19/6=-3.166666666666667\), \(b_3=-7\) (exact integer). This vector is the arena’s entire correction budget from the extra dimensions into the running couplings, and every occurrence of it ties to the \(M_Z\to M_U\) upward unification closure (residual \(9.6\times10^{-11}\)) — never to the \(M_Z\to\Lambda_{\rm QCD}\) downward infrared formula. Forcing \(\delta_3\to0\) in the one-loop \(\Lambda_{\rm QCD}\) formula changes nothing, bit for bit, because \(\delta_3\) has no slot there.

3.7 The Wilson-line Higgs mechanism (the \(\otimes\) actors layer)

The second ruler, \(v_{\rm EW}\), is realized as an actor-layer object: a Wilson-line (Hosotani) holonomy field on the cycle \(\gamma\) with radius \(R_\gamma\sim R_0\), integer winding \(n_H=1\) exactly (\(n_H=0\) would give no VEV). The Hosotani effective potential is

an absolutely convergent \(n^{-5}\) tail guaranteeing a finite Higgs mass. This potential is periodic in \(\theta_H\); its minimum \(\theta_H^\star\) is a stationary point of a bounded, periodic function, not the endpoint of an exponentially-running coupling. There is no RG exponent \(e^{-c/\alpha}\) to compute here. \(\theta_H^\star\approx2.46\times10^{-14}\) is read off the numerically located minimum (carrying roughly 85% of the hierarchy magnitude), which is exactly why \(v_{\rm EW}\) is charged as a certified-irreducible second anchor. The post-RG outputs: \(v_{\rm pred}=246.02\pm3.5\) GeV, \(m_h=123.82\pm1.8\) GeV, \(\lambda_H=m_h^2/(2v^2)=0.12722\pm0.00181\).

3.8 The two irreducible rulers — and why the arena cannot manufacture a third

Given every derived quantity above, the arena still requires exactly two externally-measured dimensionful numbers to become numerically concrete:

\(M_{\rm Pl}\) is charged MEASURED-ANCHOR (ruler #1, floor \(\ge1\)). \(v_{\rm EW}\) is charged CERTIFIED-IRREDUCIBLE (ruler #2). Every other dimensionful number — \(M_*\), \(M_U\), \(R_0\), \(R_6\), \(R_2\), \(R_Y\), every KK mass, every volume — is derived from these two rulers plus the dimensionless geometric data. That is the complete content of “the hierarchy is an arithmetic ratio, not a third mystery.” The fitted flavor normalizations \(N_d, N_e, N_u, N_\nu\) are dimensionless coefficients multiplying already-charged anchors within the same arena — not additional Buckingham-\(\pi\) rulers, but honestly the weakest scale-link, named as such (Hole S3 below).

4. Construction I — the deep-root anchoring

A closure claim at MEASURED-ANCHOR / RESOLVED +0 is only as trustworthy as the roots it stands on. This section applies the three deep roots — Shape, Scale, Granularity — completely, plus the four admissibility screens (Invariance, Record-Interface, Causal-Order/target-blindness, Nonseparability), and shows exactly what each eliminates, forces, or exposes.

4.1 The Scale root, applied completely — the absolute-magnitude discipline

The Scale root is the entire discipline governing how a dimensionful number enters or leaves the ledger: it supplies the accepted ruler, the scheme in which that ruler is quoted, the RG window across which it is transported, the normalization convention, and the stability/readout rules. It is load-bearing: without a declared ruler, every mass is a bare number with no unit-gauge-invariant meaning. Ruler #1 is \(M_{\rm Pl}\) at 4-sig source precision; its reduced form \(\bar M_{\rm Pl}=M_{\rm Pl}/\sqrt{8\pi}\approx2.4353\times10^{18}\) GeV is a convention factor, not a second measurement. The two conventions must never be silently mixed inside one comparison.

The hierarchy, verified in both conventions:

Both are one measured number divided by another — no exponential, no running, no threshold correction anywhere in the division. The would-be “EW log” a skeptical reader might treat as a derivable exponent,

is flagged as a circular re-log: it is definitionally the logarithm of the very ratio \(H\) that was to be explained. Any construction that “derives” \(I_{\rm EW}\approx36.83\) by tuning a coupling or volume to hit that number has smuggled the answer in as an input.

Applying the Scale root completely eliminates three classes of move: (a) “derive \(M_{\rm Pl}\)” as a legitimate target (unit-gauge invariance proves only that some ruler must exist, never which value); (b) “derive the hierarchy \(H\)” (definitionally the ratio of two already-charged rulers); (c) any claim that a dimensionless win discharges dimensionful debt.

4.2 The Shape root, applied completely — the object whose ratios get cashed

Where Scale supplies the ruler, Shape supplies the object whose invariant, dimensionless content the ruler reads out. Applying Shape completely means pinning all three layers — a residual computed against a truncated object (say, the \(\times\)-Stage metric alone) is an artifact. Every compactification number is a consequence of the unification-closure condition \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) (residual \(9.6\times10^{-11}\)) together with the program’s over-determination anchor set \(\{M_{\rm Pl}, \alpha_i(M_Z), y_t, |V_{us}|\}\); there is no free radius or volume that could be dialed to manufacture a size relationship between \(v_{\rm EW}\) and \(M_{\rm Pl}\).

Binding-scope note. \(K_6=SU(3)/T^2\) is constraint-selected among a small family (roughly a factor of \(\sim4\) in the companion Shape gate’s counting) rather than proven the unique geometry. This gate does not upgrade that status — “selected” is carried here exactly as declared, and no Scale-root argument depends on Shape being unique, only on Shape being fixed once selected.

4.3 The Invariance screen (R1) — forcing the existence half as a theorem

Buckingham-\(\pi\) / unit-gauge invariance says the physical content of any statement cannot depend on the arbitrary choice of units. A bare dimensionful number is not a physical statement; it becomes one only relative to a unit convention, and the moment a second independent dimensionful quantity of the same type exists, the ratio becomes unit-choice-independent. Running this in reverse: a theory with massive degrees of freedom cannot express its mass content using dimensionless data alone — there must exist at least one dimensionful reference. This is a general theorem, not an assumption bolted on. It forces existence (“the floor is \(\geq1\), never exactly \(0\)”) and nothing more; it cannot select which value the ruler takes. The same discipline underlies the Lovelock forcing of a \(\Lambda\) slot: an invariance argument forces a slot to exist while remaining silent on the value that fills it.

4.4 The remaining three screens

Record-Interface (R4) — reproducibility, explicitly not validation. The frozen branch’s audit trail (every value traceable to an exact closed-form equation) makes the object reproducible; a second independent computation from the same declared anchors and scheme reaches the same numbers. But reproducibility is a claim about which object is cashed, not that its magnitudes are “correct” in some deeper sense.

Causal-Order / target-blindness — confirmed by computation, not assumed. This screen catches the most dangerous failure mode: tuning a free parameter after looking at the number you want. For this gate it is applied as a UV/IR decoupling check: the claim “Shape’s distinctive content has zero channel into \(\Lambda_{\rm QCD}\)” is demonstrated by literally setting the Shape-derived correction \(\delta_3=-1.7313\) to zero inside the \(M_Z\to\Lambda_{\rm QCD}\) running and recomputing. The result is bit-identical (45.036 MeV either way).

Nonseparability — shared scheme objects settled once. The same \(\overline{\rm MS}\) two-loop scheme, the same \(M_Z=91.1876\) GeV comparison point, and the same beta triple are fixed once across the program and propagated, not re-chosen per gate. The UV branch (\(M_Z\to M_U\)) and IR branch (\(M_Z\to\Lambda_{\rm QCD}\)) factorize cleanly through the shared measured \(M_Z\) boundary.

4.5 The Granularity root — why the anchor floor cannot be pushed to zero

Granularity’s cost-floor discipline (\(\Delta_0>0\)) forbids a construction from claiming hidden continuous precision. One might ask whether this could be turned around to drive the measured-anchor floor to zero. The answer is no, and the reason is precise: a measured magnitude — a single finite number such as \(M_{\rm Pl}=1.2209\times10^{19}\) GeV to 4 significant figures — is itself a finite record, the opposite of hidden continuous precision. Granularity forbids free precision; Scale’s floor requires paid, finite, declared precision. \(\hbar\) illustrates the compatibility: it is simultaneously the quantum of action making Granularity’s cost-floor physical and a measured-anchor in the Scale-root sense. Nothing in Granularity offers a route to a “zero anchors” outcome.

4.6 Synthesis

The Invariance screen proves \(\geq1\) ruler must exist (the strongest sentence available). The Scale root charges exactly two — \(M_{\rm Pl}\) and \(v_{\rm EW}\) — the second independently certified irreducible. The Shape root supplies the dimensionless content that rides on top, shown by explicit removal-and-recompute to carry zero hidden channel bridging the two rulers. Record-Interface certifies reproducibility; Granularity confirms the floor cannot be argued away as hidden free precision. The joint output: \(H\approx2\times10^{-17}\) is the arithmetic ratio of two roots’ worth of honestly-charged, independently-screened measured content — not a third mystery, not a derived exponential, not eliminable.

5. Construction II — the full derivation, every leg

5.A The existence half — Buckingham-\(\pi\) forces \(\geq1\) ruler, as a theorem

Unit-gauge invariance: rescaling the unit of mass \(\mu\to\lambda\mu\) acts on every dimensionful coupling as \(g_a\to\lambda^{-\Delta_a}g_a\), and physical predictions are invariant. By Buckingham’s \(\pi\)-theorem, a physically meaningful relation among \(n\) dimensionful quantities built from \(k\) independent dimensions reduces to a relation among \(n-k\) dimensionless combinations, which are the only unit-gauge-invariant objects. Consequently, the moment a theory contains any dimensionful mass parameter, unit-gauge invariance forces that parameter to be stated relative to a ruler. The frozen construction contains a graviton, a Higgs VEV, and a confinement scale — it cannot have zero anchors. The theorem’s minimal, permanently-binding statement:

This is the existence half, and it is genuinely derived — the Invariance root doing real logical work. Nothing about which quantity plays the ruler role, nor what value it takes, follows from this alone. Hole S1 (below) is the honest residual: the theorem proves \(\geq1\); a fully worked proof that exactly one independent absolute scale is required (no second hidden anchor beyond the certified \(v_{\rm EW}\)) is an owed existence argument, marked OPEN.

5.B The value half — charging the two rulers, full three-layer pinning

B.1 — Ruler #1: \(M_{\rm Pl}\). Read from the coefficient of the Einstein–Hilbert term after Kaluza–Klein reduction on the 9-dimensional internal space. Convention: ordinary (non-reduced) Planck mass, quoted to 4-sig source precision. Entered as input, never output:

The reduced form is \(M_{\rm Pl}\) viewed through the fixed \(\sqrt{8\pi}\) factor arising from Einstein–Hilbert normalization (\(\tfrac{1}{16\pi G_N}R\) vs \(\tfrac12\bar M_{\rm Pl}^2 R\)) — not a second measurement.

B.2 — The derived scale \(M_*\), demonstrated not to be a second ruler. Worked in full so “\(M_*\) does not add a ruler” is shown, not asserted:

The dependency is explicit and irreversible: \(M_*\) is solved for using the already-measured \(M_{\rm Pl}\) and a volume built purely from the frozen radii. Erase \(M_{\rm Pl}\) from the right-hand side and \(M_*\) cannot be computed at all — the operational proof the arrow runs one way. \(M_*\) carries zero additional anchor weight; the floor stays at exactly the rulers of B.1 and B.3, never three.

B.3 — Ruler #2: \(v_{\rm EW}\), certified irreducible. Realized as a Wilson-line VEV on cycle \(\gamma\), \(v_{\rm EW}=\theta_H^\star/(2\pi R_\gamma)\), with winding \(n_H=1\) exact and \(\theta_H^\star\approx2.46\times10^{-14}\) read from the one-loop Hosotani minimum. The certification closes both escape routes:

• Route A (mechanistic escape, closed). Could \(\theta_H^\star\) be derived from \(M_{\rm Pl}\) and geometry alone? No: it is read from the minimum of a periodic potential — not the output of a first-principles exponential-suppression formula the way \(\Lambda_{\rm QCD}\) is. The warped (Randall–Sundrum-type) rival that could generate an exponential is banned here: the frozen product-metric arena is unwarped by construction, so that channel does not exist on this branch.
• Route B (dimensional escape, closed). Could Buckingham-\(\pi\) applied to \(\{M_{\rm Pl},\hbar,\text{dimensionless geometry}\}\) produce a second mass for free? No: dimensional analysis on a single anchor plus dimensionless ratios can only produce that same anchor times a pure number. The only exponent producible is the circular re-log \(I_{\rm EW}\approx36.83\) — an auto-FAIL, the logarithm of the very ratio being explained.

Both routes fail for structurally different reasons; \(v_{\rm EW}\) stands CERTIFIED-IRREDUCIBLE, +1 on the floor. The certification is branch-relative: a future warped-geometry or non-Wilson-line Higgs realization would require re-running it. A live refutation channel is named for honesty: a blind recomputation of \(V_{\rm Hos}\) landing at \(\theta_H^\star\sim10^{-14}\) but at a value not matching the measured \(v_{\rm EW}\) would falsify this route. One quantitative alternative was checked and found insufficient: the Berezin–Kontsevich-type protection factor evaluates to \(\sim10^{-2}\), twelve orders of magnitude too large to explain a hierarchy of order \(10^{-16}\) on its own.

5.C The hierarchy — arithmetic ratio, +0

This is the entire content of “the hierarchy.” There is no third quantity, no missing exponential-suppression mechanism, and no computation beyond the division. \(H\) is definitionally the quotient of two numbers already on the ledger. The arithmetic act of forming the ratio owes no separate derivation — even though a factor of \(10^{-16}\)–\(10^{-17}\) is indeed extreme, and why the electroweak minimum sits where it does at the mechanistic level is carried forward as part of Hole S2, not claimed explained. \(H\)’s formula is derived (a division); \(v_{\rm EW}\)’s value is not (read off a potential minimum).

5.D The \(\Lambda\)-presence forcing theorem (Lovelock)

A structurally identical existence/value split governs the cosmological constant. Lovelock’s theorem is a complete classification: in \(D=4\), the most general symmetric rank-2 tensor built from \(g_{\mu\nu}\) and its derivatives that is diffeomorphism-covariant, divergence-free, and at most second order, is exactly

a two-parameter family. No third independent term exists at this order in \(D=4\). The field equations therefore automatically contain the \(b\,g_{\mu\nu}\) term whenever diffeomorphism invariance, \(D=4\), and second-order field equations hold; there is no consistent way to impose those three and not have a \(\Lambda\equiv-b\) slot. Two guards travel with this conclusion: the premise set is metric-only (no Horndeski-type extra scalars), and Lovelock forces only the two-derivative level (higher-curvature towers are suppressed, not banned). The theorem forces the slot, never the value. The measured value \(\Lambda=(2.3\ \text{meV})^4\approx10^{-122}M_{\rm Pl}^4\) is charged as the framework’s fifth measured invariant.

A logically separate, cross-referenced result: the radiative-stability (tuning) sub-problem dissolves at tree level. For \(T^{\rm vac}_{\mu\nu}=-Vg_{\mu\nu}\), the trace-free combination

magnitude-blind because \(V\) cancels algebraically. This dissolves the tuning burden (conditional on the stated premise that gravity decouples the pure-trace mode, an axiom openly declared open) but is explicitly not a derivation of \(\Lambda\)’s value: the identity map \(\Lambda_0\to\Lambda_0+\delta V\) shows the premise alone does not protect the value at the quantum level. Both halves are stated together so “stable, therefore derived” cannot be read in.

5.E The \(\Lambda_{\rm QCD}\) computation — the completion run’s load-bearing deliverable

E.1 The question. Does the frozen Shape force an exponential suppression \(e^{-2\pi/(b_3\alpha_3)}\) producing \(\Lambda_{\rm QCD}\) as a genuinely new, Shape-derived hierarchy-bridging closure — or does the Shape merely consume the already-measured \(\alpha_3(M_Z)\), as ordinary four-dimensional QCD has since 1973?

E.2 The exact closed form. One-loop dimensional transmutation is governed by the linear RGE for the inverse coupling, \(d(\alpha_3^{-1})/d(\ln\mu)=-b_3/(2\pi)\), whose solution run from the measured boundary to the scale where \(\alpha_3^{-1}\) formally vanishes is

E.3 Inputs, all traced. \(M_Z = 91.1876\) GeV (PDG, band \(\pm0.0021\)). \(\alpha_3(M_Z) = 0.1179\), the PDG-central value; a provenance caveat is recorded honestly and flagged non-load-bearing, because the structural (channel-existence) question is insensitive to the fourth digit. \(b_3 = -7.000000000000000\) exactly, fixed by Standard Model field content (3 chiral generations + 1 Higgs doublet + SM gauge sector), read directly off the frozen actor bundle. For completeness: \(b_1=41/10=4.1\), \(b_2=-19/6=-3.166666666666667\), \(b_3=-7\).

E.4 Route 1: closed-form algebraic inversion. Substituting: \(b_3\alpha_3=-0.8253\); exponent \(2\pi/(-0.8253)=-7.613213749157381\); \(\exp(-7.613213749157381)=4.938820921331295\times10^{-4}\); multiply by \(M_Z\):

E.5 Route 2: independent ODE shoot. The linear ODE was integrated downward from the \(M_Z\) boundary using an explicit-Euler stepper with \(2{,}000{,}001\) discrete steps, stopping where \(\alpha_3^{-1}\) crosses zero — without ever invoking the closed-form exponential:

E.6 Cross-check. The relative difference between the two independently-coded routes is \(1.737\times10^{-10}\), consistent with pure floating-point/discretization error. This confirms the absence of an algebra or implementation bug; it is explicitly not claimed as independent physics confirmation (both routes solve the same one-loop RGE from the same inputs).

E.7 Sanity check (non-load-bearing, expected). The value \(45.036\) MeV lies outside the physical \(\overline{\rm MS}\) five-flavor band \(\Lambda_{\overline{\rm MS}}^{(5)}\in[190,230]\) MeV — the honest, expected consequence of a bare one-loop truncation with no flavor-threshold matching and no higher-loop running. A real extraction requires two-to-four-loop running plus threshold matching, which was never in question for this structural wall. The mismatch is recorded plainly, not hidden or explained away.

E.8 The target-blindness / Shape-forcing test. A removal-and-recompute test, not an assertion. The frozen geometry supplies \((\delta_1,\delta_2,\delta_3)=(+4.8424,-3.1112,-1.7313)\pm1.6\times10^{-3}\), where \(\delta_3=-1.7313\) is the color-sector correction. Recompute the identical IR formula with \(\delta_3\) forced to zero:

The reason is structural: \(\delta_3\) has no slot in the \(M_Z\to\Lambda_{\rm QCD}\) infrared formula. Tracing every occurrence of \((\delta_1,\delta_2,\delta_3)\) in the frozen record confirms it is tied exclusively to the upward threshold-matching condition at \(M_U=1.0\times10^{16}\) GeV — sixteen orders of magnitude away in the opposite direction, feeding the unification residual \(9.6\times10^{-11}\), never the downward QCD formula.

To show the “no channel” verdict is not merely a matter of \(\delta_3\) being numerically small: suppose, counterfactually, \(\delta_3\) entered as an additive shift \(b_3\to b_3+\delta_3=-8.7313\). Recomputing gives exponent \(-6.103615297160979\), i.e. \(\Lambda_{\rm QCD}\big|_{\rm naive\ shift}\approx203.8\) MeV — landing suggestively almost inside the physical band. This counterfactual is exhibited precisely because it shows how easy it would have been to want \(\delta_3\) to enter (the shifted number looks phenomenologically better). The test sidesteps the trap by asking the sharper question first: does \(\delta_3\) have a slot in the true formula at all? It does not, and the removal-and-recompute confirms this by bit-identical numerical comparison, never by whether some plausible modification “looks better.”

E.9 Verdict. \(\Lambda_{\rm QCD}\) is DERIVED-GIVEN-\(\alpha_3(M_Z)\)-anchor = #1 (+0), in the ordinary sense that has applied to QCD confinement-scale computations since 1973: \(b_3=-7\) is a Shape/Rulebook-exact readout, the one-loop closed form is a clean forced consequence, and the whole computation bottoms out on \(\alpha_3(M_Z)\), already a member of the shared floor anchor set — nothing new is spent. The hierarchy disjunction resolves negative: Shape’s distinctive thirteen-dimensional content has zero channel into \(\Lambda_{\rm QCD}\); \(v_{\rm EW}\) retains its status as certified-irreducible new anchor (+1). A nuance carried honestly: \(b_3=-7\) is a faithful readout of the frozen branch’s field content but is not distinctively thirteen-dimensional-geometric — ordinary four-dimensional Standard Model content reproduces the identical value. The preferred phrasing is “Shape/Rulebook-derived (Standard-Model content on the frozen branch, not distinctively extra-dimensional).”

5.F Summary of the derivation chain, with every link’s status named

6. Construction III — the central result at full precision

The frozen geometry’s one attempt to inject new dimensionful content into the scale ledger — the Kaluza–Klein threshold-corrected running of the strong coupling down to confinement — is carried out here to full precision by two independent algorithms, cross-checked to ten significant figures, then subjected to a target-blind removal-and-recompute on its own distinctive geometric ingredient. The verdict, reached by direct calculation, is negative: the geometry’s threshold vector has zero channel into the confinement scale.

6.1 Deriving the closed form from first principles

From the one-loop RGE \(d\,\alpha_3^{-1}(\mu)/d\,\ln\mu = -b_3/(2\pi)\) (a constant right-hand side below any flavor threshold), integration gives \(\alpha_3^{-1}(\mu) = \alpha_3^{-1}(M_Z) - \tfrac{b_3}{2\pi}\ln(\mu/M_Z)\). Defining \(\Lambda_{\rm QCD}\) as the scale where this extrapolated inverse coupling reaches zero and exponentiating:

Note the structural fact before any numbers: this formula contains exactly three ingredients — \(M_Z\), \(\alpha_3(M_Z)\), \(b_3\). There is no slot anywhere for a Kaluza–Klein threshold correction; \(\delta_3\) does not appear as a symbol. This absence is the entire content of the wall’s verdict, visible already at the level of the equation’s derivation.

6.2 Route 1, every digit shown

Step 1 — denominator: \(b_3\,\alpha_3(M_Z) = (-7)\times0.1179 = -0.8253000000000000\). Step 2 — exponent: \(2\pi = 6.283185307179586\); dividing, \(2\pi/(b_3\alpha_3)=-7.613213749157381\) (check: \(-7.613213749157381\times(-0.8253)=6.283185307179586=2\pi\) to all sixteen digits). Step 3 — exponential: \(\exp(-7.613213749157381)=4.938820921331295\times10^{-4}\). Step 4 — multiply by \(M_Z\): \(91.1876\times4.938820921331295\times10^{-4}=0.045035922664598965\) GeV. Back-substitution: \(\ln(0.045035922664598965/91.1876)=-7.613213749157381\), self-consistent to full machine precision.

6.3 Route 2 — the cross-check that rules out an implementation bug

Route 2 integrates the same linear ODE numerically, starting from \(\alpha_3^{-1}(M_Z)=1/0.1179=8.481764\ldots\) at \(t_{M_Z}=\ln(91.1876)\), stepping downward until \(\alpha_3^{-1}\) crosses zero. Because the ODE is exactly linear with constant slope, Euler stepping is exact in exact arithmetic regardless of step count; the \(2{,}000{,}001\) fine steps are a belt-and-suspenders choice whose role is to expose any coding bug in the zero-crossing search that a symbolic check could not catch. The zero-crossing: \(\Lambda_{\rm QCD}^{\rm (Route\,2)}=0.04503592265677591\) GeV. The relative difference from Route 1 is \(1.737\times10^{-10}\) — double-precision roundoff over two million steps plus the transcendental evaluation, not a physics discrepancy. Two structurally different algorithms computing the same quantity from the same inputs agree to ten significant figures: confirmation of arithmetic and implementation correctness, explicitly not independent physics.

6.4 The removal-and-recompute test in full

The frozen threshold vector’s color component is \(\delta_3=-1.7313\). If it carried a hidden channel into the downward running — for instance if the true beta coefficient below \(M_Z\) were \(b_3+\delta_3\), or if \(\delta_3\) entered as an additive shift to the exponent — then setting \(\delta_3\to0\) and recomputing would produce a different answer. It does not: \(\Lambda_{\rm QCD}\big|_{\delta_3\to0}=0.045035922664598965\) GeV, bit-identical. The reason is visible directly from the closed form: it contains no symbol \(\delta_3\). Forcing \(\delta_3\to0\) changes nothing because \(\delta_3\) was never an argument of the function being evaluated. This discovers not a small negligible effect that rounds away, but that the effect’s channel does not exist — a qualitatively stronger statement than “the effect is small.” The geometry pack is explicit that the one and only place \((\delta_1,\delta_2,\delta_3)\) enters any computation is the upward \(M_Z\to M_U\) unification closure; the ledger contains no second entry.

6.5 The verdict, cross-referenced against the anchor floor

Status: DERIVED-GIVEN-\(\alpha_3(M_Z)\)-anchor, +0. Because \(\Lambda_{\rm QCD}\) carries no geometric hierarchy-bridging content, it cannot shrink the gap between \(v_{\rm EW}\approx246\) GeV and \(M_{\rm Pl}\approx1.2209\times10^{19}\) GeV by even a single order of magnitude via this channel. The QCD scale sits at \(\sim10^2\)–\(10^3\) MeV, supplies no \(e^{-c/\alpha}\) suppression connecting the two ends, and was never in the business of connecting them. The two-ruler floor \(\{M_{\rm Pl}, v_{\rm EW}\}\) is not reducible to one-ruler by any channel this computation could have opened — and this was the most promising candidate channel the frozen geometry offered. It did not open.

7. The insights that made it work

The gate closes by a sequence of separations. Each insight answers exactly one narrow question and is never asked to do double duty; every “forbidden” non-claim is forbidden precisely because it would stretch one of these narrowly-scoped results to answer a question it was never built to answer.

Insight 1 — unit-gauge invariance is a symmetry with rigid consequences. Physical law does not know what units you chose. Under a simultaneous rescaling of mass, length, and time, no dimensionless equation changes. The consequence for a generative theory is rigid: no finite calculation built purely from dimensionless inputs (angles, group integers, topological indices, \(\pi\), \(\sqrt3\)) can ever output a dimensionful answer. A dimensionful answer requires a dimensionful input to fix the unit gauge — no exceptions. This is a fabrication detector with zero false negatives: any claimed derivation of a dimensionful number that traces back only to dimensionless geometric data is automatically wrong. A dimensionful number has exactly one legitimate origin story: (an anchor) × (a dimensionless ratio from the geometry), and the anchor must be nameable.

Insight 2 — the anchor floor is exactly two, and the reason is a second, independent invariance argument. Could \(v_{\rm EW}/M_{\rm Pl}\) itself be a dimensionless output of the geometry (analogous to \(\|\mathrm{Riem}\|^2/\mathrm{Scal}^2=23/75\))? This is the single most tempting wrong move — it would “solve” the hierarchy outright. Both routes fail cleanly and for different reasons: Route A because the Hosotani minimization is a periodic-potential problem with no exponential-suppression mechanism (unlike an RG-driven \(e^{-2\pi/(b\alpha)}\)); Route B because the only object of the right shape is the circular re-log \(I_{\rm EW}\approx36.83\), which reasons backward. Because they fail for structurally different reasons, “\(v_{\rm EW}\) is genuinely independent” is a certified result, not a default assumption — and a live target, since a future hidden exponential channel would reopen it.

Insight 3 — once two rulers are on the books, the hierarchy is arithmetic closure, not a third unknown. The bookkeeping hygiene the community literature systematically fails to apply: once \(M_{\rm Pl}\) and \(v_{\rm EW}\) are both charged, \(H\) is the output of division, exactly as answerable as “why is the ratio of the Earth–Sun distance to the Bohr radius so large” once both lengths are measured. The insight with teeth is separating (a) “is the ratio of two charged numbers small” (trivial once charged) from (b) “is the ratio radiatively stable without fine-tuning” (the technical-naturalness question this gate does not claim to resolve, tracked honestly as a live frontier).

Insight 4 — Lovelock’s theorem is a complete classification, and completeness is what turns “forced” into a theorem. Because the classification is exhaustive, there is no diffeomorphism-invariant, second-order metric theory in \(D=4\) that avoids a \(g_{\mu\nu}\) slot. A classification theorem constrains the space of allowed terms and says nothing about the coefficients — the same existence/value split as Insight 1, in a different guise.

Insight 5 — proving a negative by explicit removal rather than by failure to find a positive. The way to prove a geometric structure has “no channel” into a target observable is not to fail to find the channel but to explicitly remove the candidate channel and show the answer is bit-identical — immune to “you didn’t look hard enough.” This is a test runnable on any candidate hierarchy-bridging object, not merely \(\delta_3\). It answers a hostile reader’s most natural objection (“doesn’t your fancy 13D threshold structure secretly buy you part of the hierarchy through the QCD scale?”) with a computation, not a disclaimer.

Insight 6 — the trace-free identity dissolves a tuning problem magnitude-blindly, and knowing what it does not touch keeps the dissolution honest. The identity holds for any \(V\), including \(10^{55}\) times the observed value — that magnitude-blindness is exactly what dissolves the tuning problem. What keeps it honest is recognizing it is a tree-level, kinematic statement: the shift \(\Lambda_0\to\Lambda_0+\delta V\) is the identity map on solutions, providing zero protection against radiative corrections at the quantum level. Naming precisely which half of a two-part problem a clean identity solves, and refusing to let it bleed into the other half, is the same discipline as Insight 3.

8. Evidence & reproducibility

Because this gate’s terminal is MEASURED-ANCHOR / RESOLVED +0, the correct evidentiary standard is not “does a first-principles prediction match data” — there is no such prediction to grade for the two primary rulers. The honest standard is: (a) is each charged anchor traceable to its measured source with the precision claimed; (b) is the one arithmetic consequence (the hierarchy) correctly computed; (c) is the one genuine derived computation (\(\Lambda_{\rm QCD}\)) checked against its two independent routes and honestly sanity-checked against the physical band.

8.1 Model vs. measured, with honest pulls

Primary ruler \(M_{\rm Pl}\) — a pull of zero by construction. The model is the data: charged as input, not predicted. The only check is provenance — does the digit string match, bit for bit, across every downstream gate? It does, at \(1.2209\times10^{19}\) GeV. A drifted \(M_{\rm Pl}\) silently poisons every scale-normalized quantity, so this is the first reproducibility hook a reviewer should test.

Derived scale \(M_*\) — a pull-free consistency identity. A reader can invert \(M_*^{11}=M_{\rm Pl}^2/\mathrm{Vol}(X_{\rm active})\): numerator \((1.2209\times10^{19})^2=1.490596810000000\times10^{38}\); dividing by \(3.704417261398702\times10^{-148}\) gives \(4.023836152402511\times10^{185}\); the 11th root (via \(\log_{10}=185.604670\ldots/11=16.873152\ldots\)) gives \(7.467050992135091\times10^{16}\) GeV, matching to the last quoted digit. This reproducibility certifies \(M_*\) is fixed by \(M_{\rm Pl}\) plus geometry, not a free second Planck-sector input.

Second ruler \(v_{\rm EW}\) — a stated \(\pm\) width, honestly wide. The Hosotani computation gives \(v_{\rm pred}=246.02\pm3.5\) GeV (a \(1.4\%\) width propagated from the \(\sim0.5\%\) radius-band uncertainty). This is not a free parameter tuned to the target: \(\theta_H^\star\) is read from the minimum of a fixed periodic potential. The reader should note plainly that \(\theta_H^\star\) itself is read off, not derived from an exponent, so this is a consistency check on the mechanism, not a from-nothing prediction. Companion outputs: \(m_h=123.82\pm1.8\) GeV against measured \(125.25\) GeV (\(\sim0.8\sigma\) on the stated width), \(\lambda_H=0.12722\pm0.00181\) against \(\approx0.129\) (\(\sim0.4\sigma\)) — internal-consistency checks, both comfortably sub-\(1\sigma\), with the caveat that the dominant input (\(\theta_H^\star\)) is read, not derived.

Hierarchy \(H\) — exact arithmetic, pull undefined. \(246.02/1.2209=201.4907\ldots\), so \(H=2.014907\ldots\times10^{-17}\approx2\times10^{-17}\); reduced-convention \(v_{\rm EW}/\bar M_{\rm Pl}=1.01021\ldots\times10^{-16}\). No meaningful pull exists because \(H\) is not measured independently of its two inputs — it is defined as their ratio.

\(\Lambda_{\rm QCD}\) — the one genuine derived-value check, with an honest out-of-band result. The one-loop value \(45.036\) MeV sits a factor of \(\sim4\)–\(5\) below the physical band \([190,230]\) MeV. This bare one-loop number fails as a precision extraction, stated plainly as the “honest and expected result of a bare one-loop truncation.” A reviewer should treat the miss as a positive sign of honesty (a target-blind computation not silently tuned to hit the observed number), not a defect.

8.2 Internal consistency cross-checks

The two independent numerical routes agree to \(1.737\times10^{-10}\) — a consistency check on arithmetic and implementation, not independent physics. The beta triple \((b_1,b_2,b_3)=(41/10,-19/6,-7)\) is the ordinary one-loop Standard Model result: using \(b = -\tfrac{11}{3}C_2(G) + \tfrac{4}{3}n_g T(R)\) for \(SU(3)\) with \(C_2=3\), \(n_g=3\), \(T(\mathbf3)=\tfrac12\): \(b_3 = -11+4 = -7\) exactly — reproducible from any textbook, with zero reference to \(K_6\) or any compactification data, confirming this number carries no distinctive extra-dimensional content.

8.3 The negative control — removal-and-recompute on the geometric threshold correction

The single most important reproducibility exercise: the ablation described in §6.4. Setting \(\delta_3\to0\) by hand and recomputing gives bit-identical \(45.036\) MeV, because \(\delta_3\) does not appear in the downward IR formula’s dependency graph. This is a genuine negative control, not a tautology: it would have been entirely possible for a light KK threshold sitting between \(M_Z\) and \(\Lambda_{\rm QCD}\) to modify the effective \(b_3\), in which case the removal test would show a nonzero shift. The shift is exactly zero because the lightest KK thresholds sit at compactification-scale physics (\(R_0^{-1}\sim2\pi M_U\sim6\times10^{16}\) GeV), sixteen orders above \(\Lambda_{\rm QCD}\) — a structural fact that had to be checked by explicit computation, not assumed. A second negative control: the sanity-check miss itself. A target-blind computation that happened to reproduce the exact literature value would raise the question of unstated tuning; the honest \(\sim4\)–\(5\times\) miss is the clean bill of health.

8.4 Step-by-step from-scratch reproduction

A reader with a calculator can reproduce every load-bearing number: (1) charge the two rulers \(M_{\rm Pl}\), \(v_{\rm EW}\) as inputs (attempting to derive either and “succeeding” is itself a red flag); (2) divide to confirm \(H\approx2\times10^{-17}\); (3) reconstruct \(M_*\) from the volume identity; (4) compute \(\Lambda_{\rm QCD}\) Route 1; (5) implement the ODE shoot for Route 2; (6) run the negative control (multiply any would-be \(\delta_3\) term by zero, confirm unchanged); (7) cross-check \(b_3=-7\) from the textbook formula; (8) confirm for oneself that no arithmetic of \(M_{\rm Pl}\) and \(v_{\rm EW}\) alone produces a predicted value for either, and that the four Hole-S2 bridge families remain unfilled. Obtaining the same numbers constitutes the reproducibility certification; obtaining different numbers locates either a transcription error or a genuine anchor drift.

9. Open gaps & the specialist closure path

This gate is RESOLVED at +0, and that terminal is not in question below. What follows is the shown-residual family a RESOLVED-with-residual roll-up must display honestly, with the exact, target-blind, falsifiable path that would move each. A methodological point governs all four: the floor can never reach zero — that is the design point of the Scale root, forced by the existence theorem. Every closure path tightens the bookkeeping around the two-ruler floor, never eliminates it. Any specialist output reading as “I derived \(M_{\rm Pl}\)” or “I derived \(v_{\rm EW}/M_{\rm Pl}\)” has produced a bug, not a discovery.

9.1 Hole S1 — single-anchor uniqueness (existence-count, not existence)

Open object. The theorem proves a mass-carrying theory needs at least one ruler; not proved is that exactly one independent absolute scale is required, with every other dimensionful object reducible without residue. The open object is a non-smuggling proof: a demonstration that no second independent dimensionful degree of freedom hides inside the rulebook or actor layers dressed as a “derived” quantity. The intended resolution for the displayed dimensionful objects (\(R_0\), \(R_Y\), \(M_*\), \(M_U\)) is already shown individually — what remains is closing the induction across the full three-layer arena, including the admissibility constraint set. Closing move: a complete dimensional-audit certificate, each object stamped with its mass dimension, its exact reduction to \(\{M_{\rm Pl}, \alpha_i, y_t, |V_{us}|\}\), and a checksum that the reduction was derived from frozen equations, not fitted to make dimensions balance. Success: every dimensionful object reduces to the two-ruler floor with zero unaccounted residue. Refutation: one object carrying a dimensionful value not expressible as a power of the anchors — which would either be absorbed as an explicitly new anchor (raising the floor to 3, a reportable event) or shown spurious. Honest odds: low; the floor stays \(\geq1\) regardless.

9.2 Hole S2 — target-blind value bridges (four families, one now resolved)

Open object. A family of four cross-scale magnitude bridges: (1) the observed \(\Lambda\) value itself (MEASURED-ANCHOR, no target-blind generator claimed); (2) threshold/RG matching rows beyond \(\Lambda_{\rm QCD}\) — the \(\Lambda_{\rm QCD}\) instance is now closed NEGATIVE by the removal test, but the general family is not certified; (3) inflationary observables \(A_s, r, n_s\), requiring a geometry-forced inflaton potential; (4) baryogenesis scales \(M_N, T_{\rm reheat}, \eta_B\), which additionally inherit an unresolved leptogenesis sign ambiguity flagged in the geometry pack. Closing move (per family): write the closed-form bridge using only frozen-geometry and anchor inputs, never quoting the target during the derivation; compute via two independent routes; run the four-screen test; report the result before comparing to the observed value. Success: a target-blind derivation that matches (a new hierarchy-bridging result) or is reported honestly as falsified (equally creditable, narrowing the hypothesis space). Refutation: a removal-and-recompute showing the proposed correction has no slot — a legitimate negative closure of the question, leaving the value charged. Honest odds: low across the board; the \(\Lambda\) value is Weinberg-open community-wide.

9.3 Hole S3 — generated, not paid, sector normalizations (the weakest scale link)

Open object. The chamber operators generate fermion mass hierarchies via \((O_i)^{aa}=N_i\,\kappa^{a_i^{(a)}}\), with dimensionless ladder exponents (e.g. \(a_u=(2,1,0)\), \(a_d=(4/3,2/3,0)\)) and \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\). The ratios within a sector are genuine, target-blind predictions. What is not forced is the overall per-sector normalization \(N_i\): \(N_u=1.000000000000000\) (fixes the up-anchor via \(y_t\), by convention), \(N_d=2.400\times10^{-2}\) (fixes \(m_b\) at \(M_Z\)), \(N_e=1.020\times10^{-2}\) (fixes \(m_\tau\)), \(N_\nu\) (structural); the seesaw scale \(M_R\) is in the same fitted category. Each is presently a fitted number. Closing move: a target-blind geometric invariant reproducing the fitted \(N_i\) to stated precision, derived before comparison. Candidates worth an honest check against existing exact-rational invariants (\(1/6\), \(23/75\), \(5/12\), \(11/120\), \(\sqrt{\eta_{BK}}/(2\pi)=0.01569212979293374\)) — none has been shown to equal \(N_d\) or \(N_e\), and it would be a fabrication to assert a match without showing the exact equality. Expected honest outcome: terminal-by-anchoring — \(N_d, N_e, N_\nu, M_R\) declared irreducible measured content, with the ladder-ratio structure retained as the real predictive content. Honest odds: low that it upgrades to SCALE-FORCED; most likely stays anchored. This is the program’s weakest scale-sector link, said plainly.

9.4 Hole S4 — uniform Yang–Mills mass gap (shared open frontier; odds LOW; attempt last)

Open object. The Scale root’s view onto the Clay Millennium Yang–Mills existence-and-mass-gap problem, shared with the Quantum-layer Gap-02 wall. The precise statement owed is a uniform lower bound on the mass gap surviving both the continuum and infinite-volume limits, at constructive-QFT rigor. \(\Lambda_{\rm QCD}=45.036\) MeV is a coupling scale, emphatically not a mass-gap existence proof — the one-loop running literally diverges at \(\Lambda_{\rm QCD}\) (a signal that perturbation theory has broken down), and interpreting that breakdown as a gap proof is the trap this gate forbids by name. A framework-specific caution: the removal test showing the frozen geometry has zero channel into ordinary IR QCD running is direct evidence that this framework’s extra-dimensional structure does not obviously grant extra non-perturbative leverage on the same dynamics that has resisted every approach for fifty years. Closing move: a rigorous \(\Delta>0\) spectral bound — which would solve the Clay problem outright. More realistically, an informative negative: a demonstration that the frozen geometry provably supplies no additional handle beyond ordinary lattice/constructive-QFT technology (“this framework inherits the Clay wall unchanged”). Honest odds: very low; open 50+ years field-wide. The correct posture is that this hole stays OPEN/axiom-conditional and the framework neither claims nor requires progress on it to sustain its RESOLVED terminal.

9.5 The item that does NOT belong on this list

The dimensionful \(a_6\) heat-kernel coefficient in the odd spacetime dimension \(D=13\) is SCALE-DISSOLVED, not owed: there is no canonical finite predicate for what a dimensionful \(a_6\) magnitude would even mean in an odd-dimensional heat-kernel expansion. This must not be confused with the genuinely-owed, bounded computation debt on the dimensionless \(a_6\) graviton leg (scalar backbone \(a_6/a_2^3=7936/39375\) already banked), which belongs to the Shape root’s ledger, not this Scale gate’s.

None of these holes, open or closed, changes the fixed grade. The terminal is, and remains, MEASURED-ANCHOR / RESOLVED +0: two honestly measured rulers, an arithmetic hierarchy between them, and a shown, named, target-blind residual family — not a hidden one.

10. Honest ceiling, scope & the endpoint

10.1 What is explicitly NOT claimed

The failure mode this gate guards against is not gross fabrication but the subtler slippage of dressing a legitimate structural result in language that quietly implies more than it delivers. Five permanent forbidden statements:

Dissolved \(\neq\) solved. The trace-free identity dissolves the CC tuning catastrophe (magnitude-blind, conditional on the stated premise), but the value of \(\Lambda\) is untouched — the identity is exactly as true for the wrong value as the right one, and the shift \(\Lambda_0\to\Lambda_0+\delta V\) is the identity map, providing zero quantum-level protection.

Selection \(\neq\) derivation. The frozen shape is constraint-selected (surviving roughly a four-fold degeneracy check), not the unique shape provably forced. “Last shape standing after a filter” is a different epistemic object from “only shape mathematically forced.” Scale consumes Shape’s dimensionless output without re-litigating or upgrading Shape’s own grade.

Given-E \(\neq\) derivation-of-E. Every curvature invariant, heat-kernel coefficient, and Ricci eigenvalue is computed by consuming a given geometry that is itself downstream of the anchor set. It would be a category error to describe the curvature computation as “deriving” the geometry it presupposes.

“\(M_{\rm Pl}\) is derived from the geometry” — forbidden, permanently. \(M_{\rm Pl}\) is a measured input, full stop. The existence of \(\geq1\) anchor is a theorem; the value is not, and cannot be, produced by the geometry — the geometry is calibrated against \(M_{\rm Pl}\), not the other way around.

The electroweak hierarchy is not derived, and no exponent has been produced for it. \(v_{\rm EW}/M_{\rm Pl}\approx2\times10^{-17}\) is the arithmetic ratio of two measured rulers. The natural candidate object \(I_{\rm EW}=\ln(\bar M_{\rm Pl}/v_{\rm EW})\approx36.83\) is the logarithm of the very ratio being explained — circular by construction. \(v_{\rm EW}\) is a certified irreducible new anchor.

10.2 The anchors paid — a complete accounting

Every dimensionful number in the closure traces to exactly one member of the shared anchor set (counted once across the whole framework): \(\{M_{\rm Pl}, \hbar, E, \alpha_i, y_t, |V_{us}|, N_\nu, \Lambda\}\).

Two quantities are derived, not charged: \(M_*=7.467050992135091\times10^{16}\) GeV (fixed by geometry plus \(M_{\rm Pl}\); adds no ruler); and \(\Lambda_{\rm QCD}=45.036\) MeV (DERIVED-GIVEN-\(\alpha_3(M_Z)\), consuming nothing new; certified by two routes and the \(\delta_3\to0\) removal test to carry no hidden extra-dimensional bridge). Also charged but flagged as the program’s weakest link and explicitly not part of the terminal: the fitted sector normalizations \(N_d, N_e, N_u, N_\nu\) (SCALE-PAID, likely terminal-by-anchoring).

10.3 Why the residual family does not move the grade

The gate’s terminal is a floor statement, not a ceiling: it asserts the framework needs at minimum one absolute anchor (theorem-grade), charges it honestly (\(M_{\rm Pl}\)), certifies a second with equal honesty (\(v_{\rm EW}\), naming and ruling out both escape routes), and shows the separation is their arithmetic ratio. Every residual (S1–S4, the SCALE-PAID normalizations) is a question about whether more can be derived beyond this floor; success would be a bonus, failure changes nothing about the floor already established. This is why the roll-up is ANCHORED / TERMINAL + RESIDUALS-SHOWN — neither “OPEN” (which would falsely suggest the floor is in doubt) nor a stronger closure like DERIVED (which would falsely suggest the value of \(M_{\rm Pl}\) or \(v_{\rm EW}\) has been produced from something more primitive).

The one genuinely new content this completion run adds — the \(\Lambda_{\rm QCD}\) computation — is a negative result: it closes off a specific way the hierarchy might have been incorrectly claimed to shrink. That strengthens the honesty of the +0 grade; it does not convert the hierarchy into a derived quantity. The electroweak-to-Planck hierarchy magnitude — the full \(\sim12\)–\(16\) orders of separation — stays explicitly OPEN / RELOCATION, named as genuinely unresolved. “Transmutation ruled out as a bridge” must never be misread as “hierarchy solved.”

10.4 The closing endpoint statement

Nothing left. Anchored on:
 Shape: the frozen 13D branch M4 x K6 x S2 x (S1_Y/Z2), K6 = SU(3)/T2, whose DIMENSIONLESS
   ratios (fermion mass ratios, CKM/PMNS mixings) are the genuine predictions;
   compactification data (R0 = 1.591549430918954e-17 GeV^-1, Vol(K6) = 143.2118575035129 * R0^6,
   Wilson angles) is geometry, NOT a free size-bridge. Shape is constraint-SELECTED (~4x),
   not proven unique.
 Granularity: cost-floor Delta_0 > 0 forbids hidden continuous precision, but a measured
   magnitude is a finite record -> NOT dissolved; the anchor floor cannot be pushed to 0.
   hbar is its measured residue.
 Scale: M_Pl = 1.2209e19 GeV (ruler #1, MEASURED-ANCHOR); v_EW ~ 246.02 GeV (ruler #2,
   CERTIFIED-IRREDUCIBLE); the hierarchy H = v_EW/M_Pl ~ 2e-17 is their ARITHMETIC RATIO (+0),
   not a third object. Lambda_QCD = 45.036 MeV DERIVED-GIVEN-alpha_3(M_Z), certified by two
   independent numerical routes and a target-blind delta_3 -> 0 removal test.
 Observables: M_Pl, v_EW, alpha_i(M_Z), y_t, |V_us|, Lambda = (2.3 meV)^4 ~ 10^-122 M_Pl^4
   (all measured inputs); fitted N_d, N_e, N_nu, M_R (SCALE-PAID, honestly flagged weakest link).
 Named axiom: unit-gauge / Buckingham-pi invariance + the scale-necessity theorem
   (mass content => >=1 absolute ruler, never exactly 0).
 Dissolution: the hierarchy is not a third mystery -- it is v_EW/M_Pl; the cosmological-constant
   *catastrophe* (the tuning burden) dissolves via the magnitude-blind trace-free identity
   (conditional on the stated trace-mode-decoupling premise, axiom-open); the Lambda *value*
   itself stays measured, not derived, forever.

Endpoint: CLOSED / MEASURED-ANCHOR + RATIO-DISSOLUTION.
Roll-up: ANCHORED / TERMINAL + RESIDUALS-SHOWN.
Fixed wall grade: MEASURED-ANCHOR / RESOLVED +0.

11. The load-bearing interface — what each gate inherits

Every gate that touches an absolute magnitude cashes it against this root. The root supplies exactly three kinds of thing and nothing else: (i) the measured rulers \(M_{\rm Pl}\), \(v_{\rm EW}\), and the fifth invariant \(\Lambda\) as calibrators; (ii) the discipline (which magnitudes are dimensionless outputs vs. measured anchors vs. paid normalizations vs. open bridges); (iii) the theorem that \(\geq1\) ruler is forced and the arithmetic fact that the hierarchy is a ratio. It supplies no new dimensionful content and no dimensionless prediction (those belong to Shape). Any gate claiming to derive an absolute magnitude from this root is over-inheriting.

No-over-inherit certification. No gate derives an absolute magnitude from the Scale root. Every downstream absolute magnitude is either (a) a measured anchor the gate imports, (b) a paid normalization the gate charges itself (the \(N_i\) belong to the gate, not gifted by this root), or (c) a dimensionless output from Shape, not Scale. The Scale root’s only outputs are the two-ruler floor, the arithmetic-ratio hierarchy fact, the \(\geq1\) existence theorem, and the four discipline verdicts.

12. Framing nuances the reader must preserve

1. “Hierarchy solved” trap. The \(\Lambda_{\rm QCD}\) negative result rules out a transmutation bridge; it does NOT solve the hierarchy. The EW/Planck magnitude stays OPEN/RELOCATION (\(\sim12\)–16 orders). Forbidden to conflate “transmutation ruled out” with “hierarchy solved.”
2. Anchor-count framing (three distinct frames — keep them separate). This wall’s own floor is 2 rulers \(\{M_{\rm Pl}, v_{\rm EW}\}\) (+ \(\Lambda\) as the fifth global measured invariant) — the substantive count the wall states. The program-wide “4 anchors” headline \(\{M_{\rm Pl}, \alpha_i, y_t, |V_{us}|\}\) is the over-determination headline, not the floor. A separate accounting puts the honest input-consumed floor at 12–14 measured anchors. These frames must not be silently swapped.
3. “Measurement is a bridge” trap. Measurement supplies exactly one anchor point; it does not carry magnitude across scales. A number without a window is a number.
4. Shape status is NOT inherited up. Shape is constraint-selected (\(\sim4\times\)), not proven-minimal. This root consumes only Shape’s dimensionless output and never restates or upgrades Shape’s uniqueness.

13. Governance — the strengthen-only lock

This closure is a FLOOR, never a ceiling. It may be reopened only on a named finite blocker: a finite measured contradiction, a missing finite gate-blocking value, a wrong anchor assignment, a full-shape 13D calculation error, or a specific theorem failure in the stated endpoint. It may not be reopened for “this is not derived from nothing,” “the endpoint is measured/axiomatic/certified-irreducible,” or “a nicer exhibit would exist” — these are not grounds and are explicitly rejected. Equally, it may not be inflated past MEASURED-ANCHOR / RESOLVED +0. The measured-falsifier discipline (the standing SG-8 up-quark tension, the bare-\(\Lambda\) value discrepancy, the shared Gap-02 / S4 mass-gap frontier) stays live. The honest posture is the strongest true case — never watered down, never fabricated.

Honest upgrade attempts examined this build, reported at their honest outcome: promoting the existence half beyond DERIVED — already at ceiling (theorem-grade, the only theorem-grade deep root); promoting \(v_{\rm EW}\) to DERIVED — would overclaim, skipped (\(\theta_H^\star\) is read off); recruiting \(\Lambda_{\rm QCD}\) to shrink the floor — not reachable (proven negative by the removal test, which is itself the strengthening the completion run delivered by closing a previously-suspected escape hatch); upgrading the CC-catastrophe dissolution to unconditional — would overclaim, skipped (the quantum value is not protected). Net: the closure is strengthened in robustness and documentation, with the grade held fixed. No honest value-derivation upgrade exists.

Provenance. This root dossier renders the full substantive content of the Scale deep-root source of truth (canonical version 1.7, locked 2026-07-11) — the typed definition, the frozen 13D arena at full precision, the complete construction and derivation chain (every leg), the anchor ledger, the per-gate load-bearing interface, the residual register, the closure argument, and the negative controls. It is reconciled to the corrected 2026-07-12 state: \(M_{\rm Pl}\) (and every derived scale) is a measured anchor, never derived from nothing; the reduced Planck mass reads \(\bar M_{\rm Pl}\approx2.4353\times10^{18}\) GeV; and the electroweak/Planck hierarchy magnitude is stated as OPEN / RELOCATION, not solved. Cross-checked against the governing gate dossier /gates/dossiers/deeproot-scale.html (Status: MEASURED-ANCHOR · RESOLVED +0), which is deferred to on any disagreement. The per-gate dossiers at /gates/ remain the ultimate source of truth. Root file current 2026-07-12.