The Scale — absolute-magnitude discipline

Scale is the honest ruler discipline of this framework: only dimensionless ratios are claimed as derived, and every absolute magnitude is charged openly to a measured ruler — and that discipline is a theorem, not a style choice. Physics does not change when you rescale your units, so a naked dimensionful number carries no invariant content. By unit-gauge (Buckingham-π) invariance — proven below, not asserted — the only things a theory can derive outright are dimensionless ratios, and the only honest way to state a magnitude is relative to a named, measured anchor. Every theory that has ever stated a magnitude obeys this theorem or hides it. This framework prints it: the genuine predictions cluster in the dimensionless column, every GeV on the site traces to a declared ruler, and this page is the public ledger of which is which.

Read this first — a formula is not a magnitude. The single most common way to over-read this theory is to see the geometry produce an expression and conclude that the number it evaluates to is derived. It usually is not. A dimensionful number is Scale-derived only when its units, normalization, scheme, energy window, cross-scale bridge, and absolute anchor are all fixed, paid, or explicitly dissolved. Scale is the discipline that audits that chain. Where the chain is incomplete, the most a formula can honestly claim is a dimensionless ratio, a scheme-dependent coefficient, a consistency check, or a measured anchor carried forward — never a bare predicted magnitude.

Thesis

Scale is the discipline that tells us which numerical claims are meaningful: dimensionless first; every dimensionful value traced to an anchor, a scheme object, a generated scale, or an explicit dissolution.

Scale is one of the program's seven deep roots (it is R5 — see The Seven Deep Roots). On this page "the Scale root" is used in its complete form: the absolute-scale anchor $M_{\rm Pl}$ at full precision, applied over the complete three-layer Shape, together with the full rulebook for ratios, dimensions, RG windows, scheme objects, normalization, stability, and observable-scale readout. A closure that sets a dimensionful magnitude from a truncated shape, a rounded anchor, or an unstated scheme is using an incomplete root and may fail — or appear to succeed — for that reason alone, with nothing right or wrong in the physics.

Honesty guard

A dimensionful number is not automatically derived by Shape. Scale decides whether a number is an observable, an exact ratio, a fitted normalization, a scheme-dependent coefficient, a consistency check, or an ill-posed predicate.

This guard is the spine of the whole page. It is a completeness upgrade of the root, not an anchor-elimination: the program still bottoms on its anchors. Shape bottoms on the frozen branch identity; Granularity bottoms on the uniform positive cost-floor posit; Scale bottoms on the absolute-scale anchor $M_{\rm Pl}$, whose value is measured, not derived. Floor $\geq 1$, forever. The discipline running through every line is the program's: anchored is not derived · selected is not forced · dissolved is not solved. This is a frozen Theory-of-Everything candidate under honest audit. The gate board stands at 33 resolved at +0 · 0 anchored at +1 · 0 open (the live scoreboard and its per-gate dossiers are the closure-of-record) — and, on the separate, permanent axis this page governs, 0 of 33 gates are physics-closed, with no status changes made here. Those two statements never collide, and this page is the theorem for why: no theory — this one or any other — can state a dimensionful prediction from zero measured rulers. What distinguishes this framework is not escaping that theorem; it is naming its rulers and charging them openly, on every row.

What Scale is — two jobs, never collapsed

Scale has two distinct jobs, and conflating them is the commonest scale error in the entire program.

1. Meaning. Is this a valid thing to ask for, at this dimension, scheme, and energy window? Some magnitudes are well-posed observables; some are scheme-dependent coefficients with no invariant value; some are ill-posed predicates that have no canonical finite answer at all.
2. Source. If it is valid, what supplies its magnitude? A measured anchor, a generated scale, a paid scheme object, a cross-scale bridge — or nothing yet (open).

A value can be meaningful but fitted. A value can be stable but not derived. A ratio can be derived while its absolute magnitude stays open. None of these collapse into "derived." The full definition the page operates under is:

Scale = absolute-magnitude discipline: one accepted / derived / paid absolute scale anchor · dimensionless-ratio discipline · unit / scheme / normalization policy · RG / window / threshold bridge · stability-vs-value split · explicit dissolution for invalid scale predicates.

The compressed root statement

Scale is the absolute-magnitude discipline of the theory: one accepted scale anchor ($M_{\rm Pl}$), plus explicit rules for ratios, dimensions, RG windows, scheme objects, normalization, stability, and observable-scale readout. The existence of at least one absolute scale is proven — a theorem forced by unit-gauge invariance (see the unit-gauge layer below); its numerical value ($M_{\rm Pl}$) is supplied by measurement. These two parts — existence and value — must never be merged: arguing that a scale is needed is not deriving the number.

Scale anchor convention — the actual object

The rest of this page argues about the scale anchor symbolically. Here is the anchor itself, stated as an object, with no part left implicit.

Declared anchor. $M_{\rm Pl}$ — the ordinary Planck mass, $M_{\rm Pl}=(\hbar c/G_N)^{1/2}$. This is not the reduced Planck mass $\bar M_{\rm Pl}=M_{\rm Pl}/\sqrt{8\pi}\approx 2.4357\times10^{18}$ GeV.

Value. $$M_{\rm Pl}=1.220900000000000\times10^{19}\ \text{GeV}.$$ (Source precision is 4 significant figures, $1.2209\times10^{19}$ GeV, PDG-derived; the trailing zeros record the declared anchor field, not measured digits — see Precision-cost discipline below.)

Units / convention. Natural units $\hbar=c=1$, GeV convention. The geometry that this anchor cashes (e.g. $M_{\rm Pl}^2=M_*^{n+2}V_K$) is convention-fixed: a reviewer preferring the reduced convention must re-read every $M_{\rm Pl}$ as $\bar M_{\rm Pl}\sqrt{8\pi}$ and rescale the left-hand side by $1/(8\pi)$ — the geometry on the right-hand side is unchanged.

Status — two halves, never merged. - Existence of at least one absolute dimensionful scale is theorem-grade (forced by unit-rescaling invariance; see the unit-gauge layer below). - The numerical value of the selected anchor is a measured input — status MEASURED, never DERIVED.

Forbidden compression. Do not merge "a scale must exist" (theorem) with "this exact value was derived" (it is measured). The first is proven; the second is read from experiment and charged as input.

For the $\geq 16$-significant-figure value, the full convention statement (ordinary vs reduced), and the volume relation it pins, see GUT Appendix A1 (full precision) (PDF). The same value and convention appear on the Shape page row A.2 (manifest hash df5976a365c3).

The unit-gauge layer — why Scale is a theorem, not just a rulebook

Scale is not a house style for reporting numbers. It is forced by an invariance: physics is unchanged under a rescaling of units (changing the GeV, the second, the kilogram). Treat that rescaling as a gauge symmetry — a unit gauge — and the rules above stop being conventions and become theorems.

Unit-gauge invariance. Under a change of units, every dimensionful number is multiplied by the corresponding conversion factor. A naked dimensionful number therefore carries no invariant content: it is gauge-dependent. Only two kinds of statement are unit-gauge invariant: (i) a dimensionless ratio (all unit factors cancel), or (ii) a dimensionful value expressed relative to an accepted anchor (the anchor transforms the same way, so the ratio-to-anchor is invariant).

This is the Buckingham-$\pi$ content stated as a closure rule. Buckingham-$\pi$ says any physical relation among $n$ quantities with $k$ independent dimensions reduces to a relation among $n-k$ dimensionless $\pi$-groups. The three consequences are exactly the page's discipline, now as a theorem rather than a preference:

1. A dimensionless invariant can be derived directly. It is a $\pi$-group; it lives in the unit-gauge-invariant sector; it needs no anchor and no bridge. (This is why the safe predictions cluster in the ratio column — SCL-A.)
2. A dimensionful magnitude requires an anchor plus a bridge plus a scheme/window. It is gauge-dependent until tied to the accepted anchor ($M_{\rm Pl}$ or a declared measured scale); the bridge carries it across scales (SCL-J); the scheme/window fixes the scheme-dependent part (SCL-E). All three, or it is not a magnitude claim.
3. No anchor → no magnitude claim, ever. Buckingham-$\pi$ forces $\geq 1$ dimensionful anchor before any dimensionful prediction can exist. This is the floor of the root (SCL-I): the anchor count is $\geq 1$, forever; it is named, not eliminated.

So the existence half of the anchor convention is not asserted — it is forced: at least one absolute dimensionful anchor is required to make any dimensionful statement unit-gauge invariant. What is measured, not forced, is which value that anchor takes. Existence: theorem. Value: MEASURED. The unit gauge is the reason those two halves can never be merged.

What the theorem buys — the evidence, framed honestly

The ruler discipline is not defensive bookkeeping. Applied without flinching, it delivers three results: it retires a decades-old "problem," it states exactly what is proven versus what is measured, and it locates the framework's genuine predictions precisely where the theorem says predictions can live.

1 — The hierarchy "problem" is the ratio of two measured rulers

Gravity's notorious weakness — why the Planck scale towers so far above the electroweak scale — has driven decades of exotic proposals, none confirmed. The ruler discipline dissolves the demand. The framework rests on a ratified two-ruler floor, $M_{\rm Pl}$ and $v_{\rm EW}$, both ordinary measured anchors exactly as the meter and the second are: the hierarchy $H = v_{\rm EW}/M_{\rm Pl}$ is then the arithmetic ratio of two already-known measurements — a number you read off the anchor table, not a third input the theory owes. Dimensional bookkeeping requires at least one independent scale and can never be satisfied by zero, so the standing demand to "derive or explain" the ratio of two rulers was a false dichotomy from the start. Proven: Deep Root: Scale — the closure dossier (RESOLVED at +0, measured-anchor terminal) · the anchors register. The honest residual, shown openly: this rests on the declared two-ruler floor, not one — anchored is not derived; $M_{\rm Pl}$ and $v_{\rm EW}$ are honest measured inputs, reframed, never conjured.

2 — Existence of a scale: proven. Its value: measured. Never merged

The unit-gauge layer above is a theorem, and it splits the anchor question into two halves with different statuses. That at least one absolute scale must exist is proven — no dimensionful statement is unit-gauge invariant without one. Which value that scale takes ($M_{\rm Pl}$) is measured — read from experiment, charged as input, never derived. The framework keeps the two halves on separate lines of every ledger, because merging them is precisely the move that lets a theory launder a measured number into a "prediction." The refusal to merge is what the rest of this page enforces, constraint by constraint.

3 — The genuine predictions cluster in the dimensionless column — exactly where the theorem says they can

The anchors are declared; the answers are not among them. From two measured flavor anchors — $y_t$ and $|V_{us}|$ — the geometry returns nineteen-plus flavor observables, with six to eight more following from $\{M_{\rm Pl},\ \alpha_i\}$: a handful in, twenty-plus out (strictly counted across the whole construction, ~4× more outputs than effective inputs). The sharpest of them are dimensionless, exactly as the theorem demands: $|V_{cb}|$ lands at 0.005σ, the Jarlskog CP invariant at 0.21σ, the electron, muon, and tau masses at 0.01–0.07σ with zero lepton inputs. And when the framework's hardest number — the up-quark mass — missed at ~4.4σ, it published the miss on its own front page, then resolved it to +0.058σ with a dimensionless factor, $1/\sqrt{6} = 1/\sqrt{|S_3|}$, fixed purely by the order of the flavor shape's six-element Weyl symmetry group and derived without numerical access to the measured value. The rescue lived in the ratio column too. Proven: the gate scoreboard · the SG-8 dossier — flavor from one constant and one angle. The honest boundary, kept deliberately: anchors in, ratios out — the ratio column is where every claimed prediction lives, and the absolute magnitudes beside them are charged, openly, to the rulers.

How Scale meets every attack layer

Scale is not a single check applied at the end. It threads through every layer of an attack on a gate or wall. The table below states what Scale does at each layer and the precision it demands there.

The same number can be honest at one layer and dishonest at another. A coefficient can be a perfectly good Layer-1 consistency check and still be an illegal Layer-5 gate-closing prediction. Scale forces you to say which.

Scale claim categories — every number gets a tag

No number appears on a gate or wall without a tag. The six categories are exhaustive and mutually exclusive.

SCALE-FREE-DERIVED
  A dimensionless invariant or exact ratio derived from object identity.
  Safest category. No anchor needed; units never enter.

SCALE-ANCHORED
  A dimensionful value obtained after an accepted or measured scale
  anchor is supplied. Honest if the anchor is named and the bridge is forced.

SCALE-PAID
  A value that uses a fitted normalization, a measured input, or a
  declared scheme object. Legitimate, but it is an input, not a prediction.

SCALE-CONSISTENCY-COEFFICIENT
  A useful internal check, not an observable and not gate-closing.

SCALE-DISSOLVED
  The requested magnitude is not a valid scheme-independent object.
  The honest answer is that the question was ill-posed, not that it was solved.

SCALE-OPEN
  The bridge, scheme, or anchor is missing. No claim yet.

A central methodological point: SCALE-DISSOLVED is not a success. It records that a magnitude that looked like a prediction was never a well-defined observable. Dissolution removes a false debt; it does not pay a real one. This is the scale analogue of the program's standing rule that dissolved is not solved.

Scale-status transition table — the only legal status transitions

A number does not jump categories for free. Each status change has a required event that must actually be exhibited; without it, the label does not move. These are the only legal transitions:

The fourth row is the one most often abused: dissolving an ill-posed magnitude (e.g. the odd-$D$ $a_6$ number, or the $\Lambda$ catastrophe-half) clears a false debt and is honest — but it does not pay the real debt or close the gate. A dissolution is a terminal that says "this was never a valid thing to ask," not "this is now answered." Status change to closed always requires a real surviving observable, never a dissolution.

The Scale constraint table (SCL-A … SCL-J)

This is the technical backbone. Every scale claim on any gate must be checkable against these ten constraints. SCL-A through SCL-H are the gap-derived constraints; SCL-I (absolute scale anchor) and SCL-J (scale-bridge rule) are the root-completeness additions that make the triad complete for wall attack.

Operational test for any number

For every numerical claim, four questions must have answers:

Is this dimensionless?
If dimensionful, what scale/scheme object gives it units?
Is that object generated, measured, paid, or dissolved?
What energy window and RG/readout rule apply?

If those answers are absent, the number is not Scale-derived — whatever the formula says.

Dimensionless-first doctrine — why ratios close before magnitudes

The safest derived objects in the whole program are dimensionless: ratios, mixing angles, indices, exact rational invariants. They are safe because they never touch the scale anchor and never depend on a unit convention — by the invariance root (R1), only dimensionless quantities are frame- and unit-invariant, so a naked dimensionful number is not even a candidate physical observable until an anchor cashes it.

This is why the genuine predictions of the candidate cluster in the dimensionless column: ratios and mixings (roughly eight per sector), not the overall scales. A fermion mass ratio can be a real output of the flavor map; the absolute sector mass requires a sector-scale policy and is paid. An exact rational heat-kernel check can be derived; the GeV-magnitude that superficially accompanies it can be ill-posed. A mixing angle can be a prediction; the energy at which it is read is set by a window.

The doctrine in one line: derive the ratio, then — and only then, with a named anchor and a named bridge — reach for the magnitude. Most of the time the ratio closes and the magnitude stays SCALE-PAID or SCALE-OPEN. That is not a weakness to hide; it is the honest shape of the result. Over-determination — many derived ratios against a few paid scales — is the actual evidence, and it is destroyed, not strengthened, by dressing paid scales up as predictions. Stated at its real strength: two measured flavor anchors in, nineteen-plus flavor observables out, with six to eight more following from $\{M_{\rm Pl},\ \alpha_i\}$ — a handful in, twenty-plus out. That arithmetic runs the wrong way for a fit, and it lives entirely in the column this page certifies as derivable.

Scale-bridge catalog — what may carry information across scales

SCL-J forbids any cross-scale claim without a named bridge. There are exactly six admissible bridge types. Anything calling itself a cross-scale derivation must be one of these (or be dissolved):

• RG flow. A running quantity carried between scales by its renormalization-group equations, in a stated scheme and to a stated order. Requires SCL-E (window) and SCL-B (scheme terminus).
• Decoupling. Heavy states integrated out at their mass threshold; their effect appears as matching coefficients below. Requires the spectrum to be read from the actual Shape, not assumed.
• Threshold rule. A finite matching correction at a mass threshold, computed from the actual KK / operator spectrum (multiplicities, reps, parities, zero modes), with omitted-mode error bounds. Requires SCL-E and SCL-H.
• Compactification map. The map from higher-dimensional geometry to 4D scales (volumes, radii, Wilson angles). This is Shape data; the map itself is part of the Shape burden unless separately paid.
• FRG map. A functional-renormalization-group flow used where perturbative RG is insufficient, with a declared regulator and truncation.
• Explicit dissolution. The honest terminus when no valid bridge exists: the cross-scale magnitude is declared not to be a scheme-independent object (SCALE-DISSOLVED).

Measurement is not a bridge across scales — it is how the anchor itself is supplied (SCL-I). It sets the one absolute scale; it does not carry a magnitude from one regime to another. Keeping measurement out of the bridge catalog prevents the trap of "measuring" an answer at one scale and calling its appearance at another scale a derivation.

Bridge certificates — every bridge fills the same template

A bridge type named is not a bridge used. Any actual cross-scale step must produce a filled bridge certificate. The template forces the omitted, the paid, and the target-blindness check into the open:

BRIDGE CERTIFICATE
  Bridge type ......... RG flow / decoupling / threshold / compactification / FRG / dissolution
  Domain scale ........ the scale the quantity is carried FROM (value + units)
  Codomain scale ...... the scale it is carried TO (value + units)
  Scheme .............. the scheme object (named; shared-or-not, see SCL-H)
  Window .............. energy window + running policy (SCL-E)
  Inputs .............. every quantity consumed (each tagged measured / paid / derived)
  Paid quantities ..... which inputs are fitted/declared (SCL-F) — the honest cost
  Generated quantities. what the bridge actually produces (and at what category)
  Omitted-mode error .. modes/states dropped, with a BOUND on their contribution
  Uncertainty prop. ... how input uncertainties propagate to the output
  Target-blindness .... can this certificate be written WITHOUT the target value? (yes/no)
  Failure mode ........ the single thing that, if false, voids the bridge

A certificate with a blank "Omitted-mode error", a "no" on target-blindness, or an unbounded failure mode is not a derivation — it is a SCALE-OPEN or SCALE-PAID claim wearing a bridge label.

Worked example — the threshold bridge (decoupling + threshold rule). A threshold-matching correction is a valid bridge only if its certificate is fully filled and all of the following hold:

• the heavy spectrum comes from Shape — the masses being integrated out are read from the actual frozen geometry (KK tower, operator spectrum), not assumed;
• multiplicities, parities, and zero-modes are explicit — every state in the matching sum is enumerated from the rep content, not summarized;
• the matching scale is declared — the scale at which heavy states are integrated out, with the running policy on each side (SCL-E);
• the omitted-mode error is bounded — truncating the tower leaves a quantified residual, not an unstated one;
• the normalization is not chosen to hit the target — the row factor is written before the observed coupling is consulted (target-blindness = yes).

If any one fails, the threshold step is SCALE-PAID at best and target-selected at worst — exactly the W19 verdict. Filling this certificate honestly is what hardens W19 (thresholds), SG-7 (RG matching), SG-6, and Gap-01 (the shared heat-kernel scheme object flows through all four — see the dependency graph below); leaving it blank is what lets a tuned row masquerade as a closure.

What Scale FORCES vs what it only SUPPORTS

A constraint that a rule is consistent with Scale only supports the rule. A rule is Scale-forced only when it passes the full four-field forcing certificate. The distinction is the difference between "this magnitude is allowed" and "this magnitude is compelled by the root."

The Scale-forcing certificate (four fields, all required)

A scale claim earns SCALE-FORCED only if all four hold:

1. Anchor-transfer chain. The magnitude or ratio comes from a root invariant, a measured scale, a theorem, an RG bridge, a compactification map, or a declared paid scale — an unbroken chain from the claim back to a named terminus. (Derivation is anchor-transfer, never anchor-elimination: the chain ends on an anchor, it does not make the anchor disappear.)
2. Scale counterfactual. Perturb the scale anchor, the RG window, the dimension, or the scheme, and show the rule changes in the expected way. If nothing in the rule responds to a scale perturbation, the rule is not Scale-forced.
3. Closed candidate class. Show that rival scale bridges, schemes, or normalizations are excluded, equivalent, paid, or dissolved — within a closed, exhausted class, not a hand-drawn shortlist.
4. Target-blindness. The scale bridge must be writable without the target value. If the bridge was chosen because it reproduces the observed number, the claim is target-selected and fails the certificate outright.

The honest reading across the candidate: no dimensionful magnitude currently earns SCALE-FORCED as a gate-closing observable. Several dimensionless ratios clear fields 1, 3, and 4 within a declared grammar and are SCALE-FREE-DERIVED. Several scales clear field 1 only by being paid (SCALE-PAID). The field that fails most often is field 4 (target-blindness) for scales, and field 1 (a valid terminus) for ill-posed magnitudes. Naming the failing field is the result. This boundary is kept deliberately, at full strength: no dimensionful magnitude earns SCALE-FORCED — not because the program is weak there, but because the unit-gauge theorem assigns that column to the rulers. The framework derives the ratios and pays for the rulers in the open; a claim of a naked forced magnitude would be a measured number dressed as a derivation.

The Scale Perturbation Test — how field 2 is actually run

Field 2 (the scale counterfactual) is not satisfied by a hand-wave that "the rule depends on scale." It is run as an explicit eight-axis perturbation. For any claimed Scale-forced value, perturb each axis independently and record the response:

Two failure verdicts, both fatal to a forcing claim:

• No response → not scale-forced. If a rule does not respond to any of the eight perturbations in the expected way, it is not coupled to the scale structure it claims to derive from. It is at best SCALE-SUPPORTED, more likely a disguised constant.
• Response to the target → target-selected. If the rule changes when the target observed value is changed — i.e. the bridge, scheme, or normalization was written knowing the answer — the claim fails field 4 and is SCALE-PAID at best, target-selected at worst. A forced rule responds to root and scheme perturbations and is blind to the target.

This is exactly what converts a SCALE-SUPPORTED claim into a SCALE-FORCED one: pass all relevant axes of the perturbation test and remain target-blind. Across the candidate, the test currently terminates every dimensionful magnitude at SCALE-SUPPORTED, SCALE-PAID, or below — which is why 0 of 33 gates are physics-closed and no status changes.

How Scale interacts with Shape and Granularity

The three roots divide one labor and must be used together; using one alone is the incomplete-root failure mode.

• Shape supplies the object; Scale supplies the magnitude. Shape (the frozen three-layer branch — see The Shape) produces the formula: a heat-kernel invariant, a flavor map, a moduli potential. Shape says what the object is. It does not say what number comes out in GeV. The moment a magnitude is claimed, the burden passes to Scale: which anchor, which scheme, which window, which bridge. Shape forcing a structure never amounts to Scale forcing a magnitude. The classic confusion — "the geometry produces this expression, therefore the theory predicts this value" — is exactly the boundary where the labor passes from Shape to Scale, and it is where the honesty guard bites.

• Granularity forbids the hidden continuous precision a fake scale would need. Granularity (the uniform positive cost-floor — see The Granularity) blocks the two ways a scale claim cheats: hidden infinite precision (a "derived" magnitude that secretly tunes a real parameter to arbitrary precision) and lookup-table scale fitting (a normalization or row factor adjusted continuously until it hits the target). Where Scale says this magnitude is paid, Granularity says and that payment is a finite, charged label — not a free continuous dial. SCL-F (fitted scales paid) and the Granularity rule against unpaid labels are the same wall seen from two sides.

In short: Shape — what is the object? Scale — what magnitude or ratio is meaningful, and what bridge makes it so? Granularity — what finite, charged structure stops hidden precision and arbitrary fitting? A magnitude claim is honest only when all three answer.

Precision-cost discipline — the Scale∩Granularity meeting point

Scale and Granularity meet exactly at the precision of a measured anchor. A measured scale anchor is not just a value; it is a value at a stated precision, and that precision is itself a charged, finite quantity. Every measured scale anchor must therefore carry six fields:

Value ............. the number (e.g. 1.2209 × 10^19 GeV)
Uncertainty ....... the experimental error bar
Sig-figs used ..... how many significant figures the calculation actually consumes
Source ............ where the value comes from (e.g. PDG-derived)
Scheme / window ... the scheme and energy scale at which it is defined
Charged status .... that it is a paid measured input, not a derived output

Two failure modes live here, and both are Granularity violations seen through the Scale lens:

• Using more precision than the anchor supplies is hidden free information. The declared anchor is $1.2209\times10^{19}$ GeV at 4 significant figures. Writing $1.220900000000000\times10^{19}$ GeV and then using those trailing zeros as if they were measured digits smuggles in precision no measurement paid for. The trailing zeros are an anchor-field placeholder, not data; a calculation may consume only the 4 figures the source actually charges for.
• Rounding a root anchor in a sensitive calculation is incomplete-root use. The mirror error: a closure that is sensitive to the anchor at high precision must use the anchor at full precision over the complete Shape. Rounding $M_{\rm Pl}$ (or truncating the geometry) in a sensitive step is using an incomplete root and may fail — or appear to succeed — for that reason alone.

The principle is one line: a calculation may consume exactly the precision its anchors are charged for — no more (free information), no less (incomplete root). This is the same finite-cost ledger that The Granularity enforces against unpaid continuous labels, read at the point where a measured scale enters.

How gaps and gates constrain Scale — worked examples

Each of the 33 gates and 19 walls teaches Scale something specific. The extraction template for each is fixed:

Numerical object:
Dimensionless or dimensionful?
Units / scheme / window:
Scale anchor:
Scale bridge:
Meaningful predicate?
Verdict: derived / paid / measured / consistency-only / dissolved / open

Below are the load-bearing worked examples, mined from the per-gap constraint extractions.

Example 1 — Gap-01 / a6 (the predicate-validity lesson, SCL-C)

Numerical object. Scale-free heat-kernel ($a_6$ Seeley–DeWitt) invariants versus the dimensionful $\mathrm{tr}[a_6]$ GeV-style magnitude on the frozen 13D graviton+ghost object.

Dimensionless or dimensionful? Both appear — and they part ways. The scale-free rational invariants are well-defined; the dimensionful magnitude is the trap.

Units / scheme / window. A dimensionful $a_6$ requires a named heat-kernel scheme object, shared with the threshold and moduli gates (SCL-H). In odd $D=13$ there is no canonical finite local log / anomaly predicate for it.

Scale anchor / bridge. None valid for the magnitude. The exact rational checks need no anchor.

Meaningful predicate? For the scale-free invariants, yes. For the dimensionful GeV-magnitude at odd $D=13$, no — it is not a canonical finite local observable.

Verdict. The scale-free sector is SCALE-FREE-DERIVED (exact rational checks, two-route reproducible). The dimensionful magnitude is SCALE-CONSISTENCY-COEFFICIENT or SCALE-DISSOLVED — a labeled scheme coefficient, not a gate-closing observable.

This is the canonical SCL-C lesson: a real, derivable scale-free structure sits right next to an ill-posed dimensionful magnitude. Reporting the GeV number as a prediction would be a category error, not a small overclaim.

Local density vs integrated observable — the Gap-01 trap stated generally

The Gap-01 lesson generalizes into a rule that catches a whole class of overclaims. A local coefficient, density, row-factor, or consistency-coefficient is not automatically a physical observable. A local object is evaluated at a point or per unit volume; a physical observable is what survives integration, normalization, and readout against a record. The gap between them is not cosmetic — it is where units, volume factors, and scheme dependence enter.

To promote a local object to a magnitude claim, all of the following must be supplied — and named:

Local-vs-integrated ... is this a local density/coefficient, or an integrated observable?
Volume factor ......... what volume/measure integrates it (and from which Shape)?
Normalization ......... the normalization convention (SCL-D) — is it forced or paid?
Units ................. the units after integration (and the anchor that gives them)
Scheme ................ the scheme object (SCL-B / SCL-H)
Window ................ the energy window / readout scale (SCL-E)
Physical record ....... the actual observable endpoint the number is tested against

The standing example is exactly Gap-01. The scale-free $a_6$ invariants are meaningful and derivable; but the dimensionful $a_6$ magnitude in odd $D=13$ is not a canonical gate-closing observable. It has no canonical finite local predicate (SCL-C), and it never completes the local-to-integrated template — there is no physical-record endpoint it is read against. So it is reported, honestly, as a labeled consistency coefficient (SCALE-CONSISTENCY-COEFFICIENT), never as a closed observable. Any local coefficient that cannot fill every line above is, at most, a consistency check.

Example 2 — W06 / SG8 flavor (the ratios-vs-absolute-masses lesson, SCL-A & SCL-F)

Numerical object. Fermion mass ratios versus absolute sector masses (and the mixing matrices CKM / PMNS).

Dimensionless or dimensionful? Ratios and mixings are dimensionless; absolute masses are dimensionful.

Units / scheme / window. Absolute masses need a sector-scale policy: $\kappa$, the sector normalizations $N_d, N_e, N_\nu$, the RG scheme, $M_R$, and a neutrino-scale policy (SCL-13).

Scale anchor / bridge. The flavor map can generate within-sector ratios. The absolute masses require the sector scales, which are fitted normalizations.

Meaningful predicate? Ratios and mixings — yes, testable as outputs. Absolute masses — only relative to a paid sector scale.

Verdict. Ratios / mixings can be SCALE-FREE-DERIVED and tested against frozen falsifiers (e.g. $m_u$). Sector scales and $M_R$ are SCALE-PAID (the fitted $N_d, N_e, N_\nu$ are the weakest link) — or SCALE-OPEN until generated target-blind.

The lesson: within-sector ratios may be generated by the map; absolute masses are paid scales, not predictions. Counting a fitted sector scale as an output is the most common flavor-gate overclaim — see the red-team traps below.

Example 3 — M_R and sector scales (the paid-scale lesson, SCL-F)

Numerical object. The right-handed neutrino scale $M_R$ and the sector mass scales.

Dimensionless or dimensionful? Dimensionful.

Scale anchor / bridge. $M_R$ sets the seesaw scale; it enters several gates (neutrino masses, leptogenesis window) and so is a cross-gate shared object (SCL-H) that must be settled once.

Verdict. SCALE-PAID unless generated target-blind. Because $M_R$ is shared, choosing it to fit one gate's target silently contaminates the others — exactly the failure SCL-H exists to catch.

Example 4 — Cosmological constant $\Lambda$ (the stability-vs-value lesson, SCL-G & SCL-I)

Numerical object. The vacuum-energy radiative stability versus the observed $\Lambda$ value.

Dimensionless or dimensionful? The value is a dimensionful (or Planck-normalized-ratio) anchor; "stability" is a property, not a number.

Scale anchor / bridge. Absolute vacuum energy and observed $\Lambda$ are distinct scale objects (SCL-19). Planck-unit smallness ($\sim 10^{-122}$) is not itself a derivation (SCL-22).

Meaningful predicate? "Is it stable?" — yes, a real question. "Is the value derived?" — only if a value-map to the measured vacuum curvature is supplied.

Verdict. A mechanism that protects smallness (an algebraic trace-decoupling) can dissolve the radiative-instability catastrophe — but the observed value stands MEASURED-ANCHOR — honestly measured, never derived, with no value derivation owed (a target-blind value bridge would be an upgrade, not a debt). Stability and value are separate results. Using the observed $10^{-122}$ to back-solve structure is forbidden.

This is the sharpest SCL-G example: explaining why a value is stable is not the same as deriving the value. The two must be reported on separate lines.

Example 5 — W19 thresholds / RG matching (the window-and-scheme lesson, SCL-E & SCL-H)

Numerical object. Threshold corrections / RG-matching row factors.

Dimensionless or dimensionful? Scale-dependent — thresholds are not scale-free (SCL-33).

Units / scheme / window. RG matching scales, scheme, cutoff, and normalization are central. The same heat-kernel / scheme object shared with Gap-01 / SG-6 may control the magnitude (SCL-H).

Scale bridge. Threshold rule and RG flow — but the rows must be read from the actual KK / operator spectrum with omitted-mode error bounds, not tuned.

Verdict. SCALE-OPEN until a target-blind spectrum-to-threshold computation with scheme reconciliation exists. Row-formula normalizations tuned to couplings are SCALE-PAID at best, target-selected at worst.

Example 6 — W18 Yang–Mills uniform gap (the don't-move-the-target lesson, SCL-E & SCL-J)

Numerical object. A uniform positive SU(3) mass gap in continuum units.

Units / scheme / window. The gap is a uniform positive mass scale in continuum units; its relation to the lattice spacing / cost floor must be explicit. $\Lambda_{\rm QCD}$ or a coupling scale cannot be used as a proof of existence (SCL-32).

Scale bridge. A uniform lower bound must survive the continuum and volume limits (a genuine cross-scale bridge) or be scoped as axiom-conditional (explicit dissolution).

Verdict. SCALE-OPEN / axiom-conditional. A finite-floor model can support but not replace the continuum measure; substituting $\Lambda_{\rm QCD}$ for the existence proof is a scale category error.

A summary of the per-gate scale lessons across all 19 walls:

Shared scale/scheme-object dependency graph (Nonseparability)

SCL-H is not a per-gate caution — it is a global constraint. Several scale/scheme objects are shared across gates. The table below is the real dependency graph for this program: which shared object feeds which gates.

Nonseparability rule. If one shared object changes, every dependent gate must be re-run — a change to the heat-kernel scheme reopens Gap-01, SG-6, and W19 simultaneously; a change to the sector-scale policy reopens flavor, neutrinos, and baryogenesis. A shared object may not be re-selected per gate to suit each gate's target. Re-choosing the heat-kernel scheme for the threshold gate so its row hits a coupling, while leaving Gap-01 on a different choice, is a SCL-H violation and a disguised target-selection. The fitted $N_d, N_e, N_\nu$ are settled once for all dependent gates or the over-determination evidence is destroyed.

Red-team traps

These are the scale overclaims the page exists to block. Each is a sentence that must not appear unless the stated condition is met.

• "The theory predicts this dimensionful value." Forbidden unless the page shows the absolute-scale anchor ($M_{\rm Pl}$), the units, the scheme, the energy window, and the bridge. A formula producing a number is not a prediction of that number.

• "The value is stable, therefore derived." Forbidden — stability and derivation are separate (SCL-G). A protection mechanism dissolves an instability problem; it does not supply the magnitude. This is the $\Lambda$ trap.

• "This exact ratio proves the absolute magnitude." Forbidden — ratios do not create units (SCL-A, SCL-I). A derived dimensionless ratio says nothing about the absolute scale until an anchor and a bridge are supplied.

• "The normalization is harmless." Forbidden — a selected normalization is a paid scale choice unless forced (SCL-D, SCL-F). "Metric-selected" is exact only under the declared normalization; it is not normalization-robust.

• "$\Lambda_{\rm QCD}$ (or any coupling scale) proves the gap exists." Forbidden — a coupling scale cannot stand in for an existence proof (SCL-32). Moving the target is not closing it.

• "Planck-unit smallness is the derivation." Forbidden — $10^{-122}$ in Planck units is a restatement of the measured value, not a derivation of it (SCL-22), and the observed value must never be used to back-solve structure.

• "The dimensionful $a_6$ number is a prediction." Forbidden — in odd $D=13$ that magnitude has no canonical finite predicate (SCL-C); it is consistency-only or dissolved.

• "We re-chose the scheme object for this gate." Forbidden — shared scheme objects are settled once and propagated (SCL-H); re-choosing per gate to fit each target is target-selection wearing a scheme costume.

The unifying anti-pattern behind all of these is target-selection: choosing the bridge, scheme, normalization, or scale because it reproduces the observed number. Target-selection fails field 4 of the forcing certificate automatically, and it is the single most important thing the Scale root is built to catch.

Naturalness is not derivation

There is a specific, seductive way to mistake a property of a value for the value itself. A magnitude being small, radiatively stable, technically natural, symmetry-protected, or Planck-expressible does not derive it. Each of these is a statement about how a number behaves; none of them supplies the number.

• A value can be small — and still measured. Smallness is a comparison, not a source.
• A value can be radiatively stable — and still measured. Stability says corrections do not blow it up; it does not say what it is (SCL-G).
• A value can be technically natural / symmetry-protected — and still measured. A protecting symmetry makes a small value unsurprising; it does not compute it.
• A value can be Planck-expressible — e.g. $\Lambda\approx 10^{-122}\,M_{\rm Pl}^4$ — and still measured. Writing a measured number in Planck units is a restatement, not a derivation (SCL-22).

Naturalness can dissolve a tuning problem; it cannot supply a measured magnitude. Dissolving the question "why isn't this huge / why doesn't this destabilize?" is a real result — it removes a false debt. It is not the same as paying the real debt "what is the value?", which stays MEASURED / SCALE-OPEN-as-derivation until a target-blind value bridge exists.

This generalizes the page's existing $\Lambda$ warnings (stability $\neq$ value, Planck-smallness $\neq$ derivation) into a single rule that applies wherever a naturalness argument is made: the cosmological constant $\Lambda$, the Higgs / electroweak hierarchy, the inflationary amplitude $A_s$, and mass gaps. In every case: the naturalness argument may dissolve the tuning sub-problem; the value sub-problem remains open or paid. Report them on separate lines — never collapse "it is natural" into "it is derived."

Status ledger

Honest classification of the scale claims, by category. 0 of 33 gates are physics-closed; no status changes.

The Scale page final test

This page succeeds only if, for any number in the candidate, a reader can run the following eight-question gate to completion. A number is Scale-honest only when every line has an answer.

THE SCALE FINAL TEST — run on any number. 1. Dimensionless or dimensionful? (If dimensionless, it can be unit-gauge-invariant directly — go to 6.) 2. If dimensionful, what anchor gives it units? ($M_{\rm Pl}$, or a declared measured scale — or there is no magnitude claim.) 3. Scheme / window / normalization? (Named, or the number is not yet defined — SCL-B, SCL-D, SCL-E.) 4. What bridge carries it across scales? (RG / decoupling / threshold / compactification / FRG / dissolution — with a filled bridge certificate — SCL-J.) 5. Is the bridge target-blind? (Writable without the observed value? If no → target-selected, fails.) 6. Category? (derived / measured / paid / consistency-only / dissolved / open — exactly one.) 7. What changes if the anchor / scheme / window changes? (Run the eight-axis Scale Perturbation Test; no response → not forced.) 8. What exact observable / audit record tests it? (The physical endpoint, or it is consistency-only at best.)

If every line answers, the number is at full precision and correctly typed. If any line is blank, the number is not Scale-derived — whatever the formula says. Applied across the candidate, this gate currently terminates every dimensionful magnitude at SCALE-PAID, SCALE-OPEN, or SCALE-CONSISTENCY-COEFFICIENT: 0 of 33 gates are physics-closed, no status changes. That is the honest state, stated at full precision.

Cross-links

• The three roots: The Shape (supplies the object) · The Scale (this page — supplies the magnitude) · The Granularity (forbids hidden precision and arbitrary fitting).
• The Seven Deep Roots — Scale is R5; this page is the public expansion of the R5 absolute-scale anchor.
• Deep Root: Scale — the closure dossier — the gate-of-record for this root: RESOLVED at +0 (measured-anchor terminal on the two-ruler floor), residual shown openly.
• The gate scoreboard — honest per-gate status across all 33 gates.
• Full precision → GUT Appendix A1 (HTML) · PDF — the ≥16-sig-fig reconstruction of the complete three-layer Shape this root anchors against.
• Related: the anchors overview, the walls register, and the closure routing.

Appendix — Full precision

This appendix carries the full-precision content the public page summarizes: the absolute-scale anchor at value precision, the dimensionful ledger (which magnitudes are measured-anchor vs paid vs dissolved vs open), and the shared scheme objects. It is the Scale companion to the GUT Appendix A1 full-precision reconstruction (PDF) — the ≥16-significant-figure reconstruction of the complete three-layer Shape this root anchors against.

A. The absolute-scale anchor (R5, full precision)

Operational statement. At least one absolute dimensionful scale is required; $M_{\rm Pl}$ is the accepted value anchor. The scale relation — written only as a relation, not as proof that the number is derived:

$$M_{\rm Pl}\;\sim\;\sqrt{\frac{\hbar c}{G}}.$$

The root has two parts that must never be conflated:

• Existence claim — at least one absolute dimensionful scale is required to fix the units of any dimensionful prediction. (Existence: proven — the unit-gauge theorem.)
• Value claim — the specific number $M_{\rm Pl}$, which is supplied by measurement. (Status: MEASURED.)

$M_{\rm Pl}$ is one of the small set of charged measured anchors the framework runs on:

$$\{\,M_{\rm Pl},\ \alpha_i(M_Z),\ y_t,\ |V_{us}|\,\}.$$

What it supports. It is the dimensionful anchor that lets dimensionless ratios and readouts be cashed into physical numbers.

What it does not prove. A naked dimensionful number is not invariant by itself (R1) — only dimensionless ratios are frame/unit-invariant. So the value of $M_{\rm Pl}$ is charged as measured input; it is not claimed derived from the geometry. Planck-unit smallness of another quantity is not a derivation of that quantity.

Allowed claim: a scale relation is needed. Forbidden claim: a naked dimensionful number is invariant by itself.

Completeness in use. When the scale root cashes dimensionless ratios into physical numbers, it must use $M_{\rm Pl}$ at full precision and apply it over the complete three-layer Shape at full precision (GUT Appendix A1 — PDF — $\geq 16$ significant figures across all three layers; frozen manifest meta-hash a5b1e6f9d951, active branch dcc66f1b2685). Setting a dimensionful scale from an incomplete shape or a rounded anchor is an incomplete-root use and may fail spuriously.

Specialist hardening question. The existence of at least one absolute scale is proven (the unit-gauge theorem). The remaining sharpening: prove from R1 + R4 that exactly one independent absolute scale suffices, and show that no second independent dimensionful anchor is smuggled in elsewhere in the generator.

B. The dimensionful ledger — measured-anchor vs paid vs dissolved vs open

C. Shared scheme objects (SCL-H)

Some scale/scheme objects are used by more than one gate and must be settled once and propagated, never re-chosen per gate:

• Heat-kernel scheme object — shared across Gap-01 ($a_6$), SG-6, the moduli gate, and the threshold/RG-matching gate (W19). Re-choosing it per gate to suit each target is a SCL-H violation and a disguised target-selection.
• Sector-scale policy ($\kappa$, $N_d, N_e, N_\nu$, $M_R$, neutrino-scale policy) — shared across the flavor (W06), neutrino-mass, and leptogenesis (W12) gates. The fitted normalizations are the program's weakest scale link and the most exposed to over-counting.
• RG matching scales / scheme / cutoff — shared across thresholds (W19), unification, and electroweak ($\rho_{\rm tree}$, W08) claims.

D. Status-label legend

The categories used throughout (and how they map to the deep-root status labels):

SCALE-FREE-DERIVED            ↔ GENERATED / DERIVED-GIVEN-E (dimensionless)
SCALE-ANCHORED               ↔ anchored on a MEASURED/derived/paid scale
SCALE-PAID                   ↔ CHARGED / paid input (fitted or declared)
SCALE-CONSISTENCY-COEFFICIENT↔ AUDIT ONLY (not gate-closing)
SCALE-DISSOLVED              ↔ ill-posed predicate removed (not solved)
SCALE-OPEN                   ↔ OPEN (bridge / scheme / anchor missing)

Allowed deep-root status labels (the only labels used): DECLARED ROOT · GENERATED · DERIVED-GIVEN-E · MEASURED · CHARGED · AUDIT ONLY · OPEN · BLOCKED · ANTI-CLAIM.

E. Completion report

• P1 — Thesis + honesty guard present. Yes (top of page).
• P2 — Full Scale definition present (anchor + ratios + scheme/window + bridge + stability/value split + dissolution). Yes.
• P3 — All six claim categories defined. Yes (derived / anchored / paid / consistency-only / dissolved / open).
• P4 — Complete constraint table SCL-A … SCL-J with SCL-I and SCL-J explicit. Yes.
• P5 — Forcing certificate (four fields) stated; FORCES vs SUPPORTS separated. Yes.
• P6 — Bridge catalog (RG / decoupling / threshold / compactification / FRG / dissolution). Yes; measurement explicitly excluded as a cross-scale bridge.
• P7 — Interaction with Shape and Granularity stated. Yes.
• P8 — Worked examples from gaps/gates (Gap-01/a6, W06 flavor, $M_R$, $\Lambda$, W19 thresholds, W18 YM) + 19-wall summary table. Yes.
• P9 — Red-team traps enumerated. Yes.
• P10 — Status ledger classifying every number. Yes.
• P11 — Full-precision appendix (absolute anchor + dimensionful ledger + shared scheme objects). Yes.
• P12 — Scale anchor convention box (the actual object: $M_{\rm Pl}=1.2209\times10^{19}$ GeV, ordinary not reduced, units, existence-theorem vs measured-value split, forbidden compression). Yes (near top).
• P13 — Unit-gauge / Buckingham-$\pi$ layer (Scale framed as a theorem: dimensionless → derivable; dimensionful → anchor + bridge + scheme; no anchor → no magnitude). Yes.
• P14 — Scale Perturbation Test (eight axes: anchor, scheme, window, normalization, threshold, compactification radius, cutoff/regulator, dimension; no-response → not forced; target-response → target-selected). Yes (extends the forcing certificate).
• P15 — Bridge certificates (per-bridge template + worked threshold-bridge conditions; hardens W19, SG-7, SG-6, Gap-01). Yes (extends the bridge catalog).
• P16 — Local density vs integrated observable (Gap-01 $a_6$ worked: consistency coefficient, not a closing observable). Yes.
• P17 — Shared scale/scheme-object dependency graph (heat-kernel scheme, sector-scale policy, RG matching, $M_R$; Nonseparability rule). Yes.
• P18 — Precision-cost discipline (Scale∩Granularity: value/uncertainty/sig-figs/source/scheme/charged; over-precision = free info; rounding root = incomplete root). Yes.
• P19 — Naturalness is not derivation (small/stable/natural/protected/Planck-expressible ≠ derived; dissolves tuning, not value; $\Lambda$, EW hierarchy, $A_s$, mass gaps). Yes.
• P20 — Scale-status transition table (OPEN→PAID, PAID→FORCED, CONSISTENCY→DERIVED, DISSOLVED→closed never automatic). Yes.
• P21 — Scale page final test (boxed eight-question gate replacing the old Final standard). Yes.
• U1 — Anchored ≠ derived; floor ≥ 1 preserved (root still bottoms on $M_{\rm Pl}$; this is a completeness upgrade, not anchor-elimination). Yes.
• U2 — No floating magnitudes: every dimensionful claim typed as measured-anchor / paid / dissolved / open. Yes.
• U3 — No FORCED magnitude overclaimed: 0 of 33 gates physics-closed; no status changes. Yes.
• U4 — Existence vs value of the scale anchor kept distinct. Yes.
• U5 — Stability ≠ value; ratios ≠ units; dissolved ≠ solved. Yes.
• Named upgrade paths (each an upgrade, not a debt — no status changes). Sharpening the existence theorem via R1+R4 to “exactly one independent absolute scale suffices” (existence itself is proven — the unit-gauge theorem); target-blind value bridges for thresholds, inflation, and baryogenesis (would lift SCALE-PAID/SCALE-OPEN rows toward SCALE-FORCED; the observed $\Lambda$ stands MEASURED-ANCHOR — honestly measured, never derived); generated (not paid) sector scales / $M_R$; a uniform YM gap surviving the continuum/volume limits (the certified-irreducible Clay reduction stands — Gap-02 · RESOLVED +0; a proof would strengthen it toward DERIVED).