[zxv]

The 112-owner rounded inverse now has an exact recovery frontier across output
precisions, a constructive decoder, and a sharp history law for repeated
recoding. Five relative output bits make the mathematical map injective.
On the same grid, every additional recoding multiplies worst-case recovery
states by $13^8$, even under fixed noncommuting changes of integer coordinates.

Detailed result and research seams · Proof ·
Exact calibration · Sources

Significance for researchers

This answers when extra precision can replace retained recovery tags and when
changing frames cannot prevent history accumulation. It gives a positive route
for callers already producing a finer output grid and a scoped obstruction to
indefinite recoverable recoding with bounded history, precision, and side
information on growing input ranges. The new ingredient is the source's binary
code and its coset weight enumerator, connecting native lifting to classical
coding theory.

Exact results

For coarse input $x\in\mathbb{Z}^{112}$, the fine output is

$$y=2^g x-M\,R \!\left(\frac{2^gM^{\mathsf T}x}{52}\right), \qquad M^{\mathsf T}M=56I.$$

Here $R$ rounds each component to the nearest integer, with half ties away from zero.
The first row is the credited same-grid theorem in
PR #5323.

Relative output bits $g$	Sharp recovery states	Packed tag bits	Added fractional slots + tag¹
0	815,730,721	30	30
1	51,008,161	26	138
2	362,401	19	243
3	4,096	12	348
4	16	4	452
5	1	0	560

¹One real coordinate across 112 uniformly widened owner fields; excludes
magnitude, scratch, and movement. Every $g\ge 5$ is injective. Merely increasing
input and output precision together leaves $g=0$.

The package also proves:

• General fiber geometry: finite exact recovery tags exist precisely when
  $M^{\mathsf T}M-\beta I$ is nonsingular for
  $K_\beta(z)=z-MR(M^{\mathsf T}z/\beta)$.
  Here $M$ is an integer matrix of full column rank, $\beta$ is a positive
  integer, and the input lattice has full rank; quotient enumeration gives a decoder.
• Sharp composition: $N$ real coordinates and $t$ same-grid recodings have
  maximum fiber $13^{8Nt}$ under arbitrary fixed known unimodular pre/post
  changes, including mixing between coordinates. Packed history needs
  $\left\lceil 8Nt\log_2 13\right\rceil$ bits: 533 at eighteen steps for one real coordinate.
• Information conservation: total uniform relative refinement $G$, a
  $b$-bit history, and side information with $S$ possible values require
  $b+\log_2 S+112NG\ge 8Nt\log_2 13$. This refined bound is necessary; its
  attainability is open. The exact one-step frontier is stronger.

Explicit attaining fibers establish each table row. A sixteen-block
code-coset rank/unrank codec avoids enumerating the largest fibers. The general
history proof uses lattice volume; stationary frames also have explicit
balanced-base-thirteen witnesses with $\lVert x\rVert_\infty\le 2(13^t-1)$.

Nearest overlapping work

Work	Relationship
PR #5323, Ali Romman	Direct predecessor: sharp same-grid recovery. New scope is relative precision and changing-frame composition.
PR #4933, Dillon Amburgey; PR #4929, David Ganz	Pinned matrix, native geometry, and physical access obligations.
PR #5035, Brian; PR #5321, Ali Romman	Original finite kernel and cheaper arithmetic. The original kernel is retained byte-for-byte; recovery-codec pricing remains additional.
PR #5333, Ali Romman; PR #5438, Mark	Physical residual and exact-grid work provide places to audit relative precision; their absolute guards do not establish this theorem's relative-grid premise.

Outside input: MacWilliams'
weight-enumerator theorem
and Forney's code/Fourier duality.
Classical duality, pigeonhole counting, and lattice volume are not claimed as
new general mathematics. Overlap coverage is recorded with
limits; this is a source-specific extension, not a global priority claim.

Evidence and limits

Normal and optimized portable replay agree exactly. Checks include 24 calls
to the unchanged bitwise kernel, complete source Gram/code facts, attaining
fiber counts and sampled reconstructions, changing-frame recovery, and
noncommuting finite windows. Five semantic mutants are rejected in both modes.
The self-contained read-only export runs with an empty PATH and no Git or
network dependency. Validation records exact commands and
the finite domains; the written proof supplies the universal claims.

python3 -B research/precision-state-frontier/verify.py
python3 -B research/precision-state-frontier/test_package.py
python3 -B research/precision-state-frontier/mutation_check.py

The main tradeoff is storage: eliminating thirty tag bits by uniformly adding
five fractional bits introduces 560 bit positions before other costs. The
history lower bound requires an unbounded or sufficiently large input range;
it need not hold in a small fixed caller. Extra observations, rereads,
nonunimodular maps, and input-dependent control require their own information
and lattice accounting. Mathematical injectivity does not convert the source's
destructive cleanup into a priced reversible program.

Open next questions

1. Bind one actual inverse call's input lattice, output grid, access, and full
   invoice to test whether relative refinement is already funded.
2. Compile and price reversible rank/unrank with owned scratch and eventual
   restoration; compare against simple quotient tags and retained-input replay.
3. Derive unequal per-owner precision frontiers and sharp mixed refined chains;
   determinant volume alone does not prove an attainable optimum.
4. Transfer the lattice/coset method to another native code, and determine
   finite-range thresholds where the repeated-history obstruction becomes active.

Scoped proof and author calibration; independent proof review and physical
caller admission remain open. No complete multiplication theorem or new κ.
zxv, with substantial OpenAI Codex assistance;
Apache-2.0, with predecessor notices retained.