Correspondence with the supplied text

Correspondence with the supplied text

Table 7 locates the expanded source’s derivations by input range, equation tags and position in this paper. The sequential sections and mathematical appendices retain the complete construction and its dependencies; the additional source–readout statements give the maps, operations and recovery proofs at those same stages.

Table 7: Supplied derivations typeset in full, with their source–readout formulations.

Accessible table transcription

Table transcription, source page 524

Input lines	Subsection of the supplied text	Tags	Where
110–226	Local activity domains, derivatives and ab-	AB1–AB12	§9
	solute budgets
229–234	The finite Wilson law is a probability mea-		§9
	sure
5057–5163	The primitive integrand and its local ex-	PRI1–PRI9	§13.24
	tension
5515–5709	A common analytic stationary graph and	AS1–AS8,US1–	§14
	its response estimates	US8
5710–5719	The Fourier normalization and the kinetic		§14
	readout
5720–5753	Analytic exterior division with the origin	ED1–ED2	§14
	jets retained
5754–5806	The two-leg correction with both momen-	TEL1–TEL4	§14
	tum blocks retained
5807–5827	Production of the fresh loop moments	LM0	§14
5828–5861	Production and transport of the inherited	LM1–LM4	§14
	loop moments
5862–6109	The one-loop drift used by the finite histo-	DR1–DR7,DP1–	§14
	ries	DP12
6110–6225	Massive auxiliary symbols and the uniform	MA1–MA12	§14
	alias trace bound
6226–6325	Constructing the common asymptotic cal-	CAL1–CAL11	§14
	endar
6326–6381	Countable local demands on the same cal-	CAL12–CAL18	§14
	endar
6382–6402	Payingtheactualcompulsory-emission	EF1–EF3	§14
	forecast
6403–6430	Uniform reserve at every step, including	CAL19–CAL21	§14
	startup
6431–6531	Inversecurvaturecoordinatesandthe	SQ1–SQ12	§14
	startup fourth coefficient
6532–6616	The actual finest-scale Wilson representa-	FST1–FST8	§14
	tive
6617–6819	Finite coupling arcs and calibrated finite	FCA1–FCA11,	§14
	histories	RSP1–RSP6,
		FCC1–FCC2
6820–6842	Exact density identity after retirement	RET1–RET2	§14
6843–6988	Finite reference beacons and weighted re-	RB1–RB8	§14
	tirement continuity
6989–7259	Derivation and transfer of the reference	PL1–PL5,GL0–	§14
	chart estimate	GL5, RBG1–RBG9
7260–7503	The local activity norm and analytic com-	LCN1–LCN11,	§14
	parison estimates	ASP1–ASP5

Accessible table transcription

Table transcription, source page 525

Input lines	Subsection of the supplied text	Tags	Where
7504–7575	Stability of finite calibration and its com-	FCB1–FCB5	§14
	parison weights
7576–7627	Uniform running of the actual coupling	RUN1–RUN6	§14
7738–7797	The protected change of variables and its	PV1–PV7	§15.5
	determinant
7947–8040	Extracting stationary path coefficients and	SH1–SH10	§15.12
	their higher jets
8399–8442	Radial Taylor variables and the free equa-	RJ1–RJ5	§16
	tion space
8443–8561	Polynomial labels, coherent frames and lo-	PF1–PF13	§16
	cal sections
8562–8713	Calibrated coefficient changes and exact so-	PC1–PC13	§16
	lutions
8714–8753	The full free Maxwell polynomial hierarchy	FM1–FM3	§16
8754–8831	Positivity and strictness of the reflected	GP1–GP4,FM4–	§16
	free pairing	FM7
8832–8997	The free low-degree stress and vector	FC1–FC15	§16
	spaces
8998–9001	Free shell charges and separation of the		§16
	Ward kernel
9002–9047	The compact stress shell identity	FS1–FS3	§16
9048–9098	Low-degree vector charges	FS4–FS10	§16
9099–9152	Fixed-displacement evaluation and the tri-	WE1–WE4	§16
	angular Wick calculation
9153–9230	Detection with a fixed exterior degree	HC0–HC4,FS11–	§16
	bound	FS13
9231–9247	Elimination of higher-degree vector and an-	FS14	§16
	gular defects
9767–9788	Uniformly bounded symmetric projections,	CVP1–CVP3	§18.9
	the calibrated equation image and the dif-
	ferential quotient
9801–9801	Uniformly bounded symmetric projections,		§18.9
	the calibrated equation image and the dif-
	ferential quotient
9805–9862	The calibrated equation image and its	CVR1–CVR7	§18.9
	silent representatives
9863–9921	Inclusions and derivatives on the same	QD1–QD6	§18.9
	physical quotient
9924–9974	Bounded slab insertions and their norm	BS1–BS4	§19.1
	derivatives
10373–10578	Removing the outer angular flux and ob-	RT0a–RT16	§19.16
	taining all Euclidean rotations
10579–10594	The order of choices in the spectral sign	RSW1–RSW2	§19.16
	walk
10595–10619	Normalized two-band diagonal estimate	DB1–DB2	§19.16

Accessible table transcription

Table transcription, source page 526

Input lines	Subsection of the supplied text	Tags	Where
10620–10674	Ward kernel transfer and the rotation ac-	RT10–RT19	§19.16
	tion
13237–13427	The finite trace candidate, its equation	TF0a–TF10	§23.1
	identity and its compensation
13428–13558	The interacting-to-Gaussian comparison	IG1–IG10	§23.1
	used for the trace
13559–13566	One calibration matrix for the three panels		§23.3
13567–13754	Common scaling maps and convergence of	WS1–WS17	§23.3
	every derivative
13755–13919	Constructing the physical stress and its	SR1–SR4,SC1–	§23.3
	shell charges	SC9
13960–13994	The full short-distance limit: monotonicity,	TLim1–TLim2	§24.2
	universal radius selection and the sandwich
14485–14503	The three finite candidates and their com-	TF11	§25.1
	mon extraction
14705–14869	Local estimates for the coefficient calcula-	LCE1–LCE6,	§25.5
	tion and the six contributions	TF12–TF20
14915–14972	The local resolvent bound	LR1–LR3	§25.9
15729–15765	Finite density surgery and the equation in-	TS11–TS13	§26.8
	sertion

Coverage and amendments. The accompanying source-to-both-papers coverage register has an entry for every heading unit of the expanded source, lines 1–15788, with its substantial arguments and dependencies. It connects the two standalone accounts without making either one supply missing mathematics for the other. The preserved input, equation tags and source ranges provide a reproducible correspondence to the complete derivation.

The following register records the formula amendments, typography clarifications and local recovery corrections, including their algebra and their domains of use.

I.1 Editorial amendments and focused recovery checks

The expanded mathematical source is reproduced unchanged in the shared release. This register distinguishes damaged-character repairs, typography clarifications, supported formula amendments and local recovery corrections, with their calculations and domains of use.

A1: principal predictor parenthesization. The extraction at line 1017 reads . The formula used in the construction is

Searchable transcription

G0,a = m(q0)−1(Ta/ga −H/U).

The multiplier convention matters here. Write), and let the opposite-momentum product replace the real-slice absolute square. To exhibit the scalar cancellation without discarding the directional rail, set

Searchable transcription

a = | pullLinea(q0)|2fa
m+m−
,
F =
X
c
fc,
T = Ta,
g = ga = m+m−a + ST.

This is a local algebraic abbreviation on the regular principal collar, where= 0; it does not rename the original coefficient. With, the corresponding covariance weight and directional rebate have the reduced form. Direct subtraction gives

Searchable transcription

E0 −r0 = m−S(FT −aH)
Ug
,
T
g −H
U

= S(TU −Hg)
m+Ug
= m−S(FT −aH)
Ug
.
S
m+

Thus with the amended parenthesization. In the real, even principal-filter convention this is the familiar calculation with and the directional rail absorbed in . The formula with the unparenthesized does not give that cancellation. The remaining nonprincipal branch is unchanged. In particular the factor cancels the longitudinal before Fourier extraction; no value at zero is assigned to that isolated singular factor. The filled principal value, its analytic collar, alias estimates and subsequent dual-predictor construction use the amended expression.

Searchable transcription

A2: finite Gaussian normalization.
The extraction at line 1317 has exponent −rh + dv; the
correct exponent in the finite integral identity is −(rh + dv). Here r is the color dimension, h the
active free-link count, f the total free tree-gauge link count, rh the active real dimension, and dv
the dormant real dimension. Thus f −h free links are frozen and the ambient fluctuation dimension
is rh + dv. Set K =
√
2π and mg = c(Kg)r, where c is the Haar chart scale. Cancellation of frozen
factors gives the exact substitution

Searchable transcription

K−(rh+dv)mf
ge−(f−h) log mg = K−(rh+dv)mh
g
= K−dvgrhch.

The extracted exponent instead gives, which differs by The dormant density remains with integral one, including = 0. This amendment does not change the separate covariance determinant. The finite active/dormant factorization, frozen-link cancellation and later normalized Gaussian readers consistently use this convention.

SR1–SR4: local recovery statements. Equation (2) uses the ordered group product exp(, neutral on the protected chart and retained offit; the reversed factors would produce a commutator. Equation (3) retains distinct base and varied covariances, including both determinant and inverse derivatives, and specializes to a fixed covariance only when they coincide. The startup allowances are simultaneous on: the forced estimate needs the lower endpoint and the regular estimate also needs the upper endpoint. Finally exact station radius matching is an equality of geometric radii. Equation (5) retains the error between actual interacting and Gaussian moments, which vanishes only in the specified limiting comparison on the same table and phase. The finite startup identity, shooting argument, radius interval construction and full trace-limit sandwich retain their original roles.

Searchable transcription

Interpreted typography and a completed implication.
Three malformed extracted expres-
sions are interpreted using their local counting or norm arguments: line 1575 becomes CK =
(na + nb)AJ; line 3218 becomes 4 pcap(4)(
pdirecttop + pathstop)t =: C1t; and line 3428 becomes
HC(QRprice)n, supported by the repeated geometric estimate at line 4020. These are disclosed
interpretations, not literal transcriptions. One further transpose-scope ambiguity in source line

Searchable transcription

815 is made explicit in both presentations. The extraction writes V = d2BT
1 F(d2B1), whereas its
preceding identities give

Searchable transcription

V = B1MB1 = B1dT
2 Fd2B1 = (d2B1)TF(d2B1),

because is symmetric. The reviewed flagship already used this whole-product adjoint. This is a type-consistent parenthesization of that local factorization, not a new positivity assumption. The packet normalization at source line 3121 prints a damaged exponent. It is read as

Searchable transcription

2π β−1/2)r, consistently with the already defined Gaussian coupling g = β−1/2
in mg(β) = c(
√

and the finite normalization calculation above. The reviewed flagship already used this reading; it is explicitly retained in the companion. Equation (4) completes the truncated projection proof: forces = 0 when 1; reversing the projections gives equality of their finite kernel dimensions. Source-range headings use the expanded extraction’s 15788-line count.

The accompanying change ledger records the original and amended expressions, exact locations in both papers, and affected uses checked. The focused checks exercise noncommuting group elements, genuinely different Gaussian covariances, both shooting-band endpoints, station geometry and the two scalar substitutions. They provide reproducible checks of the local interventions documented here; the general arguments and their scope are stated in the corresponding proofs.