> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# NIOT-06 — Floating-Point Projection, Completion, and Cross-Domain Bridges: Bidirectional Semantics from Finite Machine Representations to Numerical Information Objects

## Floating-Point Projection, Completion, and Cross-Domain Bridges: Bidirectional Semantics from Finite Machine Representations to Numerical Information Objects

**Series:** Numerical Information Object Theory (NIOT) / Numerical Information Object Theory  
**Paper:** Paper 06  
**Author:** Neo.K  
**Institution:** EveMissLab / 一言諾科技有限公司  
**Version:** v0.1  
**Date:** 2026-09-22  
**Document type:** Cross-domain bridge theory / floating-point semantics / completion–projection integration paper  
**Prerequisites:** NIOT-00, NIOT-01, NIOT-02, NIOT-03, NIOT-04, NIOT-05  
**Related existing research:** Between Finite Machines and Infinite Real Numbers: Completion–Projection Bidirectional Semantics of Floating-Point Numbers, Exact Real Numbers, and Cross-Domain Operators; ICNS, MNVP, CAIR, ISQL, Active Index Calculus  
**Status:** DRAFT / BRIDGE-SEMANTICS FOUNDATION / IMPLEMENTATION-READY  
**Canonical Source:** UTF-8 Markdown  
**Mathematical source conventions:** Inline math uses only `$...$`, and display math uses only `$$...$$`

---

# Abstract

The first five NIOT papers established numerical information objects, information probes, Unboundedly Expandable Finite Matrices, spectral discrimination, and Numerical Index Geometry. A numerical value is no longer assumed to be a single scalar; it may be understood as an information object with canonical identity, representation, address, relation, history, provenance, certificate, spectrum, and bridge endpoints. This paper returns to the earliest starting point of the series: **What exactly is a floating-point number?**

This paper argues that floating-point numbers should not be treated as "the final form of numbers themselves," but placed in a more precise position:

$$
\boxed{
f
=
\pi_{\mathrm{float}}(\mathfrak N)
}
$$

Here $f$ is a floating-point projection on a finite machine, while $\mathfrak N$ is the numerical information object to which it belongs or corresponds. This does not diminish floating-point numbers' engineering role. On the contrary, it allows floating-point numbers to be placed, for the first time, in the same cross-domain semantic framework as exact rationals, symbolic reals, decimal streams, interval enclosures, completion objects, versioned numerals, measurement states, and machine execution.

This paper formalizes cross-domain conversions as bridge objects:

$$
\boxed{
\mathfrak B
=
\left\langle
X,
Y,
\llbracket-\rrbracket_X,
\llbracket-\rrbracket_Y,
B,
\overline B,
\kappa,
\varepsilon,
\mathsf{Exc},
\mathsf{Exec},
\mathsf{Addr},
\mathsf{Cert}
\right\rangle.
}
$$

where:

- $X,Y$: the source and target domains;
- $\llbracket-\rrbracket_X,\llbracket-\rrbracket_Y$: semantic maps for the respective domains;
- $B$: the raw representation / data-layer bridge;
- $\overline B$: the semantic-layer correspondence;
- $\kappa$: the bridge type;
- $\varepsilon$: error / enclosure;
- $\mathsf{Exc}$: exception states;
- $\mathsf{Exec}$: finite-machine execution semantics;
- $\mathsf{Addr}$: the bridge endpoint / index map defined by NIOT-05;
- $\mathsf{Cert}$: the bridge certificate.

This paper reorganizes seven core bridge types:

1. **Exact Recoding**;
2. **Corecursive Expansion**;
3. **Completion Lifting**;
4. **Quotient Identification**;
5. **Rounding Projection**;
6. **Interval Enclosure**;
7. **Exception Escape**.

In addition, this paper adds an eighth type specific to NIOT:

8. **Probe-Preserving Partial Bridge**: rather than requiring the entire numerical object to be converted into another representation domain, it requires only that specified probe-relevant information remain comparable or traceable between the two domains.

This category directly responds to the series' original central insight: **We need not first convert a float to another numeral base to be able to observe its microscopic information.** Some information can be localized directly through probes, indices, bridge endpoints, and local charts.

This paper particularly emphasizes four concepts that must not be conflated:

$$
\boxed{
\text{Generation}
\neq
\text{Completion}
\neq
\text{Representation}
\neq
\text{Execution}.
}
$$

and:

$$
\boxed{
\text{Round-trip}
\neq
\text{Identity}
}
$$

With non-injectivity, non-surjectivity, rounding, quotienting, exceptions, or completion, cross-domain round-trip conversion cannot naively be treated as an inverse function.

Ultimately, this paper establishes NIOT's cross-domain principle:

$$
\boxed{
\text{Numerical information objects can be projected, bridged, and completed across representation domains,
but every crossing must preserve:
provenance, addresses, semantics, distortion, exceptions, and certificates.}
}
$$

**Keywords:** Floating-Point Numbers, Completion, Projection, Cross-Domain Bridges, rounding, exact real, interval enclosure, quotient, exception, bridge certificate, NIOT

---

# 1. The Problem: Is a Float the Entire Number?

Consider:

$$
f=0.1.
$$

In ordinary programming contexts, this may denote:

- a decimal literal;
- a binary floating-point value;
- an arbitrary-precision decimal;
- a fixed-point value;
- a rational approximation;
- a display string;
- a measurement report.

If one examines only the surface token:

$$
0.1,
$$

these may be conflated.

ICNS has already addressed "insufficient interpretation of bare numerals."

NIOT-06 further asks:

$$
\boxed{
\text{Even after interpretation is complete,
does a floating-point representation equal the entire numerical object?}
}
$$

The answer is no.

---

# 2. Float as Projection

For a numerical information object:

$$
\mathfrak N,
$$

Define:

$$
\boxed{
\pi_F:
\mathfrak X
\rightharpoonup
\mathbb F.
}
$$

where:

$$
\mathbb F
$$

denotes the set of a particular floating-point format.

For example:

$$
\mathbb F_{32},
\qquad
\mathbb F_{64}.
$$

---

# 3. Floating-Point Projection Non-Identity

In general:

$$
\boxed{
\pi_F(\mathfrak N)
\neq
\mathfrak N.
}
$$

This directly continues NIOT-01's Projection Non-Identity Principle.

---

# 4. Float Object

A concrete floating-point state can itself also establish:

$$
\boxed{
\mathfrak N_f.
}
$$

Its minimal information may include:

- sign;
- exponent;
- significand;
- format;
- rounding mode;
- exception class;
- raw bits;
- source;
- operation history.

---

# 5. Raw Bits Are Not a Value

Let:

$$
b
$$

be a bit pattern.

There is a decoding map:

$$
\boxed{
D_F:
\{0,1\}^n
\rightarrow
\mathbb F.
}
$$

However:

$$
\boxed{
b
\neq
D_F(b).
}
$$

The former is a representation, while the latter is an interpreted float value.

---

# 6. A Float Value Is Still Not the Complete Object

Even if:

$$
f=D_F(b),
$$

there is still:

$$
\boxed{
f
\neq
\mathfrak N_f.
}
$$

because $\mathfrak N_f$ may also contain:

- provenance;
- execution context;
- exception history;
- uncertainty;
- bridge provenance.

---

# 7. Four Layers of Floating-Point Semantics

This paper divides floats into four layers:

## Layer 1 — Bit Representation

$$
b\in\{0,1\}^n.
$$

## Layer 2 — Format-Decoded State

$$
f\in\mathbb F.
$$

## Layer 3 — Semantic Numerical Interpretation

$$
\llbracket f\rrbracket.
$$

## Layer 4 — Numerical Information Object

$$
\mathfrak N_f.
$$

Therefore:

$$
\boxed{
b
\neq
f
\neq
\llbracket f\rrbracket
\neq
\mathfrak N_f.
}
$$

---

# 8. Why Is the Numeral Base Not the Only Entry Point?

Traditional research often observes representation structure through:

$$
\text{decimal}
\rightarrow
\text{binary}
\rightarrow
\text{hex}
$$

as displayed above.

This is very valuable, but it is only one bridge.

NIOT permits directly using:

$$
\boxed{
P_\lambda(\mathfrak N_f)
}
$$

to probe float objects without first converting the entire representation.

---

# 9. Probe-First Principle

This paper proposes:

$$
\boxed{
\text{probe first when full conversion is unnecessary}.
}
$$

If a task requires only:

- exponent structure;
- rounding neighborhoods;
- provenance;
- exception state;
- local ULP structure;

one need not first rewrite the entire object in another numeral base.

---

# 10. Float Internal Addresses

According to NIOT-05, one can establish:

```text
/float/sign
/float/exponent
/float/significand
/float/raw_bits
/float/rounding
/float/exception
```

These are all addresses, not independent object identities.

---

# 11. ULP Address

For a float:

$$
f,
$$

one can establish:

$$
\boxed{
a_{\mathrm{ULP}}(f).
}
$$

It denotes the position of $f$ in that format's local spacing structure.

---

# 12. Neighbor Probe

Define:

$$
\boxed{
P_{\mathrm{next}}(f)
=
\operatorname{nextFloat}(f).
}
$$

and:

$$
P_{\mathrm{prev}}(f).
$$

This directly supplies local float geometry.

---

# 13. Float Local Neighborhood

One can define:

$$
\boxed{
\mathcal N_k(f)
=
\{
\operatorname{prev}^k(f),
\dots,
f,
\dots,
\operatorname{next}^k(f)
\}.
}
$$

---

# 14. A Microscopic Float World Is Not More Decimal Digits

It may be:

$$
\boxed{
\text{bit field}
+
\text{ULP neighborhood}
+
\text{rounding cell}
+
\text{exception branch}
+
\text{bridge provenance}.
}
$$

---

# 15. Generation

Define generation:

$$
\boxed{
G:
S
\rightarrow
S'
}
$$

denoting the continual generation of representations or approximants from a rule.

For example, decimal expansion:

$$
0.9,
0.99,
0.999,
\dots
$$

---

# 16. Generation Is Not Completion

An infinite generating sequence:

$$
(x_n)
$$

and its completion / limit object:

$$
x
$$

do not belong to the same representation layer.

---

# 17. Completion

If:

$$
X
$$

needs completion:

$$
\boxed{
\iota_C:
X
\rightarrow
\widehat X.
}
$$

then:

$$
\widehat X
$$

is the completion domain.

---

# 18. Completion Object

For:

$$
x\in\widehat X,
$$

one can establish:

$$
\boxed{
\mathfrak N_x^{\mathrm{comp}}.
}
$$

and preserve:

- source approximants;
- the completion rule;
- the equivalence class;
- the certificate.

---

# 19. Completion Is Not a Representation Change

If:

$$
X
rightarrow
\widehat X,
$$

this is not merely changing decimals to binary.

It may change the available mathematical domain.

---

# 20. Representation

A representation is:

$$
\boxed{
r\in\mathcal R.
}
$$

Through:

$$
\llbracket r\rrbracket
$$

it corresponds to a semantic object.

---

# 21. Execution

Execution is:

$$
\boxed{
\mathsf{Exec}(r,\mathcal M)
}
$$

where:

$$
\mathcal M
$$

is the machine / runtime.

Thus the same representation may behave differently in different execution contexts.

---

# 22. Fourfold Distinction

This paper formally retains:

$$
\boxed{
\text{Generation}
\neq
\text{Completion}
\neq
\text{Representation}
\neq
\text{Execution}.
}
$$

---

# 23. Bridge Object

Define:

$$
\boxed{
\mathfrak B
=
\left\langle
X,
Y,
\llbracket-\rrbracket_X,
\llbracket-\rrbracket_Y,
B,
\overline B,
\kappa,
\varepsilon,
\mathsf{Exc},
\mathsf{Exec},
\mathsf{Addr},
\mathsf{Cert}
\right\rangle.
}
$$

---

# 24. Bridge Domain

$$
X
$$

is the source representation / semantic domain.

---

# 25. Bridge Codomain

$$
Y
$$

is the target representation / semantic domain.

---

# 26. Raw Bridge

$$
\boxed{
B:X\rightarrow Y.
}
$$

denotes a data-layer conversion.

---

# 27. Semantic Bridge

$$
\boxed{
\overline B:
\llbracket X\rrbracket
\rightsquigarrow
\llbracket Y\rrbracket.
}
$$

---

# 28. Bridge Type

$$
\boxed{
\kappa(\mathfrak B).
}
$$

denotes the bridge type.

---

# 29. Bridge Error

$$
\boxed{
\varepsilon(\mathfrak B).
}
$$

denotes error, distortion, or enclosure.

---

# 30. Exception Semantics

$$
\boxed{
\mathsf{Exc}(\mathfrak B)
}
$$

Handles:

- overflow;
- underflow;
- NaN;
- infinity;
- invalid;
- domain escape.

---

# 31. Execution Semantics

$$
\boxed{
\mathsf{Exec}(\mathfrak B)
}
$$

Records:

- the machine format;
- instructions;
- rounding;
- the environment;
- deterministic / nondeterministic behavior.

---

# 32. Address Map

According to NIOT-05:

$$
\boxed{
\mathsf{Addr}(\mathfrak B)
:
\mathcal I_X
\rightharpoonup
\mathcal I_Y.
}
$$

---

# 33. Bridge Certificate

$$
\boxed{
\mathsf{Cert}(\mathfrak B)
}
$$

Records at least:

- the source;
- the target;
- the rule;
- the version;
- error;
- exceptions;
- replay;
- proof / test status.

---

# 34. Bridge Endpoint

Define:

$$
\boxed{
i_X
\xrightarrow{\mathfrak B}
i_Y.
}
$$

---

# 35. Exact Recoding

If representation changes but semantics is preserved exactly:

$$
\boxed{
\kappa
=
\mathrm{EXACT\_RECODING}.
}
$$

Require:

$$
\boxed{
\llbracket B(x)\rrbracket_Y
=
\llbracket x\rrbracket_X.
}
$$

---

# 36. Example: Rational Re-encoding

$$
\frac12
\leftrightarrow
0.5
$$

can be exact recoding in an exact decimal domain.

---

# 37. Corecursive Expansion

If a finite seed generates an unbounded stream:

$$
\boxed{
\kappa
=
\mathrm{CORECURSIVE\_EXPANSION}.
}
$$

For example:

$$
\frac13
\mapsto
0.333\dots
$$

---

# 38. Expansion Is Not a One-Off Finite String

A stream object must be distinguished from a finite prefix.

---

# 39. Completion Lifting

If:

$$
x_n
$$

as approximants are lifted to a completion object:

$$
\boxed{
\kappa
=
\mathrm{COMPLETION\_LIFT}.
}
$$

---

# 40. Quotient Identification

If multiple representations are quotiented into the same semantic class:

$$
\boxed{
\kappa
=
\mathrm{QUOTIENT\_IDENTIFICATION}.
}
$$

---

# 41. Example: Decimal Dual Expansion

$$
1.000\dots
$$

and:

$$
0.999\dots
$$

can be identified as the same completion value in real-number semantics.

---

# 42. But Representation Identity Still Differs

Even if:

$$
\llbracket r_1\rrbracket
=
\llbracket r_2\rrbracket,
$$

one may still have:

$$
r_1\neq r_2.
$$

---

# 43. Rounding Projection

If:

$$
x\in X
$$

is projected to a finite float:

$$
\boxed{
R_F(x)
=
f,
}
$$

then:

$$
\kappa
=
\mathrm{ROUNDING\_PROJECTION}.
$$

---

# 44. Rounding Cell

For a float:

$$
f,
$$

Define:

$$
\boxed{
C_f
=
\{
x:
R_F(x)=f
\}.
}
$$

This is a very important fiber.

---

# 45. Float Fiber

Therefore:

$$
\boxed{
R_F^{-1}(f)
=
C_f.
}
$$

Multiple exact values can project to the same float.

---

# 46. Rounding Is Non-Injective

In general:

$$
\boxed{
x_1\neq x_2
$$

However:

$$
R_F(x_1)=R_F(x_2).
$$

---

# 47. Thus a Float Cannot Uniquely Determine the Exact Value in Reverse

$$
\boxed{
f
\not\Rightarrow
x_{\mathrm{unique}}.
}
$$

---

# 48. Interval Enclosure

If, instead of returning a single approximate value, one returns:

$$
[a,b]
$$

such that:

$$
x\in[a,b],
$$

then:

$$
\boxed{
\kappa
=
\mathrm{INTERVAL\_ENCLOSURE}.
}
$$

---

# 49. Enclosure Certificate

One needs a verifiable guarantee of:

$$
\boxed{
x\in I
}
$$

as displayed above.

---

# 50. An Interval Is Not a Point Estimate

$$
\boxed{
[a,b]
\neq
\frac{a+b}{2}.
}
$$

---

# 51. Exception Escape

If a source value cannot be represented normally in the target domain:

$$
\boxed{
\kappa
=
\mathrm{EXCEPTION\_ESCAPE}.
}
$$

---

# 52. Overflow

$$
|x|>\max\mathbb F.
$$

---

# 53. Underflow

A small magnitude enters the subnormal / zero region.

---

# 54. NaN

NaN is not an ordinary real value.

It needs an independent semantic class.

---

# 55. Infinity

Machine infinity must not automatically be equated with a mathematically completed infinity object either.

---

# 56. Eighth Type: Probe-Preserving Partial Bridge

NIOT adds:

$$
\boxed{
\kappa
=
\mathrm{PROBE\_PRESERVING\_PARTIAL}.
}
$$

---

# 57. Definition

If full conversion is not needed:

$$
X\rightarrow Y,
$$

and only a probe family:

$$
\mathcal P^\star
$$

is required to remain comparable between the two domains:

$$
\boxed{
P_Y(B(x))
\approx
T_P(P_X(x)).
}
$$

a partial bridge can be used.

---

# 58. Why Is This Important?

Because:

$$
\boxed{
\text{full representation conversion}
}
$$

may be:

- expensive;
- unnecessary;
- distorting;
- irreversible.

But information needed by particular probes may admit a direct bridge.

---

# 59. Probe-Preserving Bridge Example

If one wants to know only:

$$
\operatorname{sign}(x),
$$

one need not convert the entire exact real into a float.

It suffices that:

$$
\boxed{
P_{\mathrm{sign}}
}
$$

can be bridged safely.

---

# 60. Local Bridge

More generally:

$$
\boxed{
B_U:
X|_U
\rightsquigarrow
Y|_V.
}
$$

Bridge only a local address region.

---

# 61. Bridge Need Not Be Total

$$
\operatorname{Dom}(B)
\subsetneq X
$$

is entirely admissible.

---

# 62. Partiality Must Be Explicit

One cannot treat an undefined region as an error-free region.

---

# 63. Bridge Need Not Be Injective

A rounding projection is not injective.

---

# 64. Bridge Need Not Be Surjective

Some target representations may have no source preimage.

---

# 65. Bridge Need Not Be Invertible

This is the general case.

---

# 66. Round-Trip

Consider:

$$
X
\xrightarrow{B}
Y
\xrightarrow{C}
X.
$$

---

# 67. Round-Trip Identity

Only when:

$$
\boxed{
C\circ B
=
\operatorname{id}_X
}
$$

is it an exact round trip.

---

# 68. Approximate Round-Trip

More generally:

$$
\boxed{
d_X
(
C(B(x)),x
)
\le
\varepsilon.
}
$$

---

# 69. Round-Trip Failure

If:

$$
C(B(x))
\neq x,
$$

one cannot simply say that the bridge is wrong.

It may be due to:

- rounding;
- quotienting;
- compression;
- a non-injective map;
- an exception;
- a completion mismatch.

---

# 70. Round-Trip Certificate

Record:

- the starting domain;
- the intermediate domain;
- the final domain;
- exact / approximate status;
- error;
- lost information.

---

# 71. Bridge Composition

If:

$$
\mathfrak B_{XY}
$$

and:

$$
\mathfrak B_{YZ},
$$

one can define:

$$
\boxed{
\mathfrak B_{XZ}
=
\mathfrak B_{YZ}
\circ
\mathfrak B_{XY}.
}
$$

---

# 72. Composition Error

Errors cannot simply be ignored:

$$
\boxed{
\varepsilon_{XZ}
=
\mathcal E
(
\varepsilon_{XY},
\varepsilon_{YZ}
).
}
$$

---

# 73. Composition Exception

Exception sets must also propagate:

$$
\boxed{
\mathsf{Exc}_{XZ}
}
$$

---

# 74. Bridge Path

Across multiple domains:

$$
X_0
\rightarrow
X_1
\rightarrow
\cdots
\rightarrow
X_k.
$$

---

# 75. Bridge Path Cost

Define:

$$
\boxed{
K(\gamma_B)
=
\sum_i
K(\mathfrak B_i).
}
$$

---

# 76. Bridge Path Error

$$
\boxed{
E(\gamma_B)
=
\mathcal E
(
\varepsilon_1,\dots,\varepsilon_k
).
}
$$

---

# 77. Once-Conversion Principle

If multiple bridge paths exist, one usually seeks to avoid unnecessary intermediate domains:

$$
X
\rightarrow
Y
\rightarrow
Z
$$

if an admissible direct bridge exists:

$$
X
\rightarrow
Z.
$$

---

# 78. But the Fewest Conversions Are Not Always Best

A direct bridge may have:

- larger errors;
- greater cost;
- a weaker certificate.

---

# 79. Bridge Optimization

Thus one should solve:

$$
\boxed{
\gamma^\star
=
\arg\min_\gamma
\left(
\alpha K(\gamma)
+
\beta E(\gamma)
+
\gamma R(\gamma)
\right).
}
$$

---

# 80. Bridge Graph

Establish:

$$
\boxed{
G_B
=
(V_B,E_B).
}
$$

node = representation / semantic domains.

edge = bridge.

---

# 81. Bridge Atlas

Each domain can be regarded as a chart.

Cross-domain bridges form:

$$
\boxed{
\mathcal A_B.
}
$$

---

# 82. Domain Chart

For example:

- an exact rational chart;
- a decimal chart;
- a binary float chart;
- an interval chart;
- a symbolic real chart.

---

# 83. Cross-Domain Transition

$$
\boxed{
\phi_{YX}
}
$$

corresponds to a bridge transition.

---

# 84. Bridge Consistency

Between the triple path:

$$
X\rightarrow Y\rightarrow Z
$$

and the direct path:

$$
X\rightarrow Z
$$

one can compare:

$$
\boxed{
\Delta_B
=
B_{XZ}
-
B_{YZ}\circ B_{XY}
}
$$

under applicable semantics.

---

# 85. Bridge Defect

If:

$$
\Delta_B\neq0,
$$

one must distinguish:

- an expected approximation;
- inconsistent semantics;
- an invalid conversion;
- noncommuting completion;
- rounding path dependence.

---

# 86. Path Dependence

One may have:

$$
\boxed{
B_{XZ}
\neq
B_{YZ}\circ B_{XY}.
}
$$

This is an important structure, not something that should always be regarded as a bug.

---

# 87. Bridge Holonomy

Along a closed representation path:

$$
X_0
\rightarrow
X_1
\rightarrow
\cdots
\rightarrow
X_0,
$$

one obtains:

$$
\boxed{
\mathcal H_B.
}
$$

---

# 88. Nontrivial Bridge Holonomy

If:

$$
\mathcal H_B(x)\neq x,
$$

this indicates a structural defect / distortion / state change in the cross-domain round trip.

---

# 89. Bridge Curvature Candidate

This may become a candidate definition of cross-domain representation curvature.

This paper does not directly equate it with traditional differential-geometric curvature.

---

# 90. Float Rounding Geometry

For each:

$$
f\in\mathbb F,
$$

its rounding cell:

$$
C_f
$$

can be viewed as a fiber region in the exact domain.

---

# 91. Cell Boundary

The rounding boundary between two floats:

$$
f_i,f_j
$$

is:

$$
\boxed{
\partial C_{f_i,f_j}.
}
$$

---

# 92. Tie-Breaking

If an exact value lies on a boundary, the rounding mode determines its projection.

---

# 93. Rounding Mode as Probe / Context

Therefore:

$$
\boxed{
\rho
=
\text{rounding mode}
}
$$

is a first-class parameter of the bridge context.

---

# 94. Round-to-Nearest

Is only one:

$$
\rho.
$$

---

# 95. Directed Rounding

For example:

$$
\rho_\downarrow,
\qquad
\rho_\uparrow.
$$

---

# 96. Interval Arithmetic

Directed rounding can establish certified enclosures.

---

# 97. Float Neighborhood Spectrum

For a radius of:

$$
k
$$

ULPs, one can define:

$$
\boxed{
P_k(f)
=
\mathcal N_k(f).
}
$$

This forms a discrete local spectrum.

---

# 98. Float Exponent Spectrum

One can also establish a probe family by exponent bands.

---

# 99. Significand Probe

The significand structure can be analyzed locally.

---

# 100. But Bit-Pattern Structure Is Not an Additional Ontological Truth

It is representation-dependent information.

---

# 101. Representation-Dependent Probe

Define:

$$
\boxed{
P^{(R)}
}
$$

is meaningful only in a particular representation domain.

---

# 102. Representation-Invariant Probe

If semantics is preserved across multiple representations:

$$
\boxed{
P^{(\mathrm{inv})}.
}
$$

---

# 103. Bridge-Preserved Observable

If:

$$
\boxed{
P_Y(B(x))
=
T(P_X(x))
}
$$

$P$ is called bridge-preserved.

---

# 104. Bridge-Lost Observable

If a source probe cannot be reconstructed in the target:

$$
\boxed{
P_X
\not\rightsquigarrow
P_Y.
}
$$

---

# 105. Information Loss Manifest

Each lossy bridge should output:

```text
PRESERVED:
    ...

LOST:
    ...

APPROXIMATED:
    ...

UNKNOWN:
    ...
```

---

# 106. Scalar-Preserving but Structure-Losing Bridge

One may have:

$$
\boxed{
\pi_v
\text{ preserved}
}
$$

However:

- provenance;
- precision;
- history;
- representation detail;

is lost.

---

# 107. Structure-Preserving but Scalar-Approximate Bridge

One may also have:

- an interval;
- a symbolic enclosure;

preserving stronger structure without returning a single exact scalar.

---

# 108. Bridge Information Order

One can define:

$$
\boxed{
\mathfrak B_1
\preceq
\mathfrak B_2
}
$$

if $\mathfrak B_2$ preserves at least as much task-relevant information as $\mathfrak B_1$.

---

# 109. Dominating Bridge

If:

$$
\mathfrak B_1
\preceq
\mathfrak B_2
$$

and its cost is no higher, one can say:

$$
\mathfrak B_2
$$

dominates.

---

# 110. Bridge Pareto Frontier

In:

- cost；
- error；
- preserved information;
- trust;

establish a Pareto frontier.

---

# 111. Completion vs Float

From an exact real / completed real:

$$
x\in\mathbb R
$$

to:

$$
f\in\mathbb F
$$

is usually:

$$
\boxed{
\text{completion domain}
\rightarrow
\text{finite projection domain}.
}
$$

---

# 112. Float to Real

In the reverse direction:

$$
f
\mapsto
x_f
$$

one can usually embed exactly:

$$
\boxed{
\iota_F:
\mathbb F
\hookrightarrow
\mathbb R.
}
$$

because every finite float corresponds to an exact dyadic rational.

---

# 113. But This Does Not Recover the Original Exact Source

If:

$$
x
\xrightarrow{R_F}
f,
$$

then:

$$
f
\xrightarrow{\iota_F}
x_f,
$$

in general:

$$
\boxed{
x_f
\neq
x.
}
$$

---

# 114. Thus a Float Round Trip Is Projection Followed by Embedding

$$
\boxed{
\mathbb R
\xrightarrow{R_F}
\mathbb F
\xrightarrow{\iota_F}
\mathbb R.
}
$$

---

# 115. This Is Not Identity

Unless:

$$
x
$$

was already an exactly representable value of that float format.

---

# 116. Exact-Representable Set

Define:

$$
\boxed{
E_F
=
\iota_F(\mathbb F).
}
$$

---

# 117. Exact Round-Trip Condition

If:

$$
x\in E_F,
$$

one may have:

$$
\boxed{
\iota_F(R_F(x))=x.
}
$$

under compatible rounding.

---

# 118. Otherwise Approximate

If:

$$
x\notin E_F,
$$

then:

$$
\boxed{
\iota_F(R_F(x))
=
x+\delta,
}
$$

where:

$$
\delta\neq0
$$

generally holds.

---

# 119. Error Probe

Define:

$$
\boxed{
P_{\mathrm{err}}(x)
=
\iota_F(R_F(x))-x.
}
$$

---

# 120. Relative Error

$$
\boxed{
P_{\mathrm{relerr}}(x)
=
\frac{
|\iota_F(R_F(x))-x|
}{
|x|
}
}
$$

When:

$$
x\neq0.
$$

---

# 121. ULP Error

One can instead measure the error in ULPs.

---

# 122. Error Spectrum

As magnitude / exponent / rounding mode varies:

$$
\boxed{
\varepsilon(\lambda).
}
$$

forms an error spectrum.

---

# 123. Error Coupling

Errors can couple to:

- scale;
- operation count;
- conditioning;
- representation;

as listed above.

---

# 124. Operation Chain

Consider:

$$
f_0
\xrightarrow{\circ_1}
f_1
\xrightarrow{\circ_2}
\dots
\xrightarrow{\circ_n}
f_n.
$$

---

# 125. Execution Provenance

Each step should be able to record:

$$
\boxed{
\mathsf{Exec}_i.
}
$$

---

# 126. Cumulative Error

One cannot examine only the final float.

An operation-chain certificate is needed.

---

# 127. Floating-Point History Object

Establish:

$$
\boxed{
\mathcal H_F
=
\{
(f_i,\circ_i,\rho_i,\varepsilon_i)
\}.
}
$$

---

# 128. Same Final Float, Different History

One may have:

$$
f_n^{(A)}
=
f_n^{(B)}
$$

However:

$$
\mathcal H_F^{(A)}
\neq
\mathcal H_F^{(B)}.
$$

Therefore:

$$
\boxed{
\text{same float}
\not\Rightarrow
\text{same numerical information object}.
}
$$

---

# 129. Float Fiber by History

For a final float:

$$
f,
$$

there may exist:

$$
\boxed{
F_f^{\mathrm{hist}}.
}
$$

containing all the different histories producing the same final float.

---

# 130. Probe on History Fiber

One can refine it using:

- operation-depth probes;
- error probes;
- provenance probes;

as listed above.

---

# 131. Completion History

An exact real object can also preserve:

- a Cauchy sequence;
- a Dedekind-like representation;
- a symbolic generator;
- an oracle / stream;

and other sources.

---

# 132. Same Real, Different Completion Representation

Semantic values may agree while representation identities differ.

---

# 133. Completion Fiber

Define:

$$
\boxed{
F_x^{\mathrm{comp}}
=
\{
r:
\llbracket r\rrbracket=x
\}.
}
$$

---

# 134. Completion Probe

One can ask about:

- the source construction;
- the convergence rate;
- the certificate;
- the representation cost.

---

# 135. Generation Rate

If an approximant sequence:

$$
x_n
$$

exists, one can define:

$$
\boxed{
r_n
=
d(x_n,x).
}
$$

or another convergence measure.

---

# 136. The Generation Rate Is Not the Object Value

This continues the earlier "generation rate–completion semantics" series.

---

# 137. Completion Time Is Not Mathematical Completion

Finite runtime expenditure:

$$
T
$$

is only execution cost.

One cannot treat:

$$
T
$$

as part of the completion's ontology unless the task specifically requires it.

---

# 138. Completion Certificate

Record at least:

- the source representation;
- the completion rule;
- the equivalence criterion;
- limit semantics;
- verification status.

---

# 139. Cross-Domain Probe

Define:

$$
\boxed{
P^{X\to Y}
}
$$

directly compares an invariant between source / target domains.

---

# 140. Bridge Fidelity Probe

For example:

$$
\boxed{
P_{\mathrm{fid}}
(
x,B(x)
)
}
$$

assesses the degree of semantic preservation.

---

# 141. Bridge Identity Probe

Determines whether canonical identity should continue.

---

# 142. Same Object Across Representation

If:

$$
\mathfrak N_X
$$

and:

$$
\mathfrak N_Y
$$

are merely two representation views of the same canonical numerical object, one can set:

$$
\boxed{
\iota_C(\mathfrak N_X)
=
\iota_C(\mathfrak N_Y).
}
$$

---

# 143. Different Object with Same Value

If provenance / roles differ, canonical identities may differ even when values agree.

---

# 144. Bridge Must Not Force Identity

A cross-domain bridge should not automatically merge all same-value objects.

---

# 145. Identity Policy

One needs:

$$
\boxed{
\mathsf{IDPolicy}(\mathfrak B).
}
$$

declaring:

- preserve identity;
- create a view identity;
- create a derived object;
- create a new canonical object.

---

# 146. View-Preserving Bridge

Changes only the representation:

$$
\boxed{
\text{PRESERVE\_CANONICAL\_ID}.
}
$$

---

# 147. Derived-Object Bridge

For example, a numerical value is derived by a model into a new quantity:

$$
\boxed{
\text{NEW\_DERIVED\_ID}.
}
$$

---

# 148. Bridge Addressing

Source endpoint:

$$
a_X.
$$

Target endpoint:

$$
a_Y.
$$

Bridge edge:

$$
a_B.
$$

---

# 149. Bridge Query

A runtime can ask:

> From which exact / measured / symbolic object was this float projected?

---

# 150. Reverse Bridge Query

It can also ask:

> Which representations are currently regarded as corresponding to the same canonical object?

---

# 151. Bridge Registry

Establish:

$$
\boxed{
\mathcal R_B.
}
$$

Each bridge records:

- ID;
- version;
- source;
- target;
- type;
- error;
- exceptions;
- identity policy;
- certificate.

---

# 152. Bridge Versioning

$$
\boxed{
\mathfrak B^{(v1)}
\neq
\mathfrak B^{(v2)}
}
$$

if the conversion rule changes.

---

# 153. Old Result Invalidation

A bridge-version update may require reverification of old outputs.

---

# 154. Bridge Dependency DAG

Composite conversions form a DAG.

---

# 155. Bridge Cache

The same source state + bridge version can be cached.

---

# 156. Bridge Invalidation

Changes in source / bridge / rounding context require invalidation.

---

# 157. Bridge Security

An external bridge may entail:

- an unsafe parser;
- data loss;
- code execution;
- an untrusted conversion.

Governance is needed.

---

# 158. Bridge Trust Level

Provisionally:

```text
B0 — undocumented
B1 — deterministic
B2 — replay-tested
B3 — independently reproduced
B4 — formally specified
B5 — formally verified + independently reproduced
```

---

# 159. Bridge Certificate Object

$$
\boxed{
\mathfrak C_B
=
\left\langle
id,
rule,
version,
domain,
codomain,
error,
exceptions,
replay,
trust
\right\rangle.
}
$$

---

# 160. Exactness Is Typed

One cannot merely write:

```text
exact = true
```

One should write:

$$
\boxed{
\text{exact with respect to which semantics?}
}
$$

---

# 161. Representation Exactness

Bit decoding is exact.

---

# 162. Value Exactness

The semantic value is exact.

---

# 163. Round-Trip Exactness

The source is exactly recoverable.

---

# 164. Probe Exactness

The specified probe is preserved exactly.

---

# 165. Identity Exactness

Canonical identity is preserved.

---

# 166. Different Exactness Notions Must Be Separate

$$
\boxed{
\text{exactness is multi-typed}.
}
$$

---

# 167. Error Is Also Typed

There may be:

- absolute error;
- relative error;
- ULP error;
- interval width;
- structural loss;
- provenance loss;
- address loss.

---

# 168. Error Vector

Define:

$$
\boxed{
\boldsymbol\varepsilon
=
(
\varepsilon_v,
\varepsilon_r,
\varepsilon_p,
\varepsilon_a,
\dots
).
}
$$

---

# 169. Scalar Error Is Not Enough

Even if:

$$
\varepsilon_v=0,
$$

one may still have:

$$
\varepsilon_p>0
$$

denoting provenance loss.

---

# 170. Multi-Objective Bridge Fidelity

Fidelity can therefore be:

$$
\boxed{
F_B
=
F
(
\boldsymbol\varepsilon
).
}
$$

---

# 171. Task-Relative Fidelity

Different tasks assign different weights to different error components.

---

# 172. Bridge Loss Budget

Define:

$$
\boxed{
L_B
\le
\tau_{\mathcal T}.
}
$$

---

# 173. Hard-Preserve Fields

Some fields can be assigned infinite weight:

$$
w_i=\infty.
$$

The bridge must fail if they are lost.

---

# 174. Float as Lossy Bridge Result

For a general exact real:

$$
\boxed{
R_F
}
$$

is a value-approximate, finite-storage bridge.

---

# 175. Float as Exact Object in Its Own Domain

But for:

$$
f\in\mathbb F,
$$

it is an exact object in its own finite float domain.

---

# 176. These Two Statements Are Not Contradictory

The exactness domains differ.

---

# 177. Domain-Relative Exactness

$$
\boxed{
\operatorname{Exact}(f\mid\mathbb F)=1
}
$$

does not imply:

$$
\operatorname{Exact}(f\mid\mathbb R,x)=1.
$$

---

# 178. Value Layer vs Representation Layer

The same float:

$$
f
$$

can exactly represent a dyadic rational:

$$
q_f.
$$

But if the original source has:

$$
x\neq q_f,
$$

the source approximation is not exact.

---

# 179. Information Probe on Float Without Conversion

One can directly use:

$$
P_{\mathrm{exponent}}(\mathfrak N_f),
$$

$$
P_{\mathrm{ULP}}(\mathfrak N_f),
$$

$$
P_{\mathrm{history}}(\mathfrak N_f).
$$

---

# 180. This Formalizes the Series' Starting Point

$$
\boxed{
\text{One need not first change numeral bases
to see more computable information inside a float.}
}
$$

---

# 181. Representation Conversion Is One Probe Strategy

Base conversion can be written as:

$$
\boxed{
P_{\mathrm{repr}\to R'}
}
$$

or a bridge:

$$
\mathfrak B_{R\to R'}.
$$

---

# 182. But Not the Only Strategy

There are also:

- direct field probes;
- local neighborhood probes;
- spectral probes;
- provenance probes;
- bridge fidelity probes.

---

# 183. Cross-Base Comparison

Binary, decimal, and hex can be regarded as different representation charts.

---

# 184. Base Change as Chart Transition

$$
\boxed{
\phi_{b_1\to b_2}.
}
$$

---

# 185. Base Change Does Not Change Canonical Value Identity

If exact recoding:

$$
\boxed{
\iota_C
\text{ preserved}.
}
$$

---

# 186. But Surface Structure Changes

Digit patterns, length, and local syntax all change.

---

# 187. Probe Spectrum Across Bases

One can define:

$$
\boxed{
\mathcal S_{\mathrm{base}}(b).
}
$$

---

# 188. Base-Dependent Motif

Some patterns appear only in particular bases.

They cannot directly be promoted to representation-invariant properties.

---

# 189. Invariant Probe Needed

Claims of ontological structure require verification across representations.

---

# 190. Cross-Representation Structural Alignment

Compare not string similarity, but:

- semantic roles;
- canonical identity;
- bridge provenance;
- invariant relations.

---

# 191. Bridge Structural Alignment

Define:

$$
\boxed{
\operatorname{Align}_B
(
\mathfrak N_X,
\mathfrak N_Y
).
}
$$

---

# 192. Point Alignment vs Structural Alignment

As with BCODR:

- exact token equality;
- value equality;
- representation alignment;
- structural alignment;

must be distinguished.

---

# 193. Representation Alignment

Whether two representations describe the same semantic value.

---

# 194. Structural Alignment

Whether they preserve the same higher-order mechanism / provenance / relation.

---

# 195. Example: 0.5

$$
0.5_{10}
$$

and:

$$
0.1_2
$$

are value-aligned.

If they come from the same canonical object, they may also be canonically aligned.

---

# 196. Example: Same 0.5 from Different Sources

Two instances of:

$$
0.5
$$

may be value-aligned but have different provenance.

---

# 197. Example: Interval [0.49,0.51]

It is not point-aligned with 0.5.

However:

$$
0.5\in[0.49,0.51].
$$

is an enclosure relation.

---

# 198. Typed Relation Needed

One cannot conflate:

```text
equal
contains
approximates
rounds_to
represents
```

into a single relation.

---

# 199. Bridge Relation Vocabulary

Initial version:

```text
EXACTLY_REPRESENTS
ROUNDS_TO
ENCLOSES
APPROXIMATES
COMPLETES_TO
QUOTIENTS_TO
EXPANDS_TO
EMBEDS_IN
ESCAPES_TO_EXCEPTION
PROBE_PRESERVES
```

---

# 200. Relation Is First-Class

Every relation should be addressable / versioned / certifiable.

---

# 201. Cross-Domain UEFM

The NIOT-03 matrix can span representation domains.

---

# 202. Domain-Labeled Coordinates

Index:

$$
\boxed{
i=(D,a).
}
$$

where $D$ is the domain.

---

# 203. Cross-Domain Edge

$$
\boxed{
(D_X,a_X)
\xrightarrow{\mathfrak B}
(D_Y,a_Y).
}
$$

---

# 204. Bridge Matrix

One can establish:

$$
\boxed{
M_B
}
$$

recording bridge connectivity between domains.

---

# 205. Sparse Bridge Graph

Most domain pairs have no direct bridge, so a sparse graph is appropriate.

---

# 206. Bridge Frontier

Unresolved / highly distorting / high-value conversion edges form:

$$
\boxed{
F_B.
}
$$

---

# 207. BCODR on Bridge Space

One can expand bidirectionally from:

- source capability outward;
- target constraints inward;

to localize a feasible bridge spine.

---

# 208. Bridge Responsibility Spine

$$
\boxed{
S_B.
}
$$

denotes the few bridge nodes / edges that genuinely bear cross-domain conversion responsibilities.

---

# 209. Spectral Bridge

A bridge may vary with:

$$
\lambda
$$

as displayed above.

For example, precision:

$$
p.
$$

---

# 210. Precision-Parameterized Bridge

$$
\boxed{
\mathfrak B_p.
}
$$

---

# 211. Error Spectrum

$$
\boxed{
\varepsilon(p).
}
$$

---

# 212. Cost Spectrum

$$
\boxed{
K(p).
}
$$

---

# 213. Fidelity Spectrum

$$
\boxed{
F(p).
}
$$

---

# 214. Adaptive Precision Selection

Choose:

$$
\boxed{
p^\star
=
\arg\min_p
\left(
\alpha K(p)
+
\beta\varepsilon(p)
\right).
}
$$

subject to task closure.

---

# 215. Precision Is a Probe / Bridge Axis

This makes precision more than a static field.

---

# 216. Dynamic Precision

Different epochs can use different:

$$
p_t.
$$

---

# 217. Mixed Precision

Different UEFM cells can have different precision.

---

# 218. Precision Address

$$
\boxed{
a_{\mathrm{precision}}(i).
}
$$

---

# 219. Precision Provenance

Every precision change should be recorded.

---

# 220. Precision Loss

Reducing precision requires a certificate.

---

# 221. Precision Reopen

If a task requires higher fidelity, it can be expanded again.

---

# 222. Exact Real Runtime

There is no need to materialize infinitely many digits at once.

One can preserve:

- a generator;
- an oracle;
- interval refinement;
- a symbolic rule.

---

# 223. Lazy Exact Real

$$
\boxed{
\mathfrak N_x^{\mathrm{lazy}}.
}
$$

Supplies precision on demand.

---

# 224. Request Precision

$$
\boxed{
P_p(\mathfrak N_x)
}
$$

Requests precision $p$.

---

# 225. This Is Probe-Driven Materialization

rather than storing the entire infinite expansion in RAM.

---

# 226. Exact Real and UEFM

Each epoch materializes only a finite precision block.

---

# 227. Unbounded Precision, Finite Epoch

$$
\boxed{
p_t<\infty
}
$$

but allows:

$$
\sup_t p_t=\infty.
$$

---

# 228. Isomorphic to UEFM

This is another instance of:

$$
\boxed{
\text{finite actuality}
+
\text{unbounded extensibility}
}
$$

as displayed above.

---

# 229. Infinite Expansion Need Not Be Stored

One need preserve only the generation rule and current certificate.

---

# 230. Decimal Stream

For:

$$
0.333\dots
$$

can be represented as a corecursive stream.

---

# 231. Stream Prefix

Every epoch holds only:

$$
\boxed{
s_{1:n}.
}
$$

---

# 232. Completion Identity

Stream representation and completed real identity must be distinguished.

---

# 233. Approximation Chain

$$
\boxed{
x_1,x_2,\dots,x_n.
}
$$

---

# 234. Chain Provenance

Each approximant can have an address.

---

# 235. Chain Limit Certificate

If one claims:

$$
x_n\to x,
$$

a convergence certificate is needed.

---

# 236. Limit Is Not Last Prefix

$$
\boxed{
x
\neq
x_n
}
$$

generally holds.

---

# 237. This Corrects Finite-Machine Intuition

A finite prefix cannot be treated as a completed object.

---

# 238. But Task May Only Need Prefix

If task closure is achieved at:

$$
n=N
$$

one can stop.

---

# 239. Task-Relative Completion Sufficiency

Ontology-total expansion is unnecessary.

---

# 240. Completion Probe

One can ask:

> Is the current prefix sufficient to support task $\mathcal T$?

---

# 241. Completion Stop Rule

$$
\boxed{
\varepsilon_n
\le
\tau_{\mathcal T}.
}
$$

---

# 242. Completion Reopen

When a new task requires greater precision:

$$
n
\rightarrow
n+k.
$$

---

# 243. Float as Task-Relative Projection

Floats suffice for some tasks.

---

# 244. Others Need Exact / Interval

One representation cannot be declared globally optimal.

---

# 245. Representation Selection

Define:

$$
\boxed{
R^\star
=
\arg\min_R
\left(
K(R)
+
L(R,\mathcal T)
\right).
}
$$

---

# 246. Representation Router

According to the task, AI can automatically select:

- a float;
- a decimal;
- a rational;
- an interval;
- a symbolic representation;
- an exact lazy real.

---

# 247. Router Must Be Certifiable

The reasons for selection need a trace.

---

# 248. Representation Switching

A runtime can switch representations during a task.

---

# 249. Switching Cost

$$
\boxed{
K_{\mathrm{switch}}.
}
$$

---

# 250. Switching Loss

$$
\boxed{
L_{\mathrm{switch}}.
}
$$

---

# 251. Switching Hysteresis

Avoid frequent switching back and forth between representations.

---

# 252. Representation Policy

One can set:

```text
prefer_exact
prefer_speed
prefer_interval
prefer_replayability
```

---

# 253. Policy Is Not Semantics

Policy affects only routing.

---

# 254. Canonical Object Remains Above Representation

$$
\boxed{
\mathfrak N
}
$$

should not be locked into a single representation.

---

# 255. CAIR Interface

This is isomorphic to CAIR's:

$$
\boxed{
\text{canonical object}
\neq
\text{projection}
}
$$

as displayed above.

---

# 256. ICNS Interface

ICNS handles numeral interpretation.

---

# 257. MNVP Interface

MNVP handles human-perceptible presentation.

---

# 258. ISQL Interface

ISQL can supply semantic addresses for:

- bridges;
- endpoints;
- probes;
- representations;

as listed above.

---

# 259. AIC Interface

AIC supplies active transport across indices / representation charts.

---

# 260. Bridge as Active Index Transport

$$
\boxed{
\widehat{\mathcal E}_{i_X\to i_Y}
}
$$

can be implemented by $\mathfrak B$.

---

# 261. Transport Identity Policy

Whether transport preserves canonical identity must be declared explicitly.

---

# 262. Float Probe Spectrum

One can establish:

$$
\boxed{
\mathcal S_f
=
\{
P_{\mathrm{bits}},
P_{\mathrm{exp}},
P_{\mathrm{sig}},
P_{\mathrm{ulp}},
P_{\mathrm{history}},
P_{\mathrm{error}}
\}.
}
$$

---

# 263. Float Coupling Matrix

Analyze couplings such as:

- exponent × ULP;
- history × error;
- rounding × exception;

as listed above.

---

# 264. Representation Coupling

One can also establish, between different representations:

$$
\boxed{
C_{R_iR_j}.
}
$$

---

# 265. Strong Bridge Coupling

If two representations strongly agree on task invariants, one can establish a strong coupling edge.

---

# 266. Bridge Degeneracy

Multiple representations that are completely equivalent for a task can be quotiented.

---

# 267. But Preserve Provenance

Even when quotienting, provenance and expansion paths must be preserved.

---

# 268. Bridge Closure

For a task:

$$
\mathcal T,
$$

If:

1. Source/target semantics is explicit;
2. Error is below the threshold;
3. Exceptions are resolved;
4. Identity policy is decided;
5. Required probes are preserved;
6. The certificate is valid;

then:

$$
\boxed{
\operatorname{Close}^{B}_{\mathcal T}=1.
}
$$

---

# 269. Bridge Closure Is Local

It means only that the bridge suffices for that task.

---

# 270. Bridge Reopen

A new task / new precision / new probe can reopen it.

---

# 271. Cross-Domain Closure Certificate

Output:

- domains;
- the bridge path;
- error;
- lost information;
- preserved probes;
- exceptions;
- identity policy;
- trust.

---

# 272. Minimal Runtime

The first NIOT-06 runtime can support only:

1. exact rationals;
2. float64;
3. intervals;
4. decimal strings;
5. a bridge registry;
6. error certificates;
7. probe-preserving bridges;
8. round-trip audits.

---

# 273. MVP Example A

Source:

$$
\frac{1}{10}.
$$

Bridge:

$$
\mathbb Q
\rightarrow
\mathbb F_{64}.
$$

---

# 274. Float Result

we obtain:

$$
f.
$$

---

# 275. Embed Back

$$
f
\rightarrow
q_f\in\mathbb Q.
$$

---

# 276. Compare

$$
\boxed{
q_f
\neq
\frac{1}{10}.
}
$$

generally holds.

---

# 277. Audit

Record:

- the exact source;
- float bits;
- the exact dyadic value of the float;
- absolute error;
- relative error.

---

# 278. MVP Example B

Source:

$$
\sqrt2.
$$

Target:

Interval:

$$
[1.4142,1.4143].
$$

---

# 279. Relation

$$
\boxed{
\sqrt2
\in
[1.4142,1.4143].
}
$$

---

# 280. This Is Enclosure, Not Equality

---

# 281. MVP Example C

Only a sign probe is needed.

There is no need for:

$$
\sqrt2
\rightarrow
\text{1000-digit decimal}.
$$

Directly use:

$$
\boxed{
P_{\mathrm{sign}}(\sqrt2)=+.
}
$$

This is probe-first / partial bridging.

---

# 282. MVP Example D

Source float:

$$
f.
$$

One wants to know:

> Was it projected from an exact rational?

This requires a provenance bridge, not guessing the source from the bits.

---

# 283. Same Bits, Different Provenance

Different pipelines can produce identical float bits.

Thus bits cannot uniquely recover history.

---

# 284. Information-Theoretic Irreversibility

A non-injective bridge necessarily produces preimage multiplicity.

---

# 285. Bridge Fiber

For:

$$
y\in Y,
$$

Define:

$$
\boxed{
F_y^B
=
B^{-1}(y).
}
$$

---

# 286. Fiber Cardinality

If:

$$
|F_y^B|>1,
$$

the target cannot uniquely recover the source.

---

# 287. Bridge Fiber Probe

Other information can narrow the preimage.

For example, provenance.

---

# 288. Multi-Probe Source Reconstruction

$$
\boxed{
F_y^{B,P_1,\dots,P_k}
}
$$

is progressively refined.

---

# 289. Exact Reconstruction Condition

When:

$$
|F_y^{B,\mathcal P}|=1,
$$

only then can one reconstruct uniquely in the specified information language.

---

# 290. Reconstruction Is Probe-Relative

This is entirely consistent with NIOT-02.

---

# 291. Bridge Fiber and Micro-Numerical World

Thus informational distinctions after cross-domain conversion can be viewed as:

$$
\boxed{
\text{bridge fiber structure}.
}
$$

---

# 292. Float Fiber as Microscopic Region

A float:

$$
f
$$

corresponds to an entire rounding cell in the exact domain.

---

# 293. Thus "A Float Is a Point" Holds Only in the Float Domain

In the exact source domain:

$$
\boxed{
f
}
$$

is more like the label of a projection fiber.

---

# 294. This Is Important Ontological Layering

$$
\boxed{
\text{point in target domain}
\leftrightarrow
\text{region / fiber in source domain}.
}
$$

---

# 295. Probe Can Enter the Fiber

With additional context / provenance / intervals, a rounding fiber can be refined further.

---

# 296. Fiber Refinement Matrix

UEFM can materialize:

$$
\boxed{
M_t(F_f).
}
$$

---

# 297. Fiber Spectrum

Along precision / provenance / history, establish:

$$
\boxed{
\mathcal S(F_f).
}
$$

---

# 298. Fiber Index Geometry

NIOT-05 can supply:

$$
\boxed{
\mathcal I(F_f).
}
$$

---

# 299. Bridge BCODR

Outward:

Source candidates.

Inward:

Target constraints / certificates.

Overlap between the two can localize a plausible source spine.

---

# 300. Bridge Responsibility Spine

$$
\boxed{
S_B
}
$$

can denote:

> Which source subregions may still correspond to the current target state?

---

# 301. This Is the Cross-Domain Generalization of BCODR

---

# 302. Formalization Targets

First-stage Lean / Coq targets:

1. exact recoding;
2. rounding projection;
3. float embedding;
4. round-trip conditions;
5. bridge fibers;
6. a non-injective bridge implies non-unique reverse reconstruction;
7. probe-preserving partial bridges;
8. task-relative bridge closure.

---

# 303. IEEE-Specific Formalization Can Be Later

An abstract finite float domain suffices for the initial version.

---

# 304. Theorem 1 — Projection Non-Invertibility Under Noninjective Rounding

If:

$$
R_F
$$

is non-injective, there is no function:

$$
G:\mathbb F\to X
$$

such that:

$$
G\circ R_F
=
\operatorname{id}_X
$$

holding throughout $X$.

---

# 305. Proof Sketch

If there exist:

$$
x_1\neq x_2
$$

However:

$$
R_F(x_1)=R_F(x_2)=f,
$$

then:

$$
G(f)
$$

cannot simultaneously equal:

$$
x_1
$$

and:

$$
x_2.
$$

Q.E.D.

---

# 306. Theorem 2 — Exact Float Embedding Does Not Recover Rounded Source

If:

$$
x\notin E_F,
$$

then, in general:

$$
\boxed{
\iota_F(R_F(x))
\neq
x.
}
$$

---

# 307. Theorem 3 — Probe-Preserving Bridge Need Not Be Full Bridge

If a task requires only the probe family:

$$
\mathcal P^\star,
$$

one needs only to establish:

$$
P_Y\circ B
=
T_P\circ P_X
$$

for $\mathcal P^\star$.

Not every source structure needs to be convertible.

---

# 308. Theorem 4 — Same Target Value Does Not Imply Same Source Object

If:

$$
B(x_1)=B(x_2)
$$

and:

$$
x_1\neq x_2,
$$

target equality cannot imply source identity.

---

# 309. Theorem 5 — Finite Precision Does Not Imply Finite Extensibility

At every epoch:

$$
p_t<\infty
$$

one may still allow:

$$
\sup_t p_t=\infty.
$$

---

# 310. Open Problem: Bridge Category

Can one use:

- domains as objects;
- bridges as morphisms;

to establish a category while preserving partiality / error / exceptions?

---

# 311. Open Problem: Approximate Category

One needs to handle morphisms with:

$$
\varepsilon
$$

nonzero, as displayed above.

---

# 312. Open Problem: Bridge Curvature

Can defects of closed conversion loops form a usable curvature theory?

---

# 313. Open Problem: Minimal Probe-Preserving Bridge

For a task:

$$
\mathcal T,
$$

which probes minimally need to be bridged?

---

# 314. Open Problem: Fiber Complexity

How can the structural complexity of rounding fibers / quotient fibers be quantified?

---

# 315. Open Problem: Representation Search

Among multiple representations, should one first find the most suitable representation before solving the problem?

---

# 316. Open Problem: AI Bridge Router

Can AI dynamically select a representation path according to:

- the task;
- error;
- cost;
- the certificate;

as listed above?

---

# 317. Open Problem: Cross-Model Numerical Semantics

How should different AI systems / runtimes align their representations of the same numerical object?

---

# 318. Interface with NIOT-07

The next paper will address:

$$
\boxed{
\textbf{Bidirectional Structural Localization and Responsibility Closure.}
}
$$

The central object NIOT-06 supplies to it is:

$$
\boxed{
\text{bridge fiber}
}
$$

and:

$$
\boxed{
\text{source / target responsibility fields}.
}
$$

---

# 319. NIOT-07 Will Use BCODR Beyond Geometric Proof Space

It will generalize BCODR to:

- bridge space;
- probe space;
- matrix frontiers;
- source reconstruction;
- representation alignment.

---

# 320. Six Most Important Statements of This Paper

First:

$$
\boxed{
\text{float}
\neq
\text{whole numerical object}.
}
$$

Second:

$$
\boxed{
\text{representation}
\neq
\text{completion}.
}
$$

Third:

$$
\boxed{
\text{rounding projection}
\neq
\text{invertible conversion}.
}
$$

Fourth:

$$
\boxed{
\text{same target value}
\not\Rightarrow
\text{same source identity}.
}
$$

Fifth:

$$
\boxed{
\text{full conversion is not always necessary;
probe-preserving partial bridges can be enough}.
}
$$

Sixth:

$$
\boxed{
\text{every lossy bridge must expose its fiber, loss, exception, address, and certificate semantics}.
}
$$

---

# 321. Conclusion

This paper formally reconnects NIOT's earliest intuition.

At first, we thought:

$$
\boxed{
\text{floating-point number}
=
\text{the numerical value itself}.
}
$$

More precisely, modern computation often treats it as a sufficient working object in engineering.

But NIOT can now write:

$$
\boxed{
f
=
\pi_F(\mathfrak N).
}
$$

Therefore:

$$
f
$$

is an admissible, important, precisely definable machine numerical object,

but it need not equal all the information of:

$$
\mathfrak N
$$

as displayed above.

We also no longer need to equate "seeing more information" with:

$$
\boxed{
\text{first changing numeral bases.}
}
$$

Changing numeral bases is only a special case of:

$$
\boxed{
\text{representation bridge}
}
$$

as displayed above.

We can also:

- directly probe exponents;
- directly probe ULP neighborhoods;
- directly probe provenance;
- directly probe history;
- directly probe rounding fibers;
- establish interval enclosures;
- expand along a precision spectrum;
- bridge only task-relevant probes.

Microscopic numerical worlds therefore acquire a very concrete cross-domain picture in this paper:

$$
\boxed{
\text{target point}
\leftrightarrow
\text{source fiber}.
}
$$

A finite float may be an exact point in its own domain,

but in a richer information-source domain it may correspond to:

$$
\boxed{
\text{an entire source region compressed by projection.}
}
$$

This source region can be continually refined by:

- probes;
- spectra;
- provenance;
- bridge certificates;
- BCODR;
- UEFM;

as listed above.

Thus this paper ultimately obtains:

$$
\boxed{
\text{Finite machine representation is not the endpoint of numerical research,
but a localizable, verifiable, traceable projection interface of numerical information objects.}
}
$$

Cross-domain computation should no longer be written vaguely as:

$$
x
\rightarrow
y.
$$

It should instead be written as:

$$
\boxed{
(\mathfrak N_X,i_X)
\xrightarrow{
\mathfrak B,\,
\varepsilon,\,
\mathsf{Exc},\,
\mathsf{Cert}
}
(\mathfrak N_Y,i_Y).
}
$$

This provides a complete entry point to the next paper:

$$
\boxed{
\textbf{NIOT-07 — Bidirectional Structural Localization and Responsibility Closure}
}
$$

as displayed above.

---

# Canonical Source Declaration

This file is the UTF-8 Markdown canonical source for `NIOT-06-v0.1`.

The only mathematical delimiters are `$...$` and `$$...$$`.

This paper formally integrates:
- float as projection;
- float information objects;
- generation / completion / representation / execution separation;
- exact recoding;
- corecursive expansion;
- completion lifting;
- quotient identification;
- rounding projection;
- interval enclosure;
- exception escape;
- probe-preserving partial bridges;
- bridge fibers;
- bridge endpoints;
- bridge identity policy;
- bridge certificates;
- round-trip semantics;
- error vectors;
- task-relative fidelity;
- precision spectra;
- exact-real lazy materialization;
- cross-domain UEFM;
- the bridge BCODR interface.

This paper does not claim that:
- all floating-point errors can be described by a single scalar error;
- all cross-domain conversions are invertible;
- all representations have the same information capacity;
- all notions of exactness are identical;
- all completion representations have unique forms;
- all bridge-loop defects have already formed a complete curvature theory.

Subsequent NIOT-07 should take bridge fibers, source/target responsibility fields, and BCODR generalization as its principal foundations.

===END===