Assumptions, dependency chain and source–readout contracts

Assumptions, dependency chain and source–readout contracts

Three registers collect what the reconstruction must preserve and how the conclusion depends on it. Table 3 lists the assumptions and scope controls of the supplied construction, with the lines of the supplied text at which each is stated; rows A19–A27 are those of the constructions of Sections 9, 14 and 16 and of the rotation and trace arguments. Table 4 is the chain of stages through which the public conclusion of Section 4 is reached, each with its input contract, the operations that are lifted, and the recovered output. Table 5 lists the source–readout contracts of the general transport statements, with what each readout fails to invert and what is therefore retained. Together with Table 2 they are the summary form of the hypotheses under which every recovery statement of this paper holds; none of them adds a hypothesis to the supplied construction.

Table 3: Assumptions and scope controls of the supplied construction, as preserved in this paper.

Accessible table transcription

Table transcription, source page 516

Assumption	Input lines	Preserved condition	Scope control
A01Group and	9	Closed faithful matrix subgroup	No connectedness hypothesis
representation		of U(m); specified exponential	is added.
		real Lie algebra is nonabelian
		simple.
A02Finite prior and	11–26	Independent normalized Haar	Gauge invariance/continuity
Wilson order		link prior and literal ordered	and uniform-period slope
		Wilson plaquette; insertion	refer to one family.
		support offsets depend only on
		token.
A03Volume and	26–28	Every increasing period	Auxiliary compact selections
mesh quantifiers		sequence admits common	do not weaken full inner
		refinement; suitable fixed	limit.
		periods have full inner mesh
		limit before outer period limit.
A04Field domain	30	Common dense invariant cyclic	Unique invariant vacuum is
and spectrum		word domain, Schwartz	not a positive-gap claim.
		continuity, adjoints/locality,
		proper orthochronous
		covariance and forward-cone
		PVM.
A05OPE parameters	32–45	Fixed finite words/degrees,	Radius threshold may
		M≥0, η>0, ε>0; separated	depend on fixed words/
		cluster and spectators; same	parameters; uniformity is in
		phase/truncation-independent	phases and stated
		channels.	configurations.
A06Public	47–60	Centered degree-four trace and	Does not enlarge exclusion
comparison class		finite complex combinations/	to every generalized-free
		products of stated fixed	comparison; penalty is one.
		mass-shell integrals.
A07Analytic	64–108	Four dimensions; physical	Rate-zero is subpower, not
conventions		metric/chord/color factors; odd	boundedness; thresholds
		block ratios ≥3;	independent of period.
		connected-union cover; absolute
		weighted bounds.
A08Contact	12526–	Tested rate inequality on every	Remaining branch retains
continuation	12543	allowed logarithmic path;	support t≤s, full finite
		palette continuity and	remainder; projected gain is
		constancy in assigned normals;	actual degree.
		common detector collar.
A09Weak-sequence	12755–	Same weak scaling sequence/	No interpolation to every
Gaussian	12767	frames and compact distinct	positive real scale; Ward
comparison		configurations with each fixed	sign remains fixed.
		derivative order.
A10Private Gaussian	12945–	Finite regulated potential law,	Covariance is bilinear;
marks	12973	positive semidefinite	same-slot value need not
		infinite-lattice subtraction	vanish at finite regulator; no
		matrix; independent private	subtraction inverse required.
		marks by insertion.

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Table transcription, source page 517

Assumption	Input lines	Preserved condition	Scope control
A11Laplace	14113–	Positive future-cone measures	No arbitrary signed-measure
continuity	14136	with finite transforms on full	convergence assertion.
		open dual cone; common
		polynomial budget for Schwartz
		passage.
A12Maxwell	14220–	Complex-linear isometry with	Neither surjectivity nor
isometry	14252	physical image, vacuum/	identification of every field
countertest		translation intertwining, and	required; fixed-ball variant
		precise local coherent matrix	retained.
		identity.
A13Projected and	14326–	Bounded self-adjoint	Individual averaged maps
averaged	14329	fixed-image map for projected	need no extra linearity,
variants		variant; physical inner-product	boundedness or projection
		preserving pointwise	property.
		fixed-image maps for
		probability average.
A14Near-kernel	14648–	Finite real onto map with	No prior uniform
repair	14674	nonzero one-dimensional kernel,	convergence on near-kernel
		exact convergent right inverses	direction assumed; precision
		and both weighted near-kernel	paid before correction.
		errors.
A15Mass and	14890–	Strictly positive mass through	Separate eventual endpoint
volume order	14982	volume estimates; moving	estimates suffice; no uniform
		masses confined to a fixed	mass convergence is inserted.
		positive interval after tolerance
		choice.
A16Position	15317–	Complementary scalar/stress/	Real fourth-order remainder
normalization	15408	unit projections and positive	belongs to normalized frame;
branch		adjustment factors; independent	weaker positive-ray norm
		Wilson calibration of measured	alignment transfers leading
		charge.	coefficient only.
A17Marker and	15556–	Finite surgery order fixed	Uniformity in finer cutoff/
threshold order	15647	before eventual thresholds;	volume follows retained
		common windows, two marker	threshold order; no
		slots and positive weights, exact	chamber-count tarifffactor.
		common Wilson partition.
A18Local lens	15686–	Fixed read support, measurable	Flag equality is branchwise
admissibility	15766	equivariant displacement,	exact; absent insertion
		health threshold θ/4, chord D	requires no exponent
		with 8D≤θ, finite stocks and	estimate or hard-mask
		nonwrapping.	differentiation.
A19Activity domains	110–226	Threshold 0 < θ ≤1 with	Masks are never
and budgets		equality healthy; ordered left	differentiated; no
		derivatives on healthy domains	volume-independent bound
		only; budgets at a fixed finite	on the total scalar action is
		torus; early reserve s + 3.	asserted.
A20Stationary	5515–5709	Fixed derivative order n ≥2;	Uniqueness of the stationary
corridor		counting gap and exponential	point only inside the
		weights; contraction on	corridor; off-shell values and
		∥x∥< RX, ∥v∥≤RV .	the residual are retained.

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Table transcription, source page 518

Assumption	Input lines	Preserved condition	Scope control
A21Calendar and	6226–6430	Stage data and length floors,	Later enlargements are
order of choices		then lengths, then entry delays;	monotone and preserve
		one calendar chosen before	earlier payments; estimates
		every history, cutoff, period and	on different calendars are
		observable.	not combined.
A22Calibration of	6617–6819,	Bare parameter in a closed	Existence by the
finite histories	7504–7575	interval; bottom coupling ̄β at	intermediate value theorem
		level zero; deterministic forward	only; uniqueness within the
		recipe with continuous	retained recipe; no inverse
		admissible levels.	shooting map is assumed.
A23Reference tori	6843–7259	Read radius r with 2r ≤w; five	Constants are tied to fixed
and beacons		folded reference views; compact	reference tori and are not
		coupling band; orders and stage	uniform in the physical
		parameters, then block length,	period.
		then delay.
A24Labels and	8399–8753	Simple color Lie algebra, with	The label reader is inverted
calibrated		no connectedness of the group;	at each fixed level; its
readers		finite slot family; degree and	inverse bound is not uniform
		level fixed before the eventual	in the level.
		threshold.
A25Free comparison	8754–9247	Explicit Gaussian kernel on the	Statements are exact in the
hierarchy		fixed real symbol space; one	free comparison theory; they
		exterior degree bound for all	pass to the interacting
		weights in detection.	hierarchy only by rank
			transfer along convergent
			readings.
A26Euclidean	10373–	Bands, window and tolerance	No positivity is claimed for
rotations	10674	fixed before the limit removing	the six individual band
		the outer flux; three response	readings; the unsubtracted
		signs f ≥0, g ≤0, h ≥0;	stress pair is not asserted
		contact subtraction.	conserved at coincident
			points.
A27Scaling	13237–	Selected radii r0 and scaling	No uniformity in the degree
		k ↓
comparison and	13994	maps; for every sequence s0	and no bound on the scaling
		n ↓
trace limit		a subsequence with a scaling	maps or their inverses; the
		family on the same table and	interacting theory is not
		phase; monotonicity from the	asserted to be free.
		positive-time spectral measure.

Table 4: Dependency chain of the supplied conclusion: each stage with its input contract, the operations lifted, and its recovered output.

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Table transcription, source page 518

Stage (parents)	Input contract	Operation	Recovered output
D01Finite Wilson	fixed group/	frame realization and	physical
realization	representation, finite	section; Wilson density	finite-lattice law
	lattice, normalized Haar	pushforward
D02Multiscale	positive widths,	exact one-step identities	one Wilson family
calibrated family	prescribed stage ordering,	and common	before volume
(D01)	banked histories	insertion-token transport	subsequence

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Table transcription, source page 519

Stage (parents)	Input contract	Operation	Recovered output
D03Common	requested increasing	full mesh limit then outer	common hierarchy,
continuum table	period sequence;	refined-period limit	coefficients and
(D02)	fixed-period Cauchy		finite field filtration
	bounds
D04Phases and	compression algebra,	phase readout and same	common-domain
Hilbert	character hierarchy,	heat/tube/boundary	fields, cone
reconstruction	positive Gram/heat data	construction	momentum measure
(D03)			and unique vacuum
D05Fixed OPE and	same finite homogeneous	nested replacement,	same public
contact extension	reader, basis and	tested integration and	channels across
(D03, D04)	projections	combined contact	finite truncations/
		resolution	collisions
D06Gaussian	same weak sequence, free	private marks, corrected	mean Gaussian
comparison (D03)	frames and moving	covariance, Wick	comparison with
	covariance limits	recursion and profile	mixed stress jets
		sums
D07Local equation	same served coefficient	finite marker estimates	both paid
sensor (D02, D03)	difference, chart/coupling	and weighted degree-four	near-kernel leaks
	and retained scale	spectator detection
D08Physical stress	exact divergence detector;	finite correction inverse	Ward-normalized
amplitude (D06,	same nonzero Ward shell	and near-kernel	stress and scalar
D07)	test	functional repair	coefficient −b/2
			0
D09Selected trace	selected stress, exact unit	all-distance sandwich;	trace amplitude and
ultraviolet	centering, positive	dilation/Laplace/	normalized cone
measure (D04,	spectral measure	Schwartz recovery	measure
D08)
D10Specified finite	finite mass-shell tracks	damped radial expansions	source-specified
polynomial	with fixed finite data	and product-test recovery	comparison
exclusion (D09)			exclusion
D11Specified Maxwell	physical-image	common amplitudes;	source-specified
countertests	translation isometry and	local dilation polynomial;	Maxwell/projected/
(D09)	precise local coherent	finite sector growth	averaged exclusions
	identity
D12Normalized	positive frame branch;	measured	normalized-frame
position	separated Ward columns	Wilson-calibrated charge	real fourth-order
diagnostic (D08,		and exact rational	remainder; weaker
D09)		residual	leading transfer

Table 5: Source–readout contracts of the general transport statements.

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Table transcription, source page 519

Contract	Readout	Non-invertibility and	Where
		retained data
M01Native admission	The realized branch supplies	Does not recover or	Prop. 6.1
and literal	physical configuration/history	classify all possible native
preparation	readouts only after group,	antecedents; admission
	lattice, law and analytic data	does not select Wilson law
	are specified.	or identify abstract
		vertices with physical
		sites.

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Table transcription, source page 520

Contract	Readout	Non-invertibility and	Where
		retained data
	−1
M02Finite frame fibre	π(r, A)e = rx Aery; reverse	Readout forgets frame	Thm. 7.1
and section	links are inverses.	representatives; it does not
		quotient physical gauge
		equivalence. Physical
		−1
		gauge acts by r 7→rg, A
		fixed.
M03Paths and physical	Interior frames cancel; open	A trace alone discards	§7
gauge action	paths transform at endpoints	conjugacy and other
	and loop traces read	matrix information; full
	unchanged.	link history remains
		retained.
M04Normalized Wilson	πeμ = μ	No unnormalized gauge	Thm. 7.2
	∗W ; insertion F reads
measure and	F ◦π.	volume; normalization and
insertions		integrability remain part
		of the contract.
M05Tangent operators	pT Tp Tj jT;	Vertical frame directions	Thm. A.1
	2Z =1 andZ1 =2
and quadratic forms	compositions insert pj I.	remain null. Operator
	ii =
		transport differs from
		adjoint form pullback. No
		ambient vertical coercivity
		follows.
M06Actual nonlinear	Coarse π(s, B) = P(π(r, A));	Multivalued branches	§A
predictors and	differentiated frame terms	remain tagged relations;
horizontal	cancel in the coarse readout.	no invented selection or
derivatives		bound on arbitrary
		unbounded frame rates.
	Z
M07Tested limits and	Exact scalar identity I= I	No exchange of limits,	Thm. B.1
	λ λ
quantitative	at every tag.	stronger test class,
estimates		derivative of hard masks,
		new compactness or
		mixing of unrelated
		partition functions.
M08Operations,	πσ = id at every composable	No discard of branch tags	§B
relations and	intermediate stage.	or auxiliary histories
residual histories		merely because functions
		coincide.
M09Reflection quotient,	Unitary U : [v] 7→[pv] between	Null histories are	Thm. C.1
fields and spectrum	completed reflection spaces.	quotiented; an unbounded
		field is not promoted to a
		bounded whole-space
		operator. Rank-one
		vacuum is not a positive
		spectral gap.
M10Compression	Whole-word character	Factors cannot be read	Thm. D.1
algebra and phase	evaluation; ρZ = h∗ρ.	separately without a
measure		proved product identity.
		Almost-everywhere
		statements require the
		original continuity/
		full-support upgrade.

Accessible table transcription

Table transcription, source page 521

Contract	Readout	Non-invertibility and	Where
		retained data
M11Fixed OPE reader	Applying q recovers the same	Projection Πm is not	Prop. E.1
and retained	channel reader and finite	inverted; discarded tails
remainder	projections.	are retained as residuals.
		Label inverse is not an
		inverse on unreduced
		histories.
M12Contact extensions	T(g) = T(Lg) on the stated	Punctured restriction loses	§E
	Z
with subtraction	combined integrability domain.	diagonal-supported
data		contact information;
		separately nonsummable
		pieces are not assigned
		integrals.
M13Gaussian means,	Normalized Gaussian integrals	Replacing μ by −Cb loses	§F
	∗∗
covariances and	and μ −v = Ev −P (Av + b),	the specified mean identity.
	∗
residuals	E = AP −I, A = A.	No probability covariance
		is assigned to
		nonsymmetric P.
	P
	−1 k
M14Ordered inverse,	C−A = (−S)A on	Terminal A cannot be	§F
	k≥1
determinant and	the original norm-convergence	removed/commuted;
quiet-coordinate	domain.	determinant diagnostic
data		alone is not full log
		determinant. Orthogonal
		norm invariance does not
		preserve coordinate boxes.