Prove P104 phase-uniform gauge and unavoidable completion tag

research/p104-phase-uniform-gauge/CERTIFICATE.json

@@ -0,0 +1,35 @@
+{
+  "schema_version": 1,
+  "z": 420,
+  "columns": [
+    1,
+    2,
+    416,
+    8,
+    16,
+    388,
+    64,
+    292,
+    164
+  ],
+  "translation": 430582246074561917307300,
+  "translation_word": [
+    [
+      1,
+      1
+    ],
+    [
+      1,
+      1
+    ],
+    [
+      430582246074561917307301,
+      1
+    ],
+    [
+      430582246074561917307301,
+      1
+    ]
+  ],
+  "bankwise_appended_calls": 146432
+}

research/p104-phase-uniform-gauge/PROOF.md

@@ -0,0 +1,156 @@
+# A phase-uniform gauge with an unavoidable completion tag
+
+## 1. Scope and conventions
+
+Use exactly the 104 ordered odd binary labels published by Chris R / Clear1nsight in PR3954 at `f1e2839819959cd0534d008be91ed5b79ec55a6a`, retained in `source/RESULT.json`. The physical isometry and original P104/native construction are predecessor work. This note proves a changed coordinate gauge and its operator boundary; it does not supply the missing adjacent physical producer/consumer pair or admit the predecessor scaffold.
+
+All binary vectors use the usual dot product. Coordinates are numbered from zero; tensor coordinate `(i,j)` is bit `9i+j`. Write `j=511`, `q(x)=wt(x) mod 4`, and `chi_s(x)=(-1)^(s dot x)`. The scalar field can be Q(i), extended coordinatewise to arbitrary finite arrays. Define
+
+```
+X_u f(x) = f(x+u),     Z_u f(x) = (-1)^(u dot x) f(x),
+C_u = ((1+i)I+(1-i)X_u)/2,
+F_m = product_(k=0..m-1) C_(e_k),
+R_G f(x) = f(G^-1 x).
+```
+
+Thus `C_u chi_s = i^(u dot s) chi_s` and `F_m chi_s=i^q(s) chi_s`. The latter is the retained full native transform, not the Walsh transform. Orthogonality of G alone does **not** imply that R_G commutes with F_m. Gates `C_u^-1` have the opposite exponent. For an orthogonal G,
+
+```
+R_G chi_s = chi_(G s),
+R_G X_u R_G^-1 = X_(G u),
+R_G Z_u R_G^-1 = Z_(G u),
+R_G C_u R_G^-1 = C_(G u).
+```
+
+These are exact ideal operator equations. A coordinate permutation on a large physical tape is not declared free.
+
+## 2. Explicit gauge: all product-line phases agree
+
+Let `z=420`, with support `{2,5,7,8}`, and put
+
+```
+O x = x + (x dot z) z,
+O columns = [1,2,416,8,16,388,64,292,164].
+```
+
+Because `z dot z=0`, O is involutive. Expanding `(Ox) dot (Oy)` shows orthogonality: the two cross terms cancel and the last term is zero. Since `q(z)=0`, the weight identity `q(x+y)=q(x)+q(y)+2(x dot y) mod4` gives
+
+```
+q(Ox) = q(x)+2(x dot z) mod4.                 (1)
+```
+
+The source-bound finite check verifies `q(Op)=3` on all 104 published labels. Their old weight classes were 56 of class 1 and 48 of class 3. For `K=O tensor O`, the 10,816 transformed product labels obey
+
+```
+q((Op_i) tensor (Op_j)) = 3*3 = 1 mod4.
+```
+
+K is orthogonal and bijective, so all these lines remain distinct and nondegenerate; their pairings and dimensions are unchanged. In the predecessor's positive `C_u` orientation, the normalization factor `i^(f wt(u))` is consequently the retained constant `i^(9f)` for every product line and every integer parallel-column count f. This removes the 5,376 gauge-dependent sign mismatches. It does not assert that the historical compiler already uses these coordinates or this orientation.
+
+## 3. The completion defect is necessary throughout the orthogonal class
+
+Use one orthogonal binary G in dimension nine identically on both tensor
+factors; independent row/column gauges and arbitrary 81-dimensional gauges are
+outside this quantified statement. For such G, the difference `q(Gx)-q(x)` is even and has zero polar form. Dividing by two defines a linear character, so there is a unique t with
+
+```
+q(Gx) = q(x)+2(t dot x) mod4.                 (2)
+```
+
+Diagonal products have weight class 1, so any constant product class must be 1.
+If all `Gp_i tensor Gp_j` have weight class 1, then all `q(Gp_i)` have the same odd class c, either 1 or 3: a pair of different classes has product 3. The 104 source labels span the full binary nine-space. Equivalently, the equations on these labels determine at most one t for each c. The source-bound linear constraints give exactly
+
+```
+c=3: t=420;             c=1: t=91=420+511.
+```
+
+An orthogonal G fixes the characteristic vector j, because `x dot x=j dot x` for every x. Substituting x=j in (2) forces `t dot j=0`. The second candidate has odd weight and is impossible. Thus **every** orthogonal gauge making these product phases constant must send the ports to class 3 and have the same nonzero input defect `t=420`. None can preserve F_9. This is a theorem about all orthogonal gauges for these fixed labels, not an enumeration of the orthogonal group or an obstruction to general nonorthogonal/paid intertwiners.
+
+For a general such G set `K_G=G tensor G` and
+
+```
+a = z tensor j + j tensor z
+  = 430582246074561917307300
+  = 0x5b2de90b7a4d216f49a4.
+```
+
+The quadratic difference for K_G is linear after division by two. Evaluating on the 81 basis tensors gives
+
+```
+q(K_G s) = q(s)+2(a dot s) mod4.              (3)
+```
+
+Indeed `q(Ge_i)=1+2z_i`, so the tensor basis column has weight class `1+2(z_i+z_j)`. This proves (3) on all 2^81 frequencies from orthogonality and 81 coefficient checks. The input character a is universal across this class of G. It is nonzero and even. The general conjugacy is
+
+```
+R_(K_G)^-1 F_81 R_(K_G) = F_81 X_a,
+R_(K_G) F_81 R_(K_G)^-1 = F_81 X_(K_G a).
+```
+
+Input a and output K_G a need not coincide. No general involution is assumed. For our explicit O, `Oz=z`, `Oj=j`, hence `Ka=a`, and `wt(a)=2*4*5=40`.
+
+## 4. Exact all-role tags, and the unresolved physical seam
+
+For the explicit involution K, define
+
+```
+E_in = R_K,             E_out = X_a R_K.
+```
+
+The preceding equations imply
+
+```
+F_81 E_in  = E_out F_81,
+F_81 E_out = E_in  F_81.                      (4)
+```
+
+Thus the two tags alternate under repeated **already completed uniform** F_81 operations. Two such operations return to the entrance tag. If instead the next operation insists on the original E_in rather than accepting the alternate tag, the seam map is `E_in E_out^-1=X_a`. Omitting it is not an identity; a character with `a dot s=1` is a separating input.
+
+Apply the same E_in/E_out on every role of an all-role completed call, including all 72 local dirty helper roles (or every role of a declared larger owner inventory). Since (4) is an operator equality on each independently arbitrary function, it holds on their direct sum. Helpers are not assumed zero. Uniform address maps commute with scalar role matrices D,H,E0 and with the predecessor's literal slot permutation Pi; this does not make Pi a free physical move. Encoded helper equations are **not** a fresh physical dirty-return/ownership certificate: any required return to the original coordinate chart still needs its explicit boundary maps.
+
+Both address encodings act as the identity on constant address functions.
+Consequently, a nonzero role-encoding defect already visible on constant
+functions cannot be removed by these uniform gauge/translation tags alone;
+it still requires a role-boundary transition. This observation does not
+independently certify any concurrent package's specific counterexample.
+
+This does not instantiate PR3954's row/column producer. Define F_row,j as the product of positive C gates on the old standard active basis `e_k tensor p_j`, and F'_row,j on the new standard basis `e_k tensor q_j`, where q_j=Op_j. Define the column transforms analogously using `p_i tensor e_k` and `r_i tensor e_k`, with r_i=Op_i. The row correction is `z tensor q_j`; the column correction is `r_i tensor z`. Conjugate the old bases and apply (1) to the nine coefficient frequencies to obtain
+
+```
+R_K F_row,j R_K = F'_row,j X_(z tensor q_j),
+R_K F_col,i R_K = F'_col,i X_(r_i tensor z).
+```
+
+These tags vary with the fiber. They must apply to every role in the local call, including its helpers, and reconcile on shared data owners. The alternating common-tag lemma (4) supplies no such physical matching, no free source relocation and no cost-free change of required outputs. That remains the released adjacent-call obligation.
+
+## 5. A sharp restricted tariff for a forced translation
+
+Fix one bank and a nonzero even vector b in any binary dimension where such a vector exists. The alphabet is exactly `C_u` and `C_u^-1` for nonzero **odd** u, each charged one call. No other operations, free translation, role mixing, ancilla, measurement or borrowed helper is allowed. Then the exact word length of X_b is four.
+
+For an upper bound choose e to be any unit vector. Both e and `e+b` are odd, and
+
+```
+C_e^2 C_(e+b)^2 = X_e X_(e+b) = X_b.
+```
+
+For the lower bound evaluate any shorter word on frequency j, the all-ones vector. Every allowed u has `u dot j=1`, whereas `b dot j=0`. A word of odd length has an odd exponent sum of plus/minus ones and cannot have phase one. A two-call word with equal signs has phase minus one at j. With opposite signs it is `C_u C_v^-1`; if u differs from v, some frequency separates their parities, giving phase plus/minus i instead of the real phase of X_b. If u=v it is the identity. Length zero is also the identity, while X_b is not. This excludes every length below four. Allowing a constant scalar phase does not help because the zero-frequency eigenvalue fixes that scalar to one.
+
+The written proof covers arbitrary dimension and even b. The verifier checks the constructive four-call character table; the independent checker exhausts a small three-dimensional alphabet only as corroboration. This is not a native compiler/frame admission or cross-role optimality theorem. For all orthogonal phase-uniform gauges above the tensor output defect K_G a is nonzero and even, so the same restricted theorem applies to any seam that is implemented in this exact alphabet.
+
+## 6. Complete cost decision and stopping rule
+
+Orthogonal K changes neither dimensions/ranks, the 36,608-role inventory, 208 local calls nor the predecessor's stipulated mass 2,576 and net progress 2,312 per call. Grant all gauge and tag work free and keep the same favorable contract. Re-derive the named gate:
+
+```
+W = 2*104^2+2*104*72 = 36608,    m=81,    N=104^2=10816,
+required persistent progress = mW-N = 2954432,
+remaining rank >= 2954432-208*2312 = 2473536,
+lower paid mass = 208*2576+2473536 = 3009344,
+capacity = 2965248,    excess = 44096.
+```
+
+These are inherited conditional hypotheses, not freshly admitted local/native calls or global owners. At `0<=saving<1`, each child of rank r<=81 has moment term at least r/81. The unchanged positive mass gap therefore still rejects this scaffold. Finite fully paid auxiliary banks preserve the numerator gap. Phase homogenization cannot by itself remove a recursive child.
+
+If one **chooses** an appended isolated correction on every completed bank in the restricted alphabet of section 5, its extra primitive-word bill is `4W=146432` calls. **Only if each isolated C primitive is additionally admitted at one rank unit in this same ledger** does that literal route have excess `190528`, before physical wrappers. Word length alone does not establish that native tariff compatibility. This is a conditional literal-route calculation, not a lower bound on arbitrary fusion, mixed roles, different encodings or different coverage. It is not a complete upper bound on implementation time: R_K routing, chart/phase wrappers, source acquisition, dirty returns, ownership/holding/control/tapes, finite widths/precision/recovery and inherited all-size/compiler/analytic obligations remain unknown or inherited, never zero.
+
+**NARROW / STOP.** Preserve the explicit phase-uniform gauge, its unavoidable completion character and exact restricted seam tariff. Do not expand native implementation for the unchanged failed scaffold. A genuinely changed adjacent-call contract with reconciled fiber/helper tags, actual owners and a favorable complete replacement invoice is the next meaningful task.

research/p104-phase-uniform-gauge/README.md

@@ -0,0 +1,131 @@
+# P104 phase-uniform gauge and forced completion tag
+
+**Scoped ideal-operator result; no new supplier or kappa.** For the 104 binary
+labels published in [PR3954](https://github.com/CrocSwap/integer-mult-bounds/pull/3954)
+at `f1e2839819959cd0534d008be91ed5b79ec55a6a`, the explicit orthogonal involution
+
+```
+z=420,   O x=x+(x dot z)z,
+O columns=[1,2,416,8,16,388,64,292,164]
+```
+
+sends every label to weight 3 modulo 4. All 10,816 tensor product lines then have
+weight 1 modulo 4: the predecessor's 5,376 gauge-dependent normalization
+mismatches disappear. This answers one coordinate-gauge question left open by
+PR3954, without claiming to complete its physical caller.
+
+## What is proved
+
+[PROOF.md](PROOF.md) gives the arbitrary-dimension/operator arguments and the
+source-bound finite premises. Every orthogonal nine-dimensional gauge applied identically to both tensor
+factors and making these product phases constant has the **same nonzero input quadratic defect**
+`z=420`. The alternative character `91` would violate preservation of the
+all-ones characteristic vector. This class cannot preserve the standard full
+native transform F; the result is not an enumeration of the orthogonal group.
+
+For the explicit involution `K=O tensor O`, the completion translation is
+
+```
+a=z tensor 511 + 511 tensor z
+ =430582246074561917307300,   wt(a)=40,   Ka=a.
+```
+
+With `R_K f(x)=f(K^-1 x)`, exact uniform completed-F encodings are
+`E_in=R_K` and `E_out=X_a R_K`, with
+`F E_in=E_out F` and `F E_out=E_in F`. These identities hold on independently
+arbitrary functions in all 280 local roles, including 72 dirty helpers. They do
+not assume zero helper input. General non-involutive gauges distinguish the
+input defect a from its output image K_G a.
+
+If a consumer forces a return to the original tag, its translation has exact
+minimum length four in the **restricted one-bank alphabet** of odd-line
+`C_u` and `C_u^-1`, one call each. The proof allows no other gates, role mixing,
+ancillas or free translations. This is a word tariff, not native compiler,
+rank-unit or cross-role optimality admission.
+
+The inherited favorable 81-dimensional scaffold still has lower mass
+3,009,344 versus capacity 2,965,248: **excess 44,096 even when all new gauge and
+tag work is free**. A chosen isolated correction on all 36,608 banks adds
+146,432 primitive calls. Its excess would be 190,528 only if those primitives
+were additionally admitted at one rank unit each in the same ledger. No such
+admission is inferred here. Stop expansion of this unchanged scaffold.
+
+## Evidence and reproduction
+
+The only live scientific inputs are the three package-local source exports in
+`source/`, pinned by origin commit, Git blob and SHA256 in [SOURCES.json](SOURCES.json).
+Both checkers independently hardcode/authenticate the hashes. No predecessor
+executable or unmerged foreign Git object is needed to replay this package.
+The selected record/catalogue are historical base-commit provenance only.
+
+From the repository root, with Python 3.12 and its standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-phase-uniform-gauge --check-only
+python3.12 -B scripts/research_package.py --package research/p104-phase-uniform-gauge
+```
+
+For a Git-free directory copy of this package, run `verify.py`, `independent.py`
+and `test_verify.py` directly, with `-B`; repeat with `-O -B` to check assertion
+independence. Explicit exceptions, rather than Python assertions, enforce the
+scientific checks. The registered commands are in [VERIFY.json](VERIFY.json).
+
+`verify.py` checks 104 labels, 10,816 product lines, 512 possible defect
+characters, 81 tensor coefficients and 22,680 role/basis conditions. Polarization
+extends those coefficient checks to all frequencies; no enumeration of 2^81
+addresses is claimed. The separate checker uses binary RREF and 262,144 dot
+pairs, checks a non-involutive gauge convention, and corroborates the operator
+identities with four-bit literal matrices. Those small complex-matrix entries
+and bounded fixtures are exactly representable dyadics; no numerical tolerance
+accepts an equality. Its 585 three-bit words corroborate the written universal
+four-call lower bound. `test_verify.py` runs 60 primary API/CLI rejection cases
+and eight independent damaged-source rejections across normal/optimized Python,
+plus a read-only Git-free positive fixture. These finite checks accompany the
+written proofs; they are not a proof-assistant or full multiplication audit.
+
+The source's isometry/full verifier, native-channel and selected cold replay
+are not rerun. The theorem binds to the explicitly retained published labels
+and conventions, without re-admitting the predecessor's physical hypotheses.
+
+## Open next questions
+
+The structured questions and blockers are in [TASK.json](TASK.json). This is a
+**changed-scope** follow-up to the released PR3954 gauge question, not completion
+of the catalogue's chronological correction program or the latest handoff's
+missing adjacent physical producer/consumer pair.
+
+Concurrent PR3961 studies the physical cross-axis encoding mismatch; PR3962
+studies whole-fiber coverage and helper sharing. Their refreshed PR bodies
+identify distinct contracts; this package does not claim to replay them or
+remove those obstructions by changing address coordinates.
+
+The consequential next step is one actual adjacent row/column pair: write its
+all-role entrance, intermediate and exit encodings at a named chronological
+cut, and reconcile the fiber-dependent tags `z tensor q_j` and `r_i tensor z`
+on shared data and helper owners. The uniform completed-F lemma does not do
+that. First require a changed complete invoice that clears the 44,096 deficit;
+do not expand the rejected scaffold merely because its phase mismatch is fixed.
+
+A physically viable changed contract would still need actual source frames,
+R_K routing, native/binary-hyperplane orientation and global phase, compatible
+tariffs, finite-width/signed-floor precision and recovery, physical dirty
+return, owners/holding/control/tapes, opposite orientation, all-size transfer
+and complete supplier admission. These obligations remain unknown or inherited.
+
+## Credit and license
+
+Chris R / Clear1nsight's PR3954 owns the retained P104 labels, two-axis setup,
+operator conventions and conditional scaffold gate, following PR3882/3940/3943.
+Retain the source README/PROOF attribution: 1337isnot1337 PR3894; Rohan Arun
+PR1972; Chafik Boukhalfa PR1991; Adam Gall PR1940; Josie Elliston PR1071/E8;
+Aurel Prosz/Paureel, Zhihao Chen, Dominik Scholz, icekylinx, Sagar Thakrar PR1012,
+Douglas Colkitt, Jose Manuel and Vasek PR2744, plus the retained native,
+compiler, coverage, terminal-floor and caller lineage. PR3943 owns its
+960-addition adapter; this package does not reassign that construction.
+
+The new explicit gauge, orthogonal-class character obstruction, completed-F
+boundary and restricted translation tariff are by ToikY with substantial
+OpenAI Codex assistance. Separate AI contexts implemented/reviewed the proof
+checks and failure controls; this is not independent human or kernel review.
+New work and retained source exports are under the repository's Apache-2.0
+license. No merge or selected-frontier change is requested.