
Abstract

This extension acquires the primary observer preprint, the 82-page origin-and-geometry manuscript, source learning reports, provisional curvature/retention packets, and two original ZCG engine archives. It carries the earlier proof work into a usable research workbench with six tool families, twenty case applications, live test execution, measured-data analysis, source attribution and exportable session records. The researcher can change an assumption, reproduce a counterexample and retain both the original claim and the response.

The strongest new executable evidence is concrete: all 37 original ZCG tests and all 13 exception replay cases passed; 46,144 coordinates across 512 grids had no mapping collision; 2,000 varied byte payloads round-tripped; 5,000 frame transformations recovered their local coordinates; 9,261 integer triples satisfied the tested transported-algebra laws. Adversarial single-bit mutation exercised 408 streams: 252 were rejected as malformed and 156 remained structurally valid. A trusted expected hash detects changed content in either class.

Falsification found seven proof-preimage boundary defects: oversized binary fields or invalid UTF-8 escaped the original typed validation as low-level errors. A separate correction candidate rejects all seven with ZCG-E001 while preserving all 37 original regression tests. A concurrent-test payload advertised as random actually repeated one byte because it reset its generator at every draw; the corrected corpus seeds once and spans all 256 byte symbols. The original Drive sources remain unchanged.

The current author clarification on density around origin one is preserved as a distinct empirical hypothesis. A bell distribution centred on one supports a bilateral relational reading, but the distribution’s integral, coordinate centre, density height and a physical whole’s identity are four different quantities. Shifted bells demonstrate that the bell form alone does not uniquely derive a numeric origin. The workbench therefore couples this idea to explicit emergence criteria and a measured-data tool rather than presenting a chosen centre as an experimental result.


Scope and definitions


The research objects

U=1 denotes one admitted whole or origin identity. Its coordinate projection can be (1,1) or (1,1,1) without duplicating that identity. A displacement of zero from U is a relation, not an assertion that the whole is absent. A scalar payload, an active label, a whole count, a measurement coefficient, a coordinate tuple and an existence flag are distinct types. The paired notation (−1+) retains the whole and its contrary relations; it is not assigned an unproved parser interpretation.

The transported algebra recovered from the origin manuscript is separate from the paired-whole magnitude composition requested in the conversation. The July five-state display P=(−3,−2,1,2,3), later extended origin-to-displacement integer line, September one-based codec coordinates and LAW3 register domain are not interchangeable editions. Their relations may be tested through an explicit mapping; no one edition silently certifies the others.

Boxes in a box is a heterogeneous measurement construction. Local boxes anchor bodies to a face, declare scale and direction, and retain their local identities. A containing system frame supplies a shared comparison unit. An observer is located in a declared frame or participates as a body. The outer observational rationale states purpose and selection but is not turned into another absolute measured frame. The primary preprint §4.2 was acquired directly, so the earlier learning reports now act as secondary discovery aids.

BRAINK owns governed execution, transition authority and evidence. KEX owns state geometry, identity and addressing. IL-LLM owns semantic and knowledge processes. Execution bridges, host/runtime systems, workbook automation, Sites and deployment infrastructure remain separate. The interactive workbench executes declared reference models and a checked browser codec port; it is not a replacement for this estate.


Density and curvature vocabulary

Probability density p(x) integrates to one and has inverse-coordinate units. Physical environmental density ρ has units determined by an observed quantity and its sampling volume. A Gaussian mode μ is a coordinate location, while σ is spread. These differ from U=1, the whole’s identity. Emergence must be operationalised as an observable state transition or persistence criterion before it can be tested.

A metric determines equal-distance shells. Rotation changes orientation; a change from L∞ to L1 changes the distance law. A surface embedding changes how addressed coordinates are placed in a higher-dimensional view. Smooth curvature, discrete turning angle and holonomy are distinct objects. The acquired ZCG source implements a finite run map and codec, not a curvature-transport operator. Curvature reference tools are labelled accordingly.


Normalization and evidence discipline

NS01 retain the exact authored definition, source edition and operator domain. NS02 retain identity separately from coordinate/payload. NS03 carry frame, unit, time, polarity and lineage when transferring a measurement. NS04 reject missing or singular frame data instead of guessing a scale. NS05 distinguish source axioms from empirical predictions and from a comparison model introduced here. NS06 count correlated evidence once and declare bias/covariance. NS07 retain a failed counterexample and its tested response. NS08 qualify a PASS by the executed scope and do not substitute a worker’s self-report for external verification. NS09 distinguish canonical parsing from trusted-content integrity. NS10 preserve finite capacity and reject wraparound. NS11 distinguish a curve diagram from measured physical emergence. NS12 treat absence, blocked, rejected and rehydratable as separate typed states.


Architecture and mechanism


Origin to general algebra

On the unbounded integer native domain N={1}∪{n:n≥2}∪{n:n≤−2}, define ρ(1)=0, ρ(n)=n−1 for n≥2 and ρ(n)=n+1 for n≤−2. This is bijective onto the integers. Its inverse is 1 at displacement zero, d+1 when d>0, and d−1 when d<0. Define a⊕b=ρ⁻¹(ρ(a)+ρ(b)) and a⊗b=ρ⁻¹(ρ(a)ρ(b)). Bijection transports associativity, commutativity and distributivity. The additive identity is native 1 and the multiplicative identity is native 2. For example 3⊕(−3)=1; 2⊕2=3; 3⊗3=5. The mathematical structure is isomorphic to integer arithmetic, while its symbols explicitly separate origin identity and displacement. This is not a proof that a finite five-state view is closed under these operations.

The paired-whole example (−3+)+(−14+)=(−17+) instead uses a whole record B(n) carrying magnitude n, counterparts, observer and lineage. A proposed same-orientation magnitude composition B(a)⊞B(b)=B(a+b) makes the example precise while retaining parent records. This operator is labelled a proposed completion; it is not conflated with ⊕ above. The browser calculator rejects subtraction or multiplication in this incomplete paired-operator domain.


Nested frame preservation

Represent a local point q by a frame transform f(q)=sRq+t, with positive scale s and orthogonal rotation R. For a child frame fi and shared frame F, world position is F(fi(q)). Since s>0 and R is invertible, q=fi⁻¹(F⁻¹(world)). Child identity is an independent field and is unchanged by relocation. A local separation Δq becomes sRΔq: orientation and local metric depend on frame, while the record identity remains stable. A naked magnitude without s or unit cannot support inverse interpretation. At s=0 the transform becomes singular and recovery is rejected. The outer rationale is metadata with no numeric scale and never enters the transform product.

This similarity-transform experiment supplies an executable comparison for the box construction. It neither claims that the boxes are metrically identical nor substitutes coordinate recoverability for a three-body solution. The primary preprint explicitly preserves classical non-integrability and sensitive dependence.


Observer theorem and reference quality

The primary preprint defines true value V(O), self-read value Ṽ(O|E), signed gap Γ=V(O)−Ṽ(O|E), and R(E) as mutually independent sectors, with shared method/instrument/training/provenance counted once. A3 says the types are distinct and adds monotonic decrease of |Γ| with richness. The source explicitly calls the monotonicity an authorial choice and proposes environment-enrichment experiments in which the learner is held fixed.

A useful finite observation model is y1=s+f and y2=s+kf+ε. Its observation matrix has determinant k−1, giving f=(y2−y1)/(k−1) and s=y1−f when k≠1. Exact degeneracy prevents identification; near degeneracy amplifies noise by 1/|k−1|. Complete provenance identifies what the readings mean but does not create the missing independent equation.

For an unbiased sample-mean reference with independent equal-variance errors εi, E[Γ²]=σ²/k. This follows by expanding Var((Σ εi)/k): the cross-covariance terms vanish. With common correlation c, the variance is σ²[1+(k−1)c]/k; with mean bias b, MSE adds b². Thus reference count alone is insufficient. Even under the iid assumptions, the law is an expectation, not deterministic monotonic narrowing for every realised observation. Truth=10, first reading=10 and added independent biased reading=14 gives gaps 0 and −2. The source axiom, conditional statistical prediction and this counterexample remain separate in the ledger.


Square diamond circle and curvature

For one fixed origin and substrate, d∞(x,y)=max(|x|,|y|), d1(x,y)=|x|+|y|, and d2(x,y)=sqrt(x²+y²). Their continuous radius-r shells are square, diamond and circle. On the integer lattice the square shell has 8r states: (2r+1)²−(2r−1)². The diamond shell has 4r states. At r=2 the counts are 16 and 8, so equal-radius norm shells are not the same discrete set after a rotation. A rotated square and diamond can be related in continuous geometry only with the appropriate scale/radius relation; that does not automatically preserve integer lattice membership.

For a regular n-gon inscribed in a unit circle, each exterior turn is 2π/n and total turn is 2π. Its perimeter is 2n sin(π/n), approaching 2π. At a square’s sharp corners, curvature is retained as a discrete turning-angle measure rather than assigned a finite smooth pointwise value. MIT’s discrete/smooth geometry material supplies the independent comparison [E08]; the finite derivation is checked here.

The workbench’s curvature embedding is explicitly F(x,y)=(x,y,c(x²+y²)). Projection onto the first two components is an inverse on its image, so coordinate identity survives. It does not preserve Euclidean distances. For graph z=f(x,y), Gaussian curvature is (fxx fyy−fxy²)/(1+fx²+fy²)². Substitution gives K=4c²/[1+4c²(x²+y²)]², with inverse-length-squared units when c carries inverse-length units. This reference does not establish a native ZCG transport, a sphere closure theorem or path-dependent phase holonomy.


Bell curve and environmental emergence

For p(x;μ,σ)=exp(−(x−μ)²/(2σ²))/(σ sqrt(2π)), p′(x)=−(x−μ)p(x)/σ². Its unique mode is μ, and p(μ−d)=p(μ+d). Setting μ=1 yields a symmetric bell about coordinate one; at σ=1 the peak height is about 0.39894, not one. The integral equals one for every μ and σ>0. If x is transformed by x′=ax+b, a>0, the density transforms as p′(x′)=p((x′−b)/a)/a. This proves why a peak location is frame-dependent while total probability is unchanged [E07]. The bell can be read bilaterally around an admitted origin; it does not uniquely select the numeral used to label that origin.

The author’s stronger environmental claim is meaningful as a distinct research programme: a whole’s formation and persistence may depend on local density, reference richness, coupling, retention and dispersion. The acquired formula packet includes provisional relations such as Wp=((Bp×Ct×Kc×Re)/(Dd+Ic))×En and alternatives with support minus losses. Their terms and revisions must be retained; an environmental normalisation factor is not automatically a Gaussian density or an observed emergence probability.

To attempt falsification, this work introduces a transparent comparison model da/dt=g(ρ)a−a³ with g(ρ)=ρ−1−cρ², a≥0. At a=0 the linear growth rate is g. When g>0 the positive equilibrium sqrt(g) is stable because the derivative at equilibrium is −2g. When g≤0 there is no positive equilibrium. With c=0.15, g(4)=0.6 and g(8)=−2.6. Density can therefore aid growth and then inhibit it through crowding in this stated model. This is an illustrative alternative, not a recovered native physical law and not an empirical falsification of all natural emergence.

The measured-data tool accepts independent observations with density and an emergence indicator, splits densities at the sample median, computes high/low event rates, and permutes event labels 499 times using a declared seed. The reported two-sided Monte Carlo permutation value is (extreme+1)/500. It is an unadjusted association test; causal emergence requires randomized or blocked density interventions, controlled co-factors, preserved failures and held-out conditions. Repeated readings from one specimen require cluster-aware inference. No actual physical observation is invented or bundled as measured evidence.


ZCG codec and proof correction

The acquired September engine defines finite row-major coordinates from ordinal n: x=(n mod X)+1, y=(floor(n/X) mod Y)+1, z=floor(n/(XY))+1, with capacity XYZ−1. The inverse is (z−1)XY+(y−1)X+(x−1). The reserved coordinate (1,1,1) has no run ordinal. Capacity exhaustion raises a typed error rather than wrapping. The codec stores canonical symbol/run-length records with bounds, source length and run count in a binary header. It supports byte zero and does not claim universal compression.

Proof preimages pack coefficient numerator/denominator, run length and revision as uint64, and coordinates/bounds as uint32. The original validation checks positivity and type but omits upper bounds. Consequently values of 2⁶⁴ or 2³² can reach struct.pack and escape as an untyped low-level exception. Invalid surrogate text likewise escapes strict UTF-8 encoding. The correction candidate checks field widths before packing and converts UnicodeEncodeError to InvalidStateError. No schema or valid-vector bytes change. Original archives, candidate source, the exact diff, original failure observations and regression receipts are preserved together.


Method and evaluation

The research uses three lanes: exact author-source acquisition, execution of the acquired native program, and separately defined comparison calculations. The primary preprint, origin manuscript, original ZCG archives and relevant source clarifications were read. Curvature/holonomy titles were sought using exact and broader queries; the acquired codec does not contain those operators. A saved NotebookLM shell yielded no substantive compiler source and is not treated as proof. A workbook fetch returned no readable content; that is an acquisition limit, not a finding about the workbook’s mathematics.

The earlier original KEX calculation receipt remains reproduced byte for byte. Its digest excluding its own hash field is 31ba82c00739db6ef07e724a863dc317b885622d4eb7a1360bb7702a28b79513. This extension does not overwrite that evidence. The generated_utc in the receipt is a fixed source timestamp rather than the date of the new execution.

test_research.py uses seed 20261010 and standard-library finite tests. Native unit tests and exception replay run as separate subprocesses. The candidate runs in a separate import environment so the original module cannot contaminate the correction test. Frame errors use a 1e−10 tolerance; maximum observed error is approximately 3.568e−13. The observer MSE experiment uses 12,000 trials at each of k=1,2,4,8,16. Compression comparison measures bytes, not a universal throughput advantage.

The browser model is dependency-free and source-checked against three original golden codec vectors. Its live suite runs ten reference groups plus the golden-vector group, while session exports retain parameters, results and failures. Missing or singular frame recovery blocks admission. Unknown sections return to the research map. User-entered and synthetic observations are labelled distinctly. Local storage failure is reported. Interactive behavior was checked by source/syntax and model tests; managed browser QA is unavailable because the required browser-control skill is absent. No visual browser or accessibility certification is claimed.


Results and evidence

Experiment | Executed scope | Result
Original ZCG regression | 37 source tests | 37 passed
Exception replay | 13 typed machine codes | 13 matched
Proof boundary attack | 7 oversized/text probes | Original low-level failures; candidate typed rejections
Correction regression | 37 unchanged original tests | 37 passed
Native addressing | 512 grids; 46,144 coordinates | Zero collisions; inverse verified
Varied codec corpus | 2,000 seeded payloads | All byte-exact round trips
Single-bit mutation | 408 altered streams | 252 malformed; 156 structurally valid; 408 digest changes
Transported algebra | 9,261 triples | Associativity/distributivity checks passed
Nested frames | 5,000 transforms | Max recovery error 3.568e−13
Norm shells | 30 integer radii | Square 8r; diamond 4r
Polygon convergence | 7 resolutions | Total turn 2π; perimeter error decreases
Observer model | 12,000 trials per reference count | MSE improves under declared iid model
Bell comparison | 4 centres × 3 spreads | No unique numeric origin follows from bell form
Browser/native parity | 3 original golden vectors | Exact packed bytes and decode matched


Observer MSE under declared assumptions

Reference count | Expected MSE | Measured MSE
1 | 1.0 | 1.013832
2 | 0.5 | 0.50763
4 | 0.25 | 0.25579
8 | 0.125 | 0.125166
16 | 0.0625 | 0.061987


Representation sizes

Case | Input bytes | ZCG bytes | zlib bytes
empty | 0 | 11 | 8
long-run | 4096 | 15 | 28
alternating | 4096 | 8206 | 28
all-bytes | 4096 | 8206 | 315
seeded-random | 4096 | 8192 | 4107

These comparisons show the intended boundary: the ZCG codec is reversible, but it expands alternating and random inputs relative to raw bytes and often relative to zlib. A finite non-overlapping address system and a compression advantage are separate claims. The complete corpus and sizes are preserved in audit.json.


Falsification ledger


F01 Every proof validation failure is typed

Outcome: FALSIFIED in original.

Probe field | Original outcome
run_length | error
state_revision | error
coefficient_numerator | error
coefficient_denominator | error
bounds | error
coordinate | error
coefficient_version | UnicodeEncodeError

Address: Add explicit uint64/uint32 bounds and convert invalid UTF-8 to InvalidStateError in a separate correction candidate.


F02 Concurrent random corpus exercises varied bytes

Outcome: FALSIFIED.

Original unique symbols: 1. Corrected unique symbols: 256. The original finite-codec tests still pass; the defect is the corpus’s advertised coverage, not a demonstrated codec round-trip failure.

Address: Seed once outside comprehension; added 2,000 reproducible diverse round trips.


F03 A decodable stream is necessarily unchanged

Outcome: FALSIFIED.

Raw decoding accepts a changed but canonical symbol stream. Parser correctness is preserved; trusted-content integrity needs an independent expected digest. The same sender’s replacement digest is not independent warrant.

Address: Verify a trusted out-of-band SHA-256 before decoding; a digest supplied by the same untrusted sender is insufficient.


F04 A naked magnitude survives frame transfer

Outcome: FALSIFIED.

Counterexample: local 2 at scale 10 is shared 20; reporting 2 without scale loses recoverability.

Address: Carry frame id, unit, transform, polarity, parent and lineage; reject missing/singular transforms.


F05 Equal-radius L∞ and L1 shells are just the same lattice shell rotated

Outcome: FALSIFIED.

Counterexample: radius 2 gives 16 square states and 8 diamond states; continuous area is 16 vs 8.

Address: Distinguish norm change from rigid rotation with radius/scale correction; retain substrate ids.


F06 More independent references guarantee a smaller gap in every realised case

Outcome: FALSIFIED as an unconditional numeric prediction.

Address: Keep the source A3 axiom distinct; test expected error under declared bias/covariance and estimator assumptions. Independence alone does not ensure accuracy.


F07 A bell curve alone uniquely derives coordinate origin 1

Outcome: NOT ESTABLISHED; shifted bell curves are counterexamples to uniqueness.

Address: Retain whole normalization ∫p=1, anchor identity U=1, mode location μ and physical environmental density as distinct typed quantities. Test emergence through interventions, not recentering.


F08 Increasing environmental density always favours emergence

Outcome: FALSIFIED in a declared crowding model, not a falsification of a supplied native law.

Address: Measure coupling, dispersion and crowding; density can support, impede or shift an emergence threshold.


Preserved foundational proofs


Proof R01 Ordered state distance

Because r is injective, d(a,b)=|r(b)-r(a)| is nonnegative and vanishes exactly when a=b. Absolute value gives symmetry. The real-line triangle inequality gives d(a,c)≤d(a,b)+d(b,c). Thus d is a metric on the five declared states [K2]. All finite cases were checked. Adjacent labels have unit rank distance, including -2→1.

The label differences along P are (1,3,1,1); rank differences are (1,1,1,1). Total label distance is 6 and total traversal distance is 4. The 50% excess follows only when the intended cost is one per adjacent transition. It is not a claim that ordinary scalar subtraction is incorrect or that measured energy must differ by 50%.


Proof R02 Bilateral rebound preserves one

Define m(s)=r⁻¹(6-r(s)). Applying it twice gives r⁻¹(6-(6-r(s)))=s. Its fixed point solves r(s)=6-r(s), hence rank 3 and state 1. Rebound preserves distance because |(6-r(a))-(6-r(b))|=|r(a)-r(b)| [K5]. The exact mirror is -3↔3, -2↔2 and 1↔1. Raw negation would send 1 to an inadmissible -1 in this model.

This provides a precise operator interpretation compatible with the recorded observer expression “-(1.0)==1”: a rebound operator fixes the anchor. It does not establish equality of ordinary real numbers -1 and +1. The July topology’s central anchor and the later paired-whole notation must remain separate until a declared mapping joins them.


Proof R03 Closed histories erased by net-zero projection

Let A be the adjacency matrix of the path with five vertices. Its characteristic polynomial is λ(λ²-1)(λ²-3), obtained by the path recurrence D_n=λD_(n-1)-D_(n-2), D_0=1, D_1=λ. Its eigenvalues are √3,1,0,-1,-√3. For m≥1, tr(A^(2m))=2(3^m+1); odd traces vanish because the graph is bipartite [K4]. Each trace term counts a return walk rooted at its starting vertex. Every such walk has zero net rank displacement while retaining a different path.

Exact counts are 8 at length 2, 20 at length 4, 488 at length 10, 118,100 at length 20 and 6,973,568,804 at length 40. Both the original matrix script and the separate recurrence reproduced these totals. The count includes all five possible start vertices. It must not be presented as the count for the anchor alone. If these histories are equiprobable and only net zero is retained, the lost history information is log2(C_n) bits: approximately 16.85 bits at length 20 and 32.70 bits at length 40. Different probability models change entropy but not the collisions.


Proof R04 Observer identifiability

The observer paper defines O_r=T(S,M_o,F,H,R) and a linear case y_o=M_o s+F_o f+noise [K6]. In the noiseless scalar construction y1=s+f, a single observation cannot identify s and f: for any h, (s+h,f-h) has the same output. A second independent observation y2=s+2f gives f=y2-y1 and s=2y1-y2. The corresponding block design has rank 2n rather than n. This recovery was verified for all 441 pairs s,f∈{-10,…,10}.

The result proves why observer/foreground separation needs independently informative measurements, not merely extra metadata. Noise requires uncertainty propagation; nearly identical transforms can be poorly conditioned even when their rank is full. Environment inheritance is therefore represented as a modeled influence, without asserting that every experiment must suffer an uncontrolled effect or that a controlled environment begins from ontological nothingness.


Proof R05 Retained balance and existence

One hundred distinct cancelling input pairs were serialized and recovered exactly in a full record representation. All have scalar sum zero but their native records remain distinct. This directly reproduces the information-preservation principle. The distinction e(s)=1 and v(s)=0 is satisfied by an ordinary record model. A zero displacement, zero temperature coordinate or zero outcome is therefore compatible with preserved identity.

Metrology agrees that a quantity value combines a number and reference, and that a measurement result includes relevant information and ordinarily uncertainty [E2,E3]. Zero Celsius is a valid coordinate equal to 273.15 K, not an absent sample. The master document’s ratio example is correct in its intended distinction: 20°C/10°C=2 compares offset labels, whereas 293.15 K/283.15 K≈1.0353 compares thermodynamic temperatures [K7,E4]. Phase and time require additional state information; temperature alone does not determine whether a sample is ice, liquid or transitioning.


Proof R06 Coordinate and phase retention

The earlier 1–9 XYZ learning record establishes 729 distinct ordered addresses [K8]. Their uniqueness follows from ordered-tuple equality, not from inclusion of zero. The live atlas subsequently clarifies that origin identity itself is singular 1, rather than the tuple (1,1,1). Address components and whole identity therefore remain different fields.

The foundations define a phase-tagged temperature map: C<0 maps to C-1, C>0 to C+1, and external C=0 with ice/liquid tags maps to -1/+1 respectively [K9]. Its branches have disjoint images and explicit inverses, hence form a bijection on the stated domain. This is an invertible encoding, not a complete phase diagram: actual coexistence and other phases require a richer input domain.


Proof R07 Quantum state and zero outcome

For a normalized state ψ, <ψ|ψ>=1. The Hilbert-space zero vector has norm zero and cannot represent a normalized physical pure state. A component amplitude or measurement probability can nevertheless be zero. A computational basis name |0> denotes a normalized basis state, not the zero vector [E5]. This makes the defensible statement precise: a physically admitted normalized state is not an absent vector, even when a particular projection vanishes.

Consider ψ=(|a>+e^(iφ)|b>)/√2 and an output mode u=(|a>+|b>)/√2. Its probability is |<u|ψ>|²=(1+cosφ)/2. At φ=π the probability is zero, yet ψ retains unit norm. The cancellation occurs in the output-mode amplitude; adding amplitudes attached to different orthogonal basis vectors does not erase those vectors. Source and transformation context distinguish destructive interference from absent preparation. Conventional quantum theory already makes this distinction; KEX supplies an explicit provenance requirement for its computational representation.

For a pure state, ψ and -ψ differ by a global phase and have the same density matrix: (-ψ)(-ψ)†=ψψ†. This is a physically meaningful interpretation of sign reversal preserving state identity. It does not make -1=+1 in scalar arithmetic or make a relative phase unobservable. Reversing just one branch can change interference.


Proof R08 Non-factorability and recursive capacity

The Bell reference coefficient matrix is proportional to diag(1,1), with rank two. A factorized pure state has coefficient matrix uvᵀ and rank at most one. Therefore the Bell reference state is non-factorable. This proves the reference criterion, not that every overlapping KEX frame is entangled. R3 explicitly requires a native joint-state construction and factorability test [K11].

A recursive frame with five independent choices per level has 5^n potential addresses before overlap and constraints. This is a count of combinations. It does not establish simultaneous physical storage, free access to all combinations or quantum speedup. Identity preservation can prevent duplicate carriers while retaining multiple frame memberships, but the resolution rule must say whether duplicate paths refer to one object or distinct events.


Proof R09 HEX folding and calibration

The foundations’ fold F_j(a,b)=(ab+(a XOR b)+j) mod 16 has preimage counts {8 repeated 8 times,16 repeated 4 times,24 repeated 2 times,40 repeated 2 times}. Exhaustive enumeration confirmed these counts for every j, giving Shannon entropy 3.7272170014624826 bits per nibble under uniform input [K9]. Adding j permutes labels. The fold is not injective and therefore requires extra retained state for lossless recovery. This is consistent with the projection-loss doctrine.

The archived formula k=1-1/e and ln(cosh(k))/k evaluates to 0.2970137033667769. It rounds to 0.297 at three decimal places; it is not exactly 0.297. The related uniform-continuous-input expectation follows by integrating tanh(kx) over [0,1]. Discrete hash-byte sampling requires a discrete average instead. Neither expectation makes arbitrary input streams converge to that value. The live atlas records the author’s later statement that 0.297 is a literal jitter function. The public resonance page separately imposes 0.297 as a target. Those three source meanings remain distinct; a configured target is not a discovery of an attractor.

The archived decomposition’s Hamming expression Σ_(i=0)^12 binom(256,i)/2^256 evaluates to approximately 1.1558269801063893×10^-57 for a uniformly random independent 256-bit string, rather than its stated approximation 10^-48. The tail probability alone does not establish a security proof, particularly when inputs are correlated or selected adversarially. This is a numerical correction to that source’s stated model, not a rejection of the wider architecture.

Source reconciliation: the earlier R3 quantum-formalism artifact leaves weighted native operations open. The later October continuity clarification separately reports a native simulator with complex state, phase, tensor operations and Bell checks. Those are distinct artifacts and scopes. This report does not infer the absence of all native simulator code from the R3 artifact’s open list; nor does it treat a later simulator report as acquired hardware evidence.


Extensive case applications


C01 Robot cell calibration

Purpose: Transfer a tool position across local robot, workstation and inspection frames.

Human benefit: Keep millimetres, polarity and the calibration frame with every coordinate.

Inputs: Local point (2,1); scale 10; rotation 30°; translation (4,−2).

Acceptance: Invertible transform recovers the local point and retains the same tool identity.

Attempt to falsify: Discard the frame or set scale to zero.

Response: Reject recovery with FRAME_MISSING or FRAME; acquire the calibration again.

Evidence: 5,000 separate frame round trips; maximum error in audit.

Scope: An invertible coordinate transform does not itself certify robot calibration.

Tool: nested. Source: P01.


C02 Clinical sensor readings

Purpose: Prevent a measurement from silently changing meaning across instruments.

Human benefit: The patient and measurement state remain present when a displayed reading is zero.

Inputs: Two temperature readings with declared Celsius/Kelvin scale, time and patient id.

Acceptance: Translate numeric scale while retaining patient, instrument, phase and uncertainty fields.

Attempt to falsify: Treat 0°C as absence, or merge readings lacking patient/frame identity.

Response: Use typed MEASURED_ZERO and explicit units; reject missing provenance.

Evidence: Origin/zero role separation reproduced in v1 and formalised in source.

Scope: No clinical validation or patient dataset is supplied.

Tool: nested. Source: K07.


C03 Relocatable software object

Purpose: Move backing storage without rewriting child identities.

Human benefit: Stable identity makes recovery and tracing possible after a host changes.

Inputs: Run ordinals in an 8×8×8 grid; reserve the origin coordinate.

Acceptance: Every ordinal 1–511 has a distinct coordinate and direct inverse.

Attempt to falsify: Wrap ordinal 512 into a previously occupied location.

Response: Raise capacity exhaustion; expand/version the frame instead of wrapping.

Evidence: 46,144 coordinates checked across 512 grids.

Scope: The acquired map is finite, not an infinite toroidal volume.

Tool: zcg. Source: Z01.


C04 Source and foreground separation

Purpose: Recover a signal from two differently transformed observations.

Human benefit: Declare what is measured and how environmental interference enters.

Inputs: y1=s+f; y2=s+kf. Default s=3, f=2, k=2.

Acceptance: Recover s=3 and f=2 when the determinant k−1 is nonzero.

Attempt to falsify: Use k=1, or nearly identical transforms with noise.

Response: Reject exact degeneracy; report noise amplification as k approaches one.

Evidence: 441 exact mixtures checked in both reference implementations.

Scope: Noisy recovery requires uncertainty propagation and calibrated operators.

Tool: observer. Source: K06.


C05 Telemetry self-report

Purpose: Distinguish a worker claiming completion from verified external state.

Human benefit: Users receive evidence of the effect, rather than a success flag alone.

Inputs: Truth 10; first reading 10; added independent biased reading 14.

Acceptance: Keep SELF_REPORT and MEASUREMENT distinct and examine their signed gap.

Attempt to falsify: Assume a larger reference count guarantees a smaller realised gap.

Response: Test reference bias/covariance and estimator design; preserve failed monotonic prediction.

Evidence: Gap widens from 0 to 2 in the explicit counterexample.

Scope: Independent instruments can still be biased; software replay is not external assessment.

Tool: observer. Source: P01.


C06 Signed engineering residual

Purpose: Keep whether a system over-reads or under-reads.

Human benefit: Direction tells an operator which correction to apply.

Inputs: Residuals +3 and −3.

Acceptance: Store (direction,magnitude,frame); both results remain distinguishable.

Attempt to falsify: Replace each residual solely by its square 9.

Response: Retain signed residual alongside any squared loss used for optimisation.

Evidence: Noninjectivity supplies the counterexample; source A1/A4 state the rule.

Scope: A squared objective can coexist with retained raw signed data.

Tool: algebra. Source: P01.


C07 Balanced financial flows

Purpose: Preserve activity behind a net balance.

Human benefit: An audit can reconstruct the transactions that produced a zero net result.

Inputs: Opposing flows (+20,−20), (+70,−70), with distinct event identities.

Acceptance: BALANCED retains operands, rule, time, parties and lineage.

Attempt to falsify: Store only scalar net zero and attempt to recover the original transactions.

Response: Keep an event record and project the net as a display field.

Evidence: 100 distinct cancelling pairs retained and round-tripped in v1.

Scope: Arithmetic netting stays valid; complete transaction recovery needs the record.

Tool: algebra. Source: K03.


C08 Navigation distance law

Purpose: Compare reachability shells under different move rules.

Human benefit: A robot or planner can explain why diagonal and orthogonal moves produce different paths.

Inputs: Radius 2 on the same integer substrate.

Acceptance: L∞ shell has 16 states; L1 shell has 8; state identities stay unchanged.

Attempt to falsify: Assume the norm change is a rigid rotation with the same lattice states.

Response: Declare distance rule, radius and scale separately from object orientation.

Evidence: 30 radii checked: counts 8r and 4r.

Scope: Continuous shape equivalence up to rotation/scaling does not preserve discrete shell membership.

Tool: geometry. Source: O01.


C09 Curved mesh projection

Purpose: Project addressed grid states onto a declared curved surface.

Human benefit: A visualiser can change shape while a record still identifies the same state.

Inputs: Embed (x,y) as (x,y,c(x²+y²)); vary c.

Acceptance: Recover x,y from the projection; calculate Gaussian curvature K=4c²/(1+4c²r²)².

Attempt to falsify: Treat image curvature as proof of a native curved ZCG metric.

Response: Label the embedding as a reference construction; require native transport/metric before promoting it.

Evidence: Regular polygon total turn and circle convergence tested separately.

Scope: Acquired ZCG codec contains no curvature operator; this surface does not fill that gap by assertion.

Tool: geometry. Source: O01/Z01.


C10 Circle boundary resolution

Purpose: Distinguish polygon corners from smooth circular curvature.

Human benefit: A rendering can explain discretisation error instead of pretending a coarse mesh is exact.

Inputs: Regular polygons with 4,8,16,32,64,128,256 vertices on a unit circle.

Acceptance: Total turn stays 2π while polygon perimeter approaches circle circumference.

Attempt to falsify: Assign a finite smooth curvature to a sharp square corner.

Response: Use discrete turning-angle measure at corners; use smooth curvature only on smooth segments.

Evidence: Seven resolutions passed convergence checks.

Scope: These are reference geometry tests, not physical emergence observations.

Tool: geometry. Source: O01.


C11 Density and natural whole emergence

Purpose: Test the user’s proposed environmental influence around a natural origin.

Human benefit: Measure when a coherent whole forms, persists or fails under changed conditions.

Inputs: Declare whole id U=1; vary physical density, coupling, dispersion and crowding.

Acceptance: Track a specified emergence event under controlled density interventions.

Attempt to falsify: Infer universal origin one from a Gaussian centred at one.

Response: Keep anchor identity, mode, probability normalisation and physical density as separate fields.

Evidence: Four shifted means and three spreads tested; no unique numerical centre follows.

Scope: Physical emergence is a testable hypothesis; no raw physical experiment was acquired.

Tool: density. Source: U02.


C12 Crowding and retention

Purpose: Investigate whether more density helps until crowding dominates.

Human benefit: Avoid selecting an operating point that destabilises the very structure being built.

Inputs: Illustrative da/dt=(ρ−1−cρ²)a−a³, c=0.15.

Acceptance: At ρ=4 growth coefficient is +0.6; at ρ=8 it is −2.6.

Attempt to falsify: Assume density has a universal monotonic benefit.

Response: Measure positive coupling and negative dispersion/crowding terms separately.

Evidence: The stated toy model produces the counterexample exactly.

Scope: This model is authored for testing here, not recovered as a Keddeh physical law.

Tool: density. Source: U02/P02.


C13 Unequal left/right environment

Purpose: Inspect a density profile that is not symmetric about the declared anchor.

Human benefit: Expose asymmetric environmental support without moving the identity of the whole.

Inputs: Two Gaussian components; vary left fraction and separation.

Acceptance: Report peak, left/right mass and declared anchor independently.

Attempt to falsify: Force every profile to look symmetric by recentering it.

Response: Preserve original coordinates and model parameters; mark recentering as a transform.

Evidence: Workbench provides a mixture control and shifted-centre counterexample.

Scope: A probability mixture is a comparison model; physical density needs units and sampling volume.

Tool: density. Source: U02/E07.


C14 Measured emergence data

Purpose: Analyse observations without confusing a diagram with an experiment.

Human benefit: A researcher can upload data, inspect direction of association and export the calculation.

Inputs: CSV columns density,emerged; 8–2,000 independent observations; emerged is 0 or 1.

Acceptance: Compute a median split, high/low event rates and a seeded permutation test.

Attempt to falsify: Confound density with temperature, or treat repeated readings as independent.

Response: Collect randomized/blocked interventions and cluster identifiers; do not make causal claims from this tool.

Evidence: Live 499-permutation test executes on entered data; empty input produces no empirical finding.

Scope: Unadjusted association only; cluster-aware and multivariable analysis remains outside this small tool.

Tool: density. Source: U02.


C15 Archival representation cost

Purpose: Compare codec size across different byte structures.

Human benefit: Select storage encoding using measured workload rather than universal compression claims.

Inputs: Empty, long-run, alternating, all-byte and seeded-random payloads.

Acceptance: Native ZCG round trips each case; inspect actual packed size against input and zlib.

Attempt to falsify: Claim every source shrinks because addresses begin at one.

Response: Keep reversibility separate from compression; use workload-specific comparisons.

Evidence: Five acquired-code benchmark cases recorded in audit.

Scope: Size comparisons are implementation-specific; no throughput advantage is asserted.

Tool: zcg. Source: Z01.


C16 Archive tampering

Purpose: Catch changed content even when the stream still parses.

Human benefit: A valid-looking report cannot silently substitute a different payload.

Inputs: Change the symbol in a stream encoding 100 A bytes to B.

Acceptance: Raw decoder yields 100 B bytes, while a trusted expected hash detects the change.

Attempt to falsify: Equate successful decoding with content integrity.

Response: Verify a trusted digest before decode; reject changed bytes.

Evidence: Every tested single-bit mutation changed the digest; parser rejects only malformed cases.

Scope: Hash custody matters: an untrusted sender can replace both content and its own hash.

Tool: zcg. Source: Z01.


C17 Proof format boundary

Purpose: Keep malformed proof failures inside typed exception handling.

Human benefit: Callers receive a stable error instead of an uncaught low-level packing failure.

Inputs: Run length/coefficient/revision above uint64; coordinate/bounds above uint32; invalid UTF-8.

Acceptance: Correction candidate rejects all seven probes with ZCG-E001.

Attempt to falsify: Pass an oversized integer through the original validation to binary packing.

Response: Check exact packed widths before packing and convert invalid text encoding into a typed error.

Evidence: Seven original failures reproduced; correction plus all 37 original tests passed.

Scope: Correction is a separate local candidate; source Drive archives were not changed.

Tool: zcg. Source: Z01.


C18 Distributed waiting

Purpose: Keep one blocked task from suppressing unrelated runnable tasks.

Human benefit: A user can see what is waiting while useful work continues.

Inputs: A waits for dependency X; B,C,D are runnable.

Acceptance: Run B,C,D fairly without erasing A’s identity.

Attempt to falsify: Use a global waiting node or a blocking queue head.

Response: Use dependency-local suspension, cycle detection, fairness and bounded leases.

Evidence: Reference selection sequence B,C,D,B,C,D executed.

Scope: Unique addressing alone does not prove scheduler liveness or deployment.

Tool: observer. Source: P03.


C19 Dimensional address projection

Purpose: Represent the same finite population with different coordinate factorizations.

Human benefit: Organise objects without confusing a view change with creation of new objects.

Inputs: 64 stable ids shown as 8² or 4³.

Acceptance: Cardinality remains 64; attach a declared bijection and view id.

Attempt to falsify: Infer equal adjacency, distance or physical dimensionality from equal cardinality.

Response: Check relations explicitly; cardinality equivalence is weaker than geometric equivalence.

Evidence: Source origin manuscript distinguishes observer-governed dimensional definition.

Scope: The native codec reserves an origin, so an 4³ codec grid holds 63 run slots, not 64.

Tool: zcg. Source: O01/Z01.


C20 Quantum phase projection

Purpose: Retain phase-bearing state behind a probability readout.

Human benefit: A zero-probability outcome does not erase the normalised physical state.

Inputs: Two paths with relative phase φ; p=(1+cosφ)/2.

Acceptance: At φ=π output probability is zero while total state norm remains one.

Attempt to falsify: Treat zero amplitude, basis label |0⟩ and all-zero vector as the same entity.

Response: Type full state, coefficient, basis and readout separately.

Evidence: Constructive/destructive and Bell non-factorability reference checks passed in v1.

Scope: A source reports a native simulator; that report is not a native hardware qualification.

Tool: observer. Source: K11/P03.


Limitations and open questions

The original ZCG executable slice has no curvature operator or holonomy transport. The workbench does not claim that a paraboloid projection qualifies such a system. Exact named native curvature, rotational geometry and holonomy artifacts must be acquired before their execution law is tested. The provisional curvature/retention archive is preserved as a family of distinct research objects, rather than compressed into the codec.

The box-transform experiment checks recovery of local coordinates under invertible similarity transforms; it is not a test of multi-body physical dynamics. The observer experiment checks a specified estimator and source/foreground model; it is not a universal law of cognition. The bell and crowding examples are mathematical comparison models. Physical emergence needs a defined natural whole, quantities with units, time-resolved observations, controlled interventions, failure capture and a native prediction specified before observing the outcome.

The code correction is a local candidate and has not been promoted into the source Drive archive, a governed BRAINK repository or a deployment of the native engine. The privately hosted workbench is a research tool, with a separate source/version/deployment receipt. It does not edit keddeh.com or another established estate project. A successful native Sites receipt establishes the workbench deployment; browser interaction and responsive rendering remain unverified in a browser in this environment.


Concrete next experiment for density emergence

Declare a whole W by an observable coherence/persistence criterion before selecting its coordinate label. Record its identity as U=1 and retain measured physical density ρ, sampling volume, energy/coupling, dispersion, time, boundary conditions and observer transform. Generate at least three density conditions with independent specimens or randomized repeated trials. Preserve left/right asymmetry rather than recentering it away. Compare a density-neutral baseline with a specified density-dependent model and a crowding/dispersion alternative. Hold back conditions for prediction. Record onset threshold, persistence duration, collapse rate and uncertainty. Reject the chosen density law if its preregistered direction or threshold fails outside calibration uncertainty; retain the failed model and measured state as typed evidence.

This experiment addresses the substantive author claim: environmental conditions can affect the emergence and persistence of a natural whole. It does not require proving that a coordinate numeral causes emergence. Whether the natural-whole boundary, density relation and origin law form a stronger native mechanism is the object of the experiment, not an assumed conclusion.


References

P01 An Unaccepted Evolution — primary preprint, September 2026. PRIMARY AUTHOR SOURCE. Four constraints, observer theorem, boxes §4.2, empirical predictions and declared limits. https://docs.google.com/document/d/11IjwDz_VPRWyA0samFCrOI7jLjnT-cwFH_HD2Z02izc/edit

O01 ORIGIN OF ZERO — KEDDEH SYSTEMS RESEARCH.pdf. SUPPLIED RESEARCH / CONVERSATION MANUSCRIPT. 82-page acquired text; origin-to-displacement algebra and norm-based shell relations. Not an external assessment.

Z01 ZCG Core Engine archive — 15 September 2026. ORIGINAL EXECUTABLE SOURCE. 37 tests, 13 exceptions, finite map, codec, proof preimage and golden vectors. https://drive.google.com/file/d/1U5ReU8jEMvI866ZB7Zn7PAMXQtbPCoof/view

Z02 Earlier ZCG exception boundary archive. ARCHIVED EARLIER EDITION. Keep exception-only and codec editions distinct. https://drive.google.com/file/d/1sSgc7brItA9IcIh0LINv7XW5ayiVnxOg/view

P02 KEX/BRAINK Formulas topic packet. RETRIEVAL PACKET. Wake/curvature/retention, coupling, dispersion and environmental normalisation; formulas are provisional and versioned. https://drive.google.com/file/d/1TW5tLJgSzalcZqqUeiHuS36i-Vjx_QPp/view

P03 Less Zero, Continuity and Distributed Loop Logic — 2 October. SOURCE RECONCILIATION. Zero roles, typed continuity, scheduler constraints and reported quantum mapping. https://drive.google.com/file/d/1Fsh2KYOlgcIE6QxzIXVI1GQsmjA7w_he/view

B01 Boxes in a Box — source learning and deep report. SECONDARY SOURCE LEARNING. Supports source discovery; primary preprint now acquired directly. https://drive.google.com/file/d/1nimxe8zwF9y7uHfly0kcbhNtLVtixdwb/view

E07 NIST — Normal distribution. EXTERNAL PRIMARY TECHNICAL REFERENCE. Normal PDF with independent location and scale parameters. https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm

U02 Author clarification — 10 October 2026. CURRENT AUTHOR STATEMENT. Left/right density around origin one and environmental influence on natural whole emergence.

E08 MIT, Discrete vs. Smooth Geometry, 18.095 material, Justin Solomon, 2019. https://math.mit.edu/classes/18.095/2019IAP/solomon_pset.pdf . Turning angle and distinct discrete curvature constructions.

The v1 source ledger below is retained for all preserved R01–R09 derivations. Drive IDs identify acquisition targets; equal titles do not establish canonical identity.

K1 A. Keddeh. KEX/BRAINK Zeroless Computational Standard v1.0, 18 July 2026. Drive ID 1VE4Nh2Dg0yosmTn-gyJOYVNza284FaWL. https://drive.google.com/file/d/1VE4Nh2Dg0yosmTn-gyJOYVNza284FaWL/view

K2 A. Keddeh. 01_ACTIVE_STATE_FOUNDATIONS_v1_0.docx. Drive ID 1m2ipz_7dwt6t3Qeyhmnp3hkD6FTJ01ve.

K3 A. Keddeh. 02_TYPED_ZERO_NON_ERASING_ARITHMETIC_v1_0.docx. Drive ID 1pyJOwseyunwErpxkVJzF9w_FxuuiNmtA.

K4 A. Keddeh. 03_ORDERED_TOPOLOGY_PROJECTION_LOSS_v1_0.docx. Drive ID 1a_-soAFqNqmALiVRF91fopDI2WNi4bA3.

K5 A. Keddeh. 04_BILATERAL_REBOUND_RECURSIVE_GEOMETRY_v1_0.docx. Drive ID 1wDHj3WDkHuL3dLHA832dciffqeXQwYRi.

K6 A. Keddeh. 05_OBSERVER_ANCHORED_MEASUREMENT_v1_0.docx. Drive ID 1DXX1k2tWabswLE2zZ3QOcAgwtHti3wJz.

K7 A. Keddeh. KEDDEH_THEOREM_BRAINK_MASTER_v1_9_FULL_EDITED.docx. Drive ID 1kVVQgGxP4Hva8ovekUtY7l8ipVbpk3Im.

K8 Keddeh Systems learning record. KEDDEH_MATHS_01_LEARNER_RECOUNT_SYMBOL_CLASSIFICATION_1_ORIGIN_XYZ_20261002.docx. Drive ID 1CcQDTBGVMtncxIUaSO0Ze9GE3m13mDjM. AI-assisted recount, not independent assessment.

K9 A. Keddeh. KEX_BRAINK_Mathematical_Foundations_and_Proof_Obligations_v1_0.docx. Drive ID 1x3m7DwJbqSFMK4skvzf0pWUXauM-R_gX.

K10 A. Keddeh. KEX_BRAINK_Axiomatic_Research_Protocol_v1_0.docx. Drive ID 1hLS4NR4z1OjNzOCcAKL8dtbbHwZBFiAY.

K11 A. Keddeh. KEDDEH_QUANTUM_COMPUTATIONAL_FORMALISM_TECHNICAL_PUBLICATION_R3.html, 17 August 2026. Drive ID 1eVSSqgl6aXxOrM1XKqApSFft1loXh8mG.

K12 KEX_LAW3_ZEROLESS_PROTOCOL.md. Drive ID 1_vNOhRS8s8alw7j4TRcpnTwhNf1LnP0a.

K13 ZERO_LESS_SYSTEM_DELIVERY.md. Drive ID 18JlDlpRZRRLZcwjp2J9FnuTZ2WR7_2JI. Source-reported implementation/test summary.

K14 KEDDEH_Bilateral_Proof_Engine_White_Paper.pdf. Drive ID 19eMPYlqaJPDiS_crjU-4tCioyth-V63B. Source-reported workbook measurements.

K15 kex_whitepaper_calculations.py and KEX_ACTIVE_STATE_CALCULATION_RECEIPT.json. Drive IDs 1rhQzSlMfadKXqfM3jp3qEjEzqbpaeH7f and 1Qr_Z4QGXO6gz8H9AZyxYSi6cVgDQ8bH0. Raw bytes acquired and result reproduced exactly.

K16 KEX_UNIFIED_THEOREM_FINAL.md and KEX_MATHEMATICAL_DECOMPOSITION_v1.md. Drive IDs 1lpZ9iiElwB6IHhh8OQSaom2L3_Fz3dxc and 12whHV-Zt3WOrOM-UHwkR3BFbcciXMv99. Archived calibration and probability models.

K17 monograph_source_index.md. Drive ID 1g6-uZNEStLY2s0c6ZM1ShJYFIPWckrU3. Discovery index, not substitute for monograph full text.

W1 Keddeh Systems. Research atlas Origin Relation Observation. https://www.keddeh.com/research/discoveries . Retrieved 10 October 2026 Adelaide time. Includes paired-whole definition, singular origin and reference-experiment boundaries.

W2 Keddeh Systems. https://www.keddeh.com/architecture ; /research ; /kex ; /technology ; /research/resonance ; /evidence ; /learn ; /research/learn ; homepage. Direct HTML retrieval 10 October 2026 Adelaide time.

E1 S. Breaz, T. Brzeziński, B. Rybołowicz and P. Saracco. Heaps of modules and affine spaces. https://arxiv.org/abs/2203.07268 . Primary mathematical research, affine/module correspondence.

E2 JCGM. International Vocabulary of Metrology, 1.19 quantity value. https://jcgm.bipm.org/vim/en/1.19.html . Number and reference are distinct parts of a quantity value.

E3 JCGM. International Vocabulary of Metrology, 2.9 measurement result. https://jcgm.bipm.org/vim/en/2.9.html . Relevant information and uncertainty accompany measurement.

E4 BIPM. International System of Units, 9th edition, current retrieved version 4.01. https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf . Celsius/Kelvin scale definition.

E5 J. Preskill. Quantum Information and Computation, Chapter 2 Foundations I. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf ; Chapter 3 Measurement and Evolution https://www.preskill.caltech.edu/ph229/notes/chap3.pdf . State rays, normalization, phase and measurement references.

E6 F. Giacomini, E. Castro-Ruiz and Č. Brukner. Quantum mechanics and the covariance of physical laws in quantum reference frames. https://arxiv.org/abs/1712.07207 . Physical reference frames and relational transformations; not a validation of KEX by association.

U1 Supplied Pasted text(3).txt. Conversation source for current research directive and original interpretations. The (-3+)+(-14+)=(-17+) example is also explicitly supplied in the project conversation context.

Execution artifacts: results/audit.json; original-tests.txt; corrected-regression.txt; corrected-proof-probe.txt; exception-replay.txt; workbench-validation.json. Original archives and source text extracts are preserved in the evidence package with a SHA-256 inventory. The exact code correction is correction.patch.


Revision history

Version 1.0, 10 October 2026: source acquisition, original receipt reproduction, proof reconstruction and bounded separate verification. Version 2.0, 10 October 2026: primary observer-preprint acquisition, origin/metric manuscript acquisition, ZCG native code execution, falsification ledger, seven proof-boundary corrections in a separate candidate, six interactive tool families, twenty case applications and density-emergence experiment design. Authorial definitions and distinct model editions are preserved. No owner approval or external independent assessment is recorded.